Non-perturbative Nekrasov partition function from string theory Academic research paper on "Physical sciences"

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Nuclear Physics B 880 (2014) 87-108

www.elsevier.com/locate/nuclphysb

Non-perturbative Nekrasov partition function from string theory

I. Antoniadisa1,1. Florakisb'*, S. Hoheneggera, K.S. Narainc, A. Zein Assia d

a Department of Physics, CERN — Theory Division, CH-1211 Geneva 23, Switzerland b Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, 80805 München, Germany c High Energy Section, The Abdus Salam International Center for Theoretical Physics, Strada Costiera, 11-34014 Trieste, Italy d Centre de Physique Théorique (UMR CNRS 7644), Ecole Polytechnique, 91128 Palaiseau, France Received 14 October 2013; received in revised form 18 December 2013; accepted 8 January 2014 Available online 13 January 2014

Abstract

We calculate gauge instanton corrections to a class of higher derivative string effective couplings introduced in [1]. We work in Type I string theory compactified on K3 x T2 and realise gauge instantons in terms of D5-branes wrapping the internal space. In the field theory limit we reproduce the deformed ADHM action on a general Q -background from which one can compute the non-perturbative gauge theory partition function using localisation. This is a non-perturbative extension of [1] and provides further evidence for our proposal of a string theory realisation of the Q -background.

© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

1. Introduction

Gauge theory instantons have a natural description as bound states of D-branes in string theory [2,3]. This opens the possibility to calculate non-perturbative corrections in field theory from

* Corresponding author.

E-mail addresses: ignatios.antoniadis@cern.ch (I. Antoniadis), florakis@mppmu.mpg.de (I. Florakis), stefan.hohenegger@cern.ch (S. Hohenegger), narain@ictp.trieste.it (K.S. Narain), zeinassi@cern.ch (A. Zein Assi). 1 On leave from CPHT (UMR CNRS 7644) Ecole Polytechnique, F-91128 Palaiseau, France.

http://dx.doi.org/10.1016/j.nuclphysb.2014.01.006

0550-3213/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

tree-level string amplitudes. Since the former are notoriously difficult to compute, whereas established techniques exist to obtain the latter, this provides very interesting insights into the inner structure of field theories and has lead to remarkable developments over the recent years. Furthermore, the series of works [4-7] that led to the partition function of supersymmetric gauge theories in the Q -background has triggered considerable interest on this connection.

Indeed, four-dimensional super-Yang-Mills theory can be realised on a stack of D3-branes in Type IIB string theory, with additional D(-1)-branes playing the role of instantons. Their corrections to the Yang-Mills action are captured by string disc diagrams with boundary field insertions. In [8] the effect of a non-trivial constant string background in this setup was considered, by including additional bulk vertices in the tree-level amplitudes. It was shown that the insertion of anti-self-dual graviphoton field strength tensors in the point particle limit correctly reproduces the ADHM action on an Q -background with one of its deformation parameters switched off (e.g. e+ = 0). Using localisation techniques [4-7,9-11] this allows one to compute the non-perturbative Nekrasov partition function ZNek(e+ = 0,e_).

Obtaining the partition function for a gauge theory on a generic Q -background (with e+ = 0) from string theory remains an interesting question. Phrased differently, one would like to find a modification of the anti-self-dual graviphoton background, considered in [8], giving rise to the fully deformed ADHM action in the point-particle limit of the appropriate disc diagrams. A hint for answering this question comes from a series of higher derivative one-loop couplings in the effective action of the Heterotic string compactified on K3 x T2, considered in [1]. These terms generalise a class of BPS-saturated couplings of gravitons and anti-self-dual graviphoton field strength tensors through additional couplings to the self-dual field strength tensor Ft of the vector partner of the Kahler modulus of the internal T2. In the field theory limit, these one-loop amplitudes precisely reproduce the perturbative contribution of the Nekrasov partition function on the full Q-background, i.e. with e-,e+ = 0. We therefore expect that, including ) also as a background field in the instanton computation described above, allows us to extract the fully deformed ADHM action from string theory. In this paper, we show that this is indeed the case.

We work in Type I string theory compactified on K3 x T2 and consider D9-branes together with D5-instantons wrapping K3 x T2. This setting is dual to Heterotic string theory on K3 x T2 and the corresponding background is given by anti-self-dual graviphotons and self-dual field strength tensors of the vector partner of S', which we refer to as S"-vectors in the remainder of this work. We compute all tree-level diagrams with boundary insertions that couple non-trivially to this background which, in the field theory limit, correctly reproduce the fully Q -deformed version of the ADHM action, which was used to compute Nekrasov's partition function [6,7].

This result can also be interpreted as computing gauge theory instanton corrections to the higher derivative couplings discussed in [1]. The fact that we reproduce precisely the full Nekrasov partition function can also be seen as further evidence for the proposal that these couplings furnish a worldsheet description of the refined topological string. Indeed, our results suggest that the background introduced in [1] can be understood as a physical realisation of the Q -background in string theory.

We would like to mention that various RR backgrounds were recently discussed in [12,13] where the Q -deformed ADHM action was recovered using the language of D-instantons. However, contrary to our present work, the backgrounds considered there do not involve NS-NS field strengths. Moreover, the interpretation in terms of a string effective coupling, as well as the perturbative contribution, were not considered.

The paper is organised as follows. In Section 2, we briefly review the construction of gauge theory instantons in Type I as a D5-D9 system and set the notation for the various moduli arising

as massless open string states stretching between the various D-branes. In Section 3, we calculate the tree-level (disc) diagrams with bulk insertions of S'-vectors. This is first done by introducing auxiliary fields as in [12-15,8] (see also [16,17]). We show that, in the field theory limit, the resulting effective action precisely matches the one used in [6] to derive the non-perturbative gauge theory partition function. However, correlation functions involving auxiliary fields are in general not well-defined since the latter are not BRST-closed. To this end, in Appendix B, we recover the same effective couplings by inserting only physical states in the disc diagrams, hence justifying the result of the auxiliary field calculation. In Section 4, we discuss the connection of the present work with the higher derivative couplings studied in [1]. Section 5 contains our conclusions.

2. ADHM instantons from string theory

We would like to use a string perspective to study gauge theory instantons. The latter are naturally realised as D-brane bound states and their contributions to the gauge theory action can be obtained at string tree-level. We work in Type I theory compactified on R4 x T2 x K3. Gauge instantons in a theory living on a configuration2 of N D9-branes are the D5-instantons wrapping K3 x T2. For concreteness, we summarise the brane setup in the following table.

Brane Num. X0 X1 X2 X3 X4 X5 X6 X7 X8 X9

D9 N..........

