Scholarly article on topic 'Quantum periods of Calabi–Yau fourfolds'

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Abstract of research paper on Physical sciences, author of scientific article — Andreas Gerhardus, Hans Jockers

Abstract In this work we study the quantum periods together with their Picard–Fuchs differential equations of Calabi–Yau fourfolds. In contrast to Calabi–Yau threefolds, we argue that the large volume points of Calabi–Yau fourfolds generically are regular singular points of the Picard–Fuchs operators of non-maximally unipotent monodromy. We demonstrate this property in explicit examples of Calabi–Yau fourfolds with a single Kähler modulus. For these examples we construct integral quantum periods and study their global properties in the quantum Kähler moduli space with the help of numerical analytic continuation techniques. Furthermore, we determine their genus zero Gromov–Witten invariants, their Klemm–Pandharipande meeting invariants, and their genus one BPS invariants. In our computations we emphasize the features attributed to the non-maximally unipotent monodromy property. For instance, it implies the existence of integral quantum periods that at large volume are purely worldsheet instanton generated. To verify our results, we also present intersection theory techniques to enumerate lines with a marked point on complete intersection Calabi–Yau fourfolds in Grassmannian varieties.

Academic research paper on topic "Quantum periods of Calabi–Yau fourfolds"

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Quantum periods of Calabi-Yau fourfolds

Andreas Gerhardus, Hans Jockers

Bethe Center for Theoretical Physics, Physikalisches Institut der Universität Bonn, Nussallee 12, D-53115 Bonn,

Germany

Received 23 August 2016; accepted 25 September 2016 Editor: Stephan Stieberger

Abstract

In this work we study the quantum periods together with their Picard-Fuchs differential equations of Calabi-Yau fourfolds. In contrast to Calabi-Yau threefolds, we argue that the large volume points of Calabi-Yau fourfolds generically are regular singular points of the Picard-Fuchs operators of non-maximally unipotent monodromy. We demonstrate this property in explicit examples of Calabi-Yau fourfolds with a single Kahler modulus. For these examples we construct integral quantum periods and study their global properties in the quantum Kahler moduli space with the help of numerical analytic continuation techniques. Furthermore, we determine their genus zero Gromov-Witten invariants, their Klemm-Pandharipande meeting invariants, and their genus one BPS invariants. In our computations we emphasize the features attributed to the non-maximally unipotent monodromy property. For instance, it implies the existence of integral quantum periods that at large volume are purely worldsheet instanton generated. To verify our results, we also present intersection theory techniques to enumerate lines with a marked point on complete intersection Calabi-Yau fourfolds in Grassmannian varieties.

© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction

It is well-established that non-perturbative worldsheet instanton corrections of string compact-ifications on Calabi-Yau manifolds are captured in terms of the quantum cohomology ring [1-4],

E-mail addresses: gerhardus@th.physik.uni-bonn.de (A. Gerhardus), jockers@uni-bonn.de (H. Jockers). http://dx.doi.Org/10.1016/j.nuclphysb.2016.09.021

0550-3213/© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1 which arises from a deformation of the classical intersection product. From the string worldsheet 1

2 point of view the quantum cohomology ring is identified with the chiral-anti-chiral ring of the 2

3 two-dimensional N = (2, 2) conformal field theory [5,6]. In this work we study the quantum 3

4 cohomology of Calabi-Yau fourfolds [7,8] — in particular with a single Kahler modulus. Due to 4

5 N = 2 special geometry [9-11] for Calabi-Yau threefolds the number of generators of the quan- 5

6 tum cohomology ring is essentially determined by the dimension of the Kahler moduli space, 6

7 which corresponds to the number of marginal chiral-anti-chiral operators of the two-dimensional 7

8 worldsheet theory. For Calabi-Yau fourfolds, however, the ring structure of the quantum coho- 8

9 mology ring is less constrained by target space symmetries. As a consequence, the number of 9

10 generators of their quantum cohomology ring is generically only given by the number of both 10

11 marginal and certain irrelevant chiral-anti-chiral operators. That is to say the number of genera- 11

12 tors cannot simply be deduced from the dimensionality of the Kahler moduli space. 12

13 This basic observation has an interesting immediate consequence on the level of quantum 13

14 periods, which describe quantum corrected volumes of even-dimensional cycles in Calabi-Yau 14

15 manifolds. Namely, we find that while the classical Kahler volume of certain quantum cycles 15

16 vanishes their respective quantum volume can nevertheless be non-zero. As a consequence, in 16

17 the large volume regime there are non-vanishing integral quantum periods of the form 17

18 r r 18

19 U(J) = O(e ■> J) = 0, (1.1) 19

20 in terms of the Kahler form J in flat coordinates. Such quantum periods can never occur in 20

21 Calabi-Yau threefolds as all even-dimensional quantum cycles are governed by the generators of 21

22 their Kahler moduli spaces. Similarly, as a consequence of the Jurkiewicz-Danilov theorem and 22

23 the quantum Lefschetz hyperplane theorem [12-14], this phenomenon seems difficult to realize 23

24 in smooth complete intersection Calabi-Yau fourfolds in compact toric varieties [15-18] — at 24

25 least not within the toric part of the moduli space and not for the quantum periods describable 25

26 in terms of the ambient compact toric varieties. However, for generic Calabi-Yau fourfolds the 26

27 structure of the even-degree cohomology is not entirely determined by the dimensionality of the 27

28 Kahler moduli space anymore. Therefore, the appearance of integral quantum periods purely 28

29 generated by instanton numbers may not come as a surprise. Indeed, such examples have already 29

30 appeared for complete intersection Calabi-Yau fourfolds in ambient complex Grassmannians 30

31 [19],1 and are in general expected for non-complete intersections Calabi-Yau fourfolds in toric 31

32 varieties, as recently also observed in ref. [20]. 32

33 We determine the quantum periods of Calabi-Yau fourfolds as solutions to Picard-Fuchs 33

34 differential equations. With the help of gauge theory techniques [21-25], we extract these Picard- 34

35 Fuchs differential equations of non-complete intersection Calabi-Yau fourfolds with a single 35

36 Kahler modulus for examples with a purely instanton-generated quantum period (1.1). A char- 36

37 acteristic feature of such Calabi-Yau fourfolds are non-factorizable Picard-Fuchs operators of 37

38 order six (or higher). Furthermore, due to the additional quantum period the regular singular 38

39 point of the large volume phase does not have maximally unipotent monodromy with respect 39

40 to the Picard-Fuchs operator. Hence, computing the integral quantum periods becomes more 40

41 challenging, because the integration constants are not entirely determined by the perturbative 41

42 asymptotic behavior, as — for instance — computed by the Gamma class of the Calabi-Yau 42

43 fourfold [26-31]. In addition, we use the regular singular point in Kahler moduli space, where 43

45 , 45 1 The Plücker map embeds complex Grassmannians into projective spaces as non-complete intersections. Thus these

46 Calabi-Yau fourfolds are projective varieties of the non-complete intersection type. As consequence the Jurkiewicz- 46

47 Danilov theorem and the quantum Lefschetz hyperplane theorems are not applicable. 47

1 the quantum volume of the 8-brane vanishes.2 Here, the monodromy behavior of the integral 1

2 quantum periods is determined by a Thomas-Seidel twist [33]. We demonstrate that for the an- 2

3 alyzed examples the knowledge of these two monodromies combined with numerical analytic 3

4 continuation techniques is actually sufficient to unambiguously calculate the integral quantum 4

5 periods. As a non-trivial check we establish — again with numerical analytic continuation tech- 5

6 niques — that the monodromy matrices at the remaining regular singular points in Kahler moduli 6

7 space are indeed integral as well.3 7

8 With the integral quantum periods at hand, we explicitly extract the instanton corrections en- 8

9 tering the quantum cohomology rings, which geometrically amounts to extracting genus zero 9

10 Gromov-Witten invariants. Using global properties of the quantum periods in the vicinity of sin- 10

11 gular points in quantum Kahler moduli space, we determine the generalized topological index 11

12 of the N = (2, 2) superconformal worldsheet theory [38,11]. The genus zero Gromov-Witten 12

13 invariants also define recursively the Klemm-Pandharipande meeting invariants of Calabi-Yau 13

14 fourfolds, which then allow us to enumerate genus one BPS invariants of the examined Calabi- 14

15 Yau fourfolds [39]. The intricate integrality property of these genus one invariants furnishes yet 15

16 another non-trivial check on the proposed integral quantum periods. 16

17 To further check our enumerative results, we present intersection theory techniques that allow 17

18 us to directly enumerate lines with a marked point on complete intersection Calabi-Yau fourfolds 18

embedded in Grassmannians. While these intersection calculations are developed for complete

20 20 intersection Calabi-Yau fourfolds in Grassmannians, our results easily generalize to enumerate

lines on other complete intersection varieties embedded in Grassmannians.

22 22 Finally, let us briefly remark that our findings may have phenomenological applications as

23 well. The study of global properties of quantum periods — in particular the analysis of their 23

25 monodromy behavior around singular divisors in moduli space — exhibits many characteristic 25

26 features of monodromy inflation in string cosmology [40]. In the context of Calabi-Yau fourfold 26

27 compactifications of type IIA strings to two dimensions, the quantum periods (1.1) give rise to 27

28 flux-induced superpotentials of the form 28

30 ^flux(i) = aiti +b + Winst(t) , Winst(t) = O(e2ntlfm') = 0 . (1.2) 30

31 i 31

33 Here the Kahler form J = J2i is expanded in a basis of harmonic two forms Mj. Depending 33

34 on the details of the chosen background fluxes all of the constants ai and b can either be chosen to 34

35 vanish or some of them not to vanish. Assuming further that the mirror Calabi-Yau fourfold of the 35

36 analyzed fourfold has a suitable elliptic fibration, the superpotentials (1.2) can also be interpreted 36

37 in four space-time dimensions. Then the superpotential arises from four-form fluxes in F-theory 37

38 on the elliptically-fibered mirror Calabi-Yau fourfold, where the chiral fields tj parametrize the 38

39 mirror complex structure moduli space in the vicinity of a large complex structure point. Such 39

40 large complex structure points in F-theory have been considered recently in the context of string 40

42 __42

43 2 The existence of such a singularity is predicted by the Strominger-Yau-Zaslow mirror symmetry conjecture [32]. 43

44 3 Combining numerical analytic continuation techniques with the requirement of integral monodromy matrices has for 44

instance been used extensively before in the context of the moduli spaces of Calabi-Yau threefolds [34]. For Calabi-Yau

geometries associated to hypergeometric functions a systematic treatment towards analytic continuation has recently been

46 given in refs. [35,36]. Generalizing further the methods of ref. [37] to resonant periods arising in Calabi-Yau geometries 46

47 would offer a powerful framework to study analytic continuations systematically. 47

cosmology in ref. [41],4 where the hierarchy between polynomial and exponential suppressed terms is explored.

The outline of this work is as follows: In Section 2 we introduce the necessary ingredients and establish the computational techniques to derive the integral quantum periods for the class of studied Calabi-Yau fourfolds. Moreover, we recall some properties of enumerative invariants in Calabi-Yau fourfolds relevant for this work. In Section 3 we exemplify in detail how to compute integral quantum periods and how to extract Gromov-Witten invariants. We tabulate our results for all the studied Calabi-Yau fourfold examples in Appendix A. To further confirm our results, in Appendix B we calculate genus zero Gromov-Witten invariants for Calabi-Yau fourfolds directly using intersection theory methods. Our conclusions are presented in Section 4.

2. Methodology

The aim of this section is to establish the computational tools that are necessary to analyze the quantum periods of the Calabi-Yau fourfolds studied in Section 3. We review certain aspects of the quantum cohomology ring of Calabi-Yau fourfolds. Then we recall gauged linear sigma model techniques to determine the Picard-Fuchs differential equations for the quantum periods. Next we construct the asymptotic behavior of the quantum integral periods — corresponding to B-brane central charges — in the vicinity of the large volume point and the singular locus, where the 8-brane becomes massless. Finally, we describe the numerical analytic continuation techniques that allow us to determine integral quantum periods from their global structure and their asymptotic behavior at certain singular points in the quantum Kahler moduli space.

2.1. Quantum cohomology of Calabi-Yau fourfolds

The chiral-anti-chiral ring of N = (2, 2) worldsheet theories of the Calabi-Yau manifold X is given by its quantum cohomology ring, i.e., the even-dimensional cohomology group 0k Hk,k(X) together with the cup product deformed by genus zero worldsheet instanton correction s [1,5,6,2-4].

Marginal operators of the chiral-anti-chiral ring correspond to cohomology elements of H1,1 (X). For worldsheet theories associated to Calabi-Yau threefolds all chiral-anti-chiral ring elements are generated from such marginal deformations. This is a consequence of the underlying N = 2 special geometry [9-11]. However, for Calabi-Yau manifolds of complex dimension four or greater the chiral-anti-chiral ring need not be generated just by marginal chiral-anti-chiral ring elements anymore, but may require additional generators from the higher dimensional co-homology groups Hk,k(X) for k > 1 [7,8]. We study this phenomenon of quantum cohomology rings in the context of Calabi-Yau fourfolds.

A standard technique to study the quantum cohomology rings of a compact Calabi-Yau manifold X uses a quantum version of the Lefschetz hyperplane theorem [12-14]. That is to say, the information about the quantum cohomology ring of the Calabi-Yau manifold X is inferred from the quantum cohomology of some ambient space.

Mirror symmetry furnishes another very powerful — but yet indirect method — to deduce the quantum cohomology [44-48]. For compact complete intersection Calabi-Yau manifolds in toric

4 More generally, inflationary models in string cosmology arising from F-term axion monodromies have been introduced in refs. [42,43].

varieties the Batyrev-Borisov mirror construction relates Kahler moduli induced from the ambient space to polynomial complex structure deformations given in terms of the defining complete intersection equations [15,49,50,48]. That is to say, the structure of the quantum cohomology ring is again inferred via mirror symmetry from the cohomology elements induced from some ambient toric variety.

As a consequence, for Calabi-Yau manifolds X embedded in toric ambient spaces X£ of complete fans £, one typically studies the quantum cohomology ring of those cohomology elements in 0k Hk,k(X) that are induced via pullback from the cohomology ring H*(X£) of the toric ambient space X£. The Jurkiewicz-Danilov theorem for complete compact toric varieties X£ guarantees that the entire cohomology ring H*(X£) is generated by Hx,1(X£). As a result (the part of) the quantum cohomology ring of k Hk,k(X) induced from the embedding of X into X£ is also generated by Hx,1(X). Hence for compact smooth Calabi-Yau fourfolds X embedded as complete intersections in toric varieties the part of the quantum cohomology induced from the toric ambient space is always generated by marginal operators of the chiral-anti-chiral ring.

To study the more general — and actually generic — structure of the quantum cohomology ring with additional generators apart from marginal operators, we focus on Calabi-Yau four-folds X embedded as complete intersections in compact complex ambient spaces Y, whose even-dimensional cohomology ring is not just generated by H X,1(Y). This happens for instance for non-toric GIT quotients Y, which in the physics literature arise from two-dimensional N = (2,2) non-Abelian gauged linear sigma models [51-59,20,60]. The simplest examples of this kind arise from complete intersection Calabi-Yau fourfolds X embedded in complex Grass-mannians Y [19]. Namely, for Grassmannians Y = Gr(k, n) with k > 2, the cohomology group H1,1 (Y) is generated by the Schubert cycle o1, while the cohomology group H2,2(Y) is generated by the two Schubert cycles oi,i and o2, related to oi via the relation o2 = oi,i + o2. Thus oi alone does not generate H2,2(Y); an additional generator is required. From the gauged linear sigma model point of view, such GIT quotients are obtained from two-dimensional non-Abelian gauge theories based on the gauge group U(k) [52-54,56,60].

