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Nuclear Physics B 776 [PM] (2007) 287-326

Black holes in type IIA string on Calabi-Yau threefolds with affine ADE geometries and q-deformed 2d quiver

gauge theories

R. Ahl Laamaraab, A. Belhajab c d, L.B. Drissiab, E.H. Saidiabce*

a Lab/UFR—Physique des Hautes Energies, Faculté des Sciences, Rabat, Morocco b GNPHE, Groupement National de Physique des Hautes Energies, Siège focal: FS, Rabat, Morocco c Virtual African Centre for Basic Science & Technology, Focal point, Lab/UFR-PHE, FSR, Morocco d Departamento de Fisica Teorica, Universidad de Zaragoza, 50009 Zaragoza, Spain e Académie Hassan II des Sciences & Techniques, Collège Physique-Chimie, Royaume du Maroc

Received 12 December 2006; received in revised form 27 February 2007; accepted 7 March 2007

Available online 20 April 2007

Abstract

Motivated by studies on 4d black holes and q-deformed 2d Yang-Mills theory, and borrowing ideas from compact geometry of the blowing up of affine ADE singularities, we build a class of local Calabi-Yau threefolds (CY3) extending the local 2-torus model O(m) ® O(—m) ^ T2 considered in [C. Gomez, S. Montanez, A comment on quantum distribution functions and the OSV conjecture, hep-th/0608162] to test OSV conjecture. We first study toric realizations of T2 and then build a toric representation of X3 using intersections of local Calabi-Yau threefolds O(m) ® O(—m — 2) ^ P1. We develop the 2d N = 2 linear a -model for this class of toric CY3s. Then we use these local backgrounds to study partition function of 4d black holes in type IIA string theory and the underlying q-deformed 2d quiver gauge theories. We also make comments on 4d black holes obtained from D-branes wrapping cycles in O(m) ® O(—m — 2) ^ Bk with m = (mi,...,mk) a k-dim integer vector and Bk a compact complex one dimension base consisting of the intersection of k 2-spheres S? with generic intersection matrix ¡¡j. We give as well the explicit expression of the q-deformed path integral measure of the partition function of the 2d quiver gauge theory in terms of ¡¡j. A comment on the link between our analysis and the construction of [N. Caporaso, M. Cirafici, L. Griguolo, S. Pasquetti, D. Seminara, R.J. Szabo, Topological strings, two-dimensional Yang-Mills theory and Chern-Simons theory on torus bundles, hep-th/0609129] is also given. © 2007 Elsevier B.V. All rights reserved.

* Corresponding author at: Lab/UFR—Physique des Hautes Energies, Faculté des Sciences, Rabat, Morocco. E-mail addresses: ahllaamara@gmail.com (R. Ahl Laamara), belhaj@unizar.es (A. Belhaj), lbdrissi@gmail.com (L.B. Drissi), h-saidi@fsr.ac.ma (E.H. Saidi).

0550-3213/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2007.03.047

Keywords: Black holes in string theory; OSV conjecture; q-Deformed 2d quiver gauge theory; Topological string theory

1. Introduction

Few years ago, Ooguri, Strominger and Vafa (OSV) have made a conjecture [1] relating the microstates counting of 4d BPS black holes in type II string theory on Calabi-Yau threefolds X3 to the topological string partition function Ztop on the same manifold. The equivalence between the partition function Zbrane of large N D-branes, and that of the associated 4d BPS black hole ZBH leads to the correspondence Zbrane = | Ztop |2 to all orders in N expansion. OSV conjecture has brought important developments on this link: it provides the non-perturbative completion of the topological string theory [2-14] and gives a way to compute the corrections to 4d N = 2 Bekenstein-Hawking entropy [10,15]. OSV relation has been extended in [7,16] to open topological strings, which capture BPS states data information on D-branes wrapped on Lagrangian submanifolds of the Calabi-Yau 3-folds.

Evidence for OSV proposal has been obtained by using local Calabi-Yau threefolds and some known results on 2d U(N) Yang-Mills theory [17]. It has been first tested in [2] by considering configurations of D-branes wrapped cycles O(m) © O(—m) ^ T2 for m a positive definite integer. Then it has been checked in [18] by using wrapped D-branes in a vector bundle of rank 2 non-trivial fibration over a genus g-Riemann surface O(2g + m — 2) © O(—m) ^ Sg .It has been shown in these studies that the BPS black hole partition function localizes onto field configuration which are invariant under U(1) actions of the fibers. In this way, the 4d gauge theory reduces effectively to q-deformed Yang-Mills theory on Sg. These works have been recently enlarged in [19] by considering local Calabi-Yau manifolds with torus symmetries such as local P2 = O(—3) ^ P2, local F0 = O(—2, —2) ^ P^ x P^ and ALE x C. In the last example, the Ak type ALE space is given by gluing together (k +1) copies of C2 viewed as a real two-dimensional base fibered by torus T2. Other related works have been developed in [20-24,26-29]. M-theory and AdS3/CFT2 interpretations of OSV formula have been also studied in [30].

In this paper we contribute to the program of testing OSV conjecture for massive 4d black holes on toric Calabi-Yau threefolds in connection with q-deformed quiver gauge theories in two dimensions. This study has been motivated by looking for a 2d N = 2 supersymmetric gauged linear sigma model of O(m) © O(—m) ^ T2. More precisely we consider a special class of local Calabi-Yau threefolds which combine features of both O(m) © O(—m) ^ T2 and O(m) © O(—m — 2) ^ S2, and involves toric graphs that look like affine ADE Dynkin diagrams and beyond. Recall that affine ADE geometries are known to be described by elliptic fibration over C2 and lead to N = 2 conformal quiver gauge theories in 4d space-time with gauge groups G = ]"[i U(siM), where the positive integers si are the Dynkin weights of affine Kac-Moody algebras. Borrowing the method of the above quiver gauge theories [31-33] and using the results of [2], we engineer a new class of local Calabi-Yau threefolds that enlarges further the class of CY3s used before and that agrees with OSV conjecture.

Our study gives moreover an explicit toric representation of O(m) © O(—m) ^ T2 especially if one recalls that T2 viewed as S1 x S1 does not have, to our knowledge, a simple nor unique toric realization. As we know real skeleton base of toric diagrams representing toric manifolds requires at least one 2-sphere S2 which is not the case for the simplest S1 x S1 geometry. In the analysis to be developed in this study, the 2-torus will be realized by using special linear combinations [An+\] of intersecting 2-spheres with same homology as a 2-torus. The positive integer n > 1 refers to the arbitrariness in the number of [S2]'s one can use to get the elliptic

curve class [T2]. Among our main results, we mention that 2d quiver gauge theories, associated with BPS black holes in type IIA string on the local CY3s we have considered, are classified by the "sign" of the intersection matrix ¡¡k of the real 2-spheres S2 forming the compact base

According to whether J2k ¡¡kuk > 0, J]k ¡¡kuk = 0 or J2k ¡¡kuk < 0 for some positive integer vector (uk), we then distinguish three kinds of local models. For the second case

the uk's are just the Dynkin weights of affine Kac-Moody algebras and the corresponding 2d quiver gauge theory is non-deformed in agreement with the result of [2]. The above relation corresponds then to the case where the genus g-Riemann surface is a 2-torus; i.e.

2g — 2 = 0 ^ g = 1.

For the two other cases, the gauge theory is q-deformed and recovers, as a particular case, the study of [19] dealing with ALE spaces.

On the other hand, this study will be done in the type IIA string theory set up. So we shall also use our construction to complete partial results in literature on field theoretic realization using 2d N = (2, 2) linear sigma model for affine ADE geometries with special focus on the An case. To our knowledge, this supersymmetric 2d field realization of local Calabi-Yau threefolds has not been considered before.

The organization of paper is as follows: In Section 2, we begin by recalling general features on toric graphs. Then we study the toric realizations of local T2 using the techniques of blowing up of affine ADE geometries. We study also the field theoretic realization of the non-trivial fibrations of local T2 which corresponds to implementing framing property [39]. We complete this section by making a comparison between our construction and the alternative toric realization given in [24]. In Section 3, we develop the N = 2 supersymmetric gauged linear sigma model describing the local torus geometry. This study gives an explicit field theoretic realization of geometric objects such as the surface divisors of the local 2-torus and their edge boundaries in terms of field equations of motion and vevs. In Section 4, we construct the 4d BPS black holes in type IIA string by considering brane configurations using D0-D2-D4-branes in the non-compact 4-cycles. We show, amongst others, that the gauge theory of the D-branes, which is dual to topological strings on the Calabi-Yau threefold, localizes to a "q-deformed" 2d quiver gauge theory on the compact part of the affine ADE geometry and test OSV conjecture. More precisely, we show that the usual power (2g — 2) of the weight of the deformed path integral measure for O(2g + m — 2) © O(—m) ^ Sg gets replaced, in case of 2d quiver gauge theories, by the intersection matrix ¡¡j. There, we show that the property (2g — 2) = 0 for g = 1 corresponds to the identity Y,j ¡¡jsj = 0 of affine Kac-Moody algebras. Motivated by this link, we study 4d black holes based on D-branes wrapping cycles in O(m) © O(—m — 2) ^ Bk with m = (m1,...,mk) an integer vector and where Bk is a complex one dimension base consisting of the intersection of k 2-spheres Sf with generic intersection matrix ¡¡j. In Section 5, we give our conclusion and outlook.

2. Toric realization of local T2

¡¡k = [S2] S2], ¡,k = 0,...,n.