D5 k ......

space-time~R4 T2 K3~T 4/Z2

This configuration describes instantons3 of winding number k in a gauge theory with SU(N) gauge group. In order to see this, let us describe the massless spectrum of the theory. The notation used here is summarised in Appendix A. More precisely, it is natural to decompose the 10-dimensional Lorentz group into

S0(10) ^ SO(4)st x SO(6)int ^ SO(4)st x SO(2)t2 x SO(4)k3, (2.1)

reflecting the product structure of our geometry. In this way I, J = 1,..., 10 denote indices of the full S0(10), n-, v are indices of the space-time SO(4)ST, with a, ft (a, ft) the corresponding (anti-)chiral spinor indices, while a, b denote internal S0(6)jnt indices. For S0(4)K3 ~ SU(2)+ x SU(2)-, we introduce indices A, B = 1, 2 for fields transforming in the (2,1) representation (positive chirality) and A,B = 3,4 for fields in the (1,2) representation (negative chirality). Following [15], we associate upper indices of SU(2)+ (resp. SU(2)-) with charge +1/2 (resp. -1/2) of S0(2)T2 and downstairs indices with charge -1/2 (resp. +1/2) respectively. In this way, internal indices cannot be raised and lowered with the help of e-tensors, but we have to keep track of their position.

2 Before considering the effect of D5 instantons, one may start from a consistent string vacuum, e.g. the model discussed in [18] with gauge group U(16) x U(16), where the total number of D5 branes (wrapping the space-time and T2 directions) and D9 branes is fixed to be 16 by tadpole cancellation. The stack of 16 D9 branes may be further higgsed to N < 16 by turning on appropriate Wilson lines along T2, so that one may keep N generic for the purposes of our analysis.

3 As usual, the back-reaction of the D5-instantons to the background is not considered.

In this setting, there are three kinds of open string sectors which are relevant for our subsequent discussion. They can be characterised according to the location of their endpoints.

1. 9-9 sector

The massless excitations consist of a number of N = 2 vector multiplets, each of which containing a vector field A', a complex scalar $ as well as four gaugini (AaA,AaA) in the (2,1) representation of SU(2)+ x SU(2)- with SO(2)T2 charges (1/2, -1/2) respectively. The bosonic degrees of freedom stem from the NS sector, while the fermionic ones from the R sector. These fields separately realise a Yang-Mills theory living on the four-dimensional space-time.

2. 5-5 sector

These states are moduli (i.e. non-dynamical fields) from a string perspective, due to the instantonic nature of the corresponding D5-branes. Indeed, the states in this sector cannot carry any space-time momentum because of Dirichlet boundary conditions in all directions except along K3 x T2. From the Neveu-Schwarz (NS) sector, we have six bosonic moduli, which we write as a real vector a1 and a complex scalar x. From the point of view of the SYM theory living on the world-volume of the D9-branes, a1 corresponds to the position of gauge theory instantons. From the Ramond sector, we have eight fermionic moduli, which we denote as MaA, A.

3. 5-9 and 9-5 sectors

Also this sector contains moduli from a string point of view. From the NS sector, the fermionic coordinates have integer modes giving rise to two Weyl spinors of SO(4)ST which we call (ma ,coa). Notice that these fields all have the same chirality, which in our case is anti-chiral, owing to a specific choice of reflection rules in the orbifold construction (see [15] for a similar discussion in a related setup). From a SYM point of view, these fields control the size of the instanton. In the R sector, fields are half-integer modded giving rise to two fermions ('A, 'A) transforming in the (2,1) representation of SU(2)+ with positive charge under SO(2)T2.

For the reader's convenience, the field content is compiled in Table 1. As was discussed in [14,15], the string tree-level effective action involving only the massless vector multiplets of the 9-9 sector exactly reproduces, in the field theory limit, the pure N = 2 super-Yang-Mills theory with SU(N) gauge group. Inclusion of the remaining moduli fields gives rise to the ADHM action describing instantonic corrections with instanton4 number k. Therefore, this setup provides a stringy description of the gauge theory instantons.

Furthermore, by coupling the theory to a constant anti-self-dual graviphoton background [8], the resulting effective action coincides with the ADHM action in the Q -background used in [6] in the case where one of the deformation parameters vanishes (say, = 0). While the ADHM action is exact under a nilpotent Q-symmetry, the latter is still present after the deformation with e+. Hence, one can use localisation techniques in order to compute the instanton partition function [6].

From a practical perspective, this deformation is obtained by computing string disc diagrams with bulk insertions of the anti-self-dual graviphoton (Fig. 1). Due to its anti-self-duality, the

4 When taking the field theory limit, one should pay attention to the dimensionality of the various fields. In particular, a rescaling of the ADHM moduli is necessary in order for the field theory limit to be well-defined as an appropriate double scaling limit in which gYM is held fixed.

Table 1

Overview of the massless open string spectrum relevant for the disc amplitude computation. We display the transformation properties under the groups SO(4)st x SU(2)+ x SU(2)-, while Ct2 is the charge under SO(2)t2. The last two columns denote whether the field is bosonic or fermionic and the sector it stems from.

Sector Field SO(4)st CT 2 SU(2)+ x SU(2)- Statistic R/NS

9-9 A'x (1/2,1/2) 0 (1,1) boson NS

AaA (1/2, 0) 1/2 (2, 1) fermion R

Aà A (0,1/2) -1/2 (2, 1) fermion R

<P (0, 0) -1 (1,1) boson NS

5-5 (1/2, 1/2) 0 (1,1) boson NS

X (0, 0) -1 (1,1) boson NS

MaA (1/2, 0) 1/2 (2, 1) fermion R

^-à A (0, 1/2) -1/2 (2, 1) fermion R

5-9 (0, 1/2) 0 (1,1) boson NS

^A (0, 0) 1/2 (2, 1) fermion R

Fig. 1. Three-point disc diagrams with graviphoton bulk-insertion. Diagram (a) involves two boundary insertions from the 5-5 sector, whereas diagram (b) two insertions from the 5-9 sector. While the whole boundary of diagram (a) lies on the D5-branes, diagram (b) lies partly on the D9- and partly on the D5-branes. Notice that the latter mixed boundary conditions appear only in the space-time directions.

only instanton contributions come from the diagrams with insertions from the 5-5 sector and no mixed diagrams5 contribute.

In the following section, we generalise this construction to the case where the background includes, in addition to the anti-self-dual graviphoton field strength, the self-dual field strength of the S"-vector. Such a background was recently introduced in [1] as a proposal for a world-sheet description of the refined topological string. We show that this generalised background reproduces the ADHM action in the presence of a general Q -background, therefore providing a non-perturbative check of the proposal of [1].

3. Refined ADHM instantons in string theory

In this section we calculate disc diagrams relevant for making contact with the instanton sector of the gauge theory using the RNS formalism. We only consider diagrams involving the

5 By mixed diagrams we refer to disc diagrams whose boundary lies on both D9- and D5-branes.

additional S'-vector insertion in the bulk. Other couplings (e.g. diagrams involving only boundary insertions) that are not affected by the latter have already been computed in the literature (see e.g. [8,12,13]). Therefore, we do not repeat these calculations here.

