On the D(–1)/D7-brane systems

M. Billò$\,{}^{a}$, M. Frau$\,{}^{a}$, F. Fucito$\,{}^{b}$, L.
Gallot$\,{}^{c}$,
A. Lerda$\,{}^{d}$ and J.F. Morales$\,{}^{b}$

a Università di Torino, Dipartimento di Fisica,

and I.N.F.N. - sezione di Torino,

Via P. Giuria 1, I-10125 Torino, Italy

b I.N.F.N. - sezione di Roma Tor Vergata

and Università di Roma Tor Vergata, Dipartimento di Fisica

Via della Ricerca Scientifica, I-00133 Roma, Italy,

c Laboratoire d’Annecy-le-vieux de Physique Théorique LAPTh

Univ. Grenoble Alpes, Univ. Savoie Mont-Blanc and CNRS,

F-74000, Annecy, France

d Università del Piemonte Orientale,

Dipartimento di Scienze e Innovazione Tecnologica

Viale T. Michel 11, I-15121 Alessandria, Italy

and I.N.F.N. - sezione di Torino,

Via P. Giuria 1, I-10125 Torino, Italy

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
We study non-perturbative effects in supersymmetric U($N$) gauge theories in
eight dimensions realized by means of D(–1)/D7-brane systems with non-trivial
world-volume fluxes turned on. Using an explicit string construction in terms
of vertex operators, we derive the action for the open strings ending on the
D(–1)-branes and exhibit its BRST structure. The space of vacua for these open
strings is shown to be in correspondence with the moduli space of generalized
ADHM gauge connections which trigger the non-perturbative corrections in the
eight-dimensional theory. These corrections are computed via localization and
turn out to depend on the curved background used to localize the integrals on
the instanton moduli space, and vanish in flat space. Finally, we show that
for specific choices of the background the instanton partition functions
reduce to weighted sums of the solid partitions of the integers.

Keywords: $\mathcal{N}=2$ SYM theories, instantons, D-branes

###### Contents

  1. 1 Introduction
  2. 2 The D7-brane system and its open strings
    1. 2.1 The D7/D7′ and D7′/D7 strings
    2. 2.2 The no-force condition
  3. 3 Adding D–instantons
    1. 3.1 The D(–1)/D(–1) strings
    2. 3.2 The D(–1)/D7 and D7/D(–1) strings
    3. 3.3 The D(–1)/D7′ and D7′/D(–1) strings
    4. 3.4 The no-force condition
  4. 4 The instanton moduli and their action
    1. 4.1 The moduli action
    2. 4.2 Introducing vacuum expectation values and the $\varepsilon$-background
    3. 4.3 The BRST structure of the moduli spectrum
  5. 5 The instanton partition function
    1. 5.1 The $k=1$ case
    2. 5.2 Explicit results at higher instanton numbers
  6. 6 ADHM construction
  7. 7 Conclusions
  8. A Notations and conventions
  9. B Technical details

##  1 Introduction

Non-perturbative effects in gauge theories can be rephrased in the language of
string theory by considering the effects induced by branes of lower dimensions
distributed along the world-volume of branes of higher dimensions. The prime
example of this construction is represented by the D(–1)/D3-brane system which
realizes the ADHM moduli space of gauge instantons in four-dimensional gauge
theories [1, 2, 3, 4, 5, 6, 7]. The massless sector of the open strings with
at least one end-point on the D(–1)-branes accounts for the moduli of the
gauge instanton solutions in four dimensions, and their effective action can
be recovered by computing string scattering amplitudes on disk diagrams
connecting the two stacks of branes [3, 4, 5, 6].

This analysis can be generalized to other brane systems. In particular, new
stringy non-perturbative corrections to four-dimensional gauge theories can be
obtained by adding Euclidean D3-branes wrapping non-trivial cycles of the
internal space. When orientifold planes are inserted, these exotic instanton
configurations [8, 9, 10, 11, 12, 13, 14, 15] can generate non-perturbative
effects in the effective action which are prohibited in perturbation theory,
like for instance certain Majorana mass terms or Yukawa couplings, which may
be relevant for phenomenological applications.

These methods can also be used to study non-perturbative effects in eight-
dimensional field theories by considering systems of D7-branes in presence of
D-instantons. Indeed, the prepotential generated by exotic instantons in
O($N$) gauge theories in eight dimensions has been studied in [16, 17] by
means of a system made of D(–1) and D7-branes on top of an orientifold
O7-plane, allowing to explicitly test the non-perturbative heterotic/Type I
string duality [18]. More general D$p$/D$p^{\prime}$ systems can be also
considered after turning on fluxes for the Neveu-Schwarz $B$-field, in such a
way that supersymmetry is restored in the vacuum [1, 19].

The peculiarity of all these exotic or higher dimensional instanton systems is
that they lack the bosonic moduli connected to the size and the gauge
orientation of the instanton configurations. Moreover, they possess extra
fermionic zero-modes, in addition to those connected with the broken
supersymmetries, which can make their contributions to the effective action
vanish if they are not properly lifted or removed. The problems related to the
presence of these extra fermionic zero-modes can be cured in various ways, for
example by adding orientifold or orbifold projections, or more generically by
considering curved backgrounds. However, in all these cases the non-
perturbative configurations always remain point-like since no bosonic moduli
can account for a non-zero size. The fundamental reason for this is that in
all these exotic configurations there are eight directions with mixed
Dirichlet/Neumann boundary conditions, differently from the standard cases in
which there are only four mixed directions.

The non-perturbative contributions to the gauge theory effective action are
obtained after an explicit evaluation of the instanton partition functions
which are expressed as integrals over the instanton moduli space. This is
possible thanks to the localization techniques introduced in [20] for the
$\mathcal{N}=2$ super Yang-Mills theories in four dimensions, building also on
previous results in [21]. In the end, the instanton partition functions are
reduced to integrals of rational functions which can be performed using
standard complex analysis methods. Actually, the poles of these rational
functions that contribute to the integrals can be put in one-to-one
correspondence with sets of Young tableaux which in turn are related to the
partitions of the integer numbers [20, 22, 23, 24, 25, 26]. When this approach
is used for gauge theories in dimensions higher than four, higher dimensional
Young tableaux appear and a connection with the planar and solid partitions of
the integers arises [24, 27]. The instanton partition functions are thus
expressed as weighted sums over integer, plane or solid partitions. Therefore,
studying these systems may be interesting not only for the string theory
applications we mentioned above, but also for mathematical reasons, related
directly to such weighted sums (the generating function of solid partitions is
not yet known in closed form) but also, for instance, in the study of
Donaldson-Thomas invariants of Calabi-Yau four-folds [28].

Recently, the instanton partition functions of a pure supersymmetric U($N$)
gauge theory in nine dimensions and of its conformal extension with a U($N$)
flavor symmetry have been analyzed from this point of view in [27, 29] and
shown to be given by a simple all-instanton plethystic exponential formula.
More recently, these systems have been generalized to non-conformal set-ups
with a U($M$) flavor symmetry in [30] where also the instanton contributions
to the chiral correlators have been computed. In all cases it turns out that
the non-perturbative sectors are described by sets of moduli that share some
features of both the standard and exotic instantons. Thus, in [27, 29] it has
been conjectured that they might arise from the open strings of a D0/D8-brane
system with a Neveu-Schwarz $B$-field and with anti-D8 branes in the
background.

The aim of this paper is to provide an explicit string theory derivation of
the moduli space and of the instanton partition functions studied in [27, 29].
We consider a D(–1)/D7-brane system in Type II B string theory and do not
introduce any anti-branes which would make the configuration unstable, but
instead turn on a magnetic flux along the world-volume of the D7-branes
111Notice that this magnetic flux is an open string background and thus is not
represented by a Neveu-Schwarz $B$-field which belongs to the closed string
sector.. More precisely, we start from a stack of $(N+M)$ D7-branes to
describe a gauge theory in eight dimensions; then we introduce a constant
magnetic field on the first $N$ D7-branes to break the gauge group to
$\mathrm{U}(N)\times\mathrm{U}(M)$, maintaining stability and supersymmetry.
Finally, we add $k$ D(–1)-branes. The open strings with at least one end-point
on the D-instantons describe the moduli space of the non-perturbative
configurations. The spectrum of the open strings that start and end on the
D(–1)-branes is standard and includes, among others, the bosonic moduli
describing the positions of the instantonic branes in the world-volume of the
seven-branes. The spectrum of the mixed strings stretching between the D(1)
and the D7-branes is instead peculiar because, despite the presence of eight
directions with mixed Dirichlet/Neumann boundary conditions, due to the
presence of the magnetic flux, it comprises a set of bosonic moduli that can
be associated to the size and orientations of the instanton configurations in
eight dimensions.

We also provide a detailed analysis of the vertex operators associated to all
moduli and use them to derive the effective instanton action from disk
amplitudes. After introducing vacuum expectation values for the scalar fields
on the D7-branes and turning on a closed-string background with Ramond-Ramond
fluxes, which is known to mimic the so-called $\Omega$-background in the
moduli space [31], we compute the instanton partition function using
localization. Our results agree with those in [27, 29]; in particular for a
suitable choice of the background the instanton partition function reduces to
a weighted sum over the solid partitions of the integers. Our derivation also
suggests a possible consistent generalization of the results of [27, 29] in
which the $\Omega$ background is less constrained; we will investigate this
possibility in a future work.

Finally, we show that the moduli space of vacua of the matrix theory defined
on the D(–1)-branes is compatible with an ADHM construction of instanton
connections in eight dimensions, which has also been recently discussed in
[32]. In the concluding section we comment on the significance of our results
for the eight-dimensional gauge theory, which receives non-perturbative
corrections from these instanton configurations only in its U(1) part and only
in curved space. Our notations and conventions, together with some more
technical material, are collected in the appendices.

##  2 The D7-brane system and its open strings

In Type II B string theory, we consider a stack of $(N+M)$ D7-branes aligned
along the directions $\mu,\nu,\ldots=1,\ldots,8$ (with Euclidean signature).
The directions 9 and 10 are, instead, transverse.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---
$-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $*$ | $*$
Table 1: The D7-branes are aligned along the first eight directions.

On the world-volume of these D7-branes there is an eight-dimensional gauge
theory with group U($N+M)$ and sixteen supercharges. The part of the effective
action which only depends on the gauge field strength can be written as

$S_{\mathrm{D7}}=S_{2}+S_{4}+\cdots~{}.$ (2.1)

Here $S_{2}$ is the quadratic Yang-Mills action in eight dimensions

$S_{2}=\frac{1}{2g_{\mathrm{YM}}^{2}}\int\\!d^{8}x~{}\mathrm{tr}\big{(}F^{2}\big{)}$
(2.2)

with a dimensionful gauge coupling constant

${g_{\mathrm{YM}}^{2}}\equiv{4\pi g_{s}(2\pi\sqrt{\alpha^{\prime}})^{4}}~{},$
(2.3)

($g_{s}$ is the string coupling and $\sqrt{\alpha^{\prime}}$ is the string
length), while $S_{4}$ is a quartic action of the form

$\displaystyle
S_{4}=-\frac{1}{4!\lambda^{4}}\int\\!d^{8}x~{}\mathrm{\operatorname{tr}}\big{(}t_{8}\,F^{4}\big{)}-\frac{\mathrm{i}\,\vartheta}{4!(2\pi)^{4}}\int\\!d^{8}x~{}\mathrm{\operatorname{tr}}\big{(}F\wedge
F\wedge F\wedge F\big{)}$ (2.4)

where

${\lambda^{4}}\equiv{4\pi^{3}g_{s}}$ (2.5)

is a dimensionless coupling, and $\vartheta$ is the vacuum angle, which in
string theory is identified with the scalar field $C_{0}$ of the RR sector
according to

$\vartheta=2\pi\,C_{0}~{}.$ (2.6)

The eight-index tensor $t_{8}$ appearing in the first term of (2.4) is such
that [33] 222For an explicit definition of $t_{8}$, see for instance Appendix
B of [34].

$\displaystyle\mathrm{tr}\big{(}t_{8}F^{4}\big{)}\\!$
$\displaystyle\\!\equiv\\!$
$\displaystyle\\!\frac{1}{16}\,t_{8}^{\mu_{1}\mu_{2}\cdots\mu_{7}\mu_{8}}\,\mathrm{tr}\big{(}F_{\mu_{1}\mu_{2}}\cdots
F_{\mu_{7}\mu_{8}}\big{)}$ $\displaystyle\\!=\\!$
$\displaystyle\mathrm{tr}\Big{(}F_{\mu\nu}F^{\nu\rho}F^{\lambda\mu}F_{\rho\lambda}+\frac{1}{2}\,F_{\mu\nu}F^{\rho\nu}F_{\rho\lambda}F^{\mu\lambda}-\frac{1}{4}\,F_{\mu\nu}F^{\mu\nu}F_{\rho\lambda}F^{\rho\lambda}-\frac{1}{8}\,F_{\mu\nu}F_{\rho\lambda}F^{\mu\nu}F^{\rho\lambda}\Big{)}~{}.$

Finally, the ellipses in (2.1) stand for $\alpha^{\prime}$ corrections
containing at least five field strengths or their covariant derivatives.

The gauge field strength $F_{\mu\nu}$ is actually part of a scalar superfield
$\Phi(x,\theta)$ in eight dimensions, defined as

$\Phi(x,\theta)=\phi(x)+\sqrt{2}\,\theta\Lambda(x)+\frac{1}{2}\,\theta\sigma^{\mu\nu}\theta\,F_{\mu\nu}(x)+\ldots$
(2.8)

where $\theta$ is the fermionic superspace coordinate, $\phi$ is a complex
scalar and $\Lambda$ is an eight-dimensional chiral fermion 333See Appendix A
for our conventions on spinors and Dirac matrices.. In terms of the superfield
$\Phi$, the quartic action (2.4) (plus its supersymmetric completion) can be
written as

$S_{4}=\tau\int\\!d^{8}x\,d^{8}\theta~{}\mathrm{tr}\big{(}\Phi^{4}\big{)}+\text{c.c.}$
(2.9)

where $\tau$ is the complexified string coupling

$\tau=C_{0}+\frac{\mathrm{i}}{g_{s}}~{}.$ (2.10)

In principle the quartic action (2.9) can receive quantum corrections and
takes the general form

$S_{4}^{\prime}=\int
d^{8}x\,d^{8}\theta\,\mathcal{F}(\Phi,\tau)+\mathrm{c.c.}$ (2.11)

where $\mathcal{F}(\Phi,\tau)$ is the (holomorphic) prepotential.