In this section, we build toric representations of the class of local 2-torus O(m) © O(—m) ^ T2 by developing the idea outlined in the introduction. There, it was observed that although,

strictly speaking, T2 = S1 x S1 is not a toric manifold (base reduced to two points), it may nevertheless be realized by gluing several 2-spheres in very special ways. Before showing how this can be implemented in the above local CY3, recall that the study of local threefold geometry

O(m) © O(—m) ^ T2 (2.1)

is important from several views. It has been used in [2] to test OSV conjecture [1] and was behind the study of several generalizations, in particular

O(m + 2g — 2) © O(—m) ^ Sg. (2.2)

The novelty brought by these class of local CY3s stems also from the use of the non-trivial rank 2 fiber O(p) © O(—m), with p = m + 2g — 2 rather than O(2g — 2) © O(0). These nontrivial fibers, which were motivated by implementing twisting by framing [39], turn out to play a crucial role in the study of 4d BPS black holes from type IIA string theory compactification. Generally, the class of local CY3s Eq. (2.2) which will be considered later (Section 4), is mainly characterized by two integers m and the genus g and may be generically denoted as follows

X(k3'k2'k1 \ g = 0, 1,...; m e Z. (2.3)

Here ki = 2 — 2g, k2 = — m and k3 = m + 2g — 2 satisfy the Calabi-Yau condition Y,i k = 0 leaving only two free integers m and g. By making choices of these integers one picks up a particular local CY3. For g = m = 0 and g = 1, m = 0 for example, we have O(0) © O(—2) ^ P1 and O(0) © O(0) ^ T2 respectively. The local CY3s (2.3) may be also viewed as a line bundle C(m+2g—2) of the complex two-dimensional divisor

[D(m,g)] = O(—m) ^ Sg. (2.4)

This local complex surface [D(mg)] has a compact curve Sg with the following intersection number

[Dmg]. [Sg] = m + 2g — 2. (2.5)

In the case where Sg is a 2-torus (g = 1), the above two integers series of local CY3 reduces to the one integer threefold series x<f1,—m,0). The previous non-compact real 4-cycles are then given by [2]

W{m,g)] = O(—m) ^ T2, (2.6)

and their intersection number with the 2-torus class [T2] is [Dm,g]. [T2] = m.

2.1. Toric realization

Our main objectives in this subsection deals with the two following: (1) Build explicit toric realizations of the local 2-torus by using particular realizations of the real 2-cycle [T2]. These realizations, which will be used later on, have been motivated from results on blowing up of affine ADE singularities of ALE spaces and geometric engineering of 4d N = 2 supper QFTs [31-33]. It gives a powerful tool for the explicit study of the special features of local 2-torus and allows more insight in the building of new classes of local Calabi-Yau threefolds for testing OSV conjecture.

Fig. 1. (a) Toric graph of a compact 2-sphere with Kahler modulus r = 0. (b) The fattening of compact 2-sphere where the circle S1 is represented. S1 shrinks at the fixed points of the U(1) action.

(2) Use the result of the analysis of point (1) to complete partial results in literature on type IIA geometry with affine ADE singularities. More precisely, we construct the 2d N = 2 super-symmetric gauged linear sigma model

S2d=2 -j d2ad4e{^ Q+e^qy«^ + Y, (2.7)

giving the field realization of local 2-torii. This construction to be developed further in Section 3, will be used for the two following:

(a) Work out explicitly the results of (q-deformed) 2d YM theories on T2 and give their extensions to 2d quiver gauge theories on the elliptic curve realized a linear combination of intersecting 2-spheres.

(b) Study partition function properties of 4d BPS black holes along the lines of [30] and ulterior studies [19,26-29] to test OSV conjecture.

2.1.1. General on toric graphs

In building toric realization of local CY3s [34-38], one encounters few basic objects that do almost the complete job in striking analogy with the work done by Feynman graphs in perturba-tive QFTs. In particular, one has:

(i) Propagators given by the toric graph of the real 2-sphere S2, see Fig. 1. It corresponds to the two points free field Green function in the language of quantum field theory (QFT). Recall that the real 2-sphere S2 is given by the compactification of the complex line C. The latter can be realized (polar coordinates) as the half line R+ with fiber S1 that shrinks at the origin. The compactification of C; i.e. P1 the complex one-dimensional projective space, is obtained by restricting R+ to a finite straight line (a segment) which is interpreted as a propagator in the language of QFT Feynman graphs [39]. Alike, we distinguish here also two situations:

(a) Finite (internal) lines which are associated with compact 2-spheres and are interpreted in terms of propagating closed string states.

(P) Infinite (external) lines with non-compact 2-spheres (discs/complex plane) and are used to implement open string state contributions to topological string amplitudes. These open string states end on D-branes.

With these propagators, one already build complex one- and two-dimensional toric manifolds as shown on Fig. 2. To construct CY3, we need 3-vertices which we discuss in the next.

(ii) The 3-vertex can be thought of locally as the intersection of the three complex lines of C3 [39]. This vertex plays a crucial role in building local CY3s. For instance O(-3) ^ P2 has three the 3-vertices corresponding to the three fixed points of the U4(1)/U(1) toric actions, see Fig. 3. The edge propagators (2-spheres) are fixed under U2(1) subgroups of U4(1)/U(1).

With these objects one can build other local CY3s. Using fat propagators and vertices we can also have a picture on their internal topology. One may also describe its even dimensional homology cycles. Real 2-cycles Ci of local CY3s are represented by linear combinations of seg-

Fig. 2. (a) Toric quiver of two intersecting 2-spheres, (b) the corresponding fat toric graph where we have also represented the circles associated with the two U(1) toric actions.

(a) (b)

Fig. 3. Toric graph of O(-3) ^ P2. The compact part is P2. It consists of three intersecting P1's and three vertices. (a) Figure (on left) represents real skeleton. (b) Figure (on right) gives its fattening.

ments and real 4-cycles D (divisors of CY3s) by 2d polygons: a triangle for P2, a rectangle for Pfiber x Pbase and so on. Compact 4-cycles have then finite size.

The power of the toric quiver realization of threefolds comes also from its simplicity due to the fact that the full structure of toric CY3 quiver diagrams is basically captured by the lines (2-cycles) since boundaries of divisors (4-cycles) are given by taking cross products of pairs of straight line generators.

Note that, though P1 is not a Calabi-Yau submanifold since its first Chern class is c1 (P1) = 2, this one-dimensional complex projective space, together with the vertex, are the basic objects in drawing the 2d graphs of toric manifolds. Note also that torii S1, T2 and T3 of CY3 appear in this construction as fibers and play a fundamental role in the study of topological string theory amplitudes [39-41].

2.1.2. Toric realizations of x3n'-m'0)

It has been shown in [2] that the one integer series of spaces x<m'-m'0) = O(m) © O(-m) ^ T2 is a toric local Calabi-Yau threefold used to check OSV conjecture. From the above study on toric graphs it follows that, a priori, one should be able to draw its corresponding toric quiver diagram. However, unlike the 2-sphere, the usual 2-torus S1 x S1 has no simple toric graph realization. Then the question is what kind of 2-torii T2 are involved in x^'-m'0)?

Here below, we would like to address this problem by using the graphic method of toric geometry. In particular, we develop a way to build toric graphs representing classes of [T2]. This will be done by realizing [T2] in terms of intersecting 2-spheres S2 by expressing [T2] as

a "sum" of intersecting 2-spheres,

[T2] -J2 ui [S?]. u e Z+,

and thinking about x<3""-m'0) as a special limit of a family of local CY3s

X3m,-m,0) ^ x(m -P P-m) = O(m) © O(-p) ^ An+1 (2.8)

where m and p are some (n + 1)-dimensional integer vectors given by

m = (mo.mo.....mn). p = (po.Po.--.Pn)- (2.9)

In this way the Calabi-Yau condition reads

Pi - mi = 2. i = 0.....n. (2.10)

which we denote formally as p - m = 2. Then, we extend the obtained results to build the toric graphs describing x(m.-p.2) and their even real homology cycles. Here it is interesting to note the two following:

(a) This construction is important since once we have the toric quivers, one can use them for different purposes. For instance, we can use the toric graphs in the topological vertex method of [39,41] to compute explicitly the partition functions of the topological string on x^-^2.

(b) The local CY3 we propose x3m-pp-m) is not exactly x<3n.-m.0) considered in [2]; it is more general. These manifolds have different Kahler moduli spaces and their U(1) x U(1) isometry groups are realized differently. To fix the idea, think about the homogeneity group of the compact geometry An+1 of Eq. (2.8) as

Un+1(l)

U(1) x U y ~ U(1) x U(1). n> 1. (2.11)

To proceed we shall deal separately with base An+\ and the two line fibers of x3m.-p.p-m). We first look for toric quivers to describe the An+1 class and O(±k0..... ±kn) independently. Then we use 3-vertex of C3 and the Calabi-Yau condition to glue the various pieces. At the end, we get the right real 3-dimensional toric graph of our local CY3s and, by fattening, the topology of x(m.-p.2). This approach is interesting since one can control completely the engineering of the toric quivers of local Calabi-Yau threefolds. Moreover, seen that O(±k0..... ±kn) are line bundles, their toric graphs are mainly the toric graphs of O(±m) which are locally given by C patches. What remains is the determination of toric quiver of An+i class. In Figs. 4 and 5, we develop two illustrating examples. The first one (Fig. 4) concerns the compact base A6 using six intersecting 2-spheres in the same spirit one uses for the blowing up of singularities of ALE complex surfaces. The second example (Fig. 5) has in addition external non-compact lines.

In what follows, we show that these toric graphs and their fattening constitutes in fact particular topologies of infinitely many possible graphs.

(a) Solving the constraint equation of2-torus homology As noted before, the question of drawing a toric graph for T2 = S1 x S1 seems to have no sense at first sight. This is because the basic (irreducible) real 2-cycle in toric geometry is P1 ~ S2 with self intersection

[P1]. [P1] = -2. (2.12)

Fig. 5. This figure represents the fattening of the toric graph of Ag with 3-vertices and external legs ending on D-branes discs.

Its toric graph is a straight line segment with length given by the size of the 2-sphere (Kahler modulus r). The 2-torus has a zero self-intersection

[T2]. [T2] = 0, (2.13)

and a priori has no simple nor unique toric diagram. Tori Tn are generally speaking associated with Un(1) phases of complex variables. For instance, on complex line C with local coordinate z, the unit circle S1 is given by |z| = 1 and the U(1) symmetry acts as z ^ eli)z. This circle and the associated U(1) symmetry are exhibited when fattening toric graphs as shown on previous figures.

To build a toric representation of T2; but now viewed as An+1, we use intersecting 2-spheres with particular combinations as

An+1 = Y * [c 1, C -p!,

(2.14)

Fig. 6. Affine simply laced ADE Dynkin diagrams. In homology language, dots represent the 2-spheres and links the intersections. In toric language, the Poincare dual of these diagrams are associated with toric graphs of 2-torii.

where the positive integers €i are obtained by solving Eq. (2.13). Denoting by Iij the intersection matrix of the 2-cycles Ci,

[C^. [Cj] = Iij, Iii = -2, then the condition to fulfill Eq. (2.13) is

(2.15)

J2 CiI'Jj = E e< Ei^' = 0, e N. (2.16)

i,j=0 i=0 \j =0 '

A solution of this constraint relation is given by taking

Iij = -Kij(g), (2.17)

as minus the generalized Cartan matrix of affine Kac-Moody algebras g. In this case, the positive integers €j are interpreted as the Dynkin weights and the topology of the 2-torus is same as the affine Dynkin diagrams. The simplest examples of simply laced affine ADE geometry are reported in Fig. 6.