We use a method similar to the one employed in [8], namely, we introduce auxiliary fields which linearise the superalgebra. As a consequence all diagrams involve only a single bulk insertion of the S' field strength tensor and two insertions of boundary fields. Even though the computation is considerably simplified in this case, one faces the problem that auxiliary fields are not physical and, therefore, not BRST-closed. String theory amplitudes involving such fields are generically not independent of the superghost picture, thus rendering the results a priori ambiguous (an example of this phenomenon is given in Section 3.1.3). Nevertheless, in our case, the correct result is obtained for the particular choice of picture, which we use in this section. Indeed, in Appendix B we show that one may repeat the calculation in a physical framework (i.e. without resorting to auxiliary fields), where picture independence is restored. The justification for our simplified computation is provided in Appendix B. There we show that, for this special choice of picture, the relevant disc diagrams factorise in a particular form which can be re-interpreted in terms of (unphysical) auxiliary fields and matches precisely the result obtained using auxiliary fields.

3.1. Vertex operators and disc diagrams

We denote the ten-dimensional bosonic and fermionic worldsheet fields collectively as X1 and ^1 respectively, with I = 1,..., 10. More precisely, we use (X1,^1), (Z, *) and (Y',x') for worldsheet fields along the four-dimensional space-time, T2 and K3 directions respectively. In the following, for simplicity, we consider an orbifold representation of K3 as T4/Z2. However, we expect our results to be valid also for a generic (compact) K3. The vertex operators relevant for the disc amplitudes involve the ADHM moduli appearing in the massless spectrum. From the 5-5 sector, we need

Va(z) = g6 a^(z)e-<l\ (3.1)

Vx(z) = X * (z)e-<p(z\ (3.2)

Vm(z) = ^ MaASa(z)SA(z)e~2<(z). (3.3)

From the 5-9 and 9-5 sectors, we use

VM) = -6 «a A(z)Sa (z)e-*M, (3.4)

Vs (z) =^ a (z)S* (z)e-<(z). (3.5)

Here A(z), A (z) are twist and anti-twist fields with conformal weight 1 /4, which act by changing the boundary conditions and g6 is the D5-instanton coupling constant. The superghost is bosonised in terms of the free boson <. In order to linearise the supersymmetry transformations, a set of auxiliary fields is introduced whose vertex operators read as follows:

Vy(z) = V2 g6Y(z)f'(z), Vy t (z) = V2 g6Yl^(z)f'(z), (3.6)

Vx(z) = g6Xa A(z)Sa * (z), Vxt (z) = g6Xt A(z)Sa *(z), (3.7)

Vx (z) = g6Xa A (z)Sa * (z), Vx t (z) = g6X T A (z)Sa *(z). (3.8)

Finally, we turn to the closed string background defined in [1]. The vertex operator of the anti-self-dual graviphoton in the (-l)-ghost picture at zero momentum is given by:

VFG(y ï) = —1— Fg

V (y,y) 8n V2

f^fv(y)e-<p{y (y ) + e-viy)^f (y)f^fv(y )

- 2e-2(<P(y)+<P(y))sa(y){o»v)apSp(y)eABSA(y)SB(y)

The vertex operator for the self-dual field strength tensor of the S'-vector in the (-l)-ghost picture and at zero momentum is given as a sum of an NS-NS part (first line) and a R-R part (second line) [1]:

VFS' (y,y) = FSv

f^fv{y)e-(p(y)V{y) + e-v(y)V{y)f^fv(y)

+ 2e-2 (<p(y')+<p(y) Sa (y)(à»v )à ? Sp (y)tA b SA (y)SB (y)

(3.10)

The disc diagrams involving the bulk insertions of the RR part of the anti-self-dual graviphoton (3.9) have already been extensively studied in the literature (see, for example, [8]). Following the analysis performed below, one can show that the NS-NS part also gives the same contribution leading to the e_-dependent part of the Q-deformed ADHM action. Here, we focus on the more interesting case of the self-dual insertions of S'-field strengths. The relevant disc diagrams correspond to the following correlators:

Dy WS' (zi,z2,y,y) = Vy t (zi)Va(z2)VFs> (y, y))Disc, (3.11)

(zi,z2 ,y, y) = (zi)Vm(z2)VFS> (y, y)>Disc, (3.12)

DMMFS' (z1,z2,y, y) = (VM(z1)VM(z2)VFS' (y, y))Disc- (3.13)

3.1.1. D5-D5 disc diagrams

We start by evaluating the amplitude (3.11) corresponding to a disc diagram whose boundary lies entirely on the stack of D5-branes and with the S'-modulus inserted in its bulk. This allows us to fix the normalisation of the S' vertex operator, which is needed to match precisely all numerical coefficients in the refined ADHM action. The amplitude we consider is given by

8n2 i dz1 dz2 dy dy

« Vy t VaVFS'>> Disc = — 1d..2 y y Dy taFS' (z1,z2,y, y). (3.14)

gYM J dVCKG

Here the double brackets define the integral of the correlation function over its worldsheet positions. Furthermore, dVCKG is the volume of the conformal Killing-group, which we parametrise as

dz1 dy dy

dVCKG =-^-—y--. (3.15)

(z1 _ y)(z1 _ y)(y _ y)

Since the vertex operator for F S consists of two parts, the NS and R ones, we can split the correlator (3.11), which appears as the integrand in (3.14), accordingly

Dy taFS' (z1,z2,y, y) = DNtSaFS' (z1,z2,y, y) + DRWS' (z1 ,z2 ,y, y). (3.16)

Starting with the NS contribution we find

= YtavFfa )r(z2)r(y)r(y))[e-v(z2)e-v{y))

- {^(zi)^(y)){f^(zi)fv(z2)fp(y)fa(y))(e-viz2]e Using the relations given in Appendix A, we find

DNSaFs< = -gnY>vFl x 48^8va(zi - y)-1(zx - y)-1(z2 - y)-1(z2 - y)-1. (3.17)

The correlator DRt p-s,, which contains the R-part of the vertex (3.10), can be computed in the same fashion and yields precisely the same answer. Furthermore, conformai symmetry still allows us to fix the position of one of the boundary insertions as well as the position of the bulk vertex in the correlator (3.16), which we choose z1 =œ and y = i. Inserting (3.16) and (3.17) into (3.14) and integrating z2 over the real line, we obtain

« Vy t Va V» Disc = -4i Tr^avF-^ ), (3.18)

where we have used 2/(2na'g6)2 = 8n2/gYM and restored the dimensionality of the fields in terms of 2na'.6

Applying the same techniques, we show that the diagram with boundary fermionic insertions (3.13) vanishes identically:

8n2 i dzi dz2 dydy

«VmVmVfs')) = — 12 y yDmmf-s,(z1,z2,y,y) = 0. (3.19)

gYM J dvCKG

3.1.2. D5-D9 disc diagrams

Let us also discuss the contributions of mixed diagrams corresponding to the correlator (3.12). In this case, the disc amplitude can be computed in the same way as in the D5-D5 sector:

8n2 f dz\ dz2 dydy

«Vxt VmVFS')) = — 12 y y Dx,mFS'(zx,z2,y, y). (3.20)

gYM dVCKG

Going through similar steps as before, we find

« Vxt V« VfS'» = 2 Tr(X; «p (a'v)Tp F'V). (3.21)

These diagrams are sufficient to compute the tree-level string effective action involving D5-instantons. In Section 3.2, we consider the field theory limit and compare the result to the ADHM action on a general Q -background.