We now introduce a (constant) background field on the first $N$ D7-branes. The
other $M$ D7-branes, which remain without background field, will be called
from now on D7′-branes. In this way the initial gauge symmetry group U($N+M$)
is broken to U($N)$ $\times$ U($M$). In particular, we consider a background
that corresponds to a constant flux on the D7-brane world-volume described by

$2\pi\alpha^{\prime}\,F^{(0)}=\begin{pmatrix}0&+f_{1}&0&0&0&0&0&0\\\
-f_{1}&0&0&0&0&0&0&0\\\ 0&0&0&+f_{2}&0&0&0&0\\\ 0&0&-f_{2}&0&0&0&0&0\\\
0&0&0&0&0&+f_{3}&0&0\\\ 0&0&0&0&-f_{3}&0&0&0\\\ 0&0&0&0&0&0&0&+f_{4}\\\
0&0&0&0&0&0&-f_{4}&0\end{pmatrix}~{}\mathbb{1}_{N\times N}~{}.$ (2.12)

In the following we will specify the four parameters $f_{1},\ldots,f_{4}$ in
such a way to preserve some supersymmetry, but for the time being we may
consider them as arbitrary 444We notice that the flux breaks Lorentz
invariance in the eight-dimensional space, so that the preserved supersymmetry
can be also viewed as a two-dimensional supersymmetry in the transverse
space.. The background (2.12) gives rise to four different types of open
strings, as represented in Fig. 1: the 7/7 strings starting and ending on the
$N$ D7-branes with fluxes, the 7′/7′ strings starting and ending on the $M$
D7′-branes without fluxes, and the 7/7′ or 7′/7 strings which start and end on
branes of different type. The 7/7 and 7′/7′ strings are untwisted and contain
exactly the same physical states of the original strings. In particular they
give rise to the same effective action (2.1) for the group U($N$) (in presence
of the background (2.12)) and for the group U($M$), respectively. The 7/7′ or
7′/7 strings, instead, are twisted and their spectrum is completely different,
depending on the values of the fluxes.

$\ldots$
---
$\ldots$
---
$N$ D7-branes
---
$M$ D7′-branes
---
Figure 1:  A background flux is turned on on the $N$ D7 branes. The D7$/$D7
and D7′/D7$\prime$ open strings are untwisted, the mixed ones (depicted as
dashed) are instead twisted.

###  2.1 The D7/D7′ and D7′/D7 strings

To analyze the spectrum of the D7/D7′ and D7′/D7 open strings, it is
convenient to group the space-time coordinates in pairs and introduce the
complex combinations

$z^{I}=\frac{x^{2I-1}+\mathrm{i}\,x^{2I}}{\sqrt{2}}\quad\mbox{and}\quad\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu_{I}=\frac{x^{2I-1}-\mathrm{i}\,x^{2I}}{\sqrt{2}}$ (2.13)

for $I=1,2,3,4,5$. The directions with $I=1,2,3,4$ are longitudinal to branes,
while the direction with $I=5$ is transverse. In this complex notation, the
background (2.12) implies that in the 7/7′ sector, the complex string
coordinates $\big{\\{}Z^{I}(z),\mkern
1.5mu\overline{\mkern-1.5muZ\mkern-1.5mu}\mkern 1.5mu_{I}(z)\big{\\}}$ and
their fermionic partners $\big{\\{}\Psi^{I}(z),\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern 1.5mu_{I}(z)\big{\\}}$ with
$I=1,2,3,4$ are twisted with a twist parameter $\theta_{I}$ given by 555In the
7′/7 sector we have to send $f_{I}$ to $-f_{I}$.

$\mathrm{e}^{2\pi\mathrm{i}\,\theta_{I}}=\frac{1-\mathrm{i}\,f_{I}}{1+\mathrm{i}\,f_{I}}~{}.$
(2.14)

This means that they have the following monodromy properties around the origin
of the world-sheet:

$Z^{I}(\mathrm{e}^{2\pi\mathrm{i}}z)=\mathrm{e}^{2\pi\mathrm{i}\,\theta_{I}}\,Z^{I}(z)~{},\qquad\mkern
1.5mu\overline{\mkern-1.5muZ\mkern-1.5mu}\mkern
1.5mu_{I}(\mathrm{e}^{2\pi\mathrm{i}}z)=\mathrm{e}^{-2\pi\mathrm{i}\,\theta_{I}}\,\mkern
1.5mu\overline{\mkern-1.5muZ\mkern-1.5mu}\mkern 1.5mu_{I}(z)~{},$ (2.15)

and

$\Psi^{I}(\mathrm{e}^{2\pi\mathrm{i}}z)=\pm\,\mathrm{e}^{2\pi\mathrm{i}\,\theta_{I}}\,\Psi^{I}(z)~{},\qquad\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern
1.5mu_{I}(\mathrm{e}^{2\pi\mathrm{i}}z)=\pm\,\mathrm{e}^{-2\pi\mathrm{i}\,\theta_{I}}\,\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern 1.5mu_{I}(z)~{}.$ (2.16)

Here $z$ is a point in the upper half complex plane and the sign in (2.16) is
$+$ for the Neveu-Schwarz (NS) boundary conditions and $-$ for the Ramond (R)
ones. In [35], the bosonic and fermionic conformal field theory for generic
twist parameters has been worked out in detail and we summarize the most
relevant features in Appendix B. Here we just recall that the monodromy
properties (2.15) of the bosonic coordinates do not allow for the existence of
zero-modes associated to momentum in these directions.

The bosonic and fermionic coordinates $\big{\\{}Z^{5}(z),\mkern
1.5mu\overline{\mkern-1.5muZ\mkern-1.5mu}\mkern 1.5mu_{5}(z)\big{\\}}$ and
$\big{\\{}\Psi^{5}(z),\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern 1.5mu_{5}(z)\big{\\}}$ are
instead untwisted. They have the standard mode expansion of untwisted fields,
but with Dirichlet/Dirichlet boundary conditions since they are transverse to
both type of branes. Due to these boundary conditions there are no bosonic
zero-modes in these directions either. Thus, when all four $\theta_{I}$’s are
different from zero, the momentum is not defined in any direction, and the
corresponding string excitations represent non-dynamical degrees of freedom.

As mentioned above, we are interested in supersymmetric configurations. This
means that the background (2.12) must be such that

$\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}\in\mathbb{Z}~{}.$ (2.17)

Of course, if $\theta_{I}=0$ for all $I$, the background vanishes and all
sixteen supercharges are conserved on the branes. If two of the twists are
zero, the system preserves eight supercharges while if only one of the
$\theta_{I}$’s vanishes the system has four conserved supercharges. If all
twists are non-zero and satisfy (2.17), the system preserves the minimal
amount of supersymmetry, namely only two supercharges are conserved. This is
the case we will consider. For simplicity, we assume that
$0<\theta_{I}<\frac{1}{2}$ for all $I$’s 666The reason to exclude the value
$\frac{1}{2}$ is merely to avoid the appearance of fermionic zero-modes in the
NS sector.. With this choice, the supersymmetry condition (2.17) becomes

$\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}=1~{}.$ (2.18)

In this set-up, the condition that selects the physical states is

$\begin{cases}\displaystyle{\widehat{N}=\frac{1}{2}\Big{(}1-\sum_{I=1}^{4}\theta_{I}\Big{)}\qquad\text{in
the NS sector}~{},}\\\\[8.53581pt] \displaystyle{\widehat{N}=0\qquad\text{in
the R sector}~{}.}\end{cases}$ (2.19)

where $\widehat{N}$ is the sum of the bosonic and fermionic number operators
(see (B.6) and (B.8)). Using the supersymmetry condition (2.18), we see that
both the NS and R physical states must actually obey

$\widehat{N}=0~{}.$ (2.20)

Since the number operator is non-negative, this requirement can be satisfied
only by the vacuum.

In the NS sector, the physical vacuum $|\Omega\rangle_{\mathrm{NS}}$ is
related to the $\mathrm{SL}(2,\mathbb{R})$ invariant Fock vacuum $|0\rangle$
by the action of four bosonic twist fields $\sigma^{I}_{\theta_{I}}(z)$ of
conformal dimensions

$h(\sigma^{I}_{\theta_{I}})=\frac{1}{2}\,\theta_{I}(1-\theta_{I})$ (2.21)

and four fermionic twist fields $s^{I}_{\theta_{I}}(z)$ of conformal
dimensions

$h(s_{\theta_{I}})=\frac{1}{2}\,\theta_{I}^{2}$ (2.22)

as follows [36]

$|\Omega\rangle_{\mathrm{NS}}=\lim_{z\to
0}\Big{(}\prod_{I=1}^{4}\sigma^{I}_{\theta_{I}}(z)\,s^{I}_{\theta_{I}}(z)\Big{)}\,\mathrm{e}^{-\varphi(z)}\,|0\rangle~{}.$
(2.23)

Here $\varphi(z)$ is the bosonic field that is used to describe the superghost
system in the bosonized formalism [37] and the coefficient $-1$ in the
exponent indicates that we have chosen the standard $(-1)$-superghost picture.
Using (2.21), (2.22) and the supersymmetry condition (2.18), one can check
that the vertex operator creating the physical NS vacuum out of the Fock
vacuum is a conformal field of total weight 1.

The fermionic twist fields $s^{I}_{\theta_{I}}(z)$ can be written in the
bosonized formalism as

$s^{I}_{\theta_{I}}(z)=\mathrm{e}^{\mathrm{i}\,\theta_{I}\,\varphi^{I}(z)}$
(2.24)

where $\varphi^{I}(z)$ is the field that bosonizes the fermionic string
coordinates $\big{\\{}\Psi^{I}(z),\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern 1.5mu_{I}(z)\big{\\}}$
according to [37]

$\Psi^{I}(z)=\mathrm{e}^{\mathrm{i}\,\varphi^{I}(z)}~{},\qquad\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern
1.5mu_{I}(z)=\mathrm{e}^{-\mathrm{i}\,\varphi^{I}(z)}~{}.$ (2.25)

Of course, analogous bosonization formulas hold also for the untwisted
coordinates $\big{\\{}\Psi^{5}(z),\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern 1.5mu_{5}(z)\big{\\}}$.
Using these expressions, it is easy to realize that the sum of the charges of
the five bosonic fields $\varphi_{I}$ appearing in the vertex operator
corresponding to the NS vacuum minus the superghost charge is an even integer,
namely

$\sum_{I=1}^{4}\theta_{I}+0-(-1)=2~{}.$ (2.26)

Denoting by $F$ this combination of charges, we have

$(-1)^{F}\,|\Omega\rangle_{\mathrm{NS}}=+|\Omega\rangle_{\mathrm{NS}}~{}.$
(2.27)

Let us now consider the R sector. In this case the physical vacuum is created
out of the Fock vacuum by the same bosonic twist fields
$\sigma_{\theta_{I}}(z)$ as before, but in the fermionic part we have the
following twist fields

$s^{I}_{\theta_{I}-\frac{1}{2}}(z)=\mathrm{e}^{\mathrm{i}\,(\theta_{I}-\frac{1}{2})\varphi^{I}(z)}$
(2.28)

for $I=1,\ldots,4$ and the spin field

$S^{\pm}(z)=\mathrm{e}^{\pm\frac{\mathrm{i}}{2}\varphi^{5}(z)}$ (2.29)

in the fifth complex untwisted direction. The presence of this spin field
implies that the R vacuum is actually a two-component SO(2) spinor.
Altogether, working in the standard $(-\frac{1}{2})$-superghost picture, in
the R sector we find two physical vacuum states which in the bosonized
formalism are given by

$|\Omega,\pm\rangle_{\mathrm{R}}=\lim_{z\to
0}\Big{(}\prod_{I=1}^{4}\sigma^{I}_{\theta_{I}}(z)\mathrm{e}^{\mathrm{i}\,(\theta_{I}-\frac{1}{2})\varphi^{I}(z)}\Big{)}\,S^{\pm}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}\,|0\rangle~{}.$
(2.30)

The combination of charges that before we denoted by $F$ takes the value 0 on
$|\Omega,+\rangle_{\mathrm{R}}$ and the value $-1$ on
$|\Omega,-\rangle_{\mathrm{R}}$. Thus we have

$(-1)^{F}\,|\Omega,\pm\rangle_{\mathrm{R}}=\pm|\Omega,\pm\rangle_{\mathrm{R}}~{},$
(2.31)

showing that $(-1)^{F}$ actually measures the chirality of R vacuum.

If we define the GSO projector as

$P_{\mathrm{GSO}}=\frac{1+(-1)^{F}}{2}~{},$ (2.32)

then, using (2.27) and (2.31), we see that it selects the vacuum in the NS
sector and the component with positive chirality in the R sector, realizing in
this way a supersymmetric physical spectrum containing just two states, the
bosonic vacuum $|\Omega\rangle_{\mathrm{NS}}$ and the fermionic vacuum
$|\Omega,+\rangle_{\mathrm{R}}$. We stress once more that these states do not
carry any momentum and thus represent non-dynamical degrees of freedom since
they do not propagate in space. Moreover, they transform in the fundamental of
U($N$) and in the anti-fundamental of U($M$), since they arise from oriented
open strings stretching between the $N$ D7-branes and the $M$ D7′-branes.

We conclude this analysis with a couple of comments. First, the GSO-projection
in the NS appears to be different with respect to the one we are used to in
the standard untwisted case when one removes the (tachyonic) vacuum and
selects states with an odd number of fermionic oscillators. However, if one
reformulates this standard GSO projection in terms of $F$, one can realize
that it is precisely given by (2.32). Secondly, the analysis we have described
for the D7/D7′ strings can be repeated for the D7′/D7 strings without any
difficulty, and also in that case one finds that the physical states are a
non-dynamical scalar, corresponding to the NS vacuum, and the chiral component
of a non-dynamical fermion, corresponding the chiral component of the R
vacuum. Of course, these states of the 7′/7 sector transform in the anti-
fundamental of U($N$) and in the fundamental of U($M$).