Therefore, there are infinitely many toric quivers realizing T2 in terms of intersecting 2-spheres. But to fix the ideas we will mainly focus on the series based on affine An Kac-Moody algebras. In this case the elliptic curve is

An+1 = J2 Ci,

€i = 1.

(2.18)

It involves (n + 1) real 2-cycles with intersection matrix Ci. Cj = -Kij, i, j = 0, 1,...,n,

Kij = Si-i,j - 2Sij + Si+j, n + 1 = 0.

(2.19)

The curve An+\ has the homology of a 2-torus realized by the intersection of (n + 1) 2-spheres with the topology of Dynkin diagram of affine An. The Kahler parameter r of the curve An+\ defined as

(2.20)

with to being the usual real Kahler 2-form, is given by the sum over the Kahler parameters ri of the 2-cycles Cl making up An+\. We have

to rl toi, I toj = Sj (2.21)

and so the Kahler modulus of An+1 is given by

■i = Y, ri' (2.22)

i=0^ i=0

r = ^ toi = } r

In this computation we have ignored the volume of the intersection points of the 2-spheres Ci and Cj since they are isolated points and moreover their volumes vanish in any case. To fix the ideas, we shall set

r > ro > ri > ••• > rn > 0, (2.23)

and for special computation, in particular when we study the path integral of the partition function on quiver gauge theory on An+1 dual to 4d black hole (Section 4), we will in general sit at the moduli space point where

ri = --. (2.24)

In all these cases, the Kahler moduli ri are positive. We shall also suppose that we are away from r = 0 describing the singularity of the curve An+1 and where full non-Abelian gauge symmetry of quiver theory is restored.

(b) 4-cycles of O(m) © O(-p) ^ An+1 Using the above realization, one can go ahead and build the 4-cycle and the local Calabi-Yau threefold. Viewed as a whole, the non-compact 4-cycle is

D : O(-p) ^ ^[Sf^, (2.25)

with toric graph as given below.

Now using Eq. (2.13), the relation [D] . [An+1] = m giving the intersection number which reads as

[D]. I][^i2] = m (2.26)

and splitting m as

m = ^(mi - 2), pi = mi - 2. (2.27)

It is not difficult to see that [D] can be decomposed as follows, see also Fig. 7.

[D] = YPi], Di = O(-pi) ^ Sf, (2.28)

Fig. 7. Toric graph of a non-compact 4-cycle of the local Calabi-Yau threefold. The compact part consists of eight intersecting 2-spheres.

with the property

[D]. [52] = mi - 2. If we take m a positive integer, then we should have

Y^mi > 2(n + 1) i=0

(2.29)

(2.30)

as required by positivity of the intersection number.

(c) Toric graph of -p'2) Similarly, we can build the toric graph of the local Calabi-Yau threefold x3m'-p'2) .Using the realization Eq. (3.20) and the splitting Eq. (2.27), the above local CY3 reads as

O(mo,...,mn) ® O(-po,..., -pn) ^ [J2 S2

with intersection matrix

[S2] .[S2] = -AU-

(2.31)

(2.32)

The fibers O(±k0,..., ±kn) carry charges under the various U(1) gauge symmetries of the individual 2-spheres Sf. The total charge is given by Eq. (2.27). With these results at hand, we are now in position to proceed forward and study the field theoretical representation of the above class of local CY3s by using the method of 2d N = 2 supersymmetric gauged linear sigma model.

i 1 1 ill I

Fig. 8. Toric graph of complex line. Dashed lines represent the circle S1 above the base of the fibration C = Ry x S1. Compactification of C into a projective line is allowed by toric geometry while it is not in the case of compactification into a 2-torus.

2.1.3. Alternative toric realization

In an interesting study [24], a formal toric geometry has been developed to construct the local elliptic curve x<f1,-m,0) as a toric Calabi-Yau manifold. Here, we review briefly the main lines of this construction and make a comment on its link with our method.

Caporaso et al. construction The basic idea of [24] in building the Caporaso et al. formal toric realization of x<3m,-m,0) may be summarized as follows:

(1) Start from the familiar toric description of the trivial Kahler geometry of the complex three dimension space C3. Let z1, z2 and z3 be the local coordinates parameterizing each of the complex lines with the typical toric diagram, shown in Fig. 8.

This graph represents the fibration C ~ Ry0 x S1; the top (zi = 0) of the half line Ry0 corresponds to the fix point of the U(1) toric action and the dashed line refers the fiber S1 with Kahler parameter proportional to |zi |. The cross product of the three lines gives the usual toric graph of the trivalent vertex of C3 ~ Ry x T3.

(2) Then, compactify one of the zi lines, say the one parameterized by z1, so that we get a local 2-torus. This toroidal compactification1 can be achieved in different, but equivalent, manners. One way is to use the conformal map,

z1 = exp(w1 + ¿01), W1,01 e R, (2.33)

and perform a Wick like rotation w1 ^ iw 1 to get the 2-torus T2. Following above mentioned reference, one may also use the polar coordinates (|zi |,0i)1^i^3 of the fibration C3 ~ R^0 x T3 and compactify one Ry0 on a circle; i.e. t

|z1| = |z1|+n —, ^eR^n e Z, (2.34)

to end, as done in [24], with the local geometry,2

O(0) © O(0) ^ T2. (2.35)

This trivial rank 2 vector bundle on the 2-torus x3(0,0,0), with Kahler modulus t, constitutes a priori a representation of the desired space as shown on Fig. 9.

1 Notice that the non-compact complex line C with a local coordinate z can be compactified into a complex one dimension projective space P1 in agreement with toric geometry constraints. The resulting toric graph has two fix points at z = 0 and z = <x>. However the compactification of the complex line into a 2-torus has no fix point and so is not allowed by toric geometry. The reason is that while the compactification of the base is usually possible, consistency along the fiber direction breaks toric geometry requirement.

2 In afuture study [43], we adapt the Caporoso et al. construction to study local Calabi-Yau threefolds with affine ADE

geometries and beyond by using the technology of toric graphs.

Fig. 9. The diagram obtained from the toric trivalent vertex C3 after compactification of zi on a real 2-torus. The resulting diagram is not toric.

Moreover, the non-trivial fibrations x3m,-m,0) with m > 0 is recovered as usual by using framing techniques.

However this construction is not so obvious as it may seem at first sight. Under the special compactification Eq. (2.34), one looses more than one feature of the original trivalent vertex which we comment below:

(i) Because of the periodicity condition (2.34), the toric representation of the compactified complex line z1 is no longer valid. Contrary to the complex line, the "origin" |z11= 0 is not fixed under the usual U(1) toric action and so the order of degeneracy of the natural U(1)3-action at the origin of C3 gets reduced down to U(1)2.

(ii) After compactification, the resulting diagram of C3 is not "trivalent" since the fixed point of the toric action on the universal covering space C3 is not fix under U(1)3. This feature can be seen as well directly on Fig. 9, which shows an apparent "tetravalent" vertex rather than a trivalent one.

Following [24], this difficulty may be overcome by considering a formal toric Calabi-Yau threefold X3(0,0,0) given by the blow up of X^0,0,0 at the origin

Z2 = Z3 = (2.36)

by a complex projective line P1. The blown up manifold X3(0,0,0) has then the fibration

Xf°'0) = U F (zi) ^ zi, (2.37)

zi eT2

F(zi)|zi =o = C2, F(0) = C2 U C2, together with the projection

m : X3(0'0'0) ^ Xf'0'0), (2.38)

and the "formal" toric graph depicted in Fig. i°. In this way, the toric action on X^0,0,0 can be lifted to X(0,0,0) so that the blown up threefold is U(i)3-equivariant.

Note in passing that although the blow up produces an inequivalent Calabi-Yau threefold, the topological string partition function on X3(0,0,0) is related to the desired one on the original background X(0,0,0). All one has to do is once we get the result for X3(0,0,0) using topological vertex method, the corresponding result for x30,0,0) is recovered by help of the projection m (2.38).

Fig. 10. The formal tori graph made of two toric trivalent vertices.

Having described the Caporaso et al. method, we turn now to make a comment on the link with our realization we have considered before.

The link with our construction First note that the method of [24] relies on two key points: compactification of one complex line into a 2-torus and blowing up by a P1. This construction may be effectively linked to ours if we give a specific interpretation to the above procedure. To fix the ideas from the beginning, think about the 2-torus of Caporaso et al. as given, in our picture, by two projective lines Pi and P^ with intersection matrix,

'?] • [Pi] =

-2 2 2 -2

[T2] = [Pi] + [P2],

satisfying

[T2] .[T2] = [Pi] • [Pi] + [P2] • [Pa] + 2[Pi] • [Pi],

(2.39)

(2.40)

(2.4i)

which vanishes identically. The first P11 is precisely the blow up used in [24] while the second one follows for the compactification of the complex line with coordinate zi.

Both steps should be associated with the 2-torus appearing in [24] as shown in Eq. (2.40). To exhibit explicitly the link between the Caporaso et al. construction and our analysis, let us consider the complex space C3 which we split as

C3 = C x C2,

(2.42)

and borrow the analysis of [24]. We have the following correspondence:

(a) The blow up by a P1 used in [24] can be given an interpretation in terms of the resolution of the orbifold singularity C2/Z2 where Z2 acts as on the local coordinates of C2 as

Z2 : Zi

i = 2, 3,

(2.43)

but leaves zi invariant. This blowing up step is shown on Fig. ii.

Fig. ii. (a) Toric graph of ALE space with su(2) singularity. (b) Blowing up of the su(2) singularity by a Pi. (c) Mirror toric diagram where the black dot represents Pi and the white ones the non-compact complex lines.

Fig. i2. (a) Toric graph of ALE space with an su(4) singularity. (b) Blowing up of the su(4) singularity by three intersecting Pi's. (c) Mirror toric non-compact complex lines.

secting Pi's. (c) Mirror toric diagram where the three black dots represent the Pi's and the two white ones refers to the

Note by the way that one can push further this observation and consider blowing up by using several intersecting Pi's. In this case, the previous correspondence can be extended to all ALE spaces [25]

(2.44)

with r standing for a given discrete subgroup of su(2). To fix the ideas, we can take r as Zk (k > 2). In this generalization, one uses obviously (k - i) blow ups as shown on the example reported in Fig. i2 and describing the blowing up of the orbifold singularity C2/Z4.