In this note we focus on Calabi-Yau fourfolds X with dim H11 (X) = 1 — that is to say with a single Kahler modulus — and with one additional non-trivial generator in H2,2(X) induced from the embedding ambient space Y, i.e., dimH2,2(Y) = 2. As mentioned before such scenarios occur for instance for Calabi-Yau fourfolds embedded as complete intersections in Grassmannian ambient spaces or flag manifolds. From a gauged linear sigma model point of view, such examples can be constructed from gauge groups U(1) x G (or discrete quotients thereof) with the semi-simple Lie group factor G. Here, the Fayet-Iliopoulos term of the Abelian gauge group factor U(1) realizes the single Kahler modulus [51], while the non-Abelian gauge group factor G can give rise to additional operators, geometrically corresponding to elements of the ambient space cohomology group H2,2(Y) [61]. An example of a Calabi-Yau fourfold X of this more general kind has been constructed in ref. [20].

Thus we determine a chiral-anti-chiral ring of a Calabi-Yau fourfold X with the ring elements 01 generating H 1,1(X) and the ring elements 02,(i) and 02,(2) furnishing two independent generators of H2,2(X).5 The general structure of the quantum product then yields

01 * = C(1)(q) 02,(1) + C(2\q) 02,(2), (2.1)

5 Strictly speaking, we are considering a subring of the entire chiral-anti-chiral ring. This subring is generated by the ring elements induced from the embedding space Y .

1 where the coefficient functions are given in terms of the worldsheet instanton action q = e2nit 1

2 with the flat coordinate t as 2

œ d2 qd

4 C(a)(q) = c(a) + £n^d^, a = 1,2. (2.2) 4

5 ,, ,d 1 - qd 5

7 Here the classical ring structure constants are defined by the cup product ф1 U ф1 = ^ас(а)ф2,(а). 7

8 The integral genus zero worldsheet instanton numbers of degree d are denote by n0ad, where the 8

9 superscript refers to a single marked point constrained to lie on the algebraic cycle class ф2,(а). 9

10 These genus zero worldsheet instanton numbers recursively define the symmetric Klemm- 10

11 Pandharipande meeting invariants mdl,d2 = md2,dl according to [39] 6 11

12 12 13 md1,d2 = 0 for di < 0 or d2 < 0 , 13

,(a)Jb)

14 mdi,d2 = ^2gabn^lnOd + mdi,d2-di + mdi-d2,d2 for di = d2 , 14

15 a,b (2.3) 15

16 d-i ' 1

17 V^ (a) (b) . ST^ (a) (b) ST^ 17

md,d =2L gabc2 n0,d + L gabno,dn0,d - ¿^ mk,d-k

a,b a,b k=i

20 Here gab is the intersection pairing gab = jхф2,(a) U ф2,(Ь), and c2a) are the expansion coefficient 20

21 of the second Chern class of the Calabi-Yau fourfold X, i.e., c2(X) = J2a с^Фъ^) viewed as 21

22 a cohomology element of H2,2(X). The genus zero invariants n0ad together with the meeting 22

23 0,d 23

invariants mdi d2 are essential to extract the integral genus one invariants ni d of the Calabi-Yau

24 ' '24

fourfold X, as all of them appear non-trivially in the multi-covering formula for the rational

25 genus one invariants Ni d given by [39] 25

28 !>, dqd = £ni, d°^f)qdé + 24 I £ gab^n™ m^d-k I log(i - qd) . 28

29 ^ 24 1 I 29

d d,t \d,a,b d,k

30 (2.4) 30

32 Here a1(l) = Y,¡\e i is the divisor function such that the integers n1,d enumerate elliptic curves 32

33 rather than BPS states; cf., with the discussion in refs. [38,62,39]. 33

34 The genus one invariants n1,d appear in the topological limit f1°p of the generalized topolog- 34

35 ical index of the N = (2, 2) superconformal worldsheet theory [38,11], which for Calabi-Yau 35

36 fourfolds with A2,1 = 0 takes the form [63,38,11,39] 36

37 , 1 д \ 37

38 ^1°p = (24 - A1,1 - 2) log n0pt + logdet^— + £ ba log Д.&. (2.5) 38

40 Here, x is the Euler characteristic and n^pt (z) denotes the fundamental quantum period with 40

41 respect to the large volume point of the Calabi-Yau fourfold X. Furthermore, the (vector-valued) 41

42 function z(t) is the mirror map of the algebraic coordinates z to the flat coordinates t. It is the

43 inverse of the (vector-valued) function 43

45 __45

46 6 Note the genus zero invariants Що^(.Ф2,(а)) of ref. [39] relate to the genus zero invariants n^d defined here with the 46

47 identity llcod^ia^ = fx ф2,(а) U (E b nOd02,(b)). 47

1 1 П я л

1 t(z) = — , (2.6) 1

2 2Ж1 Пар1 2

with a basis of 2-branes С representing the Mori cone of the Calabi-Yau fourfold X. Finally,

Aa are the factors of the discriminant locus of the quantum Kahler moduli space with rational coefficients (including the large volume divisor). The coefficients ba reflect the holomorphic ambiguity of Fjop [38,11], and they need to be determined by the boundary conditions in the quantum Kahler moduli space. Namely, in the vicinity of the large volume point t the 8 index F1top for Calabi-Yau fourfolds takes the asymptotic form [38,11] 8

10 FÍ0P = сз(X) U J + (regular). (2.7) 10

11 1 24 у ii

12 X 12

13 Here, c3(X) and J are the third Chern class and the Kahler form of the Calabi-Yau fourfold X, 13

14 respectively. Another boundary condition yields the vicinity of the divisor in the quantum Kahler 14

15 moduli space, where the volume of the 8-brane OX vanishes. There one expects the universal 15

16 asymptotic behavior [39] 16

17 top Í 17

18 ÍT = -24 log Aox + (regular), (2.8) 18

19 where Aqx is the factor of the discriminant locus that vanishes at this divisor. 19

20 If the above boundary conditions are not sufficient to fix all coefficients ba, we employ the 20

21 additional boundary conditions from further singular loci in the quantum Kahler moduli space 21

22 characterized by vanishing quantum volumes of other branes, which exhibit the same universal 22

23 asymptotic property (2.8). Employing these boundary conditions, we observe for all our exam- 23

24 ined examples that the genus one invariants at degree one and two vanish, i.e., nÍ; Í = nÍj2 = 0. 24

25 For the examples studied in this note, we will explicitly extract the above described integral 25

26 invariants for low degrees. Due to the intricate multicovering formulas (2.2) and (2.4) the con- 26

27 (a) 27

firmed integrality of the invariants n0', and nÍ d yields non-trivial consistency checks on our

28 0 ' 28 findings. Note that for Calabi-Yau fourfolds with a single Kahler modulus there are at each de-

29 gree as many genus zero Gromov-Witten invariants as there are non-trivial chiral-anti-chiral 29

30 quantum cohomology ring elements in H2,2(X). However, independently of the quantum coho- 30

31 mology ring structure there is just a single genus one BPS invariant nÍ,d because these invariants 31

32 do not depend on a marked point. 32

34 2.2. Picard-Fuchs operators via gauged linear sigma models 34

36 Our approach to extract the quantum cohomology ring is to first determine the Picard-Fuchs 36

37 differential equation that governs the quantum periods of the examined Calabi-Yau fourfold X. 37

38 Using variation of Hodge structure techniques of the holomorphic four-form ^ of the mirror 38

39 Calabi-Yau geometry furnishes a standard technique to derive the Picard-Fuchs operators for the 39

40 quantum periods. However, this approach requires a construction of the mirror Calabi-Yau four- 40

41 fold, which for non-toric ambient spaces or non-complete intersection Calabi-Yau fourfolds in 41

42 toric ambient spaces — as studied in this note — can be rather cumbersome or is even unknown.7 42

44 --44

7 Using the Plücker embedding of Grassmannians in projective spaces, the work of ref. [64] reduces the problem

45 45 of constructing a mirror Calabi-Yau fourfold to the Batyrev-Borisov mirror recipe for complete intersections in toric

46 varieties, which is further generalized to complete intersections in flag manifolds in ref. [65]. A mirror proposal has also 46

47 been presented for certain non-complete intersection Calabi-Yau manifolds in toric varieties in ref. [66]. 47

1 Here we follow a different approach that allows us to determine the Picard-Fuchs operators 1

2 directly from the sphere or hemisphere partition function of the gauged linear sigma models, 2

3 which describe the Calabi-Yau fourfolds under consideration as a geometric target space phase. 3

4 The sphere partition function ZS2 computes the exponentiated sign-reversed Kähler potential of 4

5 the Calabi-Yau variety [23], while the hemisphere partition function ZD2dD2 directly gives rise 5

6 to quantum periods for appropriate boundary conditions of the gauged linear sigma model at 6

7 3D2 [31]. Both quantities are annihilated by the Picard-Fuchs operators Li, i.e., 7

9 Li(Za,0a)Zs2 (Za) = 0 , Ci(Za,0a)Z02,aö2 (Za) = 0 , 0a = Za~ , (2.9) 9

10 dZa 10

11 in terms of the algebraic coordinates Za with a = 1,..., h1,1 (X). Thus the Picard-Fuchs op- 11

12 erators Li can be determined by the requirement to annihilate ZS2(Za) and ZD2dD2(Za). This 12

13 approach has also been employed for instance in refs. [19,20]. 13

14 As we focus on Calabi-Yau geometries with a single Kähler modulus, there is just a sin- 14

15 gle Picard-Fuchs operators L(z, 0) depending on a single algebraic coordinate z. While for 15

16 Calabi-Yau threefolds such a Picard-Fuchs operator is always of order four (due to the afore- 16

17 mentioned ring structure of the quantum cohomology ring), for Calabi-Yau fourfolds the order 17

18 of the Picard-Fuchs operator is given by8 18

20 ord L(z,0) = 4 + #(<fc). (2.10) 20

21 Here #(<b2) denotes the number of chiral-anti-chiral ring generators associated to H2,2(X) that 21

22 22 non-trivially participate in the quantum product fa * Thus the order of the Picard-Fuchs

operator L(z, 0) is at least five or higher. For the particular quantum products (2.1) studied in

this work we obtain Picard-Fuchs operators of order six.

Note that — from the A-variation of Hodge structure point of view (see for instance refs. [67, 27]) — a large volume point in quantum Kähler moduli space of a Calabi-Yau «-fold Xn is always of unipotent monodromy of index «. For one-dimensional quantum Kähler moduli spaces and from a mirror symmetry perspective this is a consequence of the Landman monodromy theorem [68] applied to the middle dimensional cohomology of the mirror Calabi-Yau «-fold X. It states that the monodromy transformation M acting on H« (X) about a singular point in the mirror complex structure moduli space is quasi-unipotent with index of at most «, i.e., (Mk — id)n+1 = 0 for some integer k. In particular, a large complex structure point in the complex

33 structure moduli space of X« — which is mirror to a large volume point in quantum Kähler 33

34 moduli space of X« — is unipotent with the maximal index «, i.e., (M — id)«+1 = 0 but (M — ^

35 id)« = 0. This implies that the Picard-Fuchs differential equation associated to X« is always 35

36 of unipotency of index « at large volume — independently of the order of the Picard-Fuchs 36

37 operator. 37

39 In particular, large volume points of Calabi-Yau fourfolds are always points of unipotency 39 of index four. Hence they furnish regular singular points of maximally unipotent monodromy of

the differential equation only for Picard-Fuchs operators of order five. This is, for instance, the case for those complete intersection Calabi-Yau fourfolds in toric varieties with a single Kähler

43 modulus studied in refs. [15,18,67,39,69]. For the examples of order six Picard-Fuchs operators 43

44 appearing in ref. [19] and studied here, the large volume points in the quantum Kähler moduli 44

45 space are not of maximally unipotent monodromy anymore. It is this property of the order six 45

47 8 The highest power of the logarithmic derivative 9 is the order of the differential operator L(9, z). 47

1 Picard-Fuchs operators, which yields the interesting structure of the quantum cohomology ring 1

2 discussed in Section 2.1. In the following we also refer to such examples as Calabi-Yau fourfolds 2

3 with Picard-Fuchs operators of non-minimal order. 3

5 2.3. B-branes, quantum periods and monodromies 5

7 In a compact Calabi-Yau manifold X of complex dimension d the topological B-branes E• 7 on X are represented by the objects in the derived category of bounded complexes of coherent

9 ь 9

sheaves Db(X) [70-72]. Furthermore, to each B-brane E• we assign a quantum period Ug• (J), which is a function of the (complexified) Kahler class J = ^a taDa in terms of the Kahler moduli ta with a = 1, ..., h1'1(X) and the generators of the Kahler cone given in terms of divisors Da .9 For stable BPS branes E• the quantum periods enjoy the interpretation of a (Kahler moduli dependent) central charge, whose magnitude is its BPS mass that enjoys also the interpretation of a calibrated quantum volume. For further details on B-branes and their notion of stability, we

refer the reader for instance to the review [73].

16 16 The quantum periods depend only on the B-brane charges, which are captured by elements

17 of the algebraic K-group K0lg(X) [74]. In this note we want to construct a basis of (torsion 17

18 free) integral quantum periods for B-branes, which corresponds to integral generators of the 18

19 torsion-free part of the algebraic K-theory group K^g(X). The asymptotic behavior of quan- 29

21 tum periods U£.y in the large volume regime of the Calabi-Yau manifold X constrains — and 21

22 for large volume points with maximally unipotent monodromy unambiguously determines — 22

23 the integration constants of the integral quantum periods Ug. (J) as solutions to the associated 23

24 system of Picard-Fuchs differential equations. In terms of the flat Kahler coordinates the large 24

25 volume asymptotics reads [30] 25

27 Ug.!(J) = eJrC(X) chE'v . (2.11) 27

28 28 X

30 Here Гс(X) is the (multiplicative) characteristic Gamma class, which for Calabi-Yau manifolds 30

31 with c1 = 0 enjoys the expansion10 31

32 1 iZ(3) 1 2 32

33 rc(X) = 1 + — C2 + Z/сз + —— (7c2 -4c4) + ..., (2.12) 33

34 24 8n 3 5760 34

35 where ck = ck(X) are the Chern classes of X. 35

36 For the Calabi-Yau manifold X (of real dimension 2d) there are some universal B-branes that 36

37 always correspond to integral generators of the K-theory group K0g(X): 37

39 • The 2d-brane of the structure sheaf OX — with the trivial Chern character ch OX = 1 — 39

40 readily yields the asymptotic quantum period 40

42 nOl (J) = f eJrc(X). (2.13) 42

43 X 43

46 9 For simplicity we assume here that the Kahler cone is generated by h1,1(X) divisors. 46

47 10 The gamma class tc(x) is based upon the series Tp(z) = e4 T(1 — 2^7). 47

1 • A collection of 2(d - 1)-branes E; associated to the Kahler cone divisors Da are given by 1

2 the complexes 2

4 e;-. 0 Ox(-Da) Ox 0 • (2.14) 4

5 Their asymptotic periods read 5

6 . 6 7 n^.y(J) = J eJTc(X)(1 - ch Ox(Da)) . (2.15) 7

• We construct a collection of 2-branes €• as follows: Given the embedded Mori cone curves

10 a 1/2 10

11 i: Ca c—> X dual to the Kahler cone divisors Da, we consider their structure sheaf Oca(Kc' ) 11