3.1.3. Picture dependence of unphysical diagrams

It is important to note that correlation functions involving auxiliary fields depend on the picture one uses for the vertex operators. This is to be expected since auxiliary fields are not BRST-closed and, hence, their use in string amplitudes can lead to ambiguous results. However, as proven in Appendix B, the specific prescription for the pictures used in the previous section leads to the correct physical result.

6 In [19,20] a different string background was studied, which involved the self-dual vector partner of the dilaton S. However, repeating the computation of the above disc diagrams using this vector instead, leads to a vanishing result. This indicates that the background [19,20] does not give rise to the deformed ADHM action on a general Q-background.

Fig. 2. Diagrammatic representation of the factorisation limit relating the correlator of physical operators with the one using an auxiliary field.

In particular, we repeat the calculation of the relevant disc diagrams, which now involve four-point functions of physical vertex operators only, and we recover the same result in an unambiguous fashion, independently of the choice of ghost picture, in accordance with the general arguments of [21]. We then show that, only for a special choice of picture, the decoupling of the longitudinal mode of FS is manifest and the four-point correlator factorises in such a way that allows one to effectively reinterpret the OPE of two physical vertices in terms of the auxiliary field used in the previous section (see [22] for an earlier discussion of similar issues). This is pictured in Fig. 2. For other choices of pictures, it turns out that the decoupling of the longitudinal mode is no longer manifest as a factorisation in any single channel and the correct result is only obtained by combining various different contributions with different factorisation structures. This prevents us from reinterpreting the amplitude in terms of a simple three-point function involving an effective (auxiliary) vertex. Put differently, only with the picture prescription used in the previous section is the auxiliary field formalism applicable.

Therefore, working with auxiliary fields provides a simple framework that is justified only through a calculation involving BRST-closed operators, as presented in Appendix B.

3.2. ADHM action and Nekrasov partition function

In this section we compare our results with the deformed ADHM action appearing in [6]. The latter describes instantons in a gauge theory with SU(N) gauge group on a general Q -background. Our first step is to relate the states arising from the superstring construction described above with the physical gauge theory fields describing, in particular, the position and the size of the instantons. Our notation follows [23] (see e.g. [24] for a review). At the core of the (undeformed) ADHM construction lies a specific ansatz for the gauge connection. Requiring this ansatz to be a solution of the Yang-Mills equations of motion gives rise to a number of constraints that can be encoded in an action principle. To obtain a deformation of this ADHM action, we finally implement the Q background as a particular space-time U(1)2 rotation, with parameters e±.

To be more specific, consider the following ansatz for the SU(N) gauge connection

(A^)uv = UUd,JJav, with UaUav = suv. (3.22)

Here we have introduced the ADHM index a = 1,..., 2k + N and u,v = 1 ...,N, with k being the instanton number7 and U is the hermitian conjugate of U. The ansatz (3.22) is a solution of

7 For notational simplicity, we mostly suppress the indices u,v in what follows.

the Yang-Mills equations if the corresponding field-strength is self-dual. To find a solution for the matrix U which has this property, we first notice that the operator Pab = Ua Ub is a projector preserving U, i.e. P2 = P and PU = U. The field strength of A^ can be written in terms of P as

F^v = d[/j,Ua (Sab - Pab)dv]Ub. (3.23)

One is thus led to write the ansatz 1 - P = A/A, where the [N + 2k] x [2k] matrix A is called the ADHM matrix, while / is an arbitrary Hermitian matrix. If we assume that A is linear in the space-time coordinates xM

Aaia = aaia + baixaca, (3.24)

and that the matrix / can be 'diagonalised' as

j p = j" *, (3.25)

then the field strength (3.23) is self-dual, being proportional to the matrix a^v, provided that the constraint

/-11 = AA, (3.26)

is satisfied. Here, the space-time coordinates are expanded in a spinor basis, i.e. xaa = xM (a^)aa and we have introduced the instanton index i = 1,...,k. In this way, the construction of the ADHM solution is reduced to finding a consistent set of matrices a and b. The symmetries of the problem allow us to write [24]

a = ( W ) , b =( °[N]x[2k]) , (3.27)

\a / y 1[2k]x[2k] J

where w and a' are [N] x [2k] and [2k] x [2k] matrices respectively. In order to make contact with the degrees of freedom appearing in the string theory setup in Section 2, we identify

w = («a )ui, a' = (aa„ )ij. (3.28)

The ADHM instanton action can finally be expressed as [6]

Sadhm = -Tr{[xf,aap]([x, apa] + e-(ax3)pa) - x««X - a«)

- (x«a - f a a)« X ; + e+[x t,aa* ](T3afa - e+ f& faf * X «}, (3.29)

where we only display the part relevant for our discussion. Here we have introduced the vev a of the N = 2 vector multiplet that higgses the SU(N) gauge group. The terms in the second line correspond to the e+-dependent deformation of the ADHM action and which we want to compare to the effective couplings of FS to the ADHM moduli. To this end, we parametrise the vev of the S' field strength using the self-dual't Hooft symbols (see Eq. (A.10)):

7y _ -c J?S' _ -c S_

After integrating out the auxiliary fields, the contribution of the diagrams (3.18) becomes (see also (B.31))

FSv = fv Fc (3.30)

« VaVaVx t VF-S,}} = - 4i Tr{ [x t,ali]avFf} = -e+ Tr{ [x t,aa$]aYa (r3)p .}. (3.31)

Similarly, after integrating out the auxiliary fields, the contribution (3.20) of the mixed diagrams can be recast as

« VjVj t V» = 2 Tr{ «a x t Jp (â )àf> F* } = e+TrJ x f(T3 )a p J}. (3.32)

Therefore, the D5-instanton world-volume theory in our background gives a six-dimensional field theory on K3 x T2 that contains the deformed ADHM couplings. In the limit where the world-volume of the D5-instantons becomes small (i.e. of order a' or smaller), we can reduce the six-dimensional field theory to a zero-dimensional one which exactly reproduces the deformed ADHM action for the four-dimensional gauge theory.