In Figs. ii and i2, we have also given the mirror diagrams that follows under mirror symmetry. Note in passing that this duality has been extensively studied in literature [3i] and is at the basis of the geometric engineering of supersymmetric QFT4 embedded in type II superstring on Calabi-Yau threefolds. One of the powers of this method is that the restriction of the mirror graphs to the compact part (black dots) is nothing but the Dynkin diagram of the underlying Lie algebra of the orbifold singularity.

(b) The compactification of the complex line zi used in [24] can be also given an interesting interpretation. It can be associated with the affinization of the underlying su(2) symmetry of the Z2 orbifold surface C2/Z2 with behavior near the origin,

uv = w .

(2.45)

As noted before this result may be here also extended to all ALE spaces. In the case of C2/Zk for instance,

uv = w

(2.46)

the compactification of line zi should be associated with affinization of underlying su(k) symmetry. The resulting geometry we get has obviously an underlying Kac-Moody sii(k) symmetry. Let us give some details.

Fig. i3. (a) Toric graph of the orbifold C3/Z2 = C x (C2/Z2). (b) Resolved singularity lifting the degeneracy of the two vertices.

Fig. i4. Blow ups of the C3/Z| involving three intersecting real 2-spheres. The compact part may be thought of as a complex two dimension projective space P2.

To exhibit Z2 symmetry of the origin of C2 we have refereed to above, it is then interesting to promote the trivalent vertex C3 to the orbifold space,

C3/Z2 = C x (C2/Z2), (2.47)

where C2/Z2 stands for the usual local ALE surface with su(2) singularity equation (2.45). The toric graph of the above space is as shown on Fig. i3(a). Because of orbifold Z2 symmetry, one clearly sees that the trivalent vertex has a twofold degeneracy. This degeneracy is lifted by performing a blow up of the su(2) singularity at the origin which lead to the toric graph.

Notice that since C3 involves three orthogonal complex planes, then the more natural orbifold one may consider is

C3/Z2. (2.48)

It describes the intersection of three su(2) singularities at the origin. These three su(2) singularities live at

(zi, 0,0), (0,Z2, 0), (0,0,Z3),

and fuse at the origin (0, 0,0) of C3. The blow up of the three singularities involves then three intersecting Pi's as shown on Fig. i4.

Focusing on Eq. (2.47) together with Fig. i3(b) and denoting by

Ui: z(ii), J?, z«

u2: z(2), z22), z(2), (2.49)

Fig. i5. Dynkin diagram of twisted affine sii(2) Kac-Moody algebra. It is also interpreted as a toric graph.

the two local patches of the two toric trivalent vertices of the figure, the gluing relations read then as

z3i).z22) = i, = 42).42\ (2.50)

z(().z(2) = i, (2.5i)

where Eq. (2.50) describes the blowing up and where Eq. (2.5i) describes the compactification of the line into a projective line. In this view, the geometry ^3(0,0,0) of [24] is realized as a local Calabi-Yau threefold whose compact part is given by the intersection of two projective lines Pi and P2 with defining equations z3() . z22) = i and z(() . z(2) = i respectively and intersection matrix as in Eq. (2.39). These relations show that this geometry has much to do with ALE space

with twisted su(2) affine geometry. In this regards, recall that Ai )(2) affine Kac-Moody algebra has the following generalized Cartan matrix A(2),

A j = ( —2) • <2.52)

and the corresponding non-simply laced generalized Dynkin diagram.

Due to the branch cut, this non-simply laced geometry has not been studied enough in literature; but a priori like for simply laced ADE Dynkin diagrams and their affine extension, A® Dynkin diagram can be interpreted as well in terms of mirror toric graphs. Each node N, i = i, 2, of Fig. i5 can be associated as usual to a real 2-sphere Sf, that is a complex projective line Pi,

N; ^ S2 - Pi, i = i, 2. (2.53)

These spheres have a matrix intersection given by

!'J = [Sf] . [S2], (2.54)

with precisely

Iij = -A ;j. (2.55)

From our approach, one concludes that the Caporaso et al. 2-torus is then associated with the

compact part of the twisted affine A1 geometry.

This construction can be generalized to ALE spaces with ADE singularities. For the case of the orbifold C2/Zk, the full resolution involves (k — i) spheres Sf with matrix intersection given by minus the Dynkin diagram of ordinary su(k)

[S2]. [S2] = —2S;j + 8;U+i) + 8;{j—i), ij = i,...,k — i. (2.56)

The third complex variable is compactified into a sphere Sk2 with intersection

[S2]. [S2] = i, [S2]. [S2—i] = i, (2.57)

and zero elsewhere. Here also, S2 is associated with the affine node of the generalized Dynkin diagram of Kac-Moody sU(k) symmetry.

In the end of this presentation we would like to make remarks: (a) While the relation between 2-torus geometry and affine geometry of ALE surfaces is natural and has been considered in many occasions, the link between affinization of ordinary Lie algebras and the compactification of the complex line z( needs however more investigation. (ft) Allowing higher order branch cuts, the above description could have a generalization to higher genus curves. By higher genus we mean the case where the Sf spheres intersect more than once; i.e. beyond Eqs. (2.56)-(2.57). This generalization would involve intersection matrices belonging to the indefinite sector of affine Kac-Moody algebras. In Section 4, we give an argument supporting the correspondence between local genus g-Riemann surfaces and indefinite Kac-Moody algebras; see Eqs. (4.82)-(4.84).

2.2. Supersymmetric field model

Here we develop the study of type IIA geometry of X(m,-p,2). To do so, we use known results on 2d N = 2 supersymmetric gauged linear sigma model formulation and take advantage of our construction to also complete partial ones on type IIA geometries based on standard affine models as well as non-trivial fibrations.

To begin, recall that in the 2d N = 2 supersymmetric sigma model framework, the field equations of motion of the auxiliary fields Da of the gauge supermultiplets Va,

= V qa <Pi - ra = 0 (2.58)

define the type IIA geometry. This method has been used in literature to deal with K3 surfaces with ADE geometries. But here we would like to extend this method to the case of the local geometry X(m,-p,2). To that purpose, we shall proceed as follows:

(1) Study first the field realization on two special examples. This analysis, which will be given in present subsection, allows to set up the procedure. Then we give useful tools and illustrate the method.

(2) Develop, as a next step, the general field theoretical 2d N = 2 supersymmetric gauged linear sigma model of Eq. (2.3i). This is a more extensive and will be given in next section. Actually this is one of the results of the present study.

2.2.1. Type IIA model for O(m) © O(-m - 2) ^ S2

To start note that for m = 0, this local Calabi-Yau threefold describes just the usual A( geometry on the complex line. The variety has been studied extensively in literature; see [42] for instance. For non-zero m the situation is, as far as we know, new and its supersymmetric linear sigma model can be obtained by considering a U(i) gauge field V and four chiral superfields with gauge symmetry

^ &'.= eiqiA®i, i = i,...,4, (2.59)

with charges

qi = (i, i, -2 - m, m)

(2.60)

satisfying the CY condition J24=i q = °. The gauge invariant superfield action S^"2 of this model reads as

S2d=2 ~ j d2a d40J2®ieqiV^i — r J d2a d40 V (2.6i)

and the field equation of motion (2.58) of the gauge field leads to

ifaii2 +i02i2 — (2 + m)\fo |2 + m|04i2 = r. (2.62)

This equation describes indeed O(m) © O(—m — 2) ^ S2. It has four special divisors fa = 0 while the base 2-sphere corresponds to

Ifaii2 +Ifa2i2 = r. (2.63)

In case where m is positive definite, this geometry can be also viewed as describing the line bundle O(—m — 2) over the weighted projective space WP2( i m). Note that for m = i, one gets the normal bundle of P2 and the local Calabi-Yau threefold coincides with

O(—3) ^ P2. (2.64)

Note also that for m = — i, one has the resolved conifold

O(—i) © O(—i) ^ P(. (2.65)

Note finally that for m = 0, —2, one has the A( geometry fibered on the complex line C.

2.2.2. Example: O(mi,m2) © O(—pi, —p2) ^ C2

Following the same method, one can also build the type IIA sigma model for this local CY3 (p; = mi + 2) based on two intersecting 2-spheres S2 and S|,

C2 = s2 + S2 (2.66)

with intersection S2 n S| = {a point P} modulo gauge transformations. The sigma model involves two U(i) gauge fields V(, V2 and five chiral superfields .Denoting by X^ X2, X3 the bosonic field components parameterizing the compact 2-cycle C2 and by Y( , Y2 the complex variables parameterizing O(—mi — 2, —m2 — 2) and O(mi,m2) respectively, the two sigma model equations are given by

iXii2 + iX2i2 — (2 + mi)i Yi i2 + miiY2i2 = ri,

iX2i2 + i X3 i2 — (2 + m2)iYii2 + m2iY2i2 = r2. (2.67)

In these equations, one recognizes two SU(2)/U(i) relations describing the 2-spheres S2 and S| associated with taking Y( = Y2 = 0; i.e.

iXii2 + iX2i2 = ri,

iX2i2 + i X3 i 2 = r2. (2.68)

Notice that Eqs. (2.67) involve five complex (i0 real) variables which are not all of them free since they are constrained by two real constraint relations (the Di and D2 auxiliary field equations of motion) and U(i) x U(i) gauge symmetry

X( = Xiexp(i^i), X2 = X2exp(idi + i$2), X3 = X3exp(i$2), (2.69)

Fig. 16. Open chain describing a typical compact base. It involves several intersecting 2-spheres with one intersection point. Figure in top involves straight lines and figure in bottom its fattening by representing the circles above the 2-spheres.

where ■&i are the two gauge group parameters. At the end one is left with (10 - 2 - 2) degrees of freedom which can be described by three independent complex variables. Notice also that the spheres S2 and S^ intersect at

(Xi,X2,X3) = (n, 0,r2). (2.70)

X2 = 0 is the fixed point under U(1) x U(1) gauge symmetry (toric action) and up to a gauge transformation, the same is valid for Xi = r1 and X3 = r2. Indeed if parameterizing Xi = |X1 |e-!y and X3 = |X3|e-i^, one can usually set y = and f = by using U(1) x U(1) gauge invariance. Then setting X2 = 0 in Eqs. (2.68), one discovers Eq. (2.70). Notice finally that this construction generalizes easily to the case of an open chain (see Fig. 16)

Cn = Y, s2, Cn. Cn = -2, (2.71)

involving several intersecting 2-spheres S2 with [S2]. [S2] = -2 and [S2]. [S2±1] = 1 and vanishes otherwise. The sigma model field equations describing this complex one dimension curve read as

|Xi |2 + |Xi+1|2 = ri, 1 < i < n, (2.72)

involving (n + 1) complex field variables Xi constrained by n complex constraint equations.