12 twisted by a spin structure K12 of Ca. Then the 2-branes C'a are given by 12

14 C;= KOCaK/2) , (2.16) 14

15 0 0 15

in terms of the K-theoretic push-forwards i: K0(Ca) — K (X). The Chern character of С•

16 a 16

17 is computed by the Grothendieck-Riemann-Roch formula

18 i* (chKl/2Td(Ca^ 18

19 ch C• = -a-= [Ca ] , (2.17) 19

20 a Td(X) L ^ ' 20

21 because chK,V2Td(Ca) = (1 + 1 Kr )(1 - 2Kr ) = 1 and for Calabi-Yau manifolds 21

22 ca 2 a 2 a 22

23 Td1 (X) = 1 c1 = 0. Here [Ca] denotes the Poincare dual cohomology class of the curve 23

24 Ca, such that its asymptotic quantum period becomes 24

26 ^¡(Л = (-1)d-1 f eJ [Ca] = (-1)d-1ta - (2.18) 26

27 X 27

28 • Finally, we consider the skyscraper sheaf Opt for 0-brane located at a point i: pt — X in the 28

29 Calabi-Yau manifold X. Employing again the Grothendieck-Riemann-Roch theorem for 29

30 the Chern character of the K-theoretic push-forward ch iipt we find the asymptotic period 30

31 „ 31

32 nOyt(J) = (-1)d eJ [pt] = (-1)d . (2.19) 32

33 J 33

35 For Calabi-Yau threefolds the above described integral quantum periods generate all central 35

36 charges associated to the torsion-free elements in K0lg(X). However, for higher-dimensional 36

37 Calabi-Yau manifolds we also need to construct algebraic cycles representing ^-branes of even 37

38 dimension p = 4, ..., 2(d — 2). In particular, for Calabi-Yau fourfolds we determine the quantum 38

39 periods of algebraic cycles of 4-branes for cohomology elements in H2,2(X) П H4(X, Z). As 39

40 such algebraic cycles depend on the details of the Calabi-Yau manifold X, we construct them 40

41 for the explicit examples studied in Section 3. 41

42 For Calabi-Yau fourfolds with Picard-Fuchs operators of non-minimal order the large volume 42

43 asymptotics of integral quantum periods does not determine all integration constants of their 43

44 solutions to the Picard-Fuchs differential equations. As a consequence there are (integral linear 44

45 combination) of quantum periods with vanishing classical terms in the large volume regime. 45

46 Such quantum periods are purely instanton generated as described in formula (1.1). Then the 46

47 integration constants must be further constrained by monodromies around other singularities 47

1 in moduli space. They are deduced from the monodromies of the associated B-branes about 1

2 singularities in moduli space, which are described by Fourier-Mukai transformations acting upon 2

3 the derived category of bounded complexes of coherent sheaves Db(X) [75-78]. This allows us 3

4 to derive the monodromy behavior of the integral quantum periods. 4

5 The Strominger-Yau-Zaslow picture of mirror symmetry for Calabi-Yau d-folds X con- 5

6 jectures a singular point tox in the quantum Kähler moduli space, where the 2d-brane — 6

7 represented by the structure sheaf OX — becomes massless [32].11 The Seidel-Thomas twist 7

8 captures the monodromy at the singular point toX, which is represented by the Fourier-Mukai 8

9 kernel [79,33] 9

11 KOx = ConeП: E,v ^E'^ OA) . (2.20) „

12 12 The Seidel-Thomas twist is interpreted as the formation of bound states between the brane £'

— adiabatically encircling the singularity toX — and the (massless) brane OX, while the index x(E', Ox) of the open strings stretching between the branes E• and OX becomes the index for

the (relative) number of formed bound states [80,77]. Therefore, on the level of quantum periods the Seidel-Thomas twist induces the monodromy transformation

18 Mt : П£.- xE, Ox)uox. (2.21) 18

19 X 19

20 Here П£. and ПoX are the quantum periods of the branes E• and OX, respectively, whereas the 20

21 index of open strings is computed by the Hirzebruch-Riemann-Roch pairing [74,81] 21

23 x(EF') = f Td(X) ch(E*v)ch(F') . (2.22) 23

24 24 X

26 We observe that the open-string index x(OX, OX) simplifies to the arithmetic genus of the 26

27 Calabi-Yau manifold X, i.e., 27

29 x(OX, Ox) = i Td(X) = Y(-1)ph0,p(X) . (2.23) 29

30 J p 30 X p

32 Thus for Calabi-Yau threefolds with SU(3) holonomy and not a subgroup thereof, we have 32

33 x(OX, Ox) = 0 for the open-string index between two 6-branes OX. Furthermore, the open- 33

34 string index between a 0-brane Opt and a 6-brane Ox computes to x(Opt, Ox) = 1. Hence, for 34

35 the dual pair of quantum periods (npt, noX) of Calabi-Yau threefolds, the Seidel-Thomas twist 35

36 yields the characteristic monodromy 36

38 м°.-\П&)1 -')(П&)- <2-24' 38

which — in the four-dimensional N = 2 effective theory of type II strings on Calabi-Yau three-

42 folds — is due to additional massless BPS blackhole states at the singularity toX [82,83]. 42

42 For Calabi-Yau fourfolds with SU(4) holonomy and not a subgroup thereof, the arithmetic 42

44 genus yields the open-string index x(OX, Ox) = 2. Hence the monodromy (2.21) of tox maps ^

46 11 In order for the 2d-brane to become massless a suitable path from the large volume point to the singularity tQX must 46

47 be specified. 47

the quantum period of the 8-brane nox to -noX. As result we find that, applying the mon-odromy transformation (2.21) twice, maps any quantum period back to itself. That is to say we find that12

Mf0x = id , (2.25)

where MtOx generates a Z2 group action on the set of all quantum periods. The Z2 monodromy around the singularity toX in Calabi-Yau fourfolds has previously been studied in refs. [84,69].

2.4. Numerical analytical continuation

Starting from the Picard-Fuchs operator L(z, 0) for the periods in the quantum Kahler moduli space — for instance to be determined by gauged linear sigma model methods described in Section 2.2 — we now describe the use of numerical analytic continuation techniques to establish the global structure of quantum periods. In particular this allows us to determine linear combinations of solutions to the Picard-Fuchs differential equations corresponding to integral quantum periods.

In a local patch Ua on the quantum Kahler moduli space in the vicinity of the origin of the algebraic coordinate za the Picard-Fuchs operator takes the form

La(Za,0a) = J2 ^ , 0a = Za— , (2.26)

k=o dZa

in terms of some polynomials h(a)(za). The integer n is the order of the Picard-Fuchs operator, which — as discussed — for Calabi-Yau fourfolds is at least five but can be greater. Note that the operator La(za, 0 a) is also well-defined in the vicinity of the origin of the algebraic coordinate zp = za - z' with z' = <x>, which allows us to rewrite the Picard-Fuchs operator in the local patch Up associated to the new algebraic coordinate zp according to

Lp(zp,0p) = zp ■ La {zp + z',(1 + jpWp) . (2.27)

Note that the prefactor znp renders the new coefficient functions h(k(zp) to be polynomial. Similarly, for z' = <x> we set zp = z-1 and have

Lp(zp,0p) = zm ■ La (z-1, -0p) , (2.28)

where m is the maximal degree of the polynomials h(k in eq. (2.26).

For any operator La(za, 0 a) there are n linearly independent solutions n^izce) to the PicardFuchs differential equation, i.e.,

La (za,0a)n(k)(za) = 0 , (2.29)

which can be determined by the Frobenius method as an infinite series expansion in the local coordinates za. As illustrated in Fig. 1, these solutions are valid within a radius of convergence around the origin of the coordinate z that is given by the distance to the closest regular singular

12 More generally, 1 + (-1)d is the arithmetic genus of any Calabi-Yau d-fold with SU(d) holonomy of dimension d > 0. Hence, at the singularity tOx we find the monodromy behavior (2.24) for odd and (2.25) for even dimensional Calabi-Yau manifolds, respectively.

Fig. 1. Partwise illustration of a Kahler moduli space, which shows three regular singular points, zi, Z2 and 43. The

solutions n^ for a = 1, 2, 3 converge within circles around za whose radii are given by the distance to the closest other za. On the overlaps of convergence areas — such as the purple shaded area for the circles around zi and Z2 — there is a GL(n, C) transformation relating the respective solutions n®. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

point. To be precise, in particular we determine solutions in the vicinity of regular singular points. This means that we allow for a pole at or a branch cute emanating from the origin within the radius of convergence. In the following we denote by the patch Ua the disk of convergence for the solutions to the Picard-Fuchs operator around the origin of the local coordinate za.

Regular singular points of the Picard-Fuchs differential equations are points in the quantum Kahler moduli space that exhibit non-trivial monodromy behavior. Let za be a regular singular point. In terms of the period vector fta = (n^, ..., n^)1 the monodromy matrix Ma is given by

na {zae2ni) = Ml • na(za), (2.30)

deviating from the identity matrix. A necessary condition for za to be a regular singular point is that

Za = 0 or Za = ~ or h^ (Za) = 0 . (2.31)

Note, however, that the converse is not true in general.13

Since the Picard-Fuchs operator is defined globally, the quantum periods as their solutions can be analytically continued over the entire quantum Kahler moduli space. Therefore, as long as the disks Ua and Up overlap, there exists a transformation matrix Aap in GL(n, C) that relates their solutions on the overlap Ua n Up as

na(Za) = Aap • ftp(zp(za)) , (2.32)

where we express the local coordinate zp in terms of za in the overlap Ua n Up.

By repeating this analytic continuation successively from patch to patch, we see that a set of quantum periods n can be analytically continued along any path (avoiding the regular singular points) in the quantum Kahler moduli space. From the analytic continuation along suitable paths we then deduce the monodromy behavior of a basis of quantum periods around regular singular points za according to eq. (2.30). As the quantum periods describe central charges of B-branes,

13 For the general theory of ordinary differential equations with regular singular points, see for instance ref. [85].

the monodromy matrix Mza (and its inverse) for the regular singular point za = 0 must actually be integral for a generating basis of integral quantum periods [81,86-89], i.e.,14

Mza e GL(n, Z) for za = 0 a regular singular point. (2.33)

Our goal is now to find a generating set of integral quantum periods together with their integral monodromy matrices Mza around regular singular points za by combining the methods of Section 2.3 with the strong integrality constraint (2.33). Then the integral quantum periods in turn allow us to extract the quantum cohomology rings and Gromov-Witten invariants discussed in Section 2.1.

In practice we perform the analytic continuations numerically. This is done by inserting n2 different points for z — chosen from the overlap and according to the prescription to be given in the next paragraph — in eq. (2.32), which gives a set of n2 linear equations for the n2 entries of Aap. If the period vectors na and np could be evaluated at a given point exactly, the results would not depend on the particular choice of the points z. However, since we approximate their value up to a certain fixed expansion order in the respective variables za and zp only, the resulting values of the periods are approximations themselves. In order to get an estimate of the error, we choose the n2 values for z randomly several times and check, how much the results fluctuate. Moreover, we perform the continuation in both directions and check, to what precision the products Aap • Apa and Apa • Aap agree with the unit matrix. A final check of the numerical precision is, whether the appropriately ordered product of all monodromy matrices — representing a contractible path of analytic continuation with respect to a fixed basis of periods — indeed equals unity.

As mentioned before the periods at a regular singular point necessarily involve functions with a branch cut, such as roots and logarithms. When choosing values for z as described in the previous paragraph, one has to ensure that they are always on a definite side of these branch cuts with respect to a chosen path of analytic continuation. In the vicinity of a particular regular singular point we work with implementations of J^z and ln z, which have their branch cuts on the negative real axis. Since all regular singular points turn out to be located on the real axis (in terms of the algebraic coordinate zLV of the regular singular point associated to the large volume limit), all branch cuts are then located on the real axis.15 Our convention is to choose all values for z above the real axis of with respect to the coordinate zLV.

Let us close this section with a practical remark: The area of convergence associated to a regular singular point always intersects with that of another regular singular point. It is thus in principle possible to analytically continue the periods at these two points directly to each other. If, however, the overlap of convergence areas is close to the border of converge for one of the points, the corresponding periods will convergence very slowly. For a high numerical precision one would hence have to expand these periods to very high orders, which is computationally expensive. In these situations it can be better, to perform the continuation in several steps via appropriately chosen regular points in between the two singular points.

14 In the context of Calabi-Yau threefolds, N = 2 special geometry restricts the monodromy action on integral quantum periods to integral symplectic transformations [9-11]. For Calabi-Yau fourfolds algebraic relations among quantum periods put similar but yet less restrictive constraints on the possible integral monodromy transformation matrices Mza [90,91,69]. It would be interesting to study the properties of these algebraic constraints systematically, so as to further develop the notion of N = 1 special geometry [92,93].

15 Note that the negative real axis is mapped to itself under f : z ^ z—1. Hence, the branch cuts of periods in the vicinity of zlv = ^ are also located on the negative real axis.

1 3. Examples 1

3 In this section we discuss in detail two examples of Calabi-Yau fourfolds with a single Kahler 3

4 modulus, whose Picard-Fuchs operators are of order six. We explicitly construct a basis of inte- 4

5 gral periods on the entire quantum Kahler moduli space and determine the monodromy matrices 5

6 in this basis. Furthermore, for these examples we work out the quantum cohomology ring and 6

7 determine the genus zero Gromov-Witten invariants. Due to the non-maximally unipotent mon- 7

8 odromy property at large volume arising from the Picard-Fuchs operators of order six there are 8

9 two independent genus zero Gromov-Witten invariants at each degree. Using the recursive defi- 9

10 nition of the Klemm-Pandharipande meeting invariants we deduce the genus one BPS invariants 10

11 as well. Their non-trivial integrality properties furnish a consistency check on our calculations. 11

12 Our results for these and further Calabi-Yau fourfold examples are tabulated in Appendix A. 12

14 3.1. Calabi-Yau fourfold X1,4 c Gr(2, 5) 14

16 We describe the Calabi-Yau fourfold Xi,4 as a complete intersection of codimension two in 16

17 the complex six-dimensional Grassmannian Gr(2, 5). The Schubert classes ak and ak-a,a with 17

18 1 < a < |_§J generate the individual cohomology groups H2k(Gr(2, 5), Z) (while the cohomol- 18

19 ogy ring H*(Gr(2, 5), Q) is generated by a1 and a2). We realize the family of Calabi-Yau four- 19

20 folds 1: X1,4 Gr(2, 5) as the zero locus of sections of the rank two bundle O(a1) © O(4a1), 20

21 such that [X1,4] = 4a12 is the class of the Calabi-Yau fourfold in Gr(2, 5). Using standard Schu- 21

22 bert calculus techniques — see, e.g., ref. [94] — together with intersection formula 22

23 if 23

24 1* a = 4 af U a, (3.1) 24

25 X14 Gr(2,5) 25

26 26 we determine the intersection numbers of the Schubert cycles on the Calabi-Yau fourfold X1,4

to be16

29 I a4 = 20 , I a2 U a1,1 = 8 , / a? U a2 = 12 , f a1 U a3 = 4 , 29

31 Xl,4 Xl,4 Xl,4 Xl,4 (3.2) 31

32 f af 1 = 4 , f a1,1 U a2 = 4 , fa2 = 8 , f a2,2 = 4 . 32

34 X1,4 X1,4 X1,4 X1,4 34

35 Combining the Lefschetz hyperplane theorem together with Poincare duality we further deduce 35

36 the relations 2a3 ~ a2,1 and a3,1 ~ a2,2 among Schubert classes on the Calabi-Yau fourfold X1,4 36

37 as well as the cohomology generators 37

38 H°(X1,4, Z) = «1» , H2(X1,4, Z) = «a1» , H4(X1,4, Z) D«au,a2» , 38

39 , , , , (3.3) 39

40 H6(Xi,4, Z) = «as» H8(XM, Z) = «a2,2»

41 . 41

42 For the middle dimensional cohomology group H 4(X^4, Z) the Lefschetz hyperplane theorem 42

43 only states that the pullback 1* acts injectively. This implies that the classes a2 and a1,1 are lin- 43

44 early independent in H 4(X1,4, Z). However, these classes are not necessarily integral generators 44

45 of H4(X1,4, Z). Finally, by adjunction the total Chern class of X1,4 reads 45

47 16 For ease of notation we denote the pullbacks i*ak and 1*0^1 ^2 als° by ak and ,k2' respectively. 47

10 11 12

20 21 22

c(Xi,4) =

c(Gr(2, 5))

аз 02 2

= 1 + (8ai,i + 7а2) - 440-3 + 1848^p

(1 + ах)(1 + 4ах) ^ 4 4

which in particular shows that the first Chern class vanishes and determines the Euler characteristic x = 1 848 of the Calabi-Yau fourfold XM.