###  2.2 The no-force condition

We have seen that the condition (2.18) guarantees a supersymmetric physical
spectrum. To complete our analysis we now show that the same condition also
implies that the partition functions of the mixed strings of the 7/7′ and 7′/7
sectors vanish, thus leading to a no-force condition.

Let us first consider the 7/7′ open strings whose partition function is

$Z_{7/7^{\prime}}=\int_{0}^{\infty}\\!\frac{dt}{2t}~{}\mathrm{Tr}_{7/7^{\prime}}\Big{(}P_{\mathrm{GSO}}\,\mathrm{e}^{-2\pi
t\,L_{0}}\Big{)}$ (2.33)

where the GSO projector is defined in (2.32) and $L_{0}$ is the total open
string Hamiltonian consisting of the orbital part and of the ghost and
superghost parts (see for instance [37] for details). The trace in (2.33) is
computed over all excitations of the 7/7′ strings with a $+$ sign in the
bosonic NS sector and a $-$ sign in the fermionic R sector. This calculation
is pretty standard (see for example [38, 34] and references therein), but we
would like to point out just a couple of subtle issues that arise in the
present case. First of all, in the 7/7′ sector, there are eight (real) twisted
directions and two (real) untwisted directions. This is precisely the case in
which the odd spin-structure $\mathrm{R}(-1)^{F}$ gives a non-vanishing
contribution to the string partition function due to a cancellation between
the fermionic zero-modes in the two untwisted directions and the bosonic zero-
modes of the superghost-sector [39, 40, 38]. Secondly, the presence of the
background (2.12) along the D7-brane world-volume modifies the contribution of
the zero-modes of the eight twisted bosonic coordinates by multiplying the
eight-dimensional volume $V_{8}$ by the factor $\prod_{I}f_{I}$.

Taking this into account and using the GSO projection (2.32), we find

$Z_{7/7^{\prime}}=\frac{NM\,V_{8}}{(4\pi^{2}\alpha^{\prime})^{4}}\,\prod_{I=1}^{4}f_{I}\\!\int_{0}^{\infty}\\!\frac{dt}{2t}~{}\frac{1}{2}\left[\prod_{I=1}^{4}\frac{\vartheta_{3}(\theta_{I}t)}{\vartheta_{1}(\theta_{I}t)}+\prod_{I=1}^{4}\frac{\vartheta_{4}(\theta_{I}t)}{\vartheta_{1}(\theta_{I}t)}-\prod_{I=1}^{4}\frac{\vartheta_{2}(\theta_{I}t)}{\vartheta_{1}(\theta_{I}t)}-1\right]$
(2.34)

where the factor $NM$ accounts for the color multiplicity of each string
excitation and the four terms in the square brackets correspond to the
contributions of the NS, NS$(-1)^{F}$, R and R$(-1)^{F}$ spin-structures,
respectively. Here, as usual, we have defined:

$\vartheta_{1}(z)\equiv\vartheta\big{[}^{1}_{1}\big{]}\big{(}\mathrm{i}z|\mathrm{i}t)~{},~{}~{}\vartheta_{2}(z)\equiv\vartheta\big{[}^{1}_{0}\big{]}\big{(}\mathrm{i}z|\mathrm{i}t)~{},~{}~{}\vartheta_{3}(z)\equiv\vartheta\big{[}^{0}_{0}\big{]}\big{(}\mathrm{i}z|\mathrm{i}t)~{},~{}~{}\vartheta_{4}(z)\equiv\vartheta\big{[}^{0}_{1}\big{]}\big{(}\mathrm{i}z|\mathrm{i}t)$
(2.35)

where $\vartheta\big{[}^{a}_{b}\big{]}\big{(}\mathrm{i}z|\mathrm{i}t)$ are the
Jacobi $\vartheta$-functions with characteristics (see for example [41]). They
satisfy the Jacobi-Riemann identity

$\prod_{I=1}^{4}\vartheta_{3}(z_{I})-\prod_{I=1}^{4}\vartheta_{4}(z_{I})-\prod_{I=1}^{4}\vartheta_{2}(z_{I})-\prod_{I=1}^{4}\vartheta_{1}(z_{I})=-2\prod_{I=1}^{4}\vartheta_{1}(Z/2-z_{I})$
(2.36)

with $Z=\sum_{I}z_{I}$. Applying this identity in our case where
$z_{I}=\theta_{I}t$ and using the periodicity properties of the Jacobi
$\vartheta$-functions, we can recast the previous identity in the following
form

$\prod_{I=1}^{4}\vartheta_{3}(\theta_{I}t)+\prod_{I=1}^{4}\vartheta_{4}(\theta_{I}t)-\prod_{I=1}^{4}\vartheta_{2}(\theta_{I}t)-\prod_{I=1}^{4}\vartheta_{1}(\theta_{I}t)=0~{},$
(2.37)

when the twist parameters satisfy the supersymmetry relation (2.18). Thus,
inserting this relation into (2.34), we find

$Z_{7/7^{\prime}}=0~{}.$ (2.38)

In a completely similar way one can show that also for the other orientation
the partition function vanishes:

$Z_{7^{\prime}/7}=0~{}.$ (2.39)

This fact, together with the vanishing of the partition function in the 7/7
and $7^{\prime}/7^{\prime}$ sectors due to the BPS property of these branes,
implies that our brane system is stable when the condition (2.18) is
satisfied, and that one can pile up an arbitrary number of D7 and D7′-branes
since the net force in all channels is zero.

##  3 Adding D–instantons

To study some non-perturbative features in the effective theory defined on the
world-volume of the D7 and D7′-branes, we add a stack of $k$ D–instantons,
i.e. D-branes with Dirichlet boundary conditions in all directions also known
as D$(-1)$-branes. This addition gives rise to new open string sectors: the
(–1)/(–1) strings starting and ending on the D-instantons, the (–1)/7 and
7/(–1) strings which stretch between the D-instantons and the D7-branes, and
finally (–1)/7′ and 7′/(–1) strings which connect the D-instantons and the
D7′-branes. We now analyze the physical states in all these sectors.

###  3.1 The D(–1)/D(–1) strings

The open strings stretching between two D(–1)-branes are untwisted and have
Dirichlet/Dirichlet boundary conditions in all ten directions. Thus, they do
not carry any momentum and describe non-propagating degrees of freedom. To
describe the physical states in this sector, it is convenient to distinguish
the eight directions, that are longitudinal to the D7 and D7′\- branes from
the two ones that are transverse, and organize the open string states in
representations of Spin(8)$\,\times\,$Spin(2).

In the NS sector, the only physical states are those with one fermionic
oscillator, corresponding to the ten-dimensional vector. Using the same
complex notation introduced in the previous section, we can pack the eight
(real) components of this vector along the D7 or D7′-branes into four complex
variables which we denote by $B^{I}$, together with their complex conjugates
$\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}$, with
$I=1,2,3,4$. Their corresponding vertex operators, in the $(-1)$-superghost
picture, are 777The normalization factors in these and in all the following
vertices are chosen for later convenience so that the final formulas look
simpler.

$V_{B^{I}}=\frac{B^{I}}{\sqrt{2}}\,\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern
1.5mu_{I}(z)\,\mathrm{e}^{-\varphi(z)}\quad\mbox{and}\quad V_{\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}}=\frac{\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}}{\sqrt{2}}\,\Psi^{I}(z)\,\mathrm{e}^{-\varphi(z)}~{}.$ (3.1)

The remaining two components of the vector, along the transverse directions to
the branes, give rise to the variables $\xi$ and $\chi$, whose vertex
operators are

$V_{\xi}=\xi\,\mkern 1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern
1.5mu_{5}(z)\,\mathrm{e}^{-\varphi(z)}\quad\mbox{and}\quad
V_{\chi}=\frac{\chi}{2}\,\Psi^{5}(z)\,\mathrm{e}^{-\varphi(z)}~{}.$ (3.2)

All these vertex operators are conformal fields of dimension 1 and possess an
even $F$-charge, as one can see using the bosonization formulas (2.25); thus
they are preserved by the GSO projection (2.32).

In the R sector the only physical state is the vacuum. This state is actually
a sixteen-component spinor of Spin(10), which the GSO projection fixes to be
anti-chiral. To describe these sixteen components we use the formalism of spin
fields [37] and introduce the notation

$S^{\pm\pm\pm\pm}(z)=\mathrm{e}^{\pm\frac{\mathrm{i}}{2}\varphi^{1}(z)}\,\mathrm{e}^{\pm\frac{\mathrm{i}}{2}\varphi^{2}(z)}\mathrm{e}^{\pm\frac{\mathrm{i}}{2}\varphi^{3}(z)}\mathrm{e}^{\pm\frac{\mathrm{i}}{2}\varphi^{4}(z)}$
(3.3)

to denote the spin fields in the first four complex directions, which have to
be used together with the spin fields $S^{\pm}(z)$ in the fifth direction
already introduced in (2.29). The physical ground state of the R sector, in
the $(-\frac{1}{2})$-superghost picture, is therefore given by

$\lim_{z\to
0}S^{\pm\pm\pm\pm}(z)\,S^{\pm}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}\,|0\rangle$
(3.4)

with a total odd number of $-$ signs in the spin fields in order to have an
even $F$-charge and be preserved by the GSO projection. It is convenient to
distinguish the cases in which the last spin field is $S^{+}(z)$ or
$S^{-}(z)$.

If we have $S^{+}(z)$, the spin fields in the first four directions must have
an odd number of minus signs. Clearly, there are eight possibilities. Using a
notation that resembles the one used for the components of the vector field in
the NS sector, we denote these eight spin fields by $S^{I}(z)$ and $\mkern
1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern 1.5mu_{I}(z)$ with
$I=1,\ldots,4$ (see (A.26) for their detailed definition). The components of
the fermionic R ground state corresponding to these fields are denoted by
$\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I}$ and
$M^{I}$, respectively, and are described by the following vertex operators of
weight 1:

$V_{M^{I}}=M^{I}\,\mkern 1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern
1.5mu_{I}(z)\,S^{+}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}\quad\mbox{and}\quad
V_{\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{I}}=S^{I}(z)\,S^{+}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I}~{}.$ (3.5)

If instead we have $S^{-}(z)$, the spin fields in the first four directions
must have an even number of minus signs and there are again eight
possibilities. We denote these spin fields by $S(z)$, $\mkern
1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern 1.5mu(z)$ and
$S^{IJ}(z)=-S^{JI}(z)$ with $I,J=1,\ldots,4$ (see (A) for details). We can
also introduce the conjugate fields $\mkern
1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern 1.5mu_{IJ}(z)$, but they are
not independent since the following relation holds

$\mkern 1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern
1.5mu_{IJ}(z)=\frac{1}{2}\,\epsilon_{IJKL}\,S^{KL}(z)~{}.$ (3.6)

Using this notation, we see that the eight components of the R ground states
with $S^{-}(z)$, which we denote by $\lambda^{IJ}$, $\lambda$ and $\eta$, are
described by the following vertex operators

$V_{\lambda^{IJ}}=\frac{\lambda^{IJ}}{\sqrt{2}}\,\mkern
1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern
1.5mu_{IJ}(z)\,S^{-}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}~{},$ (3.7)

and

$V_{\lambda}=\lambda\,\frac{\mkern
1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern
1.5mu(z)-S(z)}{\sqrt{2}\,\mathrm{i}}\,S^{-}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}~{},\quad
V_{\eta}=\eta\,\frac{\mkern 1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern
1.5mu(z)+S(z)}{\sqrt{2}}\,S^{-}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}~{}.$
(3.8)

It is easy to verify that these vertex operators are conformal fields of
weight 1. Furthermore, the fields $\lambda$ and $\eta$ are real, while the
fields $\lambda^{IJ}$ are complex with their complex conjugates $\mkern
1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu_{IJ}$ given by

$\mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern
1.5mu_{IJ}=\frac{1}{2}\,\epsilon_{IJKL}\,\lambda^{KL}~{},$ (3.9)

as a consequence of the relation (3.6).

Since there are $k$ D-instantons, all variables appearing as polarizations in
the above vertex operators are $(k\times k)$ matrices transforming in the
adjoint representation of U($k$).

###  3.2 The D(–1)/D7 and D7/D(–1) strings

We now consider the open strings stretching between the D-instantons and the
D7-branes. In this case there are two facts that one has to consider, namely
that the first four complex space directions have mixed Dirichlet/Neumann
boundary conditions and that there is a background field on the D7-branes
given by (2.12). The combined net effect is that the first four complex
directions have an effective twist parameter given by
$\big{(}\frac{1}{2}-\theta_{I}\big{)}$.

Taking this into account, one can see that the condition which selects the
physical states in the NS sector is

$\widehat{N}+\frac{1}{2}-\frac{1}{2}\,\sum_{I=1}^{4}\theta_{I}=0$ (3.10)

where $\widehat{N}$ is the total number operator. Inserting the constraint
(2.18), we see that the physical states in the NS sector must have
$\widehat{N}=0$, which is possible only for the vacuum. Thus in the spectrum
we have only one state whose polarization we denote by $w$. Its corresponding
vertex operator, in the $(-1)$-superghost picture, is

$V_{w}=\frac{w}{\sqrt{2}}\,\Big{(}\prod_{I=1}^{4}\sigma^{I}_{\frac{1}{2}-\theta_{I}}(z)~{}\mathrm{e}^{\mathrm{i}(\frac{1}{2}-\theta_{I})\varphi^{I}(z)}\Big{)}\,\mathrm{e}^{-\varphi(z)}$
(3.11)

which is a conformal field of weight 1 when the condition (2.18) is satisfied.
The same condition also guarantees that the total $F$-charge is even so that
this vertex is preserved by the GSO projection.