The ADHM action is a key ingredient to compute the full Nekrasov partition function zNek of the supersymmetric gauge theory. It can be factorised in the following form:

ZNek(e+,e_) = ZpNeerk (e+,e-)ZNipk(e+,e-). (3.33)

While the perturbative piece Z^ does not receive contributions beyond the one-loop order, the non-perturbative part ZnN.epk. is defined as a path integral over the instanton moduli space, with the integral measure given by the deformed ADHM action SADHM [6,7].

4. Interpretation and the refined topological string

In this section we interpret our results in terms of recent developments in topological string theory. Inspired by the work of Nekrasov [6,7], it has been realised that a one-parameter extension exists, known as the refined topological string, such that its partition function at genus g, in the

limit F f.^.

[6,7,9-11]

field theory limit Fg n, reproduces the partition function of a gauge theory in the Q -background

log ZNek(e+,e_) = ¿g_24n Fgn (4.1)

_ + g,n

Several proposals for a description of the refined topological string have been presented in the literature [25-28]. However, a convincing worldsheet description has turned out to be rather challenging [20,29].

Recently, we have put forward a promising proposal in [1], in terms of a particular class of higher derivative terms in the string effective action. At the component level, it involves terms of the form

Ig,n = f d4x Fg,nR(—)^vptR'—r [FG_naF—r ]g—1[F(+)paF+]n, (4.2)

for g > 1 and n > 0. Here, R(_) denotes the anti-self-dual Riemann tensor, F— the anti-self-dual field strength tensor of the graviphoton and F(+) the self-dual field strength tensor of an additional vector multiplet gauge field. In Heterotic N = 2 compactifications on K3 x T2, the latter is identified with the super-partner of the Kahler modulus of T2, while in the dual Type I setting, it is mapped to the vector partner of the S' modulus.

For n = 0, the lgn in (4.2) are BPS-saturated and were first discussed in [30]. The Fg = Fgn=0 are exact at the g-loop level in Type II string theory compactified on an elliptically fibered Calabi-Yau threefold and compute the corresponding genus g partition function of topological string theory [30]. In the dual heterotic theory, Fg starts receiving contributions at the one-loop

level [31] and, in the point particle limit, is related to the perturbative part of Nekrasov's partition function for a gauge theory on an Q -background with only one non-trivial deformation parameter. The latter is then identified with the topological string coupling.

For n > 0 the leading contribution to Fgn is still given by a one-loop amplitude in the het-erotic theory and was computed to all orders in a' in [1]. The coupling functions Fg,n in (4.2) can be compactly expressed in the form of a generating functional

In the point particle limit, the one-loop contribution to F(e_,e+) captures the perturbative part of the Nekrasov partition function (3.33) on a generic Q -background, whose deformation parameters are identified with the expansion parameters e± in (4.3). Thus, the Fg,n are one-parameter extensions of the topological amplitudes Fg which are (perturbatively) compatible with a refinement in the gauge theory limit.

The instanton computations performed in the previous sections are indeed evidence for this proposal. As we showed, also the non-perturbative contributions to the Fgn in the point particle limit are compatible with the structure expected from gauge theory and capture precisely the non-perturbative part of the full Nekrasov partition function ZNpk(e+,€-). This suggests that the couplings (4.2) indeed provide a string theoretic realisation of the full Q -background in gauge theory. This is precisely what one would expect from a worldsheet realisation of the refined topological string.

5. Conclusions

In this paper we computed gauge theory instanton corrections to the class of higher derivative couplings, introduced in [1]. These were obtained from tree-level amplitudes in Type I string theory compactified on K3 x T2, where we allowed for bulk insertions of anti-self-dual graviphotons as well as self-dual field strength tensors Ff, of the vector partner of S'. In the field theory limit, these amplitudes reproduce correctly the fully Q-deformed ADHM action, i.e. with both deformation parameters e+,e- = 0. Using localisation, the resulting path integral over the instanton moduli can be computed [6,7,9] yielding the non-perturbative gauge theory partition function.

Our work generalises the results of [8], where the deformed ADHM action in the limit e+ = 0 was reproduced from a pure graviphoton background, and extends the calculations of [12,13] for €+ = 0 by including NS-NS field strengths. Hence, building on our earlier results [1], we proved that the specific class of higher derivative string couplings (4.2) correctly reproduces, in the point particle limit, the partition function of pure N = 2 gauge theory, both perturbatively and non-perturbatively.

We would like to stress that Ff, is crucial for obtaining the correct Q-deformation of the ADHM action. For instance, we have checked that a background including self-dual field strength tensors of the vector partner of the heterotic dilaton [20], does not lead to the full deformation of the ADHM action. It remains as an interesting question what such deformations mean from a gauge theory perspective.

As a further consequence, our findings also provide strong support to our proposal that the couplings considered in [1] can be interpreted as a worldsheet description of the refined topological string. Indeed, these couplings generalise a class of BPS-saturated couplings Fg discussed in [30-32], which capture the genus g partition function of the 'unrefined' topological string.

The additional coupling to F-s, yields a one-parameter extension which corresponds to a refinement in the field theory limit, as was discussed perturbatively in [1]. Our current work indicates that this also carries over in the presence of instanton effects. Thus, these couplings satisfy a number of properties one would expect from a worldsheet realisation of the refined topological string. It would be very interesting to further study their properties.

Finally, it would be interesting to study the connection between our results and recent proposals for a realisation of the Q -background within string theory using the flux-trap background [33].

Acknowledgements

We thank C. Angelantonj, R. Blumenhagen, A. Klemm, N. Lambert, M. Marino, N. Meka-reeya and J.F. Morales for useful discussions. We would also like to thank M. Billo, M. Frau, F. Fucito and A. Lerda for pointing out their work to us and also J. Gomis for bringing [12,13] to our attention. I.F. would like to thank the CERN Theory Division and A.Z.A. would like to thank the ICTP Trieste and Max-Planck-Institut für Physik in Munich for their warm hospitality during several stages of this work. This work was supported in part by the European Commission under the ERC Advanced Grant 226371 and the contract PITN-GA-2009-237920.