The Kähler class of the ambient Grassmannian space reads J = ta1, and it canonically induces the Kähler class J on the Calabi-Yau complete intersection Xi,4. This allows us to determine the asymptotic periods according to

П£Г(0 = J etai Tc(XiA) chE'

In addition to the described canonical B-branes we find additional B-branes arising from algebraic four cycles. There is the algebraic four cycle S1 of the zero section of O(a1) © O(a1) intersected with X1,4 and there is the algebraic four cycle S2 of the zero section of the rank two universal subbundle U of Gr(2, 5) intersected with X1,4. The associated 4-brane S'g are the push-forwards i\Si for I = 1, 2. Their Chern characters are computed by the Grothendieck-Riemann-Roch theorem and read

ch S' = (а1,1 + а2) - 5аз + а2,2 ,

ch S2 = а1,1 + 1 а2,1 + 1 а2,2

For the tuple of B-branes

E' = (E*)k=o,...,5 = (°Pt, SI Si E', OX) ,

given in terms of the canonical B-branes together with the 4-branes S*, we now determine with

eq. (3.5) their asymptotic integral period vector ftasy = ( nas.y) to be

V k J k=0,...,5

Пasy (t) =

1^3 --5" t

10t2 + 20t 4t2 - 4t - 5t2 - ^

2 t 12

51^ ^12 _ 55iZ(3)t . 61 + 121 П 3 1 + 144

55iZ(3) л 3

The symmetric intersection pairing is readily computed to be

X(E', E') =

/0 0 0 0 0 1 \

0 0 0 0 1 0

0 0 20 8 10 24

0 0 8 4 -8 6

0 1 10 -8 -14 -7

1 0 24 6 -7 2 /

with the help of eq. (2.21) to be

41 which determines the monodromy matrix Mtax

42 1 0 0 0 0 0 \

43 0 1 0 0 0 0

44 Mtax = 0 0 1 0 0 0

45 0 0 0 1 0 0

46 0 0 0 0 1 0

47 -1 0 -24 -6 7 -1

(3.10)

10 11 12

20 21 22

10 11 12

20 21 22

3.1.1. Picard-Fuchs system

In ref. [19] Honma and Manabe analyze the Calabi-Yau fourfold X1i4 with gauge theory techniques as described in Section 2.2. For the quantum periods they find the order six Picard-Fuchs operator

L(z, 0) = (0 - 1)05 - 8z(20 + 1)(40 + 1)(40 + 3) (l102 + 110 + э) 0 - 64z2(20 + 1)(20 + 3)(40 + 1)(40 + 3)(40 + 5)(40 + 7) .

(3.11)

Here, z is the local algebraic coordinate in the large volume regime. In addition to the large volume limit at z = 0 there are three additional regular singular points at z = œ, z = zi and z2, where the latter two points arise from the zero locus of the discriminant factor

A(z) = 1 - 2816z - 65536z2 , (3.12)

i.e., zi « -0.043 and z2 « 3.5 ■ 10-4.

Note that the same discriminant locus (3.12) arises directly in the gauged linear sigma model description of the Calabi-Yau fourfold X1,4. In this context the discriminant locus A(z) describes the locus in the quantum-corrected Fayet-Iliopoulos parameter space with emerging non-compact strata in the gauge theory moduli space [47]. Analogously as in refs. [54,57,56] — comparing to the expression (3.12) of the discriminant — we find that all singularities arise from non-compact strata attributed to the pure Coulomb branch with no contributions from mixed Higgs-Coulomb branches. This observation carries over to all our other examples collected in Appendix A as well.

As described in Section 2.4 we are eventually interested in the monodromy matrices expressed in terms of integral periods. To this end, we first have to find a basis of solutions to the PicardFuchs equation at all singular points. The structure of these solutions is conveniently summarized by the Riemann P-symbol, which for the present example reads

0 œ Z1 Z2

(3.13)

Let us briefly recall its meaning: The first row lists the positions of the regular singular points, here given in terms of the algebraic coordinate z. To each such point z the symbol associates the six — i.e., the order of the operator — rational numbers that are written in the corresponding column below the horizontal line. For example, the symbol associates the numbers 0, 1, 2, 3, 4 and 3/2 to z1. These so called characteristic exponents are the rational roots of the indicial equation

Ci(ui,0i)u1 = O (ua) , a e Q

(3.14)

10 11 12

20 21 22

10 11 12

20 21 22

for the exponent a, where u-z is a local coordinate on a patch around z.17 The number of times that a particular solution a0 is listed in the corresponding column of the Riemann P-symbol precisely is the order to which it is a root of eq. (3.14). Now let a1 < ... <ap for 1 < p < 6 be the distinct roots of the indicial equation at z, whose respective orders are m1,..., mp such that J2mkak = 6. A set of linearly independent solutions to the Picard-Fuchs equation on a disk U-z around z is then given by

(uz) = uf (1 + О(щ)) ,

n(k,l){u~z) = П(к,0) ■ + O (Onu-z)l-

z ( z) z (2ni)1 V z)

with 1 < l < mk - 1

(3.15)

for all 1 < к < p. In the vicinity of the regular singular point z the mko solutions П U(~c'mk-1) thus transform irreducibly amongst each other when transported around Z by uz ^ uz ■ e2ni, which leads to a non-trivial monodromy due to the branch cut of the logarithm or of a root kz. Consequently, the Jordan normal form JMz of the monodromy matrix Mz is the block matrix

„2niaq

e2niaq

„2nian

(3.16)

where the Jordan block Jq is a matrix of dimension mq x mq.

It would be interesting to see how the information encoded in the Riemann P-symbol relates to the associators for systems of differential equations recently presented in ref. [37]. Developing such a relationship promises to shed light on the global analytic structure of solutions to the Picard-Fuchs differential equations.

For the present example of X1t4 the Riemann P-symbol in eq. (3.13) shows that the large volume point at z = 0 does not have maximally unipotent monodromy due to the additional solution

(z) = z (1 + O(z)) .

(3.17)

As a result, the monodromy matrix consists of two Jordan blocks rather than only one. Note that for Calabi-Yau threefolds in general and for those Calabi-Yau fourfolds with order five Picard-Fuchs operators the large volume point is always a regular singular point of maximally unipotent monodromy.

Let us now focus on the large volume point in more detail. As seen from the Riemann P-symbol (3.13) there are two regular solutions with the expansions

n01,0)(z) = Щ(г) = 1 + 72z + 47 880z2 + 54 331200z3 + (20) ( 2 625z 6702 850z2 17302 910 625z3

n02,0)(z)=z 1 + — + —9— +-16-+

(3.18)

17 Explicitly: For z = ^ we have uz = z — z, otherwise uz = z 1. How Lz(uz, (>z) can be deduced from L(z, 6) has been explained in section 2.4.

10 11 12

20 21 22

while there are four logarithmic solutions n = n01,l), l = 1,..

, 4, based upon the period n0 =

The logarithmic period n1 determines the flat coordinate t in the large volume regime

according to

t(z) =

ni(z) ln z + O(z)

(3.19)

With these ingredients a period vector in = (n0, ..., n5)r with asymptotic limit inasy as given by eq. (3.8) in general reads

n (z) = no(z)

nasy (t(z)) + O(z)

(z) (0, 0,a2,a3,a4,a5)

(3.20)

Note that by the second term on the right hand side of this equation we have added a multiple of n02,0) (z) to the at least doubly logarithmic solutions. This is possible — in fact it is necessary to make in integral — since n02,0)(z) vanishes in the asymptotic limit z ^ 0. As the additional period n02,0) relates to the existence of B-branes on the two non-trivial algebraic cycles associated to the described cohomology classes in H4(X1,4, Z) there are no such ambiguities for the quantum periods i 0 and i 1 for B-branes in higher codimension. Since the values for the integration constants a2,..., a5, cannot be fixed by large volume asymptotics, we momentarily determine them by analyzing their global properties in the quantum Kahler moduli space.

In a next step we analytically continue the period vector i to the other three singular points by the method described in Section 2.4. As a result we obtain numerical expressions for the monodromy matrices M0, M^, Mz1 and Mz2 in the large volume basis {i0, ..., n5}, which still depend on the parameters a2,..., a5. As discussed in Section 2.3 we know, however, that one of the monodromy matrices should take the form Mto^ given in eq. (3.10). For the given example, this match can only be achieved for the monodromy matrix Mz2, whose last row reads18

-1, 0, -24 -

179 10

a3 720

719 as

720 - 720

(3.21)

By matching this to the last row of the monodromy matrix Mt0x in eq. (3.10) we identify the parameters as X

a2 = a4 = 0 , a3 = 24 , a5 = 1 . (3.22)

With these values all four monodromy matrices are indeed integral and become

1 1 30 0 0 0 \

0 1 20 8 0 0

0 0 1 0 -1 0

0 0 0 1 0 0

0 0 0 0 1 -1

0 0 0 0 0 1 /

18 With the calculated numerical precision we are able to identify the exact numerical rational values. While strictly speaking this is an educated guess, the integrality of the genus zero Gromov-Witten invariants and the determined number of lines — computed independently in Appendix B via intersection theory — confirms these rational numbers.

10 11 12

20 21 22

-19 11 -670 -72 270 - 40

-40 19 -1340 - 168 540 - 80

2 1 67 8 -27 4

-3 -1 -90 -13 35 -5

2 1 66 8 -27 4

5 2 156 22 -63 9

21 10 700 80 -300 50

40 21 1400 160 -600 100

-2 -1 -69 -8 30 -5

2 1 70 9 -30 5

-2 -1 -70 -8 31 -5

-4 -2 -140 -16 60 -9

1 0 0 0 0 0 ^

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

-1 0 -24 -6 7 -1

(3.23)

Note that these results are in accord with the consistency condition M^MZ2 M0MZ1 = 1. Moreover, the monodromy matrix Mz2 indeed agrees with the expected matrix Mt0x given in eq. (3.10). According to Section 2.3 this shows that the 8-brane OX (described by the integral period n5) becomes massless at the point z2. We also observe that at z1 — which is a second point of Z2-monodromy — the brane Bz1 associated to the integral period

n« = 10no + 20ni - n2 + n3 - U4 - 2U5

(3.24)

becomes massless. The monodromy MZ1 is thus described by a Seidel-Thomas twist as in eq. (2.21) with the 8-brane OX being replaced by the brane BZ1, with a spherical open-string index x(BZ1, BZ1) = 2. This observation in fact carries over to all examples analyzed in this paper: At every point of Z2-monodromy there is a vanishing integral period and the monodromy is described by a Seidel-Thomas twist.

As anticipated in the introduction — due to the non-maximally unipotent monodromy property with respect to the large volume regular singular point of the Picard-Fuchs operator — the structure of the integral quantum periods of the Calabi-Yau fourfold X14 indeed admits integral linear combinations, which give rise to flux-induced superpotentials of the form (1.2). Namely, in terms of the flat coordinate t we find for instance the superpotentials

a*« = n (109n0 + 360ni - 12n + 30^) =

e2nit + O(e4nit)

wflUl« = (60ni - 2n2 + 5n3) = + e2nit + O(e4nit) , n0 6 4n2

(3.25)

flux(t) = n (109n0 - 12n + 30n3) = -3601 + -4-j

e2nit + O(e4nit)

Here, the integral coefficients in the presented linear combinations should be interpreted as flux quantum numbers. The leading non-perturbative terms arise from genus zero worldsheet instantons, which we study in the next subsection in the context of the quantum cohomology ring of the Calabi-Yau fourfold X14.

10 11 12

20 21 22

Table 3.1

Genus zero Gromov-Witten invariants nand n^J of the Calabi-Yau fourfold X14 associated to 02,(1) = ffl,l and 02,(2) = °2 up to degree d = 10.

4 d n01d (2)

5 1 400 520

6 2 208 240 226 480

7 3 175 466480 191 464 760

8 4 196084 534160 213 155 450 240

9 5 255 402582 828 400 277 092686601400

6 367 048 595 782193 680 397 700 706634 553 680

10 7 564 810 585 071858 496 880 611416342763 726567 800

11 8 913 929 133 261 543 393 001 760 988 670 017 271687 389 572 480

12 9 1 536 929 129 164 031 410293 358 720 1661748145 541449358296013 440

13 10 2 664 576 223 763 330 924 317 069 072 400 2 879 777 881450393 936532565 976400

20 21 22

3.1.2. Gromov-Witten invariants and quantum cohomology ring

In section 2.1 we have introduced the quantum cohomology ring of Calabi-Yau fourfolds with Picard-Fuchs operators of non-minimal order. We now explicitly determine the quantum cohomology ring and calculate the genus zero Gromov-Witten invariants nO^J of the Calabi-Yau fourfold X1,4. Furthermore, with the help of the Klemm-Pandharipande meeting invariants we also infer the genus one invariants n1,d defined in eq. (2.4).

First of all, with the classical ring structure encoded in the intersections (3.2), we determine the genus zero Gromov-Witten invariants from the identity

d2 US. (z(t)) r

= (on * o1) U ch S' , i = 1, 2 , (3.26)

dt2 n0ot(z(t))

in terms of the mirror map z(t) for the flat coordinate t. Note that this formula holds because in Gromov-Witten theory the metric for the chiral-anti-chiral operators is identified with the classical intersection pairing. Since we have previously determined the integral periods, the left hand side of this equation is known. Using the intersection numbers (3.2), the explicit Chern characters (3.6) as well as the identification 02,(1) = o1,1 and 02,(2) = o2 in the quantum product (2.1), we arrive at

d2 ^si

dt2 n0pt d2 ns2_

dt2 n0pt

= 20 + J2 d2

1 - qd

Kd +12«05),

(3.27)

Kd + 4«02d)

By expanding these equations in q we obtain two independent equations for each degree d and are thus able to identify the unknowns n0^ and n02J. We have checked integrality up to degree 50 and list the numbers up to degree 10 in Table 3.1. With the help of the recursive definition (2.3) we further deduce the associated Klemm-Pandharipande meeting invariants listed in Table 3.2.

In Appendix B we employ intersection theory techniques to directly compute the number of lines with a marked point restricted to the codimension two Schubert classes o1,1 and o2. As further explained there, these results are in agreement with the genus zero Gromov-Witten invariants nO^ = 400 and n02) = 520 at degree one. This provides for yet another independent consistency check on the linear combinations of the obtained integral quantum periods.