In the R sector, the condition for physical states implies $\widehat{N}=0$.
Thus, also in this sector the vacuum is the only physical state. Due to the R
boundary conditions in the untwisted fifth complex direction, the ground state
is degenerate and is a two-component spinor denoted by $\mu^{\pm}$. The
corresponding vertex operator in the $(-\frac{1}{2})$-superghost picture is

$V_{\mu^{\pm}}=\mu^{\pm}\,\Big{(}\prod_{I=1}^{4}\sigma^{I}_{\frac{1}{2}-\theta_{I}}(z)~{}\mathrm{e}^{-\mathrm{i}\theta_{I}\varphi^{I}(z)}\Big{)}\,S^{\pm}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}$
(3.12)

which is a conformal field of weight 1. Exploiting the supersymmetry condition
(2.18), it is easy to see that

$(-1)^{F}\,V_{\mu^{\pm}}=\pm\,V_{\mu^{\pm}}$ (3.13)

so that only $\mu^{+}$ is selected by the GSO projection. To simplify the
notation, we will denote $\mu^{+}$ simply as $\mu$.

This analysis can be easily repeated for the strings with the opposite
orientation stretching between the D7-branes and the D-instantons. The result
is that in the NS sector we have only one GSO-even physical state $\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu$ whose vertex operator
is

$V_{\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu}=\frac{\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern
1.5mu}{\sqrt{2}}\,\Big{(}\prod_{I=1}^{4}\sigma^{I}_{\theta_{I}-\frac{1}{2}}(z)~{}\mathrm{e}^{\mathrm{i}(\theta_{I}-\frac{1}{2})\varphi^{I}(z)}\Big{)}\,\mathrm{e}^{-\varphi(z)}~{},$
(3.14)

while in the R sector the only GSO-even physical state is $\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu$ whose vertex operator
is

$V_{\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu}=\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu\,\Big{(}\prod_{I=1}^{4}\sigma^{I}_{\theta_{I}-\frac{1}{2}}(z)~{}\mathrm{e}^{\mathrm{i}\theta_{I}\varphi^{I}(z)}\Big{)}\,S^{+}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}~{}.$
(3.15)

Taking into account the multiplicity of the D-instantons and of the D7-branes,
the variables $w$ and $\mu$ are $(k\times N)$ matrices transforming in the
($\mathbf{k},\mkern 1.5mu\overline{\mkern-1.5mu\mathbf{N}\mkern-1.5mu}\mkern
1.5mu$) representation of $\mathrm{U}(k)\times\mathrm{U}(N)$, while the
conjugate variables $\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern
1.5mu$ and $\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu$
are ($N\times k)$ matrices transforming in the ($\mkern
1.5mu\overline{\mkern-1.5mu\mathbf{k}\mkern-1.5mu}\mkern 1.5mu,\mathbf{N}$)
representation of the same group.

###  3.3 The D(–1)/D7′ and D7′/D(–1) strings

We now consider the strings connecting the D-instantons and the D7′-branes.
The only but important difference with the previous case is that now there is
no background field on the 7-branes so that the string fields in the first
four complex directions have standard Dirichlet/Neumann boundary conditions
without extra twists. Therefore, we can formally set $\theta_{I}=0$ in the
previous formulas. If we do this, we immediately realize that in the NS sector
the physical states must satisfy the condition

$\widehat{N}+\frac{1}{2}=0~{},$ (3.16)

which has no solution. Thus, there are no physical states in the NS part of
the spectrum. This fact was already observed in [34] where the
D$(-1)$/D7-brane system was explored in connection with the first studies of
exotic instantons in eight dimensions.

In the R sector, instead, the condition for physical states is satisfied by
the degenerate ground state that describes a two-component spinor which we
denote by $\mu^{\prime\,\pm}$. The corresponding vertex operator is

$V_{\mu^{\prime\pm}}=\mu^{\prime\pm}\,\Big{(}\prod_{I=1}^{4}\sigma^{I}_{\frac{1}{2}}(z)\Big{)}\,S^{\pm}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}$
(3.17)

where $\sigma^{I}_{\frac{1}{2}}$ is the appropriate bosonic twist field when
the $I$-th coordinate has Dirichlet/Neumann boundary conditions. This twist
field has conformal dimension $\frac{1}{8}$, and thus the vertex operator
(3.17) has total weight 1. Furthermore, it is easy to check

$(-1)^{F}\,V_{\mu^{\prime\pm}}=\mp\,V_{\mu^{\prime\pm}}~{},$ (3.18)

so that only $\mu^{\prime-}$, which we will simply call $\mu^{\prime}$,
survives the GSO projection.

For the strings with opposite orientation that stretch between the D7′-branes
and the D-instantons, we find a similar structure with no physical states in
the NS sector, and only one physical state in the R sector which we denote
$\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu^{\prime}$. Its
vertex operator is

$V_{\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu^{\prime}}=\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu^{\prime}\,\Big{(}\prod_{I=1}^{4}{\sigma}^{I}_{-\frac{1}{2}}(z)\Big{)}\,S^{-}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}$
(3.19)

where ${\sigma}^{I}_{-\frac{1}{2}}$ is the bosonic twist field corresponding
to an $I$-th coordinate with Neumann/Dirichlet boundary conditions, that is
conjugate to the twist field $\sigma^{I}_{\frac{1}{2}}$.

Taking into account the multiplicity of the branes, we have that
$\mu^{\prime}$ is a $(k\times M)$ matrix transforming the
$(\mathbf{k},\mathbf{\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu})$ representation of $\mathrm{U}(k)\times\mathrm{U}(M)$, while $\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu^{\prime}$ is $(M\times
k)$ matrix transforming the $(\mathbf{\mkern
1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu},\mathbf{M})$
representation of this group.

###  3.4 The no-force condition

We now study the partition function of the open strings with at least one end-
point on the D-instantons. In the $(-1)/(-1)$ sector, we simply have

$Z_{(-1)/(-1)}=0$ (3.20)

as a consequence of the BPS condition satisfied by the D(–1)-branes.

Let us now consider the $(-1)/7^{\prime}$ and the $7^{\prime}/(-1)$ sectors.
Recalling that the D7′-branes do not have any background field on their world-
volume, the calculation of the partition functions $Z_{(-1)/7^{\prime}}$ and
$Z_{7^{\prime}/(-1)}$ is exactly the same as the one described in detail in
[34], which we briefly recall here. For the strings starting from the
D-instantons we have

$Z_{(-1)/7^{\prime}}=kM\int_{0}^{\infty}\\!\frac{dt}{2t}~{}\frac{1}{2}\left[\Big{(}\frac{\vartheta_{2}(0)}{\vartheta_{4}(0)}\Big{)}^{4}+\Big{(}\frac{\vartheta_{1}(0)}{\vartheta_{4}(0)}\Big{)}^{4}-\Big{(}\frac{\vartheta_{3}(0)}{\vartheta_{4}(0)}\Big{)}^{4}-1\right]$
(3.21)

where the prefactor accounts for the multiplicity of the branes, and the four
terms inside the square brackets arise from the four spin structures with
eight directions with Dirichlet/Neumann boundary conditions. We notice in
particular that the second term, corresponding to the NS$(-1)^{F}$ sector,
actually vanishes since $\vartheta_{1}(0)=0$, while the fourth term,
corresponding to the R$(-1)^{F}$ spin structure, is non-zero due to the
cancellation of the fermionic zero-modes in the two directions with
Dirichlet/Dirichlet boundary conditions with the bosonic zero-modes of the
superghost system [38]. Exploiting the abstruse identity

$\vartheta_{3}(0)^{4}-\vartheta_{4}(0)^{4}-\vartheta_{2}(0)^{4}=0~{},$ (3.22)

we easily conclude that

$Z_{(-1)/7^{\prime}}=-kM\int_{0}^{\infty}\\!\frac{dt}{2t}~{}.$ (3.23)

The partition function for the other orientation corresponds to placing the
D-instantons “on the other side” of the 7-brane, which is obtained with a
parity transformation that reverses the sign of the odd spin structure
R$(-1)^{F}$ [38]. Then we have

$Z_{7^{\prime}/(-1)}=kM\int_{0}^{\infty}\\!\frac{dt}{2t}~{}\frac{1}{2}\left[\Big{(}\frac{\vartheta_{2}(0)}{\vartheta_{4}(0)}\Big{)}^{4}+\Big{(}\frac{\vartheta_{1}(0)}{\vartheta_{4}(0)}\Big{)}^{4}-\Big{(}\frac{\vartheta_{3}(0)}{\vartheta_{4}(0)}\Big{)}^{4}+1\right]~{}.$
(3.24)

The abstruse identity (3.22) now implies that 888In the T-dual set-up, namely
in the D0/D8-brane system, the vanishing of the partition function can be
interpreted as due to the creation of a fundamental string, see for example
[40, 42, 38].

$Z_{7^{\prime}/(-1)}=0~{}.$ (3.25)

Let us now consider the contribution of the open strings connecting the
D-instantons to the D7-branes. In this case, due to the presence of the
background (2.12) on the 7-branes, we have an extra twist to take into account
with respect to the previous cases. In particular, the partition function
$Z_{(-1)/7}$ is obtained from (3.21) by twisting the $\vartheta$-functions by
$-\theta_{I}t$ in each complex direction and replacing $M$ by $N$, which is
the number of D7-branes. In other words we have

$Z_{(-1)/7}=kN\int_{0}^{\infty}\\!\frac{dt}{2t}~{}\frac{1}{2}\left[\prod_{I=1}^{4}\frac{\vartheta_{2}(-\theta_{I}t)}{\vartheta_{4}(-\theta_{I}t)}+\prod_{I=1}^{4}\frac{\vartheta_{1}(-\theta_{I}t)}{\vartheta_{4}(-\theta_{I}t)}-\prod_{I=1}^{4}\frac{\vartheta_{3}(-\theta_{I}t)}{\vartheta_{4}(-\theta_{I}t)}-1\right]~{}.$
(3.26)

Since the twists satisfy the condition (2.18), the Jacobi-Riemann identity
(2.37), together with the following parity properties

$\vartheta_{1}(-z)=-\vartheta_{1}(z)~{},\quad\vartheta_{2}(-z)=+\vartheta_{2}(z)~{},\quad\vartheta_{3}(-z)=+\vartheta_{3}(z)~{},\quad\vartheta_{4}(-z)=+\vartheta_{4}(z)~{},$
(3.27)

implies that

$Z_{(-1)/7}=0~{}.$ (3.28)

The partition function for the other orientation is obtained from (3.24) by
twisting the $\vartheta$-functions with $\theta_{I}t$ and replacing $M$ with
$N$, namely

$Z_{7/(-1)}=kN\int_{0}^{\infty}\\!\frac{dt}{2t}~{}\frac{1}{2}\left[\prod_{I=1}^{4}\frac{\vartheta_{2}(\theta_{I}t)}{\vartheta_{4}(\theta_{I}t)}+\prod_{I=1}^{4}\frac{\vartheta_{1}(\theta_{I}t)}{\vartheta_{4}(\theta_{I}t)}-\prod_{I=1}^{4}\frac{\vartheta_{3}(\theta_{I}t)}{\vartheta_{4}(\theta_{I}t)}+1\right]~{}.$
(3.29)

Using the Jacobi-Riemann identity, we easily find

$Z_{7/(-1)}=kN\int_{0}^{\infty}\\!\frac{dt}{2t}~{}.$ (3.30)

Summing all contributions (3.20), (3.23), (3.25), (3.28) and (3.30), we
conclude that the total partition function of the open strings with at least
one end-point on the D-instantons is

$\displaystyle Z_{\mathrm{tot}}=k(N-M)\int_{0}^{\infty}\\!\frac{dt}{2t}~{}.$
(3.31)

If we set

$N=M~{},$ (3.32)

the total partition function vanishes and no forces are present. This no-force
condition is equivalent to requiring that the brane system be conformal
invariant.

##  4 The instanton moduli and their action

In the previous sections we have described a system made of $N$ D7-branes with
flux, $M$ D7′-branes without flux and $k$ D-instantons. The flux on the
D7-branes is subject to the condition (2.18) with all $\theta_{I}$ different
from zero in order to preserve the minimal amount of supersymmetry. For
simplicity, from now on we choose the twists to be all equal, namely

$\theta_{1}=\theta_{2}=\theta_{3}=\theta_{4}=\frac{1}{4}~{}.$ (4.1)

With this choice the four complex directions are treated symmetrically and
many formulas simplify. For example, the fermionic twist fields to be used in
the vertex operators of the (–1)/7 strings (see (3.11) and (3.12)) become

$\Sigma(z)=\mathrm{e}^{\frac{\mathrm{i}}{4}(\varphi^{1}(z)+\varphi^{2}(z)+\varphi^{3}(z)+\varphi^{4}(z))}\quad\mbox{and}\quad\mkern
1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern
1.5mu(z)=\mathrm{e}^{-\frac{\mathrm{i}}{4}(\varphi^{1}(z)+\varphi^{2}(z)+\varphi^{3}(z)+\varphi^{4}(z))}$
(4.2)

in the NS and R sectors, respectively. The same twist fields are used also in
the vertex operators of the 7/(–1) strings (see (3.14) and (3.15)), but in
this case $\Sigma(z)$ is used in the R sector and $\mkern
1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern 1.5mu(z)$ in the NS
sector. Both $\Sigma(z)$ and $\mkern
1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern 1.5mu(z)$ are conformal
fields of weight $\frac{1}{8}$. It is also convenient to define the bosonic
twist fields

$\Delta(z)=\prod_{I=4}\sigma^{I}_{\frac{1}{4}}(z)\quad\mbox{and}\quad\mkern
1.5mu\overline{\mkern-1.5mu\Delta\mkern-1.5mu}\mkern
1.5mu(z)=\prod_{I=4}\sigma^{I}_{-\frac{1}{4}}(z)$ (4.3)

to be used for the (–1)/7 and 7/(–1) strings respectively. These fields have
conformal dimension $\frac{3}{8}$ and are conjugate to each other. Similarly,
for the strings of type (–1)/7′ and 7′/(–1) we define the following pair of
conjugate twist fields

$\Delta^{\prime}(z)=\prod_{I=4}\sigma^{I}_{\frac{1}{2}}(z)\quad\mbox{and}\quad\mkern
1.5mu\overline{\mkern-1.5mu\Delta\mkern-1.5mu}\mkern
1.5mu^{\prime}(z)=\prod_{I=4}\sigma^{I}_{-\frac{1}{2}}(z)~{},$ (4.4)

which have conformal weight $\frac{1}{2}$.