Appendix A. Notations and useful results

In this appendix we summarise our conventions and notations. A.1. Indices

Let us begin by discussing our conventions for various index structures. Raising and lowering of S0(4)sT spinorial indices is achieved with the help of the epsilon-tensors e12 = e12 = +1, and e12 = q2 = -1, i.e.

fa = +eaßf ß, fa = -eaßfß, fa = -eaß f ß, fa = +ekß fß. (A.1) Furthermore we introduce the a -matrices (a»)aa and (a»)aa of S0(4)ST

(a»)aa = (1, -ia(a»fa = (1, +ia), (A.2)

which are related to one-another by raising and lowering of the spinor indices

K)^ - eßa(a»)aä** = (ä»f, (ä») ß - e^ (a^eaß = (a») . (A.3) Furthermore, we introduce the Lorentz generators a»v, a»v of 50(4)ST

(a»v)aß — 1 (a»a v - ava»)aß , (a»v)0!ß - ^ (a»av -ava»fß, (A.4)

which are symmetric in the spinorial indices (a»v)aß = +(a»v) ßa and (a»v)aß = +(a»v). Furthermore, these generators are (anti-)self-dual in the sense

(a»v )aß = +1 e»vpa(aPa)*ß, (a»v )a$ = — e»vpa(apa)aß. (A.5)

Therefore, we can use them to define (anti-)self-dual tensors respectively. In particular we write for th field

for the self-dual Fand anti-self-dual F(v ^ part of the field strength tensor of a given gauge

with Fjj±) = eIvpa(F(±))pa. Also, note the following identities:

(a"V)ap (aPaTP = 2(8Ip8Và - 8Ià8vp + eIvpà), (A.7)

(àIV)ap (àpa f p = 2(8Ip8và — 8Ià8vp — eIvpà). (A.8)

Using the above relations, one may invert (A.6) to obtain

F(+) _ I(à )ap F(+) F(-) _ 1 (à sap F{-)

iv — 8 (àiv) Fap , fiv — 8 (àiv) Fap . (A.9)

Finally, we define the (anti-)self-dual't Hooft symbols by decomposing the sigma-matrices:

(àiv)ap - inCv(Tc)ap, (àiv)ap - inCiv(Tc)aP. (A.10)

The notation we use closely follows [8]. Self-dual spin-fields of S0(4)ST are denoted Sa and the anti-self-dual ones Sa. In this notation, the graviphoton field G is anti-self-dual and the S' self-dual.

A.2. Correlators

We summarise here the main correlation functions needed for the calculation of the disc diagrams encountered in the text:

{V(Z1)V(Z2)) = z—2 ,

{Â(Z1)A(Z2)) = -Z—21/2,

{e—,(zi)e— 2 ^(Z2)e— 2 ,(Z3)) = Z—21/2Z—31/2 Z—31/4,

{*(Z1)SA(Z2)SB(Z3)) = —iA Z—21/2Z—31/2Z—31/4,

[fI(Zi)fv(Z2)fp(Z3)fà(Z4]) = 8Iv8pàZ—21Z—41 — 8IP 8Và Z—31 Z— + S^S^Z^Z-1, Ssa Z1)Sp (Z2 S (Z3)s8 (Z4)) = ^ ^8 ^ 8^Z/2Z34,

(Z12Z13Z14Z23Z24Z34 )1/2

{fI(Z1)r(Z2)Sa(Z3)Sp(Z4))

= Z13 Z24 + Z23Z14 8iveap__Z34_(à IV )ap (A11)

2 z 12(z 13z14z23z24z34)1/2 2 (z13z14z23z24)1/2

Here we have introduced the shorthand notation zij = zi - zj.

Appendix B. Picture independence of disc amplitudes

In this appendix we repeat the disc computations performed in Section 3.1 making use of physical vertex operators only.

B.1. Picture dependence of disc diagrams

The auxiliary fields introduced in (3.6)-(3.8) are not physical and, therefore, they do not lie in the BRST cohomology of the theory. Inserting such fields as vertex operators into string scattering amplitudes typically leads to an unphysical dependence on the ghost-picture [21], thus rendering the result ambiguous. To illustrate this point, consider, for instance, the unmixed diagram (3.14) and focus, for simplicity, on the NS-NS part of bulk vertex operator Vp-Si which we now insert in a different picture:

VNf a Ppfa (e-vyfa(y)& (y)fp(y) + e-*Wfp(y)fa(y )V (y)). (B.1)

The correlator can be calculated as before:

Yt,avPM€v Y tavPveM

DNS -s, (z1,Z2,y,y) a--^-2---^-2. (B.2)

YTaF- (Z1 -z )2|Z2 - z|2 (Z2 -Z )21Z1 - z|2

- z )2|Z2 - z|:

Including the factor (3.15), setting zi ^ro, y ^ i and integrating over z2 then yields a different result than (3.21):

«VytVaVFs'))ns a YlavPMev. (B.3)

Notice that only the first term in (B.2) gives a non-vanishing contribution. The result indicates that the longitudinal modes do not decouple and is due to the fact that the auxiliary field vertex operator is not annihilated by the string BRST operator.

B.2. Physical amplitudes

In order to obtain an unambiguous answer, we calculate below the disc amplitudes without auxiliary fields and show how the results of Section 3.1 can be recovered in an unambiguous fashion. In this way, any effect stemming from integrating out all the auxiliary fields of the theory is effectively taken into account.

A crucial requirement for all our scattering amplitudes is that the external fields involved be physical, i.e. that they be annihilated by the BRST operator Q = § JBRST. The BRST-current can be written in the form

JbRST = c(^matter + Tsuperghost) + cdcb + Y T^ter - Y2b, (B.4)

where the energy momentum tensors of the matter and superghost sectors, as well as their fermionic partners, take the form

1 r 1 I

Tmatter = -dX dXr - ^f dfi,

Tsuperghost = -^Ydft - ^dYP,

Tmatter = ifrdXr. (B.5)

Here we are using the following OPEs for the worldsheet and ghost fields:

i j i j dXr(x)dXJ(y) = ---^, f (x)fJ(y) =

(x - y)2' x - y'

P(x)Y(y) =--, b(x)c(y) =

x - y x - y

V "-----' \r

Fig. 3. Four-point disc diagram with bulk-insertion of the S field strength tensor and three boundary insertions stemming from the 5-5 sector of the string setup.

As we show below, the physical amplitudes take the form of contact terms pi .pj/pi.pj where pi are the momenta of the various vertex operator insertions. These contact terms give non-trivial results in the limit pi ^ 0. To be able to compute them in a well-defined manner, we keep pi generic in all intermediate steps, which also acts as a regularisation of the worldsheet integrals, and take the limit only at the end of the calculation. However, due to the nature of our vertex insertions, we cannot switch on momenta in an arbitrary fashion: since the four-dimensional space-time corresponds to directions with Dirichlet boundary conditions for the D5-instantons, none of the ADHM moduli can carry momenta along XSimilarly, the S'-vector insertions cannot carry momenta along T2 once we impose BRST invariance (i.e. transversality and decoupling of longitudinal modes). As a way out, we take all vertices to carry momenta along the K3 directions and complexify them, if necessary, to make all integrals well-defined. In fact, technically, we first replace K3 by R4 and compute an effective action term on the D-instanton world-volume T2 x R4. Since the relevant fields that appear in these couplings survive the orbifold projection (or more generally on a smooth K3 manifold they give rise to zero-modes), the corresponding couplings exist also in the case where R4 is replaced by K3.