10 11 12

20 21 22

1 Table 3.2 1

2 Klemm-Pandharipande meeting invariants m^j = mik °f the Calabi-Yau fourfold X14 up to degree four. For ease of 2

3 presentation we only list the invariants for k < l. 3

10 11 12

20 21 22

mk,l l — 1 l — 2 l — 3 / l — 4

k — 1 4 536960 2 075 384 960 1750629 048 960 1 951 117 108140160

k — 2 961 126562 880 811503 225 375 360 904 721970 681455 680

k — 3 685189180065 298 560 763 898 769 976093 842560

k — 4 851650443 220977 804 680320

Our findings are consistent with the results presented by Honma and Manabe in ref. [19]. There the quantum correlator genus zero invariants n0,d(fc,(a)) are computed, as for instance also used in ref. [39]. With the identification ^2,(1) — — H1 and <p2,(2) — 5o2 - 3or2 — H2 these invariants are related to the quantum cohomology ring invariants n0^ and n02)) according to , ,

^—j „? u +n0>2)=<>+^g,

n04(H2) — f (5^2 - 3a2) u (n0>u + ^>2) — -4n01d + <

(2) 0,d

(3.28)

We note that integrality of n0 d and «0 d implies integrality of no,d (H1) and no,d (H2), while the

converse is not true.

Finally, we want to determine the genus one invariants n1id from the quantity f[°p specified in eq. (2.5). The discriminant locus has two rational factors, namely the large volume divisor ALV = z and the discriminant factor A of eq. (3.12). From the asymptotic behavior of F1top at large volume (2.7) and at the conifold (2.8) the coefficients b1 and b2 reflecting the holomorphic ambiguity are determined to be

1 + b1 — -

C3(X) U J —

and b2 — -

With the Euler characteristic x — 1 848 we thus have

52 log z

Ftop — 74 log n0pt + log

1 dz 2ni dt

log(1 - 2816z - 65 536z2)

(3.29)

(3.30)

In the asymptotic large volume limit z ^ 0 and after reexpressing z in terms of the variable q this expression reduces to

F,v —

to^ 55 log (g) 3

55 log (g) 3

8720g 2 8709831680g3 -1 - 1 163440g2--—

+ Y, N1,d gd

(3.31)

Hence, we can read of the rational genus one invariants N1td. By the multicovering formula (2.4) these are then translated into the integral genus one invariants, the first few of which are listed in Table 3.3. We have checked integrality up to degree 50, and we observe that n1; 1 = n1i2 = 0.

10 11 12

20 21 22

10 11 12

20 21 22

Table 3.3

Integral genus one Gromov-Witten invariants ni d of

Xi,4 up to degree d = i0.

3 -3 200

4 370 i5i 480

5 4 i08 408 756 800

6 i9 279 i 69 520 232 000

7 66 08i 794 099 798 279 680

8 i94i2244i3i0522 439007 040

9 522 534 i28 i59 i84 58i 44i 465 280

i0 i 332 480 344 03i 795 460 733 665 780 608

3.2. Skew symmetric sigma model Calabi-Yau fourfold Xi,i7,7

As our second example we consider the Calabi-Yau fourfold Xiii7,7 arising as the large volume phase of a certain gauged linear sigma model [20]. It is the non-complete intersection projective variety

Xi,i7,7 = j[x,®]eP(V ® A2V*) rkrn < 2, x e ker«} n P(L) ,

(3.32)

with the vector space V = C7 and a generic 17 dimensional subspace L c V © A2V*. In the following we use the isomorphism to the incidence correspondence of ref. [20] to describe X^ 17,7 as

Xi,i7,7 ^ | (x, p) e P16 x Gr(2,7) G(x, p) = 0} . Here G(x, p) is a generic section of the rank 22 bundle B O(1) ® A2V*

■ ® {o(\) ® u *

(3.33)

(3.34)

O(1) ® A2U

in terms of the hyperplane bundle O(1) of the projective space P16 and the rank two universal subbundle U of the Grassmannian Gr(2, 7). In particular, the class [X1i17,7] of the Calabi-Yau fourfold i: X1i17,7 P16 x Gr(2, 7) becomes the top Chern class of the bundle B, i.e.,

[Xi,17,7] = C22(B) , (3.35)

which is given in terms of the hyperplane class H of P16 and the Schubert classes a2 of Gr(2, 7).19 Then — for cohomology classes i*a pulled back from the ambient space P16 x Gr(2, 7) — we compute the intersection numbers of X1,17,7 according to20

C22(B) U a =

6 xGr(2,7) Xi,i7,7

(3.36)

19 For a review on Schubert classes see for instance ref. [94].

20 For ease of notation, in the following we suppress the pullback for the cohomology class on X117 7 induced from the ambient space.

10 11 12

20 21 22

10 11 12

20 21 22

Hence, we arrive at the intersection numbers

H4 = 98 ,

o2 U o2 = 44 ,

ct2 U H2 = 65 .

(3.37)

X1,17,7

X1,17,7

X1,17,7

Note that on the variety ^1,17,7 we have in cohomology the equivalences H ~ ai (cf., ref. [20]), i6H3 ~ 49as, and 33H3 ~ 98a2,i, as well as 11H4 ~ 98a4, 21H4 ~ 98as,i, and 6H4 ~ 49a2,2. As a result we obtain the integral cohomology generators

H°(X1,17,7, Z) = «1» , H2(X1,17,7, Z) = ((H»

H4(X1,17,7, Z) d««H2,0-2»

H6(Xi,i7,7, Z) =«918H3» , H8(Xi,i7,7, Z) = «918H4» . (3.38)

Similarly as for the previously discussed Calabi-Yau fourfold, the classes H2 and a2 are integral but not necessarily integral generators of H 4(Xi,i7,7, Z). Finally, the total Chern class of the Calabi-Yau fourfold Xi,i7,7 is given by

C(X1,17,7) =

c(P )c(Gr(2,7))

= 1 + (4H2 - 2o2) - 328-+ 672-

( 2) 98 98

(3.39)

i.e., the first Chern class vanishes and x = 672 is the Euler characteristic of the Calabi-Yau fourfold Xi,i7,7.

Apart from the canonical B-branes Opt, C*[i], E• and OX, we construct the 4-branes 5J1 and S• associated to the algebraic surfaces Si and S2 of the zero sections of the rank two bundles O(\)®2 and U intersected with Xi,i7,7. The Chern characters of the constructed 4-branes is computed by the Grothendieck-Riemann-Roch theorem to be

ch S• = H2 - H3 + — H4 ,

ch S* = (H2 - 02) + 1 (H3 - H02) + 112 (H4 - 022) .

(3.40)

With respect to the B-branes £' = (E£)k=0 5 = (Opt, C*[i], SJ, S^, E•, Ox) the asymptotic

periods nasy = (n^^) I

k=0,...,5

35 49t2 + 98t , 817 + 12 1 33

36 nasy (t) = 33 12 2 1 33 2 1

37 + 4

38 49 13 - Tt - 49 12 2 1 109 4 1 229

39 V nt , 13112 + 24 1 41iZ(3) n3 t

for the Calabi-Yau fourfold Xi,i7,6 become

41iZ(3)

(3.41)

The symmetric intersection pairing is readily computed to be

x(£*, £') =

0 0 0 0 0 1

0 0 0 0 1 0

0 0 98 33 49 79

0 0 33 12 -33 12

0 1 49 -33 -30 -15

1 0 79 12 -15 2

(3.42)

10 11 12

20 21 22

which determines the monodromy matrix Mt0x with eq. (2.21) to be

Mt0x =

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

—1 0 —79 —12 15

\ —1 0 — 79 -12 15 — 1 !

(3.43)

3.2.1. Picard-Fuchs system

In ref. [20] we have calculated the two sphere partition function of a gauged linear sigma model, which in its large volume phase realizes the non-complete intersection fourfold X1t17j. From this calculation the fundamental period has been found as

IIo(z) = 1 + 9z + 469z2 + 38 601z3 + 4008 501z4 + ... , (3.44)

where z = zLv is a coordinate around the large point. This period is annihilated by the order six Picard-Fuchs operator

L(z) = + 316932(0 - 1)05 - 98z0[70045305 + 133505804 + 160908003 + 87928502 + 249 0180 + 29106] + 962754229z2 [06 - 1976 960 88305 - 10 395 509 03104

- 14 99166296903 - 10456423 60002 - 3 667 6299100 - 521 151456] + 2z3 [9 812 727 9790 6 + 53 190 263 57305 + 105 895 432 46304

+ 103 996 363 80103 + 54017188 10602 + 14078 1117470 + 1415445066]

- 2z4 [11 549 486 89606 + 46 324 321 80405 + 73 290 469 42604

+ 60 074 870 02603 + 27 353 84716902 + 6 669 746 7190 + 696 036 075] + 174z5 (16661980 6 + 6006 98105 + 10497 8190 4 + 1155107803 + 816213002 + 3 3310470 + 588 537] - 211932z6(0 + 1)5(20 + 3).

(3.45)

In addition to the singular points at z = 0 and z = there might be singularities at the zero loci of the polynomial multiplying 06 in L(z),

hL)(z) = — ( 1 — 188z — 2 368z2 + 4z3) ■

(—316 932 + 9061 178z — 9747741z2 + 105 966z3) •

(3.46)

It turns out, however, that at the zeros of the second factor in this polynomial there are six regular solutions. Consequently, these are regular points. On the other hand, the zero loci

zi «-0.084 , z2 « 592.079 , zs « 0.005 , (3.47)

of the first factor, A(z) = 1 - 188z - 2 368z2 + 4z3, are indeed singular. The Riemann P-symbol reads

10 11 12

20 21 22

0 œ zi z2 Z3

(3.48)

We note that the large volume point z = 0 again does not have maximally unipotent mon-odromy. Its structure is, in fact, the same as for the Grassmannian example discussed in the previous section: In addition to the fundamental period n0 = n0^0) given in eq. (3.44) there is second regular solution,

= z 1 +

6125z 1524635z2 210 992 845z3

(3.49)

The singly logarithmic period, n1 = nO1'1^, defines the flat coordinate t as in eq. (3.19) and the period vector n = (n0'...' n5)r is as in eq. (3.20) with the asymptotic limit nasy now given by eq. (3.41).

By an analytic continuation of n to the other four singular points we then again obtain numerical expressions for the monodromy matrices M0, , Mz1, Mz2 and Mz3 in the large volume basis {n0' ...' n5}. Among these matrices only Mz3 can possibly agree with Mto^ in eq. (3.43). Hence, we compare its last line

a2 4675 a3 a4 4675 a5

-1, 0, -79 - — ,----3 , 15 - — ,----5

99 392 99 99 4704 99

(3.50)

to the last line of Mto and deduce

a2 = a4 = 0 , a3 =

392 1568

Inserting these values indeed makes all five monodromy matrices integral and they read

(3.51)

10 11 12

20 21 22

/1 1 147 00 0 \ 36

0 1 98 33 0 0 37

0 0 1 0 -1 0 38

0 0 0 1 0 0 39

0 0 0 0 1 -1 40

0 0 0 0 0 1 / 41

99 49 13328 588 -3724 343 \ 42

196 99 26656 1176 -7448 686 43

-2 -1 -271 -12 76 -7 44

2 1 272 13 -76 7 ' 45

-2 -1 -272 -12 77 -7 46

V -4 -2 -544 -24 152 47

1 /4117 1568 478828 23520 -115248 8232

2 4809 1833 559447 27480 -134652 9618

3 Mz2 = -84 -32 -9771 -480 2352 -168

4 0 0 0 1 0 0

5 -168 -64 -19544 -960 4705 -336

6 y -441 -168 -51303 -2520 12348 -881

10 11 12

20 21 22

/ 1 0 0 0 0 0 \

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

V -1 0 -79 -12 15 -1 /

-3975 -1490 -459291 -22890 110201 -7854 \

-1757 -617 -197897 10479 46942 -3318

67 25 7728 387 -1853 132

121 47 14181 682 -3423 245

151 57 17499 867 -4205 300

\ 198 74 22862 1146 -5487 391

(3.52)

Note that Mz3 = Mt0x and that the consistency condition MœMZ2Mz3M0MZ1 = 1 is fulfilled. While this shows that the 8-brane OX becomes massless at z3, the integral periods

nBzi = 49no + 98ni - n2 + n3 - H - 2H nB =-196n0 - 229n1 + 4n2 + 8n4 + 2in5

(3.53)

vanish at z1 and z2, respectively. Hence, at each point of Z2-monodromy there is a massless brane and the monodromies are described by Seidel-Thomas twists. The regular singular point at infinity will be discussed in Section 3.2.3.

Due to the non-minimal order property of the Calabi-Yau fourfold X^ 17,7 we can find integral quantum periods, which give rise to flux-induced superpotentials of the form (1.2). In terms of the flat coordinate t we for instance have

flux(t) = TT (5753no + 19404ni - 132^ + 392^) = — flux n0 4n2

+ O(e4nit)

flux(t) = — (4851n1 - 33n2 + 98n3) = — Ho

5753 2781

e2nit + O(e4nit),

flux(t) = TT (5753no - 132n2 + 392n3) = -194041 + —-=-flux n0 4n2

e2nit + O(e4nit).

(3.54)

3.2.2. Gromov-Witten invariants and quantum cohomology ring

To determine the Gromov-Witten invariants of the Calabi-Yau fourfold X1,17,7 we insert the intersection numbers (3.37), the explicit Chern characters (3.40) as well as the identifications 02,(1) = H2 and $2,(2) = a2 into eq. (3.26). This yields the two equations

10 11 12

20 21 22

Table 3.4

Genus zero Gromov-Witten invariants n^ and n^2^ of the Calabi-Yau fourfold X117 7 associated to 02,(1) = H2 and 02,(2) = a2 up to degree d = 10.

d n0,d ' (2) n0,d

1 33 0

2 170 721

3 16126 38 255

4 1 141312 3 042676

5 100955 257 274 320 123

6 9 821360694 27 276710118

7 1028 274 636900 2 897 092 850 989

8 113 458193 073 000 323 207 209 581582

9 13 032484 062 881000 37 444 642 819 824 776

10 1545108 865 260914434 4 469 922 540366355 762

Table 3.5

Klemm-Pandharipande meeting invariants m^j = mik °f the Calabi-Yau fourfold X117 7 up to degree four. For ease

of presentation we only list the invariants for k < l.

™k,l l = 1 l = 2 l = l = 4

k = 1 60 784 240 28194 221040 23 782299 222640 26 506970 805 517 040

k = 2 13 065 863 900400 11031985 902 832240 12299 429 676016495 600

k = 3 9 314 685 486617 406000 10384 847 256692114 669 040

k = 4 11 577 959 795 730 175 108 775 920

d2 Us;

, œ d L = 98 + Ed2TJL-d (98«0!d + 65$),

dt2 n<n

dt2 no,

(3.55)

l = 33 + yd2—q—r (:

pt h 1 -

[33»ïï + 21n®) ,

from which we find the invariants n0ad and list them up to degree 10 in Table 3.4. Further, we deduce the associated Klemm-Pandharipande meeting invariants listed in Table 3.5.

Moreover, we use the quantity Fxtop specified in eq. (2.5) to determine the genus one invariants n1,d. The discriminant locus has two rational factors, these are the large volume divisor ALV = z and the discriminant factor A = 1 - 188z - 2 368z2 + 4z3. The coefficients bt and b2 reflecting the holomorphic ambiguity are from the asymptotic behavior of f!°p at large volume (2.7) and at the conifold (2.8) determined to be

1 + bi = -

C3(X) U J =

and b2 = -

(3.56)

With the Euler characteristic x = 672 we thus find

F^ = 25 log n0pt + log

1 dz\ + 38 logz log(1 - 188z - 2 368z2 + 4z3)

2ni dt

(3.57)

which in the large volume limit z ^ 0 and after reexpressing z in terms of q reduces to

10 11 12

20 21 22

10 11 12

20 21 22

Table 3.6

Integral genus one Gromov-Witten invariants «1 d of X117 7 up to degree d = 10.

d «1d

5 224 386

6 206 613 908

7 83 707 955 196

8 23 455 827 469 526

9 5 401382970402176

10 1 107 021477 254 814 128

F top_ F1 =

41log(q) 473q 13 949q2 2276 105q3

41 log (q ) 3

N1,d qd

(3.58)

This equation determines the rational genus one invariants N1yd, which by the multicovering formula (2.4) encode the integral genus one invariants n1id listed in Table 3.6. Their integrality has been checked up to degree 50.