3. More on 2d N = 2 sigma model description

So far we have considered special examples of type IIA realization of local CY3 as an introduction to the important case where the previous (compact) open chain Cn gets closed by the adjunction of an extra 2-sphere S^,

Cn ^ Cn+1 (3.1)

which we denote also as An+1; see Fig. 17.

In this section, we give the type IIA description of these kinds of local CY3s. To proceed, we start from the previous open chain Cn and add an extra 2-sphere S02,

¿n+1 = S02 + Cn, n > 2, (3.2)

Fig. 17. Toric graph of An+1 realized by using affine An Dynkin diagrams. It is obtained by adding an extra 2-sphere in same manner as we do in building affine Dynkin diagrams from ordinary ones.

with the following features,

[So] • [So] = -2, [S2] ■ [Cnl = 2. (3.3)

In this case, we have

[An+1] • [A,+i] = 0, (3.4)

as required by the homology property of the 2-torus. One of the solutions of Eq. (2.67) is given by

[So2]. [S2] = 1, [So2]. [S2] = 1,

[So2]. [S2] = 0, i = 1,n. (3.5)

Other solutions are possible and are associated with Dynkin diagrams of affine Kac-Moody algebras. To write down the explicit sigma model field equations of the geometry O(m) © O(-p) ^ An+1, we shall first write down the equations for An+1 and then give the general result.

3.1. Sigma model equations for An+1

A way to get the sigma model field equations for the 2-torus [An+1], preserving the constraint equations; in particular the dimensionality of An+1 and [An+1] ■ [An+1] = o, is to embed it in the Cn+3 with (n + 2) constraint complex relations resulting from (n + 2) real equations of motion and (n + 2) Abelian gauge symmetries. The construction is done as follows: Start from Eqs. (2.72) describing the open chain Cn together with the field equation of motion

|Zo|2 +|Z1|2 = ro, (3.6)

describing the extra 2-sphere S2 with Kahler modulus ro. The meaning of the complex variables Zo and Z1 will be specified later. Then glue S^ and Cn by implementing the constraint relations (3.5). A priori this could be achieved by setting Z1 = X1 and Zo = Xn+1 so that Eq. (3.6) becomes

|Xn+1|2 + | X112 = ro.

In this way S2 intersects once the 2-sphere S2 defined by |Xi|2 + |12 = r1 as well as S% with equation |Xn|2 + |X„+1|2 = rn. But strictly speaking there is still a problem although the resulting geometry looks having the topology of a 2-torus. This construction does not exactly work. The point is that by combining Eqs. (2.72), (3.7), we cannot have the right dimension since the (n + 1) complex variables {Xi, 1 < i < n + 1} are constrained by (n + 1) complex constraint relations. As mentioned before, we have (n + 1) real relations coming from the field equations of motion (2.72), (3.7) and an equal number following from ^(1)n+1 gauge symmetry acting on the field

Xi ^ Xie^n=1 qi#a (3.8)

with i = 1,...,n + 1 and the 's being the U(1) group with charges qa = (q1,qa, 0,...,q'a+1) as follows,

qi1 = (1, 1, 0, 0, 0,... , 0, 0, 0),

q2 = (0,1, 1, 0,0,...,0, 0,0),

q3 = (0,0, 1, 1,0,...,0, 0,0),

qn = (0, 0,0,0, 0,..., 1,1, 0),

qn+1 = (1,0,0, 0,0,...,0, 0, 1). (3.9)

This dimensionality problem can be solved in different, but a priori equivalent, ways. Let us describe below the key idea behind these solutions.

A natural way to do is to start from the complex two dimension ALE geometry with blown up An singularity and make an appropriate dimension reduction down to one complex dimension. More precisely, start from equations

Xj-1|2 - 2|Xj|2 + Xj+1|2 = rj, j = 1,...,n (3.10)

and add the following extra constraint relation reducing the dimensionality by one

|Xn|2 - 2|Xn+112 +|X0|2 = r0. (3.11)

There is also an extra U(1) gauge invariance giving charges to (Xn,Xn+1,X0). This picture involves (n + 2) complex variables constrained by (n + 1) relations. The compact geometry is determined from Eqs. (3.10)-(3.11) by restricting to the compact divisors Xi = 0. This gives

|Xi-112 + |Xi+112 = ri, i = 1,...,n,

|X0|2 + |Xn|2 = r0. (3.12)

The second way is to use the correspondence

ai ^ Sf (3.13)

between roots of Lie algebras ai, which in present analysis are given in terms of unit basis vectors of Rn+1 as

ai = ei - ei+1 (3.14)

and the Si2 2-sphere homology of ALE space with blowing up singularities. To have the 2-torus, one should consider affine Kac-Moody symmetries and use its correspondence between the

imaginary root S = a0 -J2i ai,

5 ^ T2. (3.15)

Below, we shall develop an other way to do relying on the following correspondence

e2 ^ - Xi I2,

ei • ej ^-Ify |2, (3.16)

where ei is as in (3.14). In the case where {ei} are orthogonal; i.e. ei . ej = 0, there is no Yij variable and this is interpreted as just a divisor equation. In this correspondence, roots ai are associated with cotangent bundle of P1 since by computing a2 = e2 - 2ei . ei+i + e2+1 and using above correspondence,

|Xi |2 - 2| Yi,i+112 + |Xi+112 = p. (3.17)

For Yi,i+1 = 0, one recovers the usual 2-sphere. The link between these ways initiated here will be developed in [43].

We start from Eq. (2.72) and modify it as follows:

(i) An extra complex variable X0 so that the new system involves the following complex variables {X0,X1,...,Xn+1}. The extra variable charged under the U^1) Abelian gauge symmetry,

X0 = X0e-2W°, X'1 = X^0, X2 = X2e^°. (3.18)

(ii) An extra U0 (1) gauge symmetry acting as

X0 = X0ei&0, X1= X1ei^0 (3.19)

and trivially on the remaining others.

(iii) Modify Eq. (2.72) as

-2|X0|2 + |X1|2 +|X2|2 = n, (3.20)

whose compact part X0 = 0 is just the 2-sphere |X112 + |X2|2 = r1 of Eq. (2.72), and theremain-ing (n- 1) others unchanged

|X2|2 + |X3 |2 = r2, |Xn-1|2 + |Xn|2 = rn-1,

|Xn|2 + |Xn+1|2 = Tn. (3.21)

(iv) Finally add moreover

|Xn+1|2 + |X0|2 = T0. (3.22)

These relations involves (n + 2) complex variables Xi subject to (n + 1) real constraint equations and (n + 1) U(1) symmetries. They describe exactly the elliptic curve An+1. Note that at the level Eq. (3.20) the variable X0 parameterizes a complex space C, which in the language of toric graphs, is represented by a half line. The relation (3.22) describes then the compactification of non-compact complex space C with variable X0 to the complex one projective space (real 2-sphere).

In sigma model language, this corresponds to having (n + 2) chiral superfields with leading bosonic component fields,

Xo, X1, X2, •••, Xn-1, Xn, Xn+i, (3.23)

charged under (n + 1) Maxwell gauge superfields,

Vo, Vi, V2, •••, Vn-i, Vn, (3.24)

with the Un+^1) charges qa = (q0,q{, 0,•••,qn+1) as follows,

Uo(1): q0 = (1,0, 0, 0,0,---,0, 0,1)

U1(1): q1 = (-2, 1,1,0, 0,---,0,0,0),

U2(1): qf = (0,0, 1, 1,0,---,0, 0,0),

Un-1(1): qn-1 = (0,0,0, 0, 0,---, 1,1, 0),

Un(1): qn = (1,0, 0, 0,0,^,0, 1, 1) (3.25)

Below, we use this construction to build our class of local Calabi-Yau threefold using the elliptic curve An+1 as the compact part.

3.2. Local 2-torus

Implementing the fibers O(m) © O(-m) which, in our present realization, take the form O(m0,...,mn) and O(—p0,•••, -pn) and extending the construction of Section 2.2, one can write down the sigma model relations. We have

|X1|2 + |X2|2 - (2 + m1)|X0|2 + m1|72i2 = n, |X2|2 + |X2|2 - (2 + m2)|F1|2 + m21^212 = r2,

|Xn|2 + |Xn+112 - (2 + mn)|Fn|2 + mn|Y2|2 = rn,

|Xn+1|2 + |X0|2 - (2 + m0)|Y1|2 + m0|Y2|2 = r0, (3.26)

where Y1 and Y2 are the fiber variables and carry non-trivial charges under Un+1(1) gauge symmetries.

4. Brane theory and 4d black holes in type II string

In this section we consider type string IIA compactification on the class of local Calabi-Yau threefolds constructed in previous sections

X(m,-m-2,2) = O(m) © O(-m - 2) ^ An+1- (4.1)

Then, we develop a field theoretical method to study 4d large black holes by using the 2d q-deformed quiver gauge theory living on An+1. Large black holes in four-dimensional space-time are generally obtained by using configurations of type II or M-theory branes on cycles of the

internal manifolds. In type IIA framework, an interesting issue is given by BPS configurations involving, amongst others, N D4-branes wrapping non-compact divisors of the local CY3 giving rise to a dual of topological string. Our construction follows more a less the same method used in [2]; the difference comes mainly from the structure of the internal manifold X(m,-m-2,2) and the engineering of the quiver gauge theory living on An+1.

We first study the D-brane formulation of the BPS 4d black hole in the framework of type IIA string compactification on local An+1. Then we study the reduction of N = 4 twisted topological theory on 4-cycles to 2d quiver gauge theory, represented by ADE Dynkin diagrams.