Using these notations, we summarize for future reference all physical states
of the open strings with at least one end-point on the D-instantons and their
vertex operators in Table 2.

Sector | Moduli |  Vertex Operators | Statistics | Dimensions
---|---|---|---|---
$(-1)/(-1)$ | $B^{I}~{},~{}\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}$ | $\phantom{\bigg{|}}\frac{B^{I}}{\sqrt{2}}\,\mkern 1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern 1.5mu_{I}\,\mathrm{e}^{-\varphi}~{},~{}\phantom{\bigg{|}}\frac{\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}}{\sqrt{2}}\,\Psi^{I}\,\mathrm{e}^{-\varphi}$ | bosonic | (length)-1
$\xi~{},~{}\chi$ | $\phantom{\bigg{|}}\xi\,\mkern 1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern 1.5mu_{5}\,\mathrm{e}^{-\varphi}~{},~{}\phantom{\bigg{|}}\frac{\chi}{2}\,\Psi^{5}\,\mathrm{e}^{-\varphi}$ | bosonic | (length)-1
$M^{I}~{},~{}\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I}$ | $\phantom{\bigg{|}}M^{I}\,\mkern 1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern 1.5mu_{I}\,S^{+}\,\mathrm{e}^{-\frac{1}{2}\varphi}~{},~{}\phantom{\bigg{|}}S^{I}\,S^{+}\,\mathrm{e}^{-\frac{1}{2}\varphi}\,\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I}$ | fermionic | (length)${}^{-\frac{3}{2}}$
$\lambda^{IJ}$ | $\phantom{\bigg{|}}\frac{\lambda^{IJ}}{\sqrt{2}}\,\mkern 1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern 1.5mu_{IJ}\,S^{-}\,\mathrm{e}^{-\frac{1}{2}\varphi}$ | fermionic | (length)${}^{-\frac{3}{2}}$
$\lambda$ | $\phantom{\bigg{|}}\lambda\,\frac{\mkern 1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern 1.5mu-S}{\sqrt{2}\,\mathrm{i}}\,S^{-}\,\mathrm{e}^{-\frac{1}{2}\varphi}$ | fermionic | (length)${}^{-\frac{3}{2}}$
$\eta$ | $\phantom{\bigg{|}}\eta\,\frac{\mkern 1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern 1.5mu+S}{\sqrt{2}}\,S^{-}\,\mathrm{e}^{-\frac{1}{2}\varphi}$ | fermionic | (length)${}^{-\frac{3}{2}}$
$(-1)/7$ | $w$ | $\phantom{\bigg{|}}\frac{w}{\sqrt{2}}\,\Delta\,\Sigma\,\mathrm{e}^{-\varphi}$ | bosonic | (length)-1
$\mu$ | $\phantom{\bigg{|}}\mu\,\Delta\,\mkern 1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern 1.5mu\,S^{+}\,\mathrm{e}^{-\frac{1}{2}\varphi}$ | fermionic | (length)${}^{-\frac{3}{2}}$
$7/(-1)$ | $\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu$ | $\phantom{\bigg{|}}\frac{\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu}{\sqrt{2}}\,\mkern 1.5mu\overline{\mkern-1.5mu\Delta\mkern-1.5mu}\mkern 1.5mu\,\mkern 1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern 1.5mu\,\mathrm{e}^{-\varphi}$ | bosonic | (length)-1
$\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu$ | $\phantom{\bigg{|}}\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu\,\mkern 1.5mu\overline{\mkern-1.5mu\Delta\mkern-1.5mu}\mkern 1.5mu\,\Sigma\,S^{+}\,\mathrm{e}^{-\frac{1}{2}\varphi}$ | fermionic | (length)${}^{-\frac{3}{2}}$
$(-1)/7^{\prime}$ | $\mu^{\prime}$ | $\phantom{\bigg{|}}\mu^{\prime}\,\Delta^{\prime}\,S^{-}\,\mathrm{e}^{-\frac{1}{2}\varphi}$ | fermionic | (length)${}^{-\frac{3}{2}}$
$7^{\prime}/(-1)$ | $\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu^{\prime}$ | $\phantom{\bigg{|}}\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu^{\prime}\,\mkern 1.5mu\overline{\mkern-1.5mu\Delta\mkern-1.5mu}\mkern 1.5mu^{\prime}\,S^{-}\,\mathrm{e}^{-\frac{1}{2}\varphi}$ | fermionic | (length)${}^{-\frac{3}{2}}$
Table 2: The physical moduli corresponding to open strings with at least one
end-point on the D-instantons, and their vertex operators in the canonical
superghost pictures, $(-1)$ in the NS sector and $(-\frac{1}{2})$ in the R
sector. The last two columns contain their statistics and their scaling
dimensions.

###  4.1 The moduli action

The action of the $(-1)/(-1)$ moduli can be obtained by dimensionally reducing
the $\mathcal{N}=1$ super Yang-Mills action from ten to zero dimensions and,
in the notations previously introduced, it reads as follows

$\displaystyle S_{(-1)}$
$\displaystyle=\frac{1}{g_{0}^{2}}\,\mathrm{tr}\,\bigg{\\{}\,\frac{1}{2}\big{[}B^{I},\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}\big{]}\big{[}B^{J},\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{J}\big{]}-\big{[}B^{I},B^{J}\big{]}\big{[}\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I},\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{J}\big{]}-\big{[}\xi,B^{I}\big{]}\big{[}\chi,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}\big{]}-\big{[}\chi,B^{I}\big{]}\big{[}\xi,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}\big{]}$
$\displaystyle\quad+\frac{1}{2}\big{[}\chi,\xi\big{]}^{2}+\frac{1}{2}\,\lambda\big{[}\chi,\lambda\big{]}+\frac{1}{2}\,\eta\big{[}\chi,\eta\big{]}+\frac{1}{4}\,\lambda^{IJ}\big{[}\chi,\lambda^{KL}\big{]}\,\epsilon_{IJKL}$
$\displaystyle\quad-2\,M^{I}\big{[}\xi,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{I}\big{]}+\lambda^{IJ}\Big{(}\big{[}\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I},\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{J}\big{]}-\big{[}\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{J},\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{I}\big{]}+\epsilon_{IJKL}\,\big{[}B^{K},M^{L}\big{]}\Big{)}$
$\displaystyle\quad+(\eta+\mathrm{i}\,\lambda)\big{[}\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I},M^{I}\big{]}+(\eta-\mathrm{i}\,\lambda)\big{[}B^{I},\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I}\big{]}\bigg{\\}}$
(4.5)

where

$g_{0}^{2}=\frac{g_{s}}{4\pi^{3}\alpha^{\prime\,2}}$ (4.6)

is the Yang-Mills coupling constant in zero dimensions.

The action involving the moduli of the mixed sectors is

$S_{\mathrm{mixed}}=\frac{1}{g_{0}^{2}}\,\mathrm{tr}\,\big{\\{}\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern
1.5mu\,(\chi\,\xi+\xi\,\chi)\,w-2\,\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu\,\xi\,\mu-\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu\,(\eta+\mathrm{i}\,\lambda)\,w-\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern
1.5mu\,(\eta-\mathrm{i}\,\lambda)\,\mu+\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu^{\prime}\,\chi\,\mu^{\prime}\big{\\}}~{}.$ (4.7)

Using standard conformal field theory methods, one can verify that all cubic
couplings in the above actions can be obtained by computing the 3-point
amplitudes among the vertex operators introduced in the previous section. If
one applies this method to compute the 4-point amplitudes and obtain the
quartic couplings, one encounters divergent integrals over the vertex
insertion points, as it has also been recently pointed out in [43]. To
circumvent this problem, we adopt the strategy already used in [6, 31, 34] and
introduce auxiliary fields to disentangle the quartic interactions. For our
purposes it is enough to consider the first two quartic terms in (4.5). The
minimal set of auxiliary fields which are needed to decouple these
interactions comprises seven auxiliary fields, denoted by $D$ and $D^{IJ}$
with $D^{JI}=-D^{IJ}$ ($I,J=1,\ldots,4$). Notice that $D$ is real while the
six $D^{IJ}$’s are complex with their complex conjugates $\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{IJ}$ given by

$\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{IJ}=\frac{1}{2}\,\epsilon_{IJKL}\,D^{KL}~{}.$ (4.8)

Thus $D^{IJ}$ correspond to six real degrees of freedom. Using these auxiliary
fields, to which we assign canonical scaling dimensions of (length)-2, the
first two terms of (4.5) can be replaced by

$\displaystyle\frac{1}{g_{0}^{2}}\,\mathrm{tr}\,\bigg{\\{}$
$\displaystyle\\!\frac{1}{4}\,D^{IJ}D^{KL}\,\epsilon_{IJKL}-D^{IJ}\Big{(}\big{[}\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I},\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{J}\big{]}+\frac{1}{2}\,\epsilon_{IJKL}\,\big{[}B^{K},B^{L}\big{]}\Big{)}+\frac{1}{2}\,D^{2}-\mathrm{i}\,D\big{[}B^{I},\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}\big{]}\bigg{\\}}~{}.$ (4.9)

Indeed, eliminating the auxiliary fields through their equations of motion and
using the Jacobi identity, one can show that (4.9) reduces exactly to the
first two terms of (4.5).

We observe that the cubic terms in (4.9) can be obtained by computing the
couplings among the vertex operators $V_{B^{I}}$ and $V_{\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}}$, given in (3.1),
and the following vertex operators for the auxiliary fields

$\displaystyle V_{D^{IJ}}$ $\displaystyle=-\frac{D^{IJ}}{2}\Big{(}\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern 1.5mu_{I}(z)\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern
1.5mu_{J}(z)+\frac{1}{2}\,\epsilon_{IJKL}\,\Psi^{K}(z)\Psi^{L}(z)\Big{)}~{},$
(4.10) $\displaystyle V_{D}$
$\displaystyle=-\frac{\mathrm{i}\,D}{2}\,\Psi^{I}(z)\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern 1.5mu_{I}(z)~{}.$

These vertices are in the $(0)$-superghost picture, as appropriate for
auxiliary fields, and are conformal operators of weight 1.

The vertex $V_{D}$ has a non-vanishing coupling also with the vertex operators
$V_{w}$ and $V_{\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu}$
of the mixed bosonic moduli $w$ and $\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu$; this coupling leads to
the following term

$\frac{1}{g_{0}^{2}}\,\mathrm{tr}\,\big{\\{}\\!\\!-\mathrm{i}\,D\,w\,\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu\big{\\}}$ (4.11)

that has to be added to the moduli action (4.7). Furthermore, we find
convenient to add also the term

$\frac{1}{g_{0}^{2}}\,\mathrm{tr}\,\big{\\{}\,\mkern
1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern
1.5mu^{\prime}\,h^{\prime}\,\big{\\}}$ (4.12)

where $h^{\prime}$ and its conjugate $\mkern
1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu^{\prime}$ are auxiliary
fields with scaling dimensions of (length)-2. Even if these fields look
trivial since they do not interact with any other moduli, it is useful to
introduce them for reasons that will be clear in a moment. Also these
auxiliary fields can be described by vertex operators in the $(0)$-superghost
picture, which are

$V_{h^{\prime}}=h^{\prime}\,\Delta^{\prime}(z)\,\frac{\mkern
1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern
1.5mu(z)+S(z)}{\sqrt{2}}\quad\mbox{and}\quad V_{\mkern
1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu^{\prime}}=\mkern
1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu^{\prime}\,\mkern
1.5mu\overline{\mkern-1.5mu\Delta\mkern-1.5mu}\mkern
1.5mu^{\prime}(z)\,\frac{\mkern
1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern 1.5mu(z)+S(z)}{\sqrt{2}}~{}.$
(4.13)

These operators are conformal fields of weight 1.

We now introduce ADHM-like variables by means of the following rescalings

$B^{I}\to g_{0}\,B^{I}~{},~{}~{}M^{I}\to g_{0}\,M^{I}~{},~{}~{}w\to
g_{0}\,w~{},~{}~{}\mu\to g_{0}\,\mu~{},~{}~{}\mu^{\prime}\to
g_{0}\,\mu^{\prime}~{},~{}~{}h^{\prime}\to g_{0}\,h^{\prime}$ (4.14)

with analogous ones for their conjugates, in such a way that the rescaled
bosons $B^{I}$ and $w$ have dimensions of length, the rescaled fermions $\mu$
and $\mu^{\prime}$ have dimensions of (length)${}^{\frac{1}{2}}$, and the
rescaled auxiliary field $h^{\prime}$ becomes dimensionless. After these
rescalings, adding all contributions one finds that the instanton moduli
action can be written as

$S_{\text{inst}}=\frac{1}{g_{0}^{2}}\,S_{G}+S_{K}+S_{D}$ (4.15)

where

$\displaystyle S_{G}$
$\displaystyle=\mathrm{tr}\,\bigg{\\{}\,\frac{1}{4}\,D^{IJ}D^{KL}\,\epsilon_{IJKL}+\frac{1}{2}\,D^{2}+\frac{1}{2}\big{[}\chi,\xi\big{]}^{2}+\frac{1}{2}\,\lambda\big{[}\chi,\lambda\big{]}$
$\displaystyle\quad\qquad+\frac{1}{2}\,\eta\big{[}\chi,\eta\big{]}+\frac{1}{4}\,\lambda^{IJ}\big{[}\chi,\lambda^{KL}\big{]}\,\epsilon_{IJKL}\bigg{\\}}~{},$
(4.16a) $\displaystyle S_{K}$
$\displaystyle=\mathrm{tr}\,\bigg{\\{}\\!\\!-\big{[}\xi,B^{I}\big{]}\big{[}\chi,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}\big{]}-\big{[}\chi,B^{I}\big{]}\big{[}\xi,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}\big{]}-2\,M^{I}\big{[}\xi,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I}\big{]}$
$\displaystyle\quad\qquad-2\,\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu\,\xi\,\mu+\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu\,\xi\,\chi\,w+\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu\,\chi\,\xi\,w+\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu^{\prime}\,\chi\,\mu^{\prime}+\mkern
1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern
1.5mu^{\prime}\,h^{\prime}\bigg{\\}}~{},$ (4.16b) $\displaystyle S_{D}$
$\displaystyle=\mathrm{tr}\,\bigg{\\{}\\!\\!-\mathrm{i}\,D\Big{(}\big{[}B^{I},\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}\big{]}+w\,\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern
1.5mu\Big{)}-D^{IJ}\Big{(}\big{[}\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I},\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{J}\big{]}+\frac{1}{2}\,\epsilon_{IJKL}\,\big{[}B^{K},B^{L}\big{]}\Big{)}$
$\displaystyle\qquad\quad+\lambda^{IJ}\Big{(}\big{[}\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I},\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{J}\big{]}-\big{[}\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{J},\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{I}\big{]}+\epsilon_{IJKL}\,\big{[}B^{K},M^{L}\big{]}\Big{)}$
$\displaystyle\qquad\quad+(\eta+\mathrm{i}\,\lambda)\Big{(}\big{[}\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I},M^{I}\big{]}+w\,\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu\Big{)}+(\eta-\mathrm{i}\,\lambda)\Big{(}\big{[}B^{I},\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I}\big{]}-\mu\,\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu\Big{)}\bigg{\\}}~{}.$
(4.16c)