We consider the physical four-point amplitude depicted in Fig. 3, which is of the form

D DNS t S/+ DR t S/, (B.6)

aaj ' FS aaxtFS aaxtFS v 7

DN?xtF*= (v(0)(Xi)Vt1)(x2)V(0)(x3)ViF-h-1)iz,z)VPco(y)), (B.7)

DRaxtF, = (Va(0)(xi)Va(-1)(x2)V;;0)(x3)VF-/1-1 \z,z)). (B.8)

Here V(-,1,-1) is the NS-NS part of the closed string FS' field strength tensor introduced in

(-1 -1)

(3.10), V^'' 2 is its R-R part and VPCO is the picture changing operator.8 Upon bosonising the (P, y) ghost system, introduced in (B.4), in terms of anti-commuting fields (n,H) of dimension (1, 0), as well as a scalar $ with background charge = -1/2((d$)2 + 2d2$),

P = e-(pd%, y = eVn, (B.9)

8 For the sake of clarity, we have explicitly denoted the ghost-picture of every vertex operator.

we can write VPCO = Q%. In (B.6) we have kept the PCO insertion at a fixed position y, even though the final result should not depend on y [21]. Setting y to z, Z or x2, converts the ghost picture of the corresponding vertex operators to9

CtF-'U = (Vi0)(x1 )Va(-1)(x2)V;(0)(x3)VF-;-1)(z,Z)), (B.10)

CtF-'U = (Vi0)(x1 )Va(-1)(x2)V;(0)(x3)VF-:1,0)(z,Z)), (B.11)

DNS^U = (Va(0)(x1 )VP(x2)V(0)(x3)V^-1)(z,z)), (B.12)

respectively. Using the doubling trick, we can convert the disc into the full plane with a Z2-involution, and the four-point amplitude (B.6) becomes a five-point function with vertices at (x1,x2,x3,z, Z). Here we split

V(p:h-1)(z,-z) = VpS, (z) VpS, (Z), (B.13)

where the left-right symmetrization is implicit. SL(2, R) invariance implies that we can fix three real positions, which is related to the existence of three c-ghost zero modes on the sphere. The latter are soaked up by attaching c to three dimension one vertices such that the resulting operators are BRST closed. The dimension of these vertices becomes zero and they remain unintegrated. Since the last two terms in (B.4) annihilate any operator in the (-1)-picture, any physical operator with dimension one and negative ghost picture becomes BRST invariant in this manner.10 However, when c is attached to a vertex V(0) in the zero-picture, the last two terms in (B.4) do not annihilate cV(0). In this case, the correct dimension zero BRST invariant combination is cV(0) + y V(-1). Therefore, for simplicity, we choose the zero-picture vertices to be of dimension one (such that their positions are integrated), and all the (-1)-picture vertices to be of dimension zero (such that their positions remain unintegrated).

Let us first consider the NS-NS contributions (B.7), for which the vertex operators are

Va(x1) = g6 a»(dXM - 2ip1 ■ xfM)e2ip1 'Y(x1), (B.14)

Va(x2) = g6avce-Vfve2ip2-Y(x2), (B.15) X t

Vx t (x3) = ^(dZ - 2ip3 ■ X^)e2lp3Y (x3), (B.16)

VFs, (z) = ce-(pVei(P»X>l+PY)(z), (B.17)

VF-S, (Z) =--~~7= ce-<pf^ei(-PMXM+P (Z), (B.18)

with F-v = e[MPv], and the only relevant terms in VPCO are (since the total background charge of the superghost is -2)

e^TmFatter(y) = iev (f^X» + VdZ + VdZ + x'dY' )(y). (B.19)

Here, Yi e {X6,X7,X8,X9}

parametrise the internal R4 (which we eventually replace by K3). The momenta pi are along these directions, while the momentum of VF-> is written as (PM,P),

9 In (B.6) an insertion of £ at an arbitrary position is understood, in order to soak up the £ zero mode.

10 Indeed, the first two terms in (B.4) combined together annihilate cV for any V corresponding to a dimension one Virasoro primary operator, irrespective of the ghost picture of V.

where Pis the space-time part and P is along the Y' directions. Note that after using the doubling trick, the Neumann directions (Z, Z, Y') are mapped onto themselves, whereas the Dirichlet ones pick an additional minus sign XM ^ — XM. This is consistent with the fact that the momenta along Neumann directions are conserved J2' P' + P = 0, which follows from integrating the zero modes of Y'. On the other hand, integrating over the zero modes of the Dirichlet directions X^ does not give any conservation law for PM.

The three open string vertices contain Chan-Paton labels which need to be suitably ordered. For instance, if we are interested in computing the term Tr(a^avxt), the range of integration is the following:

{for xi > x2, x3 e ]x2,xi [,

1 3 1 (B.20)

for X2 > xi, X3 e]— cc, xi[U]x2, c[.

For the other inequivalent ordering Tr(aMxtav), the range of the x3-integration is opposite. It is easy to show that the sum of these two orderings vanishes so that the amplitude is of the form Tr(aM[x f,av ]).

For definiteness, let us focus on the term Tr(a^avxt). The contraction of y and c and the contraction of the exponentials in momenta yield

Ao =j — in Tr(a^ av x|y — z|2(y — x2)

x n (xi — xj^U ixk — zi4pk'P(z — Z)-^™. (B.21)

1<i<j<3 k=1

This is a common factor that multiplies each of the remaining contractions. Now let us consider the contribution of dZ(x3) to the amplitude. This must contract with dZ(y) in VPCO and then №(y) contracts with V(z). Then fx(z) necessarily contracts with fv(x2) and from x1 only dX<x(x1) can contribute. The result is

iSvkP^(z — Z) (y — x3)2 (y — z)(x2 — Z)|x1 — z|2 Next consider the contribution of the second term in (B.16). Here, there are two separate contributions. If p3 ■ x(x3) contracts with p1 ■ x(x1), then f^(x1), fv(x2), fx(z) and a space-time fermion fa (y) from the picture changing operator must contract, leaving dXa(y) which can only contract with the momentum parts of vertices at z and z resulting in a term proportional to Pa. Notice that the term arising from the contraction of ^^ with is killed by the transver-sality condition (a necessary condition for the operator to be in the kernel of Q). The total result is

A1 =-5---r-2. (B.22)

4ip1 ■ P3 (z — z)

(x3 — z)(x3 — x1)iy — zi

(B.23)

(x2 —z)(y — x1) (x1 —z)(y — x2)_ On the other hand, if p3 ■ x(x3) contracts with x(y) in VPCO, then dY(y) contracts with momentum dependent parts of the vertices. Thus, fx(z) must contract with fv(x2) and only dX<x at x1 can contribute and one obtains

4iSvXP^(z — z)

(x3 — z)(y — x3)(x2 — z)|x1 — z|2

m ■ P 1

(B.24)

p3 ■ p1 + p3 ■ p2 + p3 ■ P + p3 ■ P

y — x1 y — x2 2(y — z) 2(y — Z)