4.1. Brane theory in X3m,-m-2,2) background

In type IIA string compactification on local Calabi-Yau threefolds x3m'-m-2'2), the effective 4d, N = 2 supersymmetric theory has massive BPS particles coming from D-branes wrapping cycles in x3m,-m-2,2). Under some assumptions, BPS states based on a special D-branes configuration may be interpreted in terms of 4d space-time black holes. This configuration involves D0-D4- and D2-D4-brane bound states but no D6 due to the reality of the string coupling constant gs. The DO-particles couple the RR type IIA 1-form A1 while the D2- and D4-branes couple to the RR 3-form C3. Their respective charges Q0, Q2a and Q4 give the following expression of the macroscopic entropy of the black hole [6,15,19]

SbH = 6CabcQtQbQliQo - 2CabQ2aQ2b) (4.2)

Cabc = &a A Mb A M

Cabc = CabcQl, Ca Cbc = • (4.3)

The above 4d black hole construction can be made more precise for our present study. Here the local threefold x3m'-m-2'2) is given by

O(mo,...,mn) ® O(-po,..., -pn) ^ , (4.4)

with pi = mi + 2. The 2-cycles basis |[C!'], i = 1,...,h1>1{X)} of H2X, Z) is given by the compact 2-spheres Sf with Kahler modulus ri and the following supersymmetric linear sigma field theoretical realization

Ci: |Xi |2 + |Xi+1|2 = ri, i = 0,...,n, (4.5)

with the identification S2 = Sf+l and n + 1 = hl,l(X). The components [Di] of the dual basis of 4-cycles of H4(x, Z) is given by the non-compact complex surfaces

[Di ] = 0(-po,...,-pn) ^ Ci = 0,...,n, (4.6)

with generic equations

|Xi |2 + | Xi+112 - (2 + mi)|Z|2 + mi | ^212 = ri, i = 0,...,n, (4.7)

where Z stands either for X0 or Y1 as given in Eqs. (3.26). These dual 2- and 4-cycles determine a basis for the (n + 1) Abelian vector fields B' = Bl(t, r) obtained by integrating the RR 3-form C3 on the 2-cycles C as shown below

B' =1 C3, I vj = j (4.8)

Under these B' Abelian gauge fields, the D2-branes in the class [C] e H2(X, Z) and D4-branes in the class [D] e H4 (X, Z) are given by

[c ] = j2 Q2i [ C ], D] = j2 q4 [Di ], (4.9)

i=0 i=0

and carry respectively Q2i (Q2i = M') electric and Q'4(Q'4 = N') magnetic charges with

N = J2 N'. (4.10)

We also have D0-brane charge Q0 that couple the extra U(1) vector field originating from RR 1-form. D6-brane charges are turned off.

Following [2], the indexed degeneracy &(Q0,Q2i,Q'4) of BPS particles in space-time with charges Q0, Q2i, Q4 can be computed by counting BPS states in the Yang-Mills theory on the D4-brane. This is computed by the supersymmetric path integral of the four-dimensional theory on D in the topological sector of the Vafa-Witten maximally supersymmetric N = 4 theory on D [2,19,44],

: f [DA] exp( —— f Tr F A F - — f, J \ 2gs J gs J

ZBrane = I [DA] exp( -—J Tr F A F - —J v A Tr F ). (4.11)

Up to an appropriate gauge fixing, this relation can be written, by using the chemical potentials <p0 = 4gL and = for D0- and D2-branes respectively, as follows

ZBrane[Ni,^0,^'] = J2 n(Q0,Mi,Ni)exp(-Q0^0 - M'(p') (4.12)

where we have used

Qo = J Tr(F A F), Mi = f Tr F A toi.

(4.13)

The above relation may be expanded in series of e-gs due to S duality of underlying N = 4 theory that relates strong and weak coupling expansions [19]. Recall that the world-volume gauge theory on the N D4-branes is the N = 4 topological U(N) YM on D. Turning on chemical potentials for D0-brane and D2-brane correspond to introducing the observables

S = — Í Tr F A F + — [to A Tr F (4.14)

2gW gs J

where w is the unit volume form of An+\. The topological theory (4.14) is invariant under turning on the massive deformation

55 = £ ?/*.2 (4.15)

which simplifies the theory. By using further deformation which correspond to a U(1) rotation on the fiber, the theory localizes to modes which are invariant under the U(1) and effectively reduces the 4d theory to a gauge theory on An+\.

4.2. 2d quiver gauge theories on An+i

Note first that from four-dimensional space-time view, the wrapped N D-branes on D = O(-p) ^ C describe a point-particle with dynamics governed by 4d N = 2 supergravity coupled to U(N) super-Yang-Mills. On the D4-branes live a (4 + 1) space-time N = 4 U(N) supersymmetric gauge theory and on its reduced topological sector one has a 4d N = 4 topolog-ical theory twisted by massive deformations.

In our present study, the 2-cycle C is represented by the closed chain An+1 with multi-toric actions and the line O(-p) is a non-trivial fiber capturing charges under these Abelian symmetries. It will be denoted as O(-p),

p = (p1,...,pn). (4.16)

As a consequence of the topology of An+1 which is given by (n + 1) intersecting 2-spheres, the previous U(N) gauge invariance gets broken down to

U(N) ^ U(N0) x U(N1) x ••• x U(Nn), (4.17)

with the group rank condition (4.10). This symmetry breaking phenomenon-requires non-zero Kahler moduli (2.23) of the various 2-spheres of the base An+1,

ri = 0, i = 0,1,...,n. (4.18)

Viewed from 4d space-time, the effective theory of type IIA string low energy limit on local CY3 with D-branes wrapping cycles is given by a 4d N = 2 supergravity coupled to 4d N = 2 quiver gauge theory with gauge group G = U(N0) x ••• x U(Nn). This is the general picture of the string low energy effective field approximation.

By requiring 4d space-time N = 2 super-conformal invariance, the vanishing conditions fii = 0 of one-loop beta functions on the gauge group factors U(Nt) put a strong constraint on the ranks Ni of these U(Ni)'s. These conditions have been approached for different purposes; in particular in the context of geometric engineering of 4d space-time N = 2 superconformal theories embedded in type IIA string theory on local CY3s with blown up affine ADE singularities. There, these conditions take the remarkable form

J^KK'jNj = 0, (4.19)

and are solved by taking Ni ranks as Ni = si M, where the si's are the Dynkin weights introduced earlier. In the case of affine An model, the si's are equal to unity and so the U(N) gauge group gets reduced, in the superconformal case, to

U(N) ^ GScft = U(M)n+1 (4.20)

N = (n + 1)M. (4.21)

On the 4-cycle D of the local Calabi-Yau threefold, the theory is a N = 4 topologically twisted gauge theory; but using the result of [2], this theory can be simplified by integrating gauge field configuration on fiber O(-p) and fermionic degrees freedom to end with a 2d bosonic quiver gauge theory on An+1. This theory has ]"[i U(Ni) as a gauge symmetry group and involves:

(i) Gauge fields Ai, i = 0,1,...,n, for each gauge group factor U(Ni) with field strength Fi = dAi + Ai A Ai,

F = £ HaF? + J2 E+F—P + E—(4.22)

ai = 1 Pi of U(Ni)

where {Hai ,E± } is the Cartan basis of U(Ni) and where pi are the positive roots3 of the ith gauge group factor. The above relation contains as a closed subset the usual UNi (1) Abelian part dAi = ENU Hai dAf where the Hai'

s are the commuting Cartan generator.

(ii) 2d scalars = @i(z, z) in the adjoint for each factor U(Ni) having a similar expansion as in (4.22); but which we reduce to its UNi (1) diagonal form

= J2 Ha®±Pi = 0, i = 0,...,n. (4.23)

These fields are obtained by integration of the 4d gauge field strengths Fi (z, Z, y, y) on the fiber O(-p),

®i(z,~z) = J d2yFi(z,Z,y,y) (4.24)

which, as usual, can be put as

$i(z,z) = j) Ai(z,z,y,y), z e S2,

where the loop Sj can be thought of as a circle at the infinity (|y | —^ to) of the non-compact fiber O(—pi) ~ C parameterized by the complex variable y.

(iii) 2d matter fields in the bi-fundamentals of the quiver gauge group living on the intersection of An+1 patches with, in general, a leg on S2 and the other on S2.

In the language of the representations of the gauge symmetry U(Ni) x U(Nj), these fields belongs to (Ni, Nj ) and describe the link between the gauge theory factors living on the irreducible 2-cycles making ^n+i.

In the language of topological string theory using caps, annuli and topological vertex [39], these bi-fundamentals can be implemented in the topological partition function thought insertion

3 We have used the Greek letter P to refer to the roots of the of the gauge group U(N). Positive roots of the U(Ni) are denoted by Pi and should not confused with simple roots ai used in the intersection matrix (Kij = ai . aj) of the 2-cycles of the base An of local CY3.

operators type

SK,Kl+1 (4.25)

using sums over representations RiRi+1 of the gauge invariance U(Ni) x U(Ni+1). This construction has been studied recently in [19] for particular classes of local CY3 such as O(-3) ^ P2 and O(-2, -2) ^ P1 x P1. We will not develop this issue here.4 Below we shall combine however field theoretical analysis and representation group theoretical method to deal with bi-fundamentals.

4.2.1. Derivation of 2d quiver gauge field action

Here we construct the 2d field action S^n+1 describing the localization of the topological gauge theory of the BPS D4-, D2-, DO-brane configurations on the non-compact divisor [D] = O(-p) ^ An+1 of the local CY3. This action can be obtained by following the same method as done for the case O(-p) ^ Sg.