In the field theory limit $\alpha^{\prime}\to 0$, or equivalently in the
strong-coupling limit $g_{0}\to\infty$, the term $S_{G}$ can be discarded and
the fields $D$ and $D^{IJ}$ become Lagrange multipliers for the bosonic
constraints

$\big{[}B^{I},\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}\big{]}+w\,\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern
1.5mu=0~{},\qquad\big{[}\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I},\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{J}\big{]}+\frac{1}{2}\,\epsilon_{IJKL}\,\big{[}B^{K},B^{L}\big{]}=0~{},$
(4.17)

while the fields $(\eta\pm\mathrm{i}\,\lambda)$ and $\lambda^{IJ}$ become
Lagrange multipliers for the fermionic constraints

$\displaystyle\big{[}\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I},M^{I}\big{]}+w\,\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu=0~{},~{}~{}~{}\big{[}\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I},\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{J}\big{]}-\big{[}\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{J},\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{I}\big{]}+\epsilon_{IJKL}\,\big{[}B^{K},M^{L}\big{]}=0~{}.$ (4.18)

In Section 6 we will show that the same equations arise from the ADHM
construction for instantons in eight dimensions. This fact allows us to regard
the open strings ending on the D(–1) branes as instanton moduli.

###  4.2 Introducing vacuum expectation values and the
$\varepsilon$-background

We now generalize the action (4.15) by introducing the interactions with
external backgrounds.

From the string theory point of view, these backgrounds correspond either to
vacuum expectation values of massless fields propagating on the world-volume
of the D7 and D7′-branes, or to vacuum expectation values of bulk fields in
the closed string sectors. Therefore, one can obtain their interactions with
the instanton moduli by following the procedure described in [6, 31, 16],
namely by computing the correlation functions among the vertex operators of
the moduli and the vertex operators of the external background fields.
However, there is also an alternative route to obtain these interactions which
exploits the symmetries of the brane system.

Let us give some details. The moduli actions (4.16) are invariant by
construction under the symmetry group U($k$) of the D-instantons, and the
moduli $\chi$ and $\xi$ can be interpreted as the parameters of infinitesimal
U($k$) transformations 999The use of $\chi$ or $\xi$ is fixed by a neutrality
condition with respect to the charge in the fifth complex plane of the initial
string construction.. In our conventions, these transformations are

$\mathsf{T}_{\chi}[\,\bullet\,]=\begin{cases}-[\chi,\bullet]~{}~{}\mbox{if
$\bullet$ is an adjoint modulus}~{},\\\ -\chi\bullet~{}~{}\mbox{if $\bullet$
is a fundamental modulus}~{},\\\ ~{}\bullet\chi~{}~{}\mbox{if $\bullet$ is an
anti-fundamental modulus}~{},\end{cases}$ (4.19)

and similarly for $\mathsf{T}_{\xi}$. Using this notation, for example the
terms $-\big{[}\xi,B^{I}\big{]}\big{[}\chi,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}\big{]}$ and $\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu^{\prime}\,\chi\,\mu^{\prime}$ appearing in (4.16b) can be rewritten
respectively as

$-\mathsf{T}_{\xi}[\,B^{I}\,]\,\mathsf{T}_{\chi}[\,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}\,]\quad~{}\mbox{and}\quad-\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu^{\prime}\,\mathsf{T}_{\chi}[\,\mu^{\prime}\,]~{}.$ (4.20)

All other $\chi$\- and $\xi$-dependent terms in (4.16) can be treated in the
same way.

The idea is to extend this approach to all symmetries of the brane system. The
actions (4.16) are invariant under the U($N$) transformations of the $N$
D7-branes and under the U($M$) transformations of the $M$ D7′-branes, in which
only the mixed moduli transform according to the representations defined by
the open string construction. The actions (4.16) are also invariant under the
SU(4) transformations related to the rotations in the four complex planes
indexed by $I$. Let us denote by $\varepsilon_{I}$ the parameters of such
rotations and define the quantities

$\varepsilon=\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}~{},$
(4.21)

and

$\varepsilon_{IJ}=\varepsilon_{I}+\varepsilon_{J}-\frac{\varepsilon}{2}~{}.$
(4.22)

Then, it is easy to show that the actions (4.16) are invariant under the
following transformations

$\displaystyle(B^{I},M^{I})$
$\displaystyle\to\quad\mathrm{e}^{+\mathrm{i}\,\varepsilon_{I}}\,(B^{I},M^{I})~{},$
(4.23) $\displaystyle(\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I},\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I})$
$\displaystyle\to\quad\mathrm{e}^{-\mathrm{i}\,\varepsilon_{I}}\,(\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I},\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I})~{},$
$\displaystyle(\lambda^{IJ},D^{IJ})$
$\displaystyle\to\quad\mathrm{e}^{+\mathrm{i}\,\varepsilon_{IJ}}\,(\lambda^{IJ},D^{IJ})~{},$

with all other moduli unchanged, provided $\varepsilon=0$. This is a SU(4)
symmetry whose origin is quite clear. Indeed, the initial SO(8) symmetry of
the D-brane system is broken to SO(7) on the D-instantons [21, 16], and this
symmetry is further reduced to SO(6)$\,\simeq\,$SU(4) by the complex structure
we have introduced. Therefore, the four parameters $\varepsilon_{I}$ subject
to the condition $\varepsilon=0$ can be understood as the parameters of the
SU(4) transformations along the Cartan directions (see Appendix A).

We now exploit the U($N$), U($M$) and SU(4) symmetries and replace the U($k$)
transformations $\mathsf{T}_{\chi}$ and $\mathsf{T}_{\xi}$ according to

$\displaystyle\mathsf{T}_{\chi}[\,\bullet\,]{}$
$\displaystyle\to~{}\mathsf{T}_{(\chi,a,m,\varepsilon_{I})}[\,\bullet\,]~{},$
(4.24) $\displaystyle\mathsf{T}_{\xi}[\,\bullet\,]{}$
$\displaystyle\to~{}\mathsf{T}_{(\xi,\overline{a},\mkern
1.5mu\overline{\mkern-1.5mum\mkern-1.5mu}\mkern
1.5mu,\overline{\varepsilon}_{I})}[\,\bullet\,]$

where $\mathsf{T}_{(\chi,a,m,\varepsilon_{I})}$ and
$\mathsf{T}_{(\xi,\overline{a},\mkern
1.5mu\overline{\mkern-1.5mum\mkern-1.5mu}\mkern
1.5mu,\overline{\varepsilon}_{I})}$ denote
$\mathrm{U}(k)\times\mathrm{U}(N)\times\mathrm{U}(M)\times\mathrm{SU}(4)$
infinitesimal transformations parametrized by $\chi$, $a$, $m$ and
$\varepsilon_{I}$, and by $\xi$, $\overline{a}$, $\mkern
1.5mu\overline{\mkern-1.5mum\mkern-1.5mu}\mkern 1.5mu$ and
$\overline{\varepsilon}_{I}$ respectively, in the appropriate representations.
For example, applying this rule the terms in (4.20) get shifted and become

$\displaystyle-\mathsf{T}_{(\xi,\overline{a},\mkern
1.5mu\overline{\mkern-1.5mum\mkern-1.5mu}\mkern
1.5mu,\overline{\varepsilon}_{I})}[\,B^{I}\,]\,\mathsf{T}_{(\chi,a,m,\varepsilon_{I})}[\,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}\,]$
$\displaystyle=-\big{(}\\!-\big{[}\xi,B^{I}\big{]}+\overline{\varepsilon}_{I}\,B^{I}\big{)}\big{(}\\!-\big{[}\chi,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}\big{]}-\varepsilon_{I}\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}\big{)}~{},$ (4.25)
$\displaystyle-\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu^{\prime}\,\mathsf{T}_{(\chi,a,m,\varepsilon_{I})}[\,\mu^{\prime}\,]$
$\displaystyle=-\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu^{\prime}\,(-\chi\,\mu^{\prime}+\mu^{\prime}\,m)~{}.$

Proceeding systematically in this way, we can generate new terms in the moduli
action. Taking for simplicity the U($N$) and U($M$) parameters only along the
Cartan directions, these new terms correspond to shifting the actions (4.16)
as follows 101010Notice that no dependence on $\mkern
1.5mu\overline{\mkern-1.5mum\mkern-1.5mu}\mkern 1.5mu$ arises with this
method, since there are no terms depending both on $\xi$ and on the primed
moduli in the original action.

$\displaystyle S_{G}~{}$
$\displaystyle\to~{}S^{\prime}_{G}=S_{G}+\operatorname{tr}\,\bigg{\\{}\sum_{I<J,K<L}\\!\\!\varepsilon_{IJ}\,\lambda^{IJ}\,\lambda^{KL}\,\epsilon_{IJKL}\bigg{\\}}~{},$
(4.26a) $\displaystyle S_{K}~{}$
$\displaystyle\to~{}S^{\prime}_{K}=S_{K}+\mathrm{tr}\,\bigg{\\{}2\sum_{I}\Big{(}\varepsilon_{I}\,B^{I}\,[\xi,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}]+\overline{\varepsilon}_{I}\,B^{I}\,[\chi,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}]+\overline{\varepsilon}_{I}\,\varepsilon_{I}\,B^{I}\,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}-\overline{\varepsilon}_{I}\,M^{I}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I}\Big{)}$
$\displaystyle\qquad\qquad\qquad-2\,a\,\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu\,\xi\,w-2\,\mkern
1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5mu\,\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu\,(\chi-a)\,w+2\,\mkern
1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5mu\,\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu\,\mu-\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu^{\prime}\,\mu^{\prime}\,m\bigg{\\}}~{},$ (4.26b) $\displaystyle
S_{D}~{}$ $\displaystyle\to~{}S^{\prime}_{D}=S_{D}~{}.$ (4.26c)

As mentioned above, these new terms can also be obtained from the interactions
that the instanton moduli have with the background fields on the D7 and
D7′-branes or with the bulk fields of the closed string sectors. In
particular, the terms with $a$, $\mkern
1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5mu$ and $m$ arise when a
vacuum expectation value is given to the adjoint scalars $\phi$ and $\mkern
1.5mu\overline{\mkern-1.5mu\phi\mkern-1.5mu}\mkern 1.5mu$ on the $N$ D7-branes
(see (2.8)) and to the analogue scalars on the $M$ D$7^{\prime}$-branes. One
can explicitly check that all such terms arise from the correlators among the
vertex operators of the instanton moduli and the vertex operators of these
adjoint scalars. The latter have the same structure as the vertices of $\xi$
and $\chi$ in (3.2), and thus it is not surprising that they produce similar
couplings in the moduli action.

The terms in (4.26) that depend on $\varepsilon_{I}$ and
$\overline{\varepsilon}_{I}$ can instead be generated by introducing a
graviphoton background corresponding to the 3-form field strength
$\mathcal{W}=d(C_{2}-\tau\,B_{2})$ of Type II B string theory 111111Here
$C_{(2)}$ and $B_{(2)}$ are the 2-forms in the R-R and NS-NS sectors,
respectively, and $\tau$ is the axio-dilaton (2.10).. In our set-up, the
3-form field strength has both holomorphic and anti-holomorphic components
along the fifth complex direction, namely it is of the form
$\mathcal{W}_{\mu\nu z}$ and $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{\mu\nu\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu}$ and is skew-symmetric
along the $\mu$ and $\nu$ directions:

$\displaystyle\mathcal{W}_{12z}=\varepsilon_{1}~{},\quad\mathcal{W}_{34z}=\varepsilon_{2}~{},\quad\mathcal{W}_{56z}=\varepsilon_{3}~{},\quad\mathcal{W}_{78z}=\varepsilon_{4}~{},$
(4.27a) $\displaystyle\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{12\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu}=\overline{\varepsilon}_{1}~{},\quad\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{34\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu}=\overline{\varepsilon}_{2}~{},\quad\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{56\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu}=\overline{\varepsilon}_{3}~{},\quad\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{W}\mkern-1.5mu}\mkern 1.5mu_{78\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu}=\overline{\varepsilon}_{4}~{}.$ (4.27b)

As shown in [31], to obtain the $\varepsilon_{I}$-dependent terms of the
moduli action, it is enough to consider the R-R part of the 3-form field-
strength $\mathcal{W}$, whose vertex operator was discussed in detail in [16].
In the notation for the spin fields we are adopting now, the vertex operator
for the holomorphic components is proportional to

$\displaystyle V_{\mathcal{W}}(z,\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)$
$\displaystyle=\sum_{I}\Big{(}\varepsilon_{I}-\frac{\varepsilon}{2}\Big{)}\,\Big{[}\mkern
1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern
1.5mu_{I}(z)\,S^{+}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}\times\widetilde{S}^{I}(\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu)\,\widetilde{S}^{+}(\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu)\,\mathrm{e}^{-\frac{1}{2}\widetilde{\varphi}(\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)}$ (4.28)
$\displaystyle\qquad\qquad\quad-S^{I}(z)\,S^{+}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}\times\widetilde{\mkern
1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern 1.5mu}_{I}(\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu)\,\widetilde{S}^{+}(\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu)\,\mathrm{e}^{-\frac{1}{2}\widetilde{\varphi}(\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)}\Big{]}$

where the tilded and untilded operators denote the right and left-moving parts
respectively. On the other hand, the vertex for the anti-holomorphic part is
proportional to