3.2.3. The regular singular point at infinity

From the Riemann P-symbol in eq. (3.48) we see that the structure of solutions at the singular point z = is similar to that at the large volume point z = 0. Namely, there are two non-logarithmic solutions, which in terms of w = z-1 enjoy the expansions

a,0)(w)-

(w) = w (1 + 21 w + 2 989w2 + 714549w3 + 217 515 501 w4 + ...) ,

n^2'0)(w) = w3/^1 +

10085w 782127w2 379 170 123 893 w3

+ --- + -- + .•

88 200

(3.59)

Moreover, there are four logarithmic solutions n^ for l = 1,..., 4. As opposed to the second non-logarithmic period at large volume — h02'0) given in eq. (3.49) — the additional solution

has a branch cut arising from the square root of the solution. This already indicates that

z = is not large volume limit of a smooth Calabi-Yau fourfold.

Let us now look at the integral_period vector n = (no, rii, n2, n3, n4, n5)r, which is related to the integral period vector ft by the SL(5, Z) transformation S according to

(3.60)

41 /nA / 105 98 -2 -1 -4 -8 n0

42 fix -49 -56 1 0 2 5 n1

43 fl2 -1498 -1400 28 14 53 89 n2

44 f3 -648 -615 12 6 22 34 n3

45 ft 4 -330 -243 6 5 12 16 n4

46 \n5/ V 196 229 -4 0 -8 -2V J n5

47 5

10 11 12

20 21 22

10 11 12

20 21 22

By an analytic continuation of this period vector to z = (x* we find that it corresponds to the

following linear combination of solutions n^ /

(k,l).

/M>\ n1 M2 f3 M4

0 _ 29 2

275iZ(3) + 8n3 73 192

275iZ(3) 8n 3

0 87 2 87 4 87 8

0 0 0 0

7 2n 2 _21_

4n 2 7 8n 2

/n(1,0)\ n

(1,1) œ

(1,2) œ

(1.3) œ

y n(2,0) nœ

(3.61)

Hence, we deduce in the limit w ^ 0 the asymptotic behavior for n to be

n (s) =

s2 + 53 s - 29

87 s2-

29 „ 4 16 s

2 57 8 61

-87 s2 + 10s - 61 +

+ 22 s2_ + 8 s

275iZ(3) 8n3 s +

275iZ(3) f 8n3 73 192

(3.62)

in terms of the flat coordinate 1 n1(w)

s(w) =

(3.63)

2ni ll0(w)

In the newly defined integral basis n the monodromy matrices at z = and z2 transform into

1 1 70 33 6 9 / 1 0 0 0 0 0

0 1 87 39^ 20 9 0 1 0 0 0 0

0 0 1 1 1 , MZ2 = 0 0 1 0 0 0

0 0 0 -1 -2 3 -1 0 0 0 1 0 0

0 0 0 0 1 -1 0 0 0 0 1 0

0 0 0 0 0 1 V -1 0 0 0 0 -1

while the intersection pairing becomes

(3.64)

Sx(£', £')ST =

0 0 0 0 0 1

0 0 0 0 1 0

0 0 174 87 70 0

0 0 87 44 32 0

0 1 70 32 32 0

1 0 0 0 0 2

(3.65)

We observe that in terms of the transformed intersection pairing the monodromy matrix MZ2 has the characteristic form of the Seidel-Thomas twist (2.21) with respect to the structure sheaf of a geometric target space. However, by the structure of the quantum periods in the vicinity of w = 0, this target space cannot be a smooth Calabi-Yau fourfold for various reasons. Firstly, as can be seen from eq. (3.61) — apart from the logarithmic branch cut — there is also a square root branch cut appearing in one doubly logarithmic, the triply logarithmic and quadruply logarithmic

10 11 12

20 21 22

10 11 12

20 21 22

Table 3.7

Rational genus zero invariants nA and associated to the doubly logarithmic periods n 2 and 113.

d (A) n(B) n0,d

1/2 - 14

1 7 569 3 781

3/2 - 167

2 735 324 367 662

5/2 - 23 647

3 129 395187 258 790207 4

7/2 - 18 828 027 4

4 29 766 479 280 14 883 239 640

9/2 - 9280 303 369 8

5 7 978 989 505 959 15957 978 988 271 4

quantum periods. This square root branch cut, however, does not conform with the singularity behavior of quantum volumes of cycles in a large volume phase. Secondly, if the target space were a smooth Calabi-Yau fourfold, the leading asymptotic term 19í4 + ... in the quadruply

logarithmic period would encode the degree k of the Calabi-Yau fourfold according to Ks4 +____

This yields, however, the non-integral coefficient k =

On the other hand, due to the discussed similarities to a large volume phase, it is conceivable that the target space enjoys an interpretation as a singular Calabi-Yau variety — possibly with a singularity in codimension two, which could explain the square root branch cut starting in one of the doubly logarithmic quantum periods. Having such a geometric picture in our mind, we naively extract an instanton expansion from the doubly logarithmic integral periods n2 and n3 according to

87 2 53 29 ^

= — s 2 H--s---+> ;

' 2 2 2 ^ d=1

l(A) 0,d

Li2 (e2nis'd) ,

n3 = 87s2 + 11s - 57 + E«S2LÍ2 (e2nis-d'2) .

(3.66)

The leading numbers of this expansion are listed in Table 3.7. Note that the doubly logarithmic solution without the square root branch cut yields a conventional genus zero instanton expansion with integral invariants n0AJ for integral d. The other doubly logarithmic solution, however, yields instanton invariants nB arising also at half instanton degrees, which reflects the square root

branch cut behavior of this quantum period. Moreover, the invariants nB in general are rational numbers with powers of two in their denominators. It would be interesting to give a geometric interpretation of all the large volume like features, potentially as speculated in terms of a singular Calabi-Yau fourfold variety.

4. Conclusions

In this work we have studied the Gromov-Witten theory on Calabi-Yau fourfolds, emphasizing the role of non-marginal chiral-anti-chiral operators in the associated quantum chiral

10 11 12

20 21 22

1 rings. We established and demonstrated explicitly that the number of chiral-anti-chiral opera- 1

2 tors of conformal weight (2,2) — i.e., operators corresponding to generators of the middle- 2

3 dimensional cohomology group of the Calabi-Yau fourfold — yields the number of indepen- 3

4 dent genus zero Gromov-Witten invariants with a single marked point at each degree. We 4

5 argued that for Calabi-Yau fourfolds with a single Kahler modulus such examples arise from 5

6 the Picard-Fuchs operators of quantum periods with non-minimal order. Namely, the regu- 6

7 lar singular point associated to the large volume limit is not a regular singular point with 7

8 maximally unipotent monodromy of the associated Picard-Fuchs operator. Our explicit exam- 8

9 ples of this phenomenon were constructed from non-complete intersection projective varieties 9

10 or from complete intersections in non-toric ambient spaces. To deduce their quantum coho- 10

11 mology rings we calculated the integral quantum periods with the help of numerical analytic 11

12 continuation techniques. Furthermore, we computed the monodromy matrices about all reg- 12

13 ular singular points in quantum Kahler moduli space with respect to the established integral 13

14 basis. Finally, we determined the Klemm-Pandharipande meeting invariants and the genus one 14

15 BPS invariants for the analyzed Calabi-Yau fourfolds. The confirmed integrality property of 15

16 these invariants furnished a non-trivial check on the deduced quantum cohomology rings. As 16

17 a further check on our results, we independently verified the genus zero Gromov-Witten in- 17

18 variants at degree one entering the quantum cohomology ring with intersection theory meth- 18

20 20 We established that the large volume asymptotics of quantum periods admitted purely instan-

ton generated integral linear combinations. As briefly mentioned, this observation may prove

22 22 useful in string cosmology for F-term monodromy inflation scenarios [40-43]. Moreover, such

instanton generated quantum periods are interesting from an open-closed string duality point of view [95], which — in certain geometric situations — relates closed-string quantum periods of Calabi-Yau fourfolds to open-string quantum periods of Calabi-Yau threefolds with branes [95,96,84]. Identifying purely instanton generated open-string quantum periods would hence es-

27 tablish stable brane configurations in Calabi-Yau threefolds at large volume. The absence of ^

29 perturbative terms in the expansion of open quantum periods would imply that the associated 29

30 open-closed deformation space were obstructed by closed sphere and open disk instanton effects 30

31 only. Such setups promise interesting enumerative interpretations in terms of real and Ooguri- 31

32 Vafa invariants in compact Calabi-Yau geometries [97-100]. 32

33 We would like to point out an observation that the Picard-Fuchs operators of some of our 33

34 Calabi-Yau fourfold examples — namely for some of those given as complete intersections in 34

35 ambient Grassmannian spaces — exhibit intriguing algebraic properties. That is to say that the 35

36 fundamental periods factorize into the Hadamard product of two new fundamental periods that 36

37 are solutions to a Calabi-Yau threefold and elliptic curve Picard-Fuchs differential equations of 37

38 fourth and first order, respectively.21 For instance, the fundamental period (3.18) of the example 38

39 discussed in Section 3.1 enjoys the expansion 39

:; ™=eec:)2(2:)(:)2("=ms«.^,. :

43 :_0 k~0 43

44 with the fundamental periods 44

47 21 We are thankful to Gert Almkvist for pointing and explaining this factorization property. 47

10 11 12

20 21 22

о = ££

nlfc) =

n=0 k=0 v 7 v 7 v 7 v 7 n=0

and the Hadamard product (f + g)(z) = ^nanbnzn defined in terms of the series expansions

f(z) = Y.n anzn and g(z) = Y.n bnzn. In particular, the fundamental period nCY3(z) is the so-

lution to the fourth order Picard-Fuchs operator [101,102]22

LCY3 = 04 - 4z(40 + 1)(40 + 3)(1102 + 110 + 3) - 16z2(40 + 1)(40 + 3)(40 + 5)(40 + 7) ,

with maximally unipotent monodromy point at z = 0. It gives rise to the integral genus zero Gromov-Witten invariants 920, 50 520, 5 853 960, ..., cf., ref. [102]. It would be interesting to find a geometric interpretation for these Hadamard factorization of such Calabi-Yau fourfolds, perhaps along the lines of ref. [103].

Finally, let us mention that the non-minimal order property of the analyzed Picard-Fuchs operators for the Calabi-Yau fourfold periods may also exhibit interesting features from a modular form perspective, see, e.g., refs. [104,105]. At least, we expect that a better understanding of global properties of the quantum Kahler moduli space should simplify the required derivation of integral quantum periods.

Acknowledgements

We would like to thank Gert Almkvist, Rolf Kappl, Albrecht Klemm, Peter Mayr, Dave Morrison, Urmi Ninad, Thorsten Schimannek, Stephan Stieberger and Eva Silverstein for discussions and correspondences. A.G. is supported by the Graduate School BCGS and the Studienstiftung des deutschen Volkes.

Appendix A. Tabulated results of analyzed examples

In this appendix we tabulate the data that specifies the quantum periods and monodromy structure for several Calabi-Yau fourfolds with a order six Picard-Fuchs operator. We also list the leading genus zero Gromov-Witten invariants generating the quantum cohomology rings and the genus one BPS invariants of these Calabi-Yau fourfolds. Two of these examples — with their tables listed in Appendix A.1 and Appendix A.7 — are discussed thoroughly in the main text in Section 3.1 and Section 3.2, respectively. The data of the remaining examples is calculated analogously.

A.1. Calabi-Yau fourfold Xi,4 с Gr(2, 5)

10 11 12

20 21 22

Picard-Fuchs operator:

L(z) = (0 - 1)05 - 8z(20 + 1)(40 + 1)(40 + 3) ^1102 + 110 + 3) 0 - 64z2 (20 + 1)(20 + 3)(40 + 1)(40 + 3)(40 + 5)(40 + 7)

47 22 Compare with example AESZ 51 in ref. [101] and the online Calabi-Yau database [102].

10 11 12

Discriminant locus: A(z) = 1 - 2 816z - 65 536z2 Regular singular points: z = 0 z = œ

z = ZOX(= Z2) * 3.5 • 10-4 Z = z1 * -0.043

Riemann P-symbol:

Intersection pairing:

0 0 0 0 0 1

0 0 0 0 1 0

0 0 20 8 10 24

0 0 8 4 -8 6

0 1 10 -8 -14 -7

1 0 24 6 -7 2

Large volume asymptotics: /

nasy (t) =

10t2 + 20t + 107 4t2 - 4t + 7

- f t3 - 5t2 - f t - 42 +

55iZ(3) n 3

514+37 2 _ 61 + 121

55iZ(3). . 7 n 3 . + 144

Monodsromy matrices:

17 1 1 30 0 0 0 \ / -19 -11 -670 -72 270 - 40

18 0 1 20 8 0 0 -40 -19 - 1340 168 540 -80

Mq = 0 0 1 0 -1 0 Mœ 2 1 67 8 - 27 4

19 0 0 0 1 0 0 = -3 -1 -90 -13 35 -5

20 0 0 0 0 1 -1 2 1 66 8 - 27 4

21 0 0 0 0 0 1 / V 5 2 156 22 - 63 9 /

22 21 10 700 80 -300 50 1 0 0 0 0 0 \

23 40 21 1400 160 -600 100 0 1 0 0 0 0

24 Mz1 = 2 -1 69 -8 30 -5 M 0 0 1 0 0 0

2 1 70 9 -30 5 Ox = 0 0 0 1 0 0

25 2 -1 70 -8 31 -5 0 0 0 0 1 0

26 V- 4 -2 - 140 -16 60 -9 -1 0 - 24 -6 7 - 1

Generators of cohomology ring:

(1 ; 01 ; C11,a2 ; 03 ; 022 » c H00 (X) © ...© H44(X)

Total Chern character:

03 02 2

c(X) = 1 + (8o1,1 + 7o2) - 440^- + 1 848—

33 Intersection numbers:

34 01 1.01 1 = 4 , 01 1.02 = 4 , 02.02 = 8

Zeros of integral quantum periods:

H5 = 0 at zoX . n® = 10n0 + 20n1 - n2 + n3 - n4 - 2n5 = 0 at z1

400 208 240 175 466 480 196084 534 160 255 402 582 828 400 367 048 595 782193 680 564810585 071858496 880 913 929 133 261 543 393 001 760 1536929129164 031410293 358 720 2 664 576 223 763 330 924 317 069 072 400

520 226 480 191464 760 213 155 450240 277 092686601400 397 700706634 553 680 611416342763 726567 800 988670017271687389 572480 1661748145 541 449 358 296013 440 2 879 777 881 450 393 936532 565 976400

10 11 12

20 21 22

Genus one Gromov-Witten invariants n\ j

10 11 12

20 21 22

-3 200 370151480 4 108 408 756 800 19 279 169 520232 000 66081794 099 798 279 680 194 122441310 522439 007 040 522 534 128 159 184 581 441 465 280 1 332 480 344 031 795 460 733 665 780 608

A.2. Calabi-Yau fourfold X2,3 c Gr(2 , 5)

Picard-Fuchs operator:

L(z) = (0 - 1)05 - 6z(20 + 1)(30 + 1)(30 + 2) ( 1102 + 110 + 3) 0 - 36z2 (20 + 1)(20 + 3)(30 + 1)(30 + 2)(30 + 4)(30 + 5)

Discriminant locus: A(z) = 1 - 1 188z - 11 664z Regular singular points: z = 0 z = œ z = zOX z = Z1 ^-0.10

Riemann P-symbol:

; 8.3 • 10-4

Intersection pairing:

0 0 0 0 0 1

0 0 0 0 1 0

0 0 30 12 15 26

0 0 12 6 -12 5

0 1 15 -12 -16 -8

1 0 26 5 -8 2

Large volume asymptotics: /

n asy (t) =

15t2+30t + 77

6t2 - 6t+4

_5t3 _ ^t^ _ 47» 45iZ(3)

5 2 1 4 1 8 + n 3 514 _L 2212 _ 45iZ(3) . 23 41 + 8 1 n 3 ' + 96

Monodromy matrices:

1 1 45 0 -5 0 \ / -25 -13 -1015 -46 390 -50

0 1 30 12 0 0 -60 -29 -2400 -132 930 -120

0 0 1 0 -1 0 Mœ = 2 1 80 4 -31 4

0 0 0 1 0 0 -3 -1 -105 -8 40 -5

0 0 0 0 1 -1 2 1 79 4 -31 4

0 0 0 0 0 1 / V 5 2 184 13 -72 9

25 12 1020 48 -420 60 / 1 0 0 0 0 0

60 31 2550 120 -1050 150 0 1 0 0 0 0

-2 -1 -84 -4 35 -5 mzox = 0 0 1 0 0 0

2 1 85 5 -35 5 0 0 0 1 0 0

-2 -1 -85 -4 36 -5 0 0 0 0 1 0

-4 -2 -170 -8 70 -9 V -1 0 -26 -5 8 -1

Generators of cohomology ring:

(l ; 01 ; *u,*2; 03 ; °f) e HX) ® H4A{X)

Total Chern character:

03 02 2

c(X) = 1 + (6o1 1 + 5o2) - 360— + 1188-^

Intersection numbers:

01,1-01,1 = 6 , 01,102 = 6 ,

0202 = 12

Zeros of integral quantum periods:

ns = 0 at zoX , nB = 12n0 + 30n^ - n2 + n3 - n4 - 2n5 = 0 at zi

Genus zero Gromov-Witten invariants n0\ (left) and n02j (right)

150 34 635 12266460 5 755 894 980 3 144 906174 450 1895113 546937 010 1222 482269 477 448 870 829 123 506 499 521 864 000 584 369 804 499 128 982 030 870 424 582 414 793 779 873 760 931 825

210 38 175 13 599 540 6352627 620 3 462780142950 2083 385152900350 1342443 529 699 952610 909 737 222 891 667 295 200 640780961536667 529 927 090 465 334 861 886 835 590 355 227 325

Genus one Gromov-Witten invariants n\ j

-40 6629 085 33 762 865 500 72983 984 748 600 111703 298 516011620 143 677197 771963 884 280 167 307 680 280 218 203 241 460 183 135 579 515 334 103 668 439 662

A.3. Calabi-Yau fourfold X13 3 C Gr(2, 6)

Picard-Fuchs operator:

L(z) = (0 - 1)05 - 3z(29 + 1)(30 + 1)(3Ö + 2) ( 1302 + 130 + 4^ 0 - 27z2(30 + 1)(30 + 2)2(30 + 4)2(30 + 5)z2

Discriminant locus: A(z) = 1 - 702z - 19 683z2 Regular singular points:

z=0 z = œ

z = zoX = 729-1 z = z1 = -27-1

Riemann P-symbol:

'0 TO z1 zoX

0 1 3 0 0

0 2 3 1 1

0 2 3 2 2

0 4 3 3 3

0 4 3 4 4

1 5 3 3 2 3 2

Intersection pairing:

2 /0 0 0 0 0 1

3 0 0 0 0 1 0

0 0 42 15 21 41

4 X = 0 0 15 6 -15 8

5 0 1 21 -15 -20 -10

6 1 0 41 8 -10 2

Large volume asymptotics: /

n asy (t) =

21t2 + 42t +

15,2 2

t2 - i51 + 5

7t3 21 12 61 1 47 . 213iZ(3) -7t - Tt - Tt - "8 +--7T3—

10 11 12

20 21 22

114 , 33 f2 _ 213if(3) J_ 4t + 8 t 4n 3 t + 1«

4n 3 16

Monodromy matrices:

1 1 63 0 0 0 \ / -62 -22 -3360 -300 1029 105 \

0 1 42 15 0 0 -126 -41 -6720 -645 2058 210

0 0 1 0 -1 0 Mœ 3 1 160 15 -49 5

0 0 0 1 0 0 = -4 -1 -200 -22 60 -6

0 0 0 0 1 -1 3 1 159 15 -49 5

U 0 0 0 0 1 / V 7 2 359 38 -110 11

43 21 2730 210 -966 126 \ 1 0 fr 0 0 0 Q ^

84 43 5460 420 -1932 252 0 1 0 0 0 0

-2 -1 -129 -10 46 6 Mz 0 0 1 0 0 0

2 1 130 11 -46 6 OX = 0 0 0 1 0 0

-2 -1 -130 -10 47 -6 0 0 0 0 1 0

V- -4 -2 -260 -20 92 -11 -1 0 -41 -8 10 -1

Generators of cohomology ring:

(1 ; 01 ; „1,1,0-2 ; 12 ; 022 ) € HQ,Q(X) © ... ® H^'^(X)

Total Chern character:

03 02,2

c(X) = 1 + (601 1 + 4o2) - 42^-3 + 1368—2—

Intersection numbers:

„1,1.01,1 = 6 , 01,1.02 = 9 ,

02.02 = 18

Zeros of integral quantum periods: ns = 0 at zoX ,

ng = 2in0 + 42n^ - n2 + n3 - n4 - 2n5 = 0 at zi

45 129

11 169 13 731

2334015 2977 203

670339 377 843 149 973

222531477 228 278 449 436724

81416926226097 101484 761783 937

31861797197 835 564 39 609 507 515 035 620

13 104 024 227 969 549 085 16258171900604 949 897

5 598 901 286 610 753 390 696 6935 937 444 307 917 236 520

2 465 575 949 291 932 283 056 560 3 050652167 218 394 830016340

Genus one Gromov-Witten invariants n\ j

0 0 20

117 369 1 111542426 2030 821680 744 2190 254 867 538 498 1859 490 547 470080 793 1 386 159 363 843 011 650 458 955 211 114 503 390 944 999 069

10 11 12

20 21 22

10 11 12

20 21 22

A.4. Calabi-Yau fourfold X1222 C Gr(2, 6)

Picard-Fuchs operator:

L(z) = (6 - 1)65 - 4z(26 + l)3 ^1362 + 130 + 4

6 - 48z2(26 + 1)2(20 + 3)2(36 + 2)(36 + 4)

Discriminant locus: A(z) = 1 - 416z - 6 912z2 Regular singular points: z = 0 z = to

z = ZoX = 432-1 z = Z1 = -16-1

Riemann P-symbol:

0 TO z1 zoX

0 1 2 0

0 1 2 1 1

0 2 3 2 2

0 4 3 3 3

0 3 2 4 4

1 3 2 3 2 3 2

Intersection pairing:

0 0 0 0 0 1 \

0 0 0 0 1 0

0 0 56 20 28 50

0 0 20 8 -20 9

0 1 28 -20 -22 -11

1 0 50 9 -11 2 )

Large volume asymptotics:

n asy (t) =

28t2 + 56t +

10t2 - 10t + 35

- 2813 - 14t2 - 18t - f + ^

214 _i_ 13 t2 _ 43iZ(3) 47 3 ' + 3 1 n 3 ' + 144

Monodromy matrices:

1 1 84 0 0 0 \

0 1 56 20 0 0

0 0 1 0 -1 0

0 0 0 1 0 0

0 0 0 0 1 -1

0 0 0 0 0 1

f -83 -29 -5572 -400 1512 -140 \

-168 -55 -11144 -860 3024 -280

3 1 199 15 -54 5

-4 -1 -248 -23 66 -6

3 1 198 15 -54 5

7 2 446 39 -121 11

m 168 \ / 1 0 0 0 0 0

57 28 4592 280 -1456

112 57 9184 560 -2912 336 0 1 0 0 0 0

-2 -1 -163 -10 52 -6 MzOX = 0 0 1 0 0 0

2 1 164 11 -52 6 0 0 0 1 0 0

-2 -1 -164 -10 53 -6 0 0 0 0 1 0

\ -4 -2 -328 -20 104 -11 -1 0 -50 11 -1

V -9

Generators of cohomology ring:

(1 ; „1 ; „1,1,0-2 ; 16 ; ^ ) e H 0'0(X) © ...© H 4'4(X)

Total Chern character:

„3 „2 2 c(X) = 1 + (5„1,1 + 3„2) - 34^—3 + 888 —

Intersection numbers:

„1,1.01,1 = 8 , „1,1 .„2 = 12 , „2 .„2 = 24

Zeros of integral quantum periods: ns = 0 at zoX ,

n® = 28n0 + 56n^ - n2 + n3 - n4 - 2n5 = 0 at zi

10 11 12

20 21 22

Genus zero Gromov-Witten invariants ni^d (left) and (right)

3710 4 662

456996 601 308

77 744 208 100674 808

15 262779 768 19 647 842 856

3 300 982396086 4 230686 882 622

763 420 513 970084 975 446610603 036

185 520589 035 937 760 236505 646336207 216

46 831421841938 832444 59 596 808 422 526 994 692

12 183 382 927 032 659 991 892 15 482698161874 509 215 956

Genus one Gromov-Witten invariants n\ d

17 898 60657 824 65 864 201 248 43 546640994 304 22541684 709 460 560 10173 360 305 632 854 080 4 221 177 321952488 663 680

A.5. Calabi-Yau fourfold Xj5 2 C Gr (2 , 7)

Picard-Fuchs operator:

L(z) = + 9(0 - 1)05 - 6z0 (31005 + 91904 + 88403 + 47602 + 1320 + 15^

- 4z2 (2131106 + 7895105 + 15439504 + 18054403 + 12108602 + 425460 + 6048^

- 8z3 (20 + 1) (5756105 + 24937204 + 41227303 + 31058102 + 1043880 + 11691)

- 16z4(20 + 1)(20 + 3) (1050104 + 2013803 + 1309602 + 26760 - 154^ + 1184z5 (0 + 1)3(20 + 1)(20 + 3)(20 + 5)

Discriminant locus:

A(z) = 1 - 228z - 4 624z2 + 64z3

Regular singular points:

z = œ

z = zoX ^ 0.004 z = z1 s -0.053 z = z2 s 72.3

Riemann P-symbol:

0 œ z1 z2 zOx

0 1 2 0 0 0

0 1 1 1 1

0 1 2 2 2

0 1 3 3 3

0 3 2 4 4 4

1 5 2 3 2 3 2 3 2

Interssection pairing:

0 0 0 0 0 1

0 0 0 0 1 0

0 0 84 28 42 70

0 0 28 10 -28 11

0 1 42 -28 -28 -14

1 0 70 11 -14 2

Large volume asymptotics:

n asy (t) =

42t2 + 84t +

t2 _ 491 _ 35 , 91iZ(3)

- 14t3 - 21t2 - 491 - 35 + ^

7 t4 , 21 f2 _ 91if(3) _65_ 2 ' + 4 ' 2n 3 ' + 192

14t - 14t + 12

Monodromy matrices:

1 1 126 0 00 \ / - 533 -43 -36946 -3626 6902 -252

0 1 84 28 00 1008 -83 -70252 -6916 13244 -504

0 0 1 0 -1 0 13 1 897 89 -167 6

0 0 0 1 0 0 Mœ = -14 -1 -966 -99 182 -7

0 0 0 0 1 -1 16 1 1078 110 -195 6

u 0 0 0 0 1 / V 42 2 2772 294 -490 13 /

/ 85 42 9996 504 -2940 294 97 0 5824 672 896 0

168 85 19992 1008 -5880 588 420 1 25480 2940 - 3920 0

-2 -1 -237 -12 70 7 Mz2 -3 0 -181 -21 28 0

2 1 238 13 -70 7 = 12 0 728 85 112 0

-2 -1 -238 -12 71 -7 0 0 0 0 1 0

\ -4 -2 -476 -24 140 -13 V -9 0 -546 -63 84 1

1 0 0 0 0 0 \

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

-1 0 -70 -11 14 -1

Generators of cohomology ring:

(1 ; ; a1,1,a2 ; g ; ^ ) * H00 (X) ® H44 (X)

Total Chern character:

c(X) = 1 + (5a1 1 + 2o2) - 364 — + 846

1,1 2 28 10

Intersection numbers:

ff1,1.ff1,1 = 10 , °1,1.°2 = 18 , °2.°2 = 38

Zeros of integral quantum periods:

05 = 0 at zoX , UBz = 42n0 + 84n1 - n2 + n3 - n4 - 2n5 = 0 at z1 nBz2 = 32n0 + 140n1 - n2 + 4n3 - 3n5 = 0 at z2

Genus zero Gromov-Witten invariants ni1, (left) and riSdi (right)

-10 46

1 009 1 499

66 436 111 012

6611218 10644 996

744 513 554 1 186 881 242

92436 371702 146004 322 222

12248 099 597 230 19 229 229169 542

1704 064 096112480 2663 089 251024 164

246133 929 404 316702 383 301240195 065 542

36625 042233 637 069 635 56 876037 388 681 122 041

Genus one Gromov-Witten invariants n\ d

0 0 0 175 1 251 544 1 106013 132 502633 629 368 165747 820001414 458 876986698 03032 11434 511768 888 583 676

A.6. Calabi-Yau fourfold Xj8 C Gr(2 , 8)

Picard-Fuchs operator:

L(z) = + 121(0 - 1)05 - 22z0 (43805 + 209404 + 171003 + 95002 + 2750 + 33j

+ z2( - 83931306 - 247166105 - 403755604 - 449730403 - 309394802 - 11587400

- 180048) - 2z3 (574675406 + 2647066605 + 5118422404 + 5048047003 + 2629533502 + 66848430 + 604098) - 4z4 (408188406 + 1489448405 + 1882590304 + 747203003

- 369883902 - 40998390 - 993618) + 56z5 (2959206 + 25596005 + 80644804 + 127278703 + 108840302 + 4834310 + 87609) + 1568z6 (0 + 1)3(20 + 3)(40 + 3)(40 + 5)

Discriminant locus:

A(z) = (1 + 16z)(1 - 136z + 16z2) Regular singular points: z=0 z = TO

Riemann P-symbol:

0 TO z1 z2 zoX

z = z1 = -16" z = z2 ^ 8.5

0 4 0 0 0

0 1 1 1 1

0 1 2 2 2

0 1 3 3 3

0 5 4 4 4

1 3 3 3 3

2 2 2 2

10 11 12

20 21 22

Intersection pairing:

0 0 0 0 0 1

0 0 0 0 1 0

0 0 132 42 66 102

0 0 42 14 -42 14

0 1 66 -42 -36 -18

1 0 102 14 -18 2

Large volume asymptotics: /

n asy (t) =

66t2 + 132t + 21t2 - 21t + 12

3 _ -3-3,2 _ 69, _ 47 42iZ(3) n 3

-22t3 - 33t2 - f1 - 47 +

11,4 , 2512

r + ¥ ^ -

42if(3). 115 n 3 1 + 288

Monodromy matrices:

1 1 198 0 0 0 \ / -1109 -67 -111054 -10010 17610 -462

0 1 132 42 0 0 -2100 -131 -211308 -19026 33780 -924

0 0 1 0 -1 0 M to = 17 1 1699 154 -269 7

0 0 0 1 0 0 -18 -1 -1800 -167 288 -8

0 0 0 0 1 -1 20 1 1968 182 -305 7

0 0 0 0 0 1 ) V 50 2 4848 462 -738 15

133 66 23760 924 -6336 528

264 133 47520 1848 -12672 1056

-14 96

15 -96

-14 97

-1 -359 1 360

-2 -1 -360

-4 -2 -720 -28

Mzox =

0 0 0 0 0

-8 8 -8 -15 /

/ 157 0 14040 1456 -1872 0\

648 1 58320 6048 -7776 0

-3 0 -269 -28 36 0

12 0 1080 113 -144 0

\ -9 0 -810 -84

\-1 0 -102 -14 18 -1 /

Generators of cohomology ring: °3 . °2, 2 48 ' IT

(1 ' „1 ' „1, 1, „2 ' g ' € HQ ■ Q(X) ® ...® H4 • 4(X)

Total Chern character:

c(X) = 1 + (5„1 1 + „2) - 336„^ + 636 _ 1 1 2 48_14

z = zoX ^ 0.007

1 Intersection numbers:

2 01 1.01 1 = 14 , a1 1.02 = 28 , 02.02 = 62

Genus zero Gromov-Witten invariants n0\ (left) and ni2), (right)

-20 28

222 462

7 564 18 732

433 184 999 488

27132712 61 606 888

1 883975,918 4 190 840 486

138 861570764 305141892524

10734197 390 880 23 363 298 862176

860337105 561204 1859 026775 810036

70983 785 067 825 508 152499 803 765 006068

Genus one Gromov-Witten invariants n d

3 Zeros of integral quantum periods:

4 05 = 0 at zoX ,

5 ÖR = 66n0 + 132n1 - n2 + n3 - n4 - 2n5 = 0 at z1

6 nBz = 52n0 + 216n1 - n2 + 4n3 - 3n5 = 0 at z2

10 11 12

20 21 22

24 14 591 4 331039 882540559 145 991 147911 21275 702 877 573

0 0 0 0 528 360 424 446 616 816

A.7. Skew symmetric sigma model Calabi-Yau fourfold X1, 17,7

Picard-Fuchs operator:

L(z) = +316932(0 - 1)05 - 98z0 (70045305 + 133505804 + 1609 08003 + 87928502 + 2490180 + 29106) + 962754229z2(06 - 197696088305 - 10395 50903104 - 1499166296903 - 10456423 60002 - 36676299100 - 521 151456) + 2z3(981272797906 + 53 190263 57305 + 105 89543246304 + 103 996363 80103 + 54017 18810602 + 140781117470 + 1415 445066) - 2z4(1154948689606 + 4632432180405 + 73 290 469 42604 + 60 074 870 02603 + 27 353 847 16902 + 6 669 746 7190 + 696 036 075) + 174z5(1 66619806 + 600698105 + 1049781904 + 1155107803 + 816213002 + 3 3310470 + 588 537) - 211932z6 (0 + 1)5 (20 + 3)

Discriminant locus:

A(z) = (1 - 188z - 2368z2 + 4z3)

Regular singular points:

z=0 z =

z = ZoX(= Z3) ^ 0.005 z = z1 s -0.084 z = z2 s 592

Riemann P-symbol:

0 TO z1 z2 zoX

0 1 0 0 0

0 1 1 1 1

0 1 2 2 2

0 1 3 3 3

0 1 4 4 4

I1 2 3 3 3

10 11 12

20 21 22

Intersection pairing:

0 0 0 0 0 1

0 0 0 0 1 0

0 0 98 33 49 79

0 0 33 12 -33 12

0 1 49 -33 -30 -15

1 0 79 12 -15 2

Large volume asymptotics:

nasy(t) =

49t2 + 98t + 82

_49 t3 _ 49 t2 _ 109 t 22y ,

t a ~ i , I OA +

229 _i_ 41if(3)

2214 4. 2 _ :

101 + OA 1

'-t A- —

L + IS

Monodromy matrices:

M„ =

1 1 147 0 00 \ 99 49 13328 588 -3724 343

0 1 98 33 00 196 99 26656 1176 -7448 686

0 0 1 0 -1 0 Mz1 -2 1 271 -12 76 -7

0 0 0 1 0 0 = 2 1 272 13 -76 7

0 0 0 0 1 -1 -2 -1 - 272 -12 77 -7

0 0 0 0 0 1 / -4 -2 - 544 -24 152 -13

4117 1568 478828 23520 -115248 8232 \ 1 0 0 0 0 0

4809 1833 559447 27480 -134652 9618 0 1 0 0 0 0

-84 -32 -9771 480 2352 -168 0 0 1 0 0 0

0 0 0 1 0 0 MZOx = 0 0 0 1 0 0

-168 -64 -19544 960 4705 -336 0 0 0 0 1 0

V- 441 -168 -51303 2520 12348 -881 -1 0 -79 -12 15 -1

/ -3975 -1490 -459291 -22890 110201 -7854

-1757 -617 -197897 -10479 46942 -3318

67 25 7728 387 -1853 132

121 47 14181 682 -3423 245

151 57 17499 867 -4205 300

198 74 22862 1146 -5487 391

Generators of cohomology ring:

' hi H4. s

98 ' 98

1 ; H ; a2,H

e H°>0(X) © H4'4(X)

Total Chern character:

c(X) = 1 + (4H2 - 2ao) - 328-+ 672-

2 98_98

Intersection numbers:

a2.a2 = 44 , a2.H2 = 65 , H2.H2 = 98

Zeros of integral quantum periods:

n = 0 at zoX ,

ng = 49n0 + 98n1 - n2 + n3 - n4 - 2n5 = 0 at z1 ,

nB = -196n0 - 229n1 + 4n2 + 8n4 + 21n5 = 0 at z2

33 170 16126 1 141312 100955 257 9 821360694 1028 274 636900 1 134 58193 073 000 13 032484 062 881000 1545108 865 260914434

0 721 38 255 3 042 676 274 320 123 27 276710118 2 897 092 850 989 323 207 209 581582 37 444 642 819 824 776 4 469 922 540366355 762

33 12_ 33 t , 33

Genus one Gromov-Witten invariants n\ j

224 386

6 206613 908 6

7 83 707 955196 7

8 23 455 827 469 526 8

9 5 401382970402176 9

1 107 021477 254 814128

12 Appendix B. Lines on Calabi-Yau fourfolds 12

14 To verify the computed integral quantum periods and the deduced quantum cohomology ring, 14

15 we here enumerate the number of lines with a marked point located on a codimension two 15

16 algebraic cycle in the studied Calabi-Yau fourfolds, which arise as complete intersections in 16

17 Grassmannian spaces Gr(2 , n) for various choices of n. Note that the presented derivation gen- 17

18 eralizes to other complete intersection varieties embedded into general Grassmannians Gr(k , n) 18

19 as well, and this appendix is rather independent from the main text. 19

20 The moduli space M1 of lines with a marked point in the ambient Grassmannian variety 20

21 Gr(2, n) is the flag variety Fl(1, 2, 3, n), whose points are the flags V1 c V2 c V3 c Vn of com- 21

22 plex vector spaces with Vi ~ Cl. For such a flag the two-dimensional quotient vector space 22

23 V3/V1 describes the projective line P(V3/V1). The points in P(V3/V1) are the one-dimensional 23

24 subvector spaces A1 c V3 /V1, which canonically define two planes V1 © A1 to be identified with 24

25 points in the Grassmannian variety Gr(2, n). Furthermore, the subvector space A1 = V2/V1 cor- 25

26 responds to the marked point on the projective line that maps to the two plane V1 © V2/V1 ~ V2 26

27 in Gr(2, n). It defines the evaluation map of the marked point 27

28 ev1: M1 ^ Gr(2,n),V1 c V2 c V3 c Vn ^ V2 . (B.1) 28

30 We realize the flag variety M1 ~ Fl(1, 2, 3, n) in terms of the nested fibrations of projective 30

31 spaces [106]: 31

32 U1 © Q1 U2 © Q2 U3 © Q3 32

33 111 33

34 ill (B.2) 34

35 P(Vn) P(Q1) P(Q2) 35

37 Here, U1, U2 and U3 are the universal line bundles of the (fibered) projective spaces, whereas Q1, 37 Q2 and Q3 are their respective quotient bundles of dimension (n - 1), (n - 2) and (n - 3), i.e.,

39 U1 © Q1 = Vn, U2 © Q2 = n*Q1 , U3 © Q3 = n2*Q2 . (B.3) 39

40 Let M2 be the moduli space of lines with two marked points in Gr(2, n) given by the fibration 40

42 P(n|Uz ©U3) -► M2 42

43 f I /R A ^ 43

f I . (B.4)

45 M 1 45

46 The projection f to the base M1 is the forgetful map that removes the second marked point, 46

47 whereas its evaluation map reads 47

1 ev2 : M2 ^ Gr(2,n) ,(A1,V1 c V2 c V3 c Vn) ^ V1 © A1 , (B.5) 1

3 in terms of the one-dimensional vector space A1 for the points of the projective fibers and the 3

4 flag V1 c V2 c V3 c Vn for the base point in M1. 4

The cohomology of the flag variety M1 — as given in the nested fibration (B.2) — becomes

6 [106] 6

7 H *(M1, Q) = Q[H1,H2,H3,^1),...,^n-3)]/I. (B.6) 7

9 The generators of the cohomology ring arise from the Chern classes of the bundles over the 9

nested fibration (B.2) as

11 H1 = -7T2*n*C1(U1) , H2 = -7T2*C1(U2) , H3 = -C1(U3) , 11

12 (B.7) 12

13 = ci(Q3), i = 1,...,n - 3 , 13

14 (i) 14 where H1, H2, and H3 are the hyperplane classes of the (fibered) projective spaces and ' the

Chern classes of the quotient bundle over the last fibered projective space P(Q2). The ideal I is

17 generated by the homogeneous terms (with respect to the form degree of the generators) in the 17

18 expression 18

19 1 - (1 - H1)(1 - H2)(1 - H3)(1 + f3(1) +...+ f3("-3)). (B.8) 19

21 Note that the relations in the ideal I determine the cohomology classes ^ ' in terms of the 21

22 hyperplane class generators H1, H2, and H3. Furthermore, the total Chern class of the quotient 22

23 bundles n2fn1*Q1 and n|Q2read 23

24 1 24

25 ^21n1c(Q1) = 1 + §1(1) + ... + fi"-1) = --— e H*(M1), 25

26 1 - H1 26

27 n1c(Q2) = 1 + ^ + ... + f2"-2) = (1 * H) e H 1(M1). 27

28 (1 - H1)(1 - H2) 28

29 Now we want to enumerate the number of lines on Calabi-Yau fourfolds, which for our class 29

30 of examples are given as complete intersections Xk1,...,ka (with ki > 1) embedded in the Grass- 30

31 mannian spaces Gr(2, n) as the zero locus of a generic section in O(k1a1) ©... © O(kaa1). Since 31

32 dimC Gr(2, n) = 2(n - 2) and c1 (Gr(2, n)) = n a1, we obtain four-dimensional Calabi-Yau va- 32

33 rieties in Gr(2, n) only for 33

35 a = 2(n - 4), n = k1 +... + ka. (B.9) 35

36 In the next step, we impose the complete intersection constraints on the level of the moduli 36

37 space M1. We observe that the line bundles O (ki a1) induce on M1 the vector bundles 37

39 B(k) = fi ev2 O(ko1). (B.10) 39

40 These bundles are explicitly determined to be 40

41 ; r , ^ 41

42 B(k) = Symk \n\n'2U\ ® (n*U2 ФЫъ)\ , (B.11) 42

43 in terms of symmetrized tensor products of the rank two bundle n*U1 ® U2 ФU3) on M]_. 43

44 By construction the zeros of induced sections on B(k) describe the loci in Gr(2, n), where the 44

45 entire projective line of M1 vanishes. Thus the zero locus of the induced section of the bundle 45

46 B(k1) Ф ... Ф Ba on M1 describes the moduli space of lines with a single marked point of the 46

47 Calabi-Yau variety Xkl..,ka. 47

10 11 12

20 21 22

Table B.1

The table enumerates lines with marked points on the codimension two (pulledback) Schubert cycles 01 and 02 for the listed Calabi-Yau fourfolds embedded as complete intersections in Grassmannians. These number are calculated from the derived intersection formula (B.13), and the results correctly relate with eq. (B.14) to the genus zero Gromov-Witten invariants at degree one tabulated in Appendix A.

Calabi-Yau fourfold N(01,1 ) N(02)

Xi,4 C Gr(2, 5) 3 680 5 760

*2,3 C Gr(2, 5) 2160 3 420

X13 3 C Gr(2,6) 1431 2 727

X12 22 C Gr(2, 6) 1072 2 064

X15 2 C Gr(2,7) 728 1568

Xi8 C Gr(2, 8) 504 1 176

To enumerate genus zero Gromov-Witten invariants at degree one on Xk1,..,ka, it remains to restrict the marked point on the lines to one of the codimension two (pulled-back) Schubert classes o1;1 or o2 in Xk1,..,ka. On the moduli space M1 these classes become ev^ o1;1 and ev^ o2, respectively. Note that the quotient bundle QGr(2,n) of Gr(2, n) pulls back to ev^ QGr(2,n) — n-2*Q2, which — due to c(Qor(2,n)) = 1 + 01 + 02 + ... — implies together with eqs. (B.3) and (B.6) that ev1 o1 = H1 + H2 and ev1 o2 = H2 + H22 + H1H2. Thus, with o2 = o2 + ou we find

ev1 01,1 = Hi H2

ev1 02 = H2 + H22 + H1H2 .

(B.12)

With all the necessary ingredients assembled, we now count the number of lines with its marked point restricted to a codimension two Schubert cycle in Xk1,..,kn according to

N(aii) = J ctop(B(ki)) U ...U ctop(B(ka)) U evi ^1,1

N(02) = J ctop(B(ki)) U ... U ctop(B(ka)) U evi ct2 .

(B.13)

Here ctop denotes the top Chern class of the bundles B(ki), which by construction have rank ki + 1. Thus — imposing the Calabi-Yau fourfold conditions (B.9) — the integrand becomes an element of H(3n-6,3n-6)(M1), which indeed represents a top form on M1 because dimC M1 = 3n - 6. The numbers of lines N(o1t1) and N(o2) obtained in this way compare to the genus zero Gromov-Witten invariants n01) and n02) of the quantum cohomology ring as (cf., Section 3.1 and Appendix A)

N(ou) N(02)

01,1.01,1 01,1.02

01,1.02 02.02

i(2) 0,1,

(B.14)

in terms of the intersection pairings of the Schubert cycles o1; 1 and o2 on the Calabi-Yau fourfold

Xk1,...,ka.

In this work we explicitly analyze the Calabi-Yau fourfolds X1i4, X2,3, X13 3, X12 22, X15,2, and X18 (with the obvious notation for repeated indices and the corresponding embedding space

10 11 12

20 21 22

1 Gr(2, n) determined through Calabi-Yau fourfold conditions (B.9)). For these Calabi-Yau four- 1

2 folds we explicitly count the number of lines according to eq. (B.13) as listed in Table B.1. For 2

3 all our examples we find agreement with the genus zero Gromov-Witten invariants at degree one 3

4 listed in Appendix A. This furnishes another non-trivial check on the deduced linear combina- 4

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