One starts from Eq. (4.14) describing the gauge theory on the N D4-branes wrapping D with D0-D4 and D2-D4 bound states

f Tr(F A F) + — f Tr F A m. (4.26)

J gs J

^4d = -— ^

[D] [D]

In this equation, the parameters gs and 0 are related to the chemical potentials and for D0-and D2-branes respectively as = and = ^g-0. The field F is the 4d U(N) gauge field

strength F = dA + A A A. It is a Hermitian 2-form with gauge connexion A. The field 1-form A reads in the local coordinates {z, z, y, y}, with (z, z) parameterizing the base An+1 and (y, y) for the fiber, as follows

A = Azdz + Aydy + Az dz + Ay dy,

A^ = A^(z,z,y,y), = z,y,z,y. (4.27)

Moreover, like A, the 2-form field F is valued in the Lie algebra of U(N) gauge symmetry and so can be expanded as

F = j2 HaFa + j2 E+F-p + E-F+p, (4.28)

a=1 positive roots p of U(N )

where {Hai ,Ep±} is the Cartan basis of U(Ni). The above relation contains as a closed subset the usual UN(1) Abelian part

dA = J2 HadAa, [Ha,Hn] = 0, (4.29)

4 As argued in [19], the matter fields localized at the intersection point Pi of the 2-spheres S2 and S?+i corresponds

to inserting the operator V = XItc Tr^ Vi 1 Tr^ Vi+i) with Vi = exp(i*P(j) — i / A(i)) and Vi+i) = e'®(i+1) where the integral contour is a small loop around Pi .

which plays a crucial role in the computation of Wilson loops. In Eq. (4.26), w is the 2-form on the compact cycle An+\ on which the D2-branes lives and is normalized as

« = 1. (4.30)

On the other hand, using Eqs. (2.18)-(2.21), we can put the right-hand side of the above relation5 as follows

w) - 2>:i I w )• (4.31)

f« ?(/«) - 2 £(/

S2 i=J S2nS2

Note that since fS2nS2 w vanishes for the intersections Sf n S2- which are given by a set of sepa-

rated points, we can simplify this expression further into

f« = 1 È-ri = 1' r = 0. (4.32)

A i=0 An

This shows that on An+1 = ^n=0 Ci, the Kahler form splits as w = 1 ^n=o rwi with fct W2 =

The next step is to perform integration on the fiber variables y and y. The topological theory (4.26) localizes to modes which are invariant under U^1 (1) symmetries and effectively reduces to a gauge theory on the base An+^ Let us give details by working out explicitly these steps: (i) First, we have

j Tr(F A F) = j Tr^ j F A F^j

[D4] zeAn+1 O(-p)—z

= 2 j Tr(0F), (4.33)

where we have set

$(z,z) = J d2y F(y, y ,z,z) (4.34)

O(-p)—z

and where we have restricted F to its values in the UN(l) Abelian subalgebra Eq. (4.29) in order to put @(z, z) in the Wilsonian form

$(z,z) = f A(y,y,z,z), z e An+1- (4.35)

S 1y| —^z, iyi—OT

The same thing can be done for the second term of Eq. (4.26). We find

j « A (Tr F) = j «Tr(0). (4.36)

[D] An+i

We have used the formula A ijb K = AK + fgK — Ja^b K for Kahler modulus of two intersecting surfaces.

The 4d action (4.26) reduces then to the following 2d one

¿2d = — f Tr(0F) + — f m Tr 0. (4.37)

gs J gs J

An+1 An+1

Now using the fact that An+1 = YTi=0 Pi combined with Eq. (4.31), we see that, depending on the patches of An+1 where the Wilson field 0 is sitting, we get either adjoint 2d scalars 0i or bi-fundamentals 0ij as shown below

0(z) = 0i(z), z e P1,

0(z) = 0ij(z), z e P1 n P1. (4.38)

Note that the 0i fields are valued in UNi(1) maximal Abelian group. They parameterize the maximal TNi torii of the Lie group U(Ni). So they should be compact and undergo periodicity conditions. This means that the linear expansion

0i =£ 0ai Hai, 0ai ~ Tr(Hfli 0i), (4.39)

should be understood as

Ui = exp i0i, i = 0,...,n, (4.40)

and so the 2d field components 0ai are constrained as

0ai = 0ai + 2nmai, mat e Z, (4.41)

leaving Ui invariant.

Now substituting Eq. (4.38) in the relation (4.37), we obtain after implementing the hermitic-ity condition 0 = 1 (0 + 0) and F = 2 (F + F), the following

J Tr(0F) = Y f Tr(0iFi) - 8 J2 f Tr(0ijFji+ c.c ), (4.42)

An i P1 i=jP1nP]

where we have set Fji = Fij and where we have disregarded the terms 0ij Fij transforming in the (Nf2, N®2) representation of U(Ni) x U(Nj). These terms do not preserve the Abelian subsym-metry Un+1 (1) of the quiver gauge group U(N0) x ••• x U(Nn). These 2d field configurations have a group theoretical interpretation. They correspond to splitting the adjoint representation of U(N) with N = No +-----+ Nn in terms of representation of U(No) x ••• x U(Nn)

Adj U(N) = j2Adj U(Ni) ®j2(Ni,Nj). (4.43)

i=0 i=j

The terms of the first sum are associated with 0i (z) while the other are associated with 0ij. Obviously since in present case only Pi1 n Pi1 1 which are non-trivial, there are no bi-fundamentals 0ij for j = i ±1.

For the term fA m Tr(0) (4.37), we get

f Tr(0)M = it'-if

Tr(0i)Mi. (4.44)

i=0 p11

Let us first discuss these configurations separately and then give the general result.

Adjoint 2d scalars Putting Eq. (3.22) back into (4.26) and focusing on the patches Pi by substituting $diag = Y,L $L(z), we get the diagonal part of the topological 2d quiver gauge field action

Sdiag = Y, Si (4.45)

Si = - f Tr($iF) + -- f Tr $l + f Tr $¡, (4.46)

gs J gs - J 2gs J

pi pi sf

and where we have added the topologically invariant point-like observables Tr $L2 at the points

1 _F._— .

z e Pi. Upon integrating out fermions and adjoint scalars using $i = —p—L and following [2],

this topologically twisted theory is equivalent to the bosonic 2d Yang-Mills theory

/1 - YM p

-2Tr( Ff) - /Tr Fl (4.47)

2gi gi p1 p1

with Yang-Mills gauge coupling constants gYM L = gL and -LYM = -L terms given by

g2 = pgsl, -YM = (4.48)

L r L r

Here the rL's are the Kahler parameters of the 2-sphere constituting An+1. Note that these gauge coupling constants and -L's are not completely independent and are related amongst others as

bH» = WM = -. (4.49)

gs g2 i

6s L=0 i=0

The first equation should be compared with the standard relation -1 = Yn=0 g—2 appearing in the geometric engineering of quiver gauge theories. s

2d bi-fundamentals To get the field action describing the contribution of bi-fundamentals, it is interesting to proceed in steps as follows:

Start from the topological field action on the 4-cycle [D4],

f Tr(F A F) + — f

S4d = — I Tr(F A F) + — I Tr F A « (4.50)

[D4] [D4]

and think about F as a field strength valued in the maximal non-Abelian gauge group U(No + ----+ Nn). Then expand the real field F as

F = E Fi +E(F22 + F2i), F2i = W) (4.51)

i=o i<2

with Fji = (Fij). The Fi's are the real field strengths of the gauge fields Ai valued in the adjoints of U(Ni) factors with generators {Tai}

Fi =J2 FfiTal, i = 1 Fi = (Fi). (4.52)

ai = 1

The Fij's are the field strengths of the gauge fields Aij valued in the Lie algebra associated to the coset

U(N0 +----+ Nn)

(4.53)

U(No) x... xU(Nn)

Obviously the group U(N0 +-----+ Nn) is not a full gauge invariance of the N = 4 topological

gauge theory since the gauge fields Aij part get non-zero masses mij ~ (ri - rj) after breaking U(N0 + ••• + Nn) down to U(N0) x ••• x U(Nn).

The next step is to use the same trick as before by integrating partially over the variables of the fiber O(-p0,..., -pn). We get

¿4d = — f Tr(0i A F) + — f Tr A w

gS ■ J gS ■ J

' P,1 ' P,1

- f Tr(^ij A F)- f Tr &ij A w. (4.54)

gS ■/■ J gS ■ / ■ J

'=J p1npl 1=] P11nP1

Now using the expansion (4.51) and the property

Tr(TaiTb.) - SabSij (4.55)

we have

Tr№ A F) = Tr№ A Fi), Tr(0ij A F) = Tr(0ij A Fji),

Tr($ij) = 0. (4.56)

Then we can bring Eq. (4.54) into the following reduced form

¿2d = g-^ J Tr№ A Fi) + — Y f Tr ®i A w + 2g- J Tr ®2

' P,1 ' P,1 S2

- £ £ f WjA j+ £ 2gs f

Tr(0j A Fji) +> ^ Tr(<Pij$ji), (4.57)

'=j p1 nP1 '=j P1 nP1

where we have added the typical mass deformations 2gj Tr and by analogy 2Pj Tr(@ij@ji ) with some pij integers which a priori should be relatedS to the pi degrees of theS line bundle. Integrating out the scalar fields, one ends with

¿2d =-£/¿Tr(F2) '-^¡Tr Fif G-TrFjFji)

0YM f „ /• 1

~2 ~2 J ' ^ J 2G2

P1 '=] P1nP1 J

2g2 i i g2

(4.58)

where g2 and 0YM are as in (4.48) and where

G22 = ^^ vol(P1 n P1)• (4.59)

Note that for the base An+1 realizing the elliptic curve in terms of intersecting 2-spheres, the intersection P1 n P1 is given by a finite and discrete set of points P2j of An+1. These points have

zero volumes vol(P1 n P1) and so ij

g22 ~ 0^ (4.60)

In the present case where An+1 is taken as Yi P*, we have (n + 1) intersection points Pi,i+1. The non-zero intersection numbers is between neighboring spheres Pi1 and Pi1 1 ,

([p1] n [P] ]) = 2,i±1- (4.61)

Implementing this specific data, the last term of Eq. (4.58) reduces then to a sum of integrals over the following field densities

t~2 Tr(|Fi,i+112), Fn,n+1 = Fn,o, (4.62)

which diverge as long as |Fi,i+1|2 = 0. This property is not strange and was in fact expected. It has the behavior of a Dirac function one generally use for implementing insertions. To exhibit this feature, denote by P1 n P!1+1 = {Pi}, the points where the 2-spheres intersect. Then we have

f fGGfTr(iFi,i+1i2)=/ Gs(p—Pi)Tr(iFi,i+1i2) (4.63)

PjnPj+1 ' P? '

where S(P — P+) is a Dirac delta function. Combining the above results, one ends with the following field action of the 2d bosonic quiver gauge theory describing the brane configuration on the non-compact 4-cycle [D4] = O(—p0,..., —pn) — An+1 of the local CY3,

SAn = itf (¿^F) + F + 2G2 S(P — Pi) Tr(iF-'+1i20 • (4.64)

i=0 p1 1 1 1

In this relation, the coupling constants gf and G\ are expressed in terms of the string coupling gs, the Kahler moduli of the 2-spheres of the base An+1 and the degrees of the fiber O(—p0,^„, —pn). The Fi's are the U(Ni) gauge field strengths and

S(P — Pi) x Fi,i+1 (4.65)

are insertion operators in bi-fundamental representations and are needed to glue the spheres. The

last term may be rewritten in different forms. For example like x Tr(|Fi,i+1(Pi)|2) and

it depends on the P* 's.

4.2.2. Path integral measure in 2d q-deformed quiver gauge theory

Here we want to study the structure of the measure in the path integral description of the partition function of the quiver gauge field action S2d Eqs. (4.57), (4.64) which we rewrite as

v f 1 f 9YM f

— £ {¡s] Tr *2 + -T.j ™ a ^ + £ 9grJ T a .