The total correlation function is thus

VN _S, = Ao(Ai + A2 + A3), (B.25)

which must be integrated over x1 and x3. Note that all the terms in A1, A2 and A3 come with one power of space-time momentum P1 which is exactly what is required to obtain a coupling to the field strength of the closed string gauge field. However, both A2 and A3 are quadratic in the momenta along the Y' directions and they can only contribute to the amplitude in the zero-momentum limit if the integration over x1 and x3 gives a pole of the form 1/(pa ■ pb). Clearly, A0 ■ A3 cannot provide such a pole (we are assuming a generic value of y in the complex plane, i.e. Im(y) = 0). On the other hand, the integral over x3 for A0■ A2 gives a pole of the form 1 /(p1 ■ p3). Performing the x3 integral in both the regions (B.20) yields precisely the same result, hence, the x1 integral over the entire real line reads

AoA2 = — Tr\aßavx dx1 -ÏZ-ÎLe, X " P^ - P^

0 2 LMv^y 1 xi - zi2,_(X2-z)(y - xi) _

(B.26)

where we have set all the momenta along the Y' directions to zero since there are no singularities in the remaining x1 integral. Notice that A0 ■ A2 alone does not lead to a gauge invariant answer. As for the A0A1 term, the x3 and x1 integrals have no singularities and therefore the momenta along the Y directions can be set to zero. The resulting x3 integral for both regions (B.20) gives

precisely the same result:

/1/1 ^ r tl /" J (z -z) (y -z)(x1 - x2) „uevx

A0A1 = --6Tr\alxavx 11 dxi--ek--—--PlSVA. (B.27)

J |xi - z|2 (x2 -z)(y - xi)

Adding the two terms (B.26) and (B.27), we see that the result is gauge invariant. Performing the x1 integration yields11:

{{VaVaVxvNI ))= -2' Tr^x t]ek(PlSvl - PVSl1). (B.28)

Finally, let us consider the R-R contributions (B.8). The vertex operators are the same as above, except for the S'-vector part which is given by

V- -2\z, z) = PPce-2S,SAe'(PX+PY)(z)

x eAB (a $ ce-2 S? SB e'(-P X+P Y (z). (B.29)

Since the total superghost charge of the vertices is -2, there is no need for a picture changing operator. The total charge in the torus plane implies that only p3 ■ x&(x3) in (B.16) contributes so that only p1 ■ xfl(x1) in (B.14) contributes. This term is proportional to p1 ■ p3. Once again the integral over x3 gives a pole 1/(p1 ■ p3) in the channel x3 ^ x1. Performing the integrals over x1 and x3 as above leads to the same result:

{{Va Va Vx VN)) = {{Va Va Vx VR,)). (B.30)

11 Notice the additional factor of 2 due to the left-right symmetrization in the closed string vertex.

(1) (2) (3)

Fig. 4. Factorisation channels of the disc. The diagram at the top illustrates the four-point function of physical vertices with an insertion of a PCO. The diagrams at the bottom depict various choices for the position of the latter. In case (1), the result takes the form of a contact term, leading to an effective auxiliary field vertex operator insertion on the boundary of the disc. Similarly, case (2) can be re-expressed as a sum of two contact terms, while in case (3) no such interpretation is possible.

In order to compare with the result obtained with the auxiliary vertices, we sum over the inequiv-alent orderings of the open vertex operators, which yields

« VaVaVxVFs,)) = — 4i Tr[x ^a^avF'f. (B.31)

Consequently, the use of auxiliary fields with the picture prescription of Section 3.1 leads to the correct physical results.

B.3. Channel factorisation and auxiliary fields

Another way to see that the use of vertex operators ultimately yields the correct physical result is as follows. Consider the NS-NS contribution (B.7) and take the limit where y goes to the points z, z and x2, respectively, corresponding to using different pictures for the associated operators. The three cases below are illustrated in Fig. 4.

1. y ^ z

In this case A0A1 = 0, whereas A0A3 still cannot produce any pole in the momenta. On the other hand, A0A2 simplifies to

4ip1 ^ p3-—-(SvkP^ — 8^Pv). (B.32)

x3 — x1 (x3 — z)(x1 —z)

Notice that the longitudinal mode manifestly decouples. Looking at the vertices (B.14) and (B.16), we recognise that p3 ■ p1/(x3 — x1) appears from contracting p1 ■ x(x1) with p3 ■ x(x3). The pole 1/p1 ■ p3 appears from x3 ^ x1. Therefore, in this limit, the result is effectively reproduced by the OPE of the vertices at x1 and x3, resulting in an effective vertex tyf-v at x1. This is why in the (—1,0)-picture for the NS-NS part of VfS', the auxiliary vertex VYt in (3.6) gives the correct answer.

2. y ^ x2

In this case again A0A1 vanishes. From A0A2, only the kinematic structure S^lPv survives, with the same answer as above. However the p3 ■ p2/(x3 — x2) factor in A0A3 contributes to the other kinematic structure, SvlP,x. The final result is of course the same but the total result comes from two different factorisation limits x3 ^ x1 and x3 ^ x2 for the two different kinematic structures.

3. y ^ z

In this case A0A3 vanishes, but both the remaining terms contribute. In particular it is not clear if the A0A1 term can even be thought of as a contact term.

Hence, only for y ^z can the result be understood as the factorisation in a single channel, such that it can be effectively reproduced by replacing Va(x1) and Vx(x3) by their OPE, which is simply the auxiliary vertex VYt used in Section 3.1.

Finally, in the R-R contributions (B.8), the 'contact term' appears only from the channel x3 ^ x1 and the result can be obtained by a three-point function involving the vertices at x2, z, z and an auxiliary vertex at x1. The analysis of the mixed D5-D9 diagrams is very similar and leads to the same conclusion, that is, in the (—1, 0)-picture for the NS-NS part and (—1/2, — 1/2)-picture for the RR part of VFs<, the entire result comes from contact terms in a single channel and the result can be reproduced by a three-point function involving the auxiliary vertex VXt (3.8).

As mentioned above, even though the calculations are performed on the target space R4 x T2 x R4, the couplings we have obtained are non-vanishing for non-trivial momenta only along the first R4 (space-time). When we compactify R4 x T2 x R4 ^ R4 x T2 x K3, these couplings are unchanged up to possible a' corrections. However, the latter are irrelevant in the field theory limit that we take in order to compare with the non-perturbative part of the Q -deformed gauge theory partition function. In addition, we have focused here on the gauge theory coming from D9-branes for which the relevant D-instanton is the D5-brane wrapping the internal space. However, it is straightforward to extend our calculation to other setups by applying T-duality. For example, for a gauge theory realised by D5-branes wrapping T2, the relevant D-instanton is the D1-brane wrapped on T2. The corresponding couplings can be obtained from the above calculations by performing four T-dualities along the Y' directions.

References

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