$\displaystyle V_{\overline{\mathcal{W}}}(z,\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)$
$\displaystyle=\sum_{I<J}\overline{\varepsilon}_{IJ}\,\Big{[}\mkern
1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern
1.5mu_{IJ}(z)\,S^{-}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}\times\widetilde{S}^{IJ}(\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu)\,\widetilde{S}^{-}(\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu)\,\mathrm{e}^{-\frac{1}{2}\widetilde{\varphi}(\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)}\Big{]}$ (4.29)

where $\overline{\varepsilon}_{IJ}$ is defined in analogy with (4.22). Setting
$\varepsilon=\overline{\varepsilon}=0$ and computing mixed open/closed string
amplitudes on disks with insertions of the moduli vertex operators on the
boundary and of the vertex operators $V_{\mathcal{W}}$ and
$V_{\overline{\mathcal{W}}}$ in the interior 121212See for example [44] for
details on mixed amplitudes involving R-R backgrounds., one can correctly
reproduce all terms in (4.26) that depend on $\varepsilon_{I}$ and
$\overline{\varepsilon}_{I}$. If we release the constraint $\varepsilon=0$,
the following extra term

$\frac{1}{g_{0}^{2}}\,\mathrm{tr}\,\big{\\{}\mathrm{i}\,\varepsilon\,(\lambda\,\eta-D\,\xi)\big{\\}}$
(4.30)

is generated by the vertex (4.28). As we will see in the next section, this
term plays a crucial role when one integrates over the instanton moduli space
and its presence turns out to be necessary in the intermediate steps, even if
the end result will not exhibit any _explicit_ dependence on $\varepsilon$.
The full moduli action we will consider is therefore

$S_{\text{inst}}^{\prime}=\frac{1}{g_{0}^{2}}\,\Big{(}S^{\prime}_{G}+\mathrm{tr}\,\big{\\{}\mathrm{i}\,\varepsilon\,(\lambda\,\eta-D\,\xi)\big{\\}}\Big{)}+S^{\prime}_{K}+S^{\prime}_{D}$
(4.31)

with $S^{\prime}_{G}$, $S^{\prime}_{K}$ and $S^{\prime}_{D}$ given in (4.26).

###  4.3 The BRST structure of the moduli spectrum

We now show that all moduli, except for $\chi$, can be organized in doublets
with respect to a BRST charge $\mathcal{Q}$ which is a real combination of two
super-symmetries preserved in the brane system. More specifically, we take

$\mathcal{Q}=\frac{Q+\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}{\sqrt{2}}$ (4.32)

where

$Q=\oint\frac{dz}{2\pi\mathrm{i}}\,S(z)\,S^{-}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}\qquad\mbox{and}\qquad\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu=\oint\frac{dz}{2\pi\mathrm{i}}\,\mkern
1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern
1.5mu(z)\,S^{-}(z)\,\mathrm{e}^{-\frac{1}{2}\varphi(z)}$ (4.33)

with $S(z)$ and $\mkern 1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern
1.5mu(z)$ given in (A.22).

To obtain a BRST transformation of the form

$\delta X\,\equiv\,\mathcal{Q}\,X=Y~{},$ (4.34)

we can follow the methods described in [6, 16], namely we evaluate the
commutator of $\mathcal{Q}$ with the vertex operator for $Y$ and from the
resulting expression read the form of the BRST variation according to

$\Big{[}\mathcal{Q},V_{Y}(w)\Big{]}=V_{\delta X}(w)~{}.$ (4.35)

We can explicitly verify that there is no fermionic vertex operator which
under the action of $\mathcal{Q}$ can reproduce the structure of the vertex
for $\chi$. This implies that

$\mathcal{Q}\,\chi=0~{},$ (4.36)

so that the interpretation of $\chi$ as a parameter for the U($k$)
transformations can be maintained. On the contrary, we find

$\mathcal{Q}\,\xi=\eta~{}.$ (4.37)

The choice of the BRST charge $\mathcal{Q}$ introduces therefore an asymmetry
between $\chi$ and $\xi$, which up to this point played a quite similar role.

To obtain the BRST transformation of $Y$, we impose the nilpotency of
$\mathcal{Q}$ up to symmetries, namely

$\mathcal{Q}\,Y=\mathcal{Q}^{2}\,X=\mathsf{T}_{(\chi,a,m,\varepsilon_{I})}[\,X\,]~{}.$
(4.38)

Following this method, for the moduli of the $(-1)/(-1)$ sector we explicitly
find

$\displaystyle\mathcal{Q}\,B^{I}=M^{I}~{},\qquad\mathcal{Q}\,M^{I}=-\big{[}\chi,B^{I}\big{]}+\varepsilon_{I}\,B^{I}~{},$
(4.39) $\displaystyle\mathcal{Q}\,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}=\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{I}~{},\qquad\mathcal{Q}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I}=-\big{[}\chi,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}\big{]}-\varepsilon_{I}\,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu^{I}~{},$
$\displaystyle\mathcal{Q}\,\lambda^{IJ}=D^{IJ}~{},\quad~{}\mathcal{Q}\,D^{IJ}=-\big{[}\chi,\lambda^{IJ}\big{]}+\varepsilon_{IJ}\,\lambda^{IJ}~{},$
$\displaystyle\mathcal{Q}\,\lambda=D~{},\qquad\quad~{}\mathcal{Q}\,D=-\big{[}\chi,\lambda\big{]}~{},$
$\displaystyle\mathcal{Q}\,\xi=\eta~{},\qquad\quad~{}~{}\mathcal{Q}\,\eta=-\big{[}\chi,\xi\big{]}~{}.$

with the understanding the repeated indices are not summed. For the mixed
moduli, instead, we obtain

$\displaystyle\mathcal{Q}\,w=\mu~{},\qquad\mathcal{Q}\,\mu=-\chi\,w+w\,a~{},$
(4.40) $\displaystyle\mathcal{Q}\,\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu=\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu~{},\qquad\mathcal{Q}\,\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu=\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu\,\chi-a\,\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu~{},$
$\displaystyle\mathcal{Q}\,\mu^{\prime}=h^{\prime}~{},\quad~{}~{}\mathcal{Q}\,h^{\prime}=-\chi\,\mu^{\prime}+\mu^{\prime}\,m~{},$
$\displaystyle\mathcal{Q}\,\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu^{\prime}=\mkern
1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern
1.5mu^{\prime}~{},\quad~{}~{}\mathcal{Q}\,\mkern
1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu^{\prime}=\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu^{\prime}\,\chi-m\,\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu^{\prime}~{}.$

These BRST transformations allow us to build doublets of the form
$(X,\mathcal{Q}\,X)$. They are

$(B^{I},M^{I})~{},~{}~{}~{}(\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I},\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{I})~{},~{}~{}~{}(\xi,\eta)~{},~{}~{}~{}(\lambda^{IJ},D^{IJ})~{},~{}~{}~{}(\lambda,D)$
(4.41)

in the $(-1)/(-1)$ sector, and

$(w,\mu)~{},~{}~{}~{}(\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern
1.5mu,\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu)~{},~{}~{}~{}(\mu^{\prime},h^{\prime})~{},~{}~{}~{}(\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu^{\prime},\mkern
1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu^{\prime})$ (4.42)

in the mixed sectors. Furthermore, we can check that the action (4.31) is BRST
invariant and $\mathcal{Q}$-exact. Indeed,

$\mathcal{Q}\,S^{\prime}_{\mathrm{inst}}=0~{},\quad
S^{\prime}_{\mathrm{inst}}=\mathcal{Q}\,\Lambda$ (4.43)

where

$\displaystyle\Lambda$
$\displaystyle=\mathrm{tr}\,\bigg{\\{}\,\frac{1}{g_{0}^{2}}\,\Big{(}\frac{1}{4}\,\lambda^{IJ}D^{KL}\,\epsilon_{IJKL}+\frac{1}{2}\,\lambda\,D-\frac{1}{2}\,\eta\big{[}\chi,\xi\big{]}-\mathrm{i}\,\varepsilon\,\lambda\,\xi\Big{)}$
(4.44) $\displaystyle\qquad~{}~{}~{}+\xi\Big{(}\big{[}B^{I},\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I}\big{]}+\big{[}\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I},M^{I}\big{]}+w\,\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu-\mu\,\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern
1.5mu\Big{)}+\overline{a}\big{(}\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu\,\mu-\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu\,w\big{)}$
$\displaystyle\qquad~{}~{}~{}-\lambda^{IJ}\Big{(}\big{[}\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I},\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{J}\big{]}+\frac{1}{2}\,\epsilon_{IJKL}\,\big{[}B^{K},B^{L}\big{]}\Big{)}-\mathrm{i}\,\lambda\Big{(}\big{[}B^{I},\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}\big{]}+w\,\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu\Big{)}$
$\displaystyle\qquad~{}~{}~{}-\sum_{I}\overline{\varepsilon}_{I}\,\big{(}B^{I}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I}-M^{I}\,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}\big{)}+\frac{1}{2}\,\mu^{\prime}\,\mkern
1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern
1.5mu^{\prime}+\frac{1}{2}\,h^{\prime}\,\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu^{\prime}\bigg{\\}}~{}.$

We observe that the dependence of the moduli action on $\overline{a}$,
$\overline{\varepsilon}_{I}$ and $\varepsilon$ is entirely encoded in the
fermion $\Lambda$ on which the BRST charge acts. This fact will be important
in the following, and in particular it suggests that the end-result of the
localization process will not carry any _explicit_ dependence on these
parameters.

##  5 The instanton partition function

The effects of the D-instantons in the D7/D7′ system are encoded in the gran-
canonical partition function

$Z=\sum_{k=0}^{\infty}q^{k}\,Z_{k}$ (5.1)

where $q=\mathrm{e}^{2\pi\mathrm{i}\tau}$ is the instanton fugacity and
$Z_{k}$ is the partition function in the $k$-instanton sector (with the
convention that $Z_{0}=1$). The latter is defined by the following integral

$Z_{k}=\int\\!d\mathcal{M}_{k}~{}\mathrm{e}^{-S^{\prime}_{\mathrm{inst}}}~{}.$
(5.2)

where $\mathcal{M}_{k}$ denotes the set of moduli with $k$ instantons.
Actually, as we have seen in the previous section, all such moduli except for
$\chi$ are paired in BRST doublets each one containing a boson and a fermion.
We have in fact the following structure

$d\mathcal{M}_{k}=d\chi(dB^{I}\,dM^{I})(d\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}\,d\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{I})(d\lambda^{IJ}\,dD^{IJ})(d\lambda\,dD)(d\xi\,d\eta)(dw\,d\mu)(d\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu\,d\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu)(d\mu^{\prime}\,dh^{\prime})(d\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu^{\prime}\,d\mkern
1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu^{\prime})~{}.$ (5.3)

Therefore, any rescaling of the doublets cancels between bosons and fermions
and does not affect the end result. We exploit this feature and rescale all
doublets with a factor of $1/x$. In the limit $x\to\infty$ a drastic
simplification occurs since only the quadratic terms of the action
$S^{\prime}_{\mathrm{inst}}$ are relevant in the limit, while the cubic and
quartic terms become subleading and can be neglected. In other words the
moduli action becomes

$\displaystyle S^{\prime}_{\mathrm{inst}}$
$\displaystyle=\frac{1}{x^{2}g_{0}^{2}}\,\mathrm{tr}\,\bigg{\\{}\,\frac{1}{4}\,D^{IJ}D^{KL}\,\epsilon_{IJKL}+\frac{1}{2}\,D^{2}+\frac{1}{2}\big{[}\chi,\xi\big{]}^{2}+\frac{1}{2}\,\lambda\big{[}\chi,\lambda\big{]}+\frac{1}{2}\,\eta\big{[}\chi,\eta\big{]}$
$\displaystyle\qquad\qquad\quad+\frac{1}{4}\,\lambda^{IJ}\big{[}\chi,\lambda^{KL}\big{]}\,\epsilon_{IJKL}+\sum_{I<J,K<L}\\!\\!\varepsilon_{IJ}\,\lambda^{IJ}\,\lambda^{KL}\,\epsilon_{IJKL}+\mathrm{i}\,\varepsilon\,(\lambda\,\eta-D\,\xi)\bigg{\\}}$
$\displaystyle~{}~{}~{}+\frac{1}{x^{2}}\,\mathrm{tr}\,\bigg{\\{}\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu^{\prime}\,\big{(}\chi-m\big{)}\,\mu^{\prime}+\mkern
1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern
1.5mu^{\prime}\,h^{\prime}-2\,\mkern
1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5mu\,\big{(}\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu\,(\chi-a)\,w-\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu\,\mu\big{)}$ (5.4)
$\displaystyle\qquad\qquad\quad+2\sum_{I}\overline{\varepsilon}_{I}\Big{(}B^{I}\,[\chi,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}]+\,\varepsilon_{I}\,B^{I}\,\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}-\,M^{I}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{I}\Big{)}\bigg{\\}}+O(x^{-3})~{}.$

Notice that this quadratic action is invariant under the transformations
(4.23) even for $\varepsilon\not=0$. In other words, in the scaling limit we
are considering, the SU(4) symmetry of the initial action could be promoted to
a U(4) symmetry.

The integrations over all moduli but $\chi$ are straightforward since all
integrals are Gaussian and can be readily performed. From the primed doublets
$(\mu^{\prime},h^{\prime})$ and $(\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu^{\prime},\mkern
1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu^{\prime})$ we obtain
131313Here and in the following we report the results of the integrations up
to numerical factors that can always be reabsorbed in the overall
normalization of the instanton partition function.