+ £ v1 f Tr№j+1®,+u) - — £ f Tr(0,.,+1 a F,+1.,). (4.66)

. 2gs J ¡s . J

i i + 1 i i+1

We will give arguments indicating that bi-fundamentals contribute as well to the deformation of path integral measure and in a very special manner. More precisely, we give an evidence that adjoints and bi-fundamentals altogether deform the measure by the quantity

n Ni _

J(qo,.,qn№ =n n - ^ ]qi [*a; - $b} j ~hj. (4.67)

ij =0 ai<bi =1

In this relation, the $ai 's are as in Eq. (4.39) and where Iij is the intersection matrix of the 2-spheres of the An+1 base. It is equal to minus the generalized Cartan matrix of affine An.

To begin recall that the partition function ZYM(Sg) of topological 2d q-deformed U(M) YM on a genus g-Riemann surface Sg is given by

1 r / m \ m 2

^YM(^g) = ± ft U^a ] [AH(ta)]2-2gU eg YM h(*bFb+0*b+2 *b\ (4.68)

\a=1 f b=1

Here the Qa's are the diagonal values of U(M) unitary gauge symmetry and where/'n^ [DQa ] denotes the path integral with points with AH(Qa) — 0 omitted. In this relation, AH(Qa) is given by

Au(fa) — fi

a<b— 1

M \ . f fa - fa 2sin

(4.69)

and is invariant under the periodic changes (4.41). It can take the following form

[Ah(4)]2g-2 = n ([*ab]q)2"2g (4.70)

2i(fa - 0b) r , ( x -x) -gs Xab =-, [x]q = (q2 - q 2), q = e gs. (4.71)

Using this relation, we see that, on each 2-sphere Sf of An+1, the correction to the path integral measure is

ai<bi — 1

2sir/-

which by setting qi = exp(-^ -f ) can be put in the equivalent form

Js2 = EI - ^]«)2, i = 0,...,n. (4.72)

ai<bi = 1

The power 2 in the right hand of above relation can be interpreted in terms of the entries of the intersection matrix Iii = -Kii of the ith 2-spheres of An+\. This property is visible on Eq. (4.68) where the power 2 - 2g (Euler characteristics) is just the self-intersection of the Riemann surface Sg. As such, the above relation can be put in the form

Js2 = n (Vt^i - <Pbi - &bi ]qi)K", Kii= 2, i = 0,...,n, (4.73)

ai<bi = 1

at least for unitary gauge groups. For other gauge groups, the situation could be different and deserves more study. The above relation is obviously very suggestive, it lets understand that this feature is a special property of a more general situation where appears the intersection number. More precisely, the structure of the deformation of the path integral measure for local Calabi-Yau threefolds with some base B made of 2-cycles Ci with intersection matrix Iij = [Ci ]. [Cj ] should be as follows

Jb = n n (v[^i - 0bi]® [*aj - 0bj j ■ (4-74)

i,j a <bi=1

In our concern, the 2d manifold is given by the base manifold An+1 of the local Calabi-Yau threefold. In this case the intersection matrix of the 2-cycles is given by Iij = -Kij with Kij being the Cartan matrix of the A-series of finite dimensional Lie algebras. So the partition function of the q-deformed 2d quiver gauge theory reads in general as

„ ,,ANt ]

^■Quiver ■

; / n Ni

v i=0 ai = 1

x ( n n - ®bi]qi [<paj - $bj ]qjrj\ e-SQuiver, (4.75)

\i,j=0 ai<bi = 1 /

where SQuiver is the action given by Eq. (4.64)-(4.66). Of course, here An+1 is an elliptic curve and so one should have JB = 1. This condition can be turned around and used rather as a consistency condition to check the formula (4.74). Indeed, for the case of the 2-torus T2 = S1 x S1, we know that

[T2]. [T2] = 0, (4.76)

and so no q-deformation in agreement with Eq. (4.68). The same property is valid for [An+1]. But at this level, one may ask what is then the link between the two realizations T2 = S1 x S1 and [An+1] = Y, n=oj The answer is that in the second case the role of the condition

(2 - 2g) = 0

(4.77)

that is obeyed by S1 x S1 is now played by the vanishing property,

j2k'jsj=0

(4.78)

for affine Kac-Moody algebras (with sj = 1 for affine An). Let us check that Jau+1 Eq. (4.74) is indeed equal to unity. We will do it in two ways:

First consider the simplest case given by the superconformal model with gauge symmetry as in Eq. (4.20) and specify the Kahler parameters at the moduli space point where all the 2-spheres have the same area (r; = n+i). In this case the quantity [<a; - <bi 1® is independent of the details of An+1 and so the above formula reduces to

i=0Z. j=0 Kij

J|C+F1T(<) = n < - <b 1q

(4.79)

\a<b=1

which is equal unity (J (<) = 1) due to the relation J2"j=0 Kij = 0. In the general case where the gauge group factors are arbitrary and for generic points in the moduli space, the identity (4.74) holds as well due to the same reason. For the instructive case n = 2 for instance, we have

JQr<) = n <0 - <b0iJ n <1 - <b1 q

\a0<b0=1 / N2

a1 <b1 =1 2

X i J"] [<a2 - <b2 ]q2

\a2<b2 = 1

/ N0 N1

x( II [<a0 - <b0]«> EI [<a1 - <b11

\a0<b0 = 1 N1

a1 <b1 = 1

M n [<a1 - <b11q1 n [<a2 - <b21

\a1<b1=1 a2<b2 = 1

/ N2 N0

X ( n [<a2 - <b2 ]q2 n [<a0 - <b0 ]q0

(4.80)

\a2<b2 = 1

a0<b0=1

As we see, the diagonal terms of the first line of the right-hand side of the above relation are compensated by the off diagonal terms. Thus JQF(<) reduces exactly to unity and so the 2d quiver gauge theory is not deformed. Nevertheless, one should keep in mind that this would be a special property of a general result for 4d black holes obtained from BPS D-branes in type IIA superstring moving on the following general local Calabi-Yau threefolds

O(m) © O(-m - 2) ^ Bk.

(4.81)

Here m = (m1,...,mk) is an integer vector and Bk is a complex one dimension base consisting of the intersection of k 2-spheres S;2 with some intersection matrix I;j. Using Vinberg theorem [32,45-47], the possible matrices I;j may be classified basically into three categories. In the language of Kac-Moody algebras, these correspond to:

(i) Cartan matrices of finite dimensional Lie algebras satisfying

J2lijUj> 0 (4.82)

for some positive integer vector (uj). In this case the resulting 2d quiver gauge theory is q-deformed. This theory has been also studied in [19].

(ii) Cartan matrices for affine Kac-Moody algebras including simply laced ADE ones

J^I'JUJ = 0, (4.83)

where now the uj's are just the Dynkin weights. In this case, the 2d quiver gauge theory is un-deformed due to the identity Yj Iijsj = 0 where the sj's are the Dynkin weights.

(iii) Cartan matrices for indefinite Kac-Moody algebras where the intersection matrix satisfies the condition

J^IijUj< 0, (4.84)

for some positive integer vector (uj). Here the 2d quiver gauge theory is q-deformed. 5. Conclusion

In this paper, we have studied 4d black holes in type IIA superstring theory on a particular class of local Calabi-Yau threefolds with compact base made up of intersections of several 2-spheres. This study aims to test OSV conjecture for the case of stacks of D-brane configurations on CY3 cycles involving q-deformed 2d quiver gauge theories with gauge symmetry G having more than one U(Ni) gauge group factor. The class of local threefolds we have considered with details is given by x(m,-m-2,2) = O(m) © O(-m - 2) ^ An+1 where m is a (n + 1) integer vector (m0,...,mn). The mi components capture the non-trivial fibration of rank 2 line bundles of the local CY3. They also define the Un+1(1) charges of the corresponding chiral superfields in the supersymmetric gauged linear sigma model field realization. The compact elliptic curve An+1 is generally given by (n + 1) intersecting spheres according to affine Dynkin diagrams. As far as the construction of this specific class of local Calabi-Yau threefolds is concerned, we devote a main part of the second section to first describe our geometry realization; then we compare it to the Caporaso et al. method [24].

Our study has been illustrated in the case of the local affine An model; but may a priori be extended to the other affine models especially for DE simply laced series and beyond. Black holes in four dimensions are realized by using D-brane configurations in type IIA superstring compactified on X(m,-m-2,2). The topological twisted gauge theory on D4-brane wrapping 4-cycles in the local CY3s is shown to be reduced down to a 2d quiver gauge theory on the base An+1 and agrees with OSV conjecture. This agreement is ensured by the results on O(2g + m - 2) © O(-m) ^ Sg obtained in [2,18]. It is interestingly remarkable that bi-fundamentals and adjoint scalars contribute to the deformed path integral measure with opposite powers and compensate in the case of affine geometries as shown in the example (4.80).

In developing this analysis, we have taken the opportunity to complete partial results on the 2d N = 2 supersymmetric gauged linear sigma model realization of the resolution of affine singularities and the local Calabi-Yau threefold with non-trivial fibrations. We have also given

comments on other black hole models testing OSV conjecture. They concern the class of 4d black holes with D-branes wrapping cycles in local threefolds with complex one dimension base manifolds Bk (4.81) classified by Vinberg theorem [45]. The latter is known to classify Kac-Moody algebras in three main sets: (i) ordinary finite dimensional, (ii) affine Kac-Moody and (iii) indefinite set.

In the end, we would like to note that the computation given here can be also done by using the topological vertex method. Aspects of this approach have been discussed succinctly in present study. More details on this powerful method as well as other features related to 2d quiver gauge theories and topological string will be considered elsewhere.

Acknowledgements

This research work has been done in several steps, at Rabat—Morocco, at ICTP—Italy, and at Faculdad de Ciencias, Universidad de Zaragoza, Spain. A.B. and E.H.S. would like to thank Departamento de Fisica Teorica, Zaragoza for kind hospitality. They also thank Manuel Azorey, Luis. J. Boya, Jose L. Cortes, Pablo Diaz, Sergio Montagnez and Antonio Segui for fruitful discussions. A.B. is supported by Ministerio de Educacion y Cienca, Spain, under grant FPA 2006-02315. L.B.D. thanks "Le programme de la bourse d'excellence, CNRST, Rabat". E.H.S. would like to thank ICTP for hospitality and Senior Associate Scheme for kind generosity. He thanks also N. Chair for discussions. This research work is supported by the program Protars III D12/25.

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