$\prod_{r=1}^{M}\mathrm{Pf}_{\mathbf{k}}\big{(}\chi-m_{r}\big{)}$ (5.5)

where $m_{r}$ are the eigenvalues of $m$ and the Pfaffian is computed in the
fundamental representation of U($k$). Integrating over the mixed doublets
$(w,\mu)$ and $(\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern
1.5mu,\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu)$ yields

$\prod_{u=1}^{N}\Big{(}\mathrm{Det}_{\mathbf{k}}\big{(}\chi-
a_{u}\big{)}\Big{)}^{-1/2}$ (5.6)

where $a_{u}$ are the eigenvalues of $a$ and the determinant is computed in
the fundamental representation of U($k$). We observe that in order to get this
result it is essential to have the $\overline{a}$ parameter in the action
which provides an effective “mass” for the fermions $\mu$ and $\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu$. However, this
$\overline{a}$ dependence cancels out in the final result between the
numerator and the denominator.

The integration over the adjoint doublet $(\lambda^{IJ},D^{IJ})$ provides

$\prod_{I<J=1}^{3}\mathrm{Pf}_{\mathbf{adj}}\big{(}\chi-\varepsilon_{IJ}\big{)}$
(5.7)

while from the integration over $(B^{I},M^{I})$ and $(\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I},\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I})$ we get

$\prod_{I=1}^{4}\Big{(}\mathrm{Det}_{\mathbf{adj}}\big{(}\chi-\varepsilon_{I}\big{)}\Big{)}^{-1/2}~{}.$
(5.8)

In this case it is essential to have the $\overline{\varepsilon}_{I}$ terms in
the action since they provide a “mass” for the fermions $M^{I}$ and $\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{I}$. This is necessary
to guarantee a non-vanishing result but, as before, the final answer (5.8)
does not depend on these parameters.

The last integrations to be performed are those on the doublets $(\lambda,D)$
and $(\xi,\eta)$. One can explicitly verify that these Gaussian integrations
yield just a non-vanishing numerical constant. In fact, the entire dependence
on $\chi$ and $\varepsilon$ cancels between the contribution of the fermions
$\lambda$ and $\eta$ and the contribution of the bosons $D$ and $\xi$. Again
it is important in the intermediate steps to keep the
$\varepsilon$-dependence; in particular the term $\varepsilon\,\lambda\,\eta$
in the action acts as an effective “mass” for the components of $\lambda$ and
$\eta$ along the Cartan directions and it is crucial to have such a mass in
order to have a non-zero result. However, the final result does not depend on
$\varepsilon$, which can therefore be safely set everywhere to zero at the end
or kept at a non-zero value, if one wishes to do so.

Collecting all terms, we have

$Z_{k}=\int
d\chi~{}~{}\frac{\prod_{r=1}^{M}\mathrm{Pf}_{\mathbf{k}}\big{(}\chi-
m_{r}\big{)}}{\prod_{u=1}^{N}\Big{(}\mathrm{Det}_{\mathbf{k}}\big{(}\chi-
a_{u}\big{)}\Big{)}^{1/2}}~{}\frac{\prod_{I<J=1}^{3}\mathrm{Pf}_{\mathbf{adj}}\big{(}\chi-\varepsilon_{IJ}\big{)}}{\prod_{I=1}^{4}\Big{(}\mathrm{Det}_{\mathbf{adj}}\big{(}\chi-\varepsilon_{I}\big{)}\Big{)}^{1/2}}~{}.$
(5.9)

Exploiting the invariance of the integrand under U($k$) transformations, we
can bring $\chi$ to the Cartan sub-algebra and, after introducing the
Vandermonde determinant, integrate over the $k$ eigenvalues $\chi_{i}$
($i=1,\ldots,k)$. Thus, we can rewrite (5.9) in the following way

$Z_{k}=\int\prod_{i=1}^{k}d\chi_{i}~{}\prod_{i=1}^{k}\frac{\prod_{r=1}^{M}(\chi_{i}-m_{r})}{\prod_{u=1}^{N}(\chi_{i}-a_{u})}\,\prod_{i,j=1}^{k}\frac{\chi_{ij}^{1-\delta_{ij}}\,\prod_{I<J=1}^{3}(\chi_{ij}-\varepsilon_{IJ})}{\prod_{I=1}^{4}(\chi_{ij}-\varepsilon_{I})}$
(5.10)

with $\chi_{ij}=\chi_{i}-\chi_{j}$. For $\varepsilon=0$, the formula (5.10)
agrees with the one proposed in [29] where it was conjectured to possibly
derive from a system with branes and anti-branes. This instanton partition
function has been recently considered also in [30, 32].

From now on we set $\varepsilon=0$ and for simplicity present explicit results
only for $N=M$. In this case the instanton partition function $Z_{k}$ becomes
dimensionless ($\chi_{i}$, $m_{u}$, $a_{u}$ and $\varepsilon_{I}$ have all
dimensions of a mass) and the theory behaves as if it were conformal. With
these positions we can rewrite (5.10) as follows

$Z_{k}=\mathcal{V}^{\,k}\int\prod_{i=1}^{k}d\chi_{i}~{}\prod_{i=1}^{k}\prod_{u=1}^{N}\frac{(\chi_{i}-m_{u})}{(\chi_{i}-a_{u})}\,\prod_{i<j=1}^{k}\frac{\chi_{ij}^{2}\,\prod_{I\not=J=1}^{4}(\chi_{ij}-\varepsilon_{IJ})}{\prod_{I=1}^{4}(\chi_{ij}-\varepsilon_{I})}$
(5.11)

where

$\mathcal{V}=-\frac{(\varepsilon_{1}+\varepsilon_{2})(\varepsilon_{1}+\varepsilon_{3})(\varepsilon_{2}+\varepsilon_{3})}{\varepsilon_{1}\,\varepsilon_{2}\,\varepsilon_{3}\,\varepsilon_{4}}$
(5.12)

with $\varepsilon_{4}=-\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}$. Even
if it is not at all obvious, we will show that $Z_{k}$ in (5.11) depends on
$a_{u}$, $m_{u}$ and $\varepsilon_{I}$ only through the combination
$\mathcal{V}\mathcal{M}$ where

$\mathcal{M}=\sum_{u=1}^{N}(m_{u}-a_{u})~{}.$ (5.13)

Moreover, we will find that our explicit results for $Z_{k}$ are compatible
with the small $q$ expansion of

$Z=\big{[}M(-q)\big{]}^{\mathcal{V}\mathcal{M}}$ (5.14)

where

$M(q)=\prod_{n=1}^{\infty}\frac{1}{(1-q^{n})^{n}}=1+q+3q^{2}+6q^{3}+13q^{4}+24q^{5}\ldots$
(5.15)

is the MacMahon function which counts the number of the planar partitions of
the integers. This result was also derived in [27] using a conjectured
plethystic exponential formula.

From (5.14) we may deduce that the non-perturbative prepotential is

$\mathcal{F}=\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}\varepsilon_{4}\,\log
Z=-(\varepsilon_{1}+\varepsilon_{2})(\varepsilon_{1}+\varepsilon_{3})(\varepsilon_{2}+\varepsilon_{3})\,\mathcal{M}\,\log
M(-q)~{}.$ (5.16)

###  5.1 The $k=1$ case

At $k=1$ the instanton partition function is

$Z_{1}=\mathcal{V}\int\\!d\chi~{}\prod_{u=1}^{N}\frac{(\chi-m_{u})}{(\chi-
a_{u})}~{}.$ (5.17)

As usual, the integral over $\chi$ is computed by moving the poles $a_{u}$ in
the upper-half complex plane and using the residue theorem after closing the
integration path at infinity [21, 20]. The result is simply

$Z_{1}=\mathcal{V}\,\sum_{u=1}^{N}\frac{\prod_{v=1}^{N}(a_{u}-m_{v})\phantom{\big{|}}}{\prod_{v\not=u=1}^{N}(a_{u}-a_{v})\phantom{\big{|}}}=-\mathcal{V}\mathcal{M}~{}.$
(5.18)

###  5.2 Explicit results at higher instanton numbers

To evaluate $Z_{k}$ at a generic instanton number $k$, we follow the
prescription of [21]. This corresponds to move again the poles $a_{u}$ in the
upper-half complex plane, assign to each $\varepsilon_{I}$ a small imaginary
part in such a way that

$\mathrm{Im}(\varepsilon_{4})\gg\mathrm{Im}(\varepsilon_{3})\gg\mathrm{Im}(\varepsilon_{2})\gg\mathrm{Im}(\varepsilon_{1})\gg\mathrm{Im}(a_{u})~{},$
(5.19)

and then integrate over each $\chi_{i}$ in lexicographic order by closing each
contour in the upper-half plane. With this prescription 141414This is fully
equivalent to the so-called Jeffrey-Kirwan prescription [45]., the poles that
contribute to the result can be put in one-to-one correspondence with an
$N$-array of four-dimensional Young tableaux $\\{Y_{u}\\}$ (with
$u=1,\ldots,N$) containing a total number of $k$ boxes. Each box in the
diagram $Y_{u}$ is labeled by four positive integers
$(n_{1},n_{2},n_{3},n_{4})$ and is associated to a pole for one of the
integration variables in (5.10) located at

$\chi^{(n_{1},n_{2},n_{3},n_{4})}=a_{u}+(n_{1}-1)\,\varepsilon_{1}+(n_{2}-1)\,\varepsilon_{2}+(n_{3}-1)\,\varepsilon_{3}+(n_{4}-1)\,\varepsilon_{4}~{}.$
(5.20)

For example, for $N=1$ and $k=2$ we have

$Z_{2}=\mathcal{V}^{2}\\!\int\\!d\chi_{1}d\chi_{2}~{}\,\frac{(\chi_{1}\\!-\\!m)(\chi_{2}\\!-\\!m)(\chi_{1}\\!-\\!\chi_{2})^{2}\prod_{I\not=J=1}^{4}(\chi_{1}\\!-\\!\chi_{2}\\!-\\!\varepsilon_{I}\\!-\\!\varepsilon_{J})}{(\chi_{1}\\!-\\!a)(\chi_{2}\\!-\\!a)(\chi_{1}\\!-\\!\chi_{2}\\!-\\!\varepsilon_{1})(\chi_{1}\\!-\\!\chi_{2}\\!-\\!\varepsilon_{2})(\chi_{1}\\!-\\!\chi_{2}\\!-\\!\varepsilon_{3})(\chi_{1}\\!-\\!\chi_{2}\\!-\\!\varepsilon_{4})}$
(5.21)

and we see that the poles for $(\chi_{1},\chi_{2})$ are at

$(a,a+\varepsilon_{1})~{},\quad(a,a+\varepsilon_{2})~{},\quad(a,a+\varepsilon_{3})~{},\quad(a,a+\varepsilon_{4})~{},$
(5.22)

which indeed correspond to the four four-dimensional Young tableaux with two
boxes. These four tableaux are also associated to the four solid partitions of
2. This is a general feature of the $N=1$ theory in which the poles that
contribute to the integrals of $Z_{k}$ are in one-to-one correspondence with
the solid partitions of $k$.

For $N=2$ and $k=2$, instead, the poles for $(\chi_{1},\chi_{2})$ turn out to
be at

$(a_{1},a_{2})~{},\quad(a_{u},a_{u}+\varepsilon_{1})~{},\quad(a_{u},a_{u}+\varepsilon_{2})~{},\quad(a_{u},a_{u}+\varepsilon_{3})~{},\quad(a_{u},a_{u}+\varepsilon_{4})$
(5.23)

with $u=1,2$. The first pole location corresponds to a pair of tableaux with
one box each; the other locations describe pairs of tableaux, one with two
boxes and the other with no boxes. This example can be systematically
generalized to other values of $N$ and $k$.

The evaluation of the residues is straightforward, even if tedious. When we
sum up all contributions, remarkable simplifications occur and, as
anticipated, the dependence on $a_{u}$, $m_{u}$ and $\varepsilon_{I}$ in the
final result is only through the combination $\mathcal{V}\mathcal{M}$. We have
explicitly checked this fact up to $k=5$, and found

$\displaystyle Z_{2}$
$\displaystyle=\frac{1}{2}\,(\mathcal{V}\mathcal{M})^{2}+\frac{5}{2}\,\mathcal{V}\mathcal{M}~{},$
(5.24) $\displaystyle Z_{3}$
$\displaystyle=-\frac{1}{6}\,(\mathcal{V}\mathcal{M})^{3}-\frac{5}{2}\,(\mathcal{V}\mathcal{M})^{2}-\frac{10}{3}\,\mathcal{V}\mathcal{M}~{},$
$\displaystyle Z_{4}$
$\displaystyle=\frac{1}{24}\,(\mathcal{V}\mathcal{M})^{4}+\frac{5}{4}\,(\mathcal{V}\mathcal{M})^{3}+\frac{155}{24}\,(\mathcal{V}\mathcal{M})^{2}+\frac{21}{4}\,\mathcal{V}\mathcal{M}~{},$
$\displaystyle Z_{5}$
$\displaystyle=-\frac{1}{120}\,(\mathcal{V}\mathcal{M})^{5}-\frac{5}{12}\,(\mathcal{V}\mathcal{M})^{4}-\frac{115}{24}\,(\mathcal{V}\mathcal{M})^{3}-\frac{163}{12}\,(\mathcal{V}\mathcal{M})^{2}-\frac{26}{5}\,\mathcal{V}\mathcal{M}~{}.$

These results allow us to infer the formula (5.14).

Let us remark that, once individuated the poles through the prescription
(5.19), the evaluation of the residues does not introduce any sign ambiguity
in summing up the contributions associated to the various partitions of the
type considered in [27, 29]. A particular sign choice is automatically picked
up, and is exactly the one that leads to the correct exponentiation of the
result. This choice is in agreement with the one pointed out in [30].

We mentioned above that the poles contributing to the integrals of $Z_{k}$ for
the rank 1 theory are in one-to-one correspondence with the solid partitions
of $k$. However, it is the generating function of the planar partitions $M(q)$
that appears in the end result (5.14). To understand this fact, let us
specialize to $N=1$ with $a=0$ and $m=\varepsilon_{4}$, and then take the
limit $\varepsilon_{4}\to 0$. This limit clearly implies that
$\mathcal{V}\mathcal{M}\to 1$. It turns out that in this scaling all
contributions to $Z_{k}$ corresponding to Young tableaux which contain at
least one box with $n_{4}>1$ are set to zero, while all contributions
associated to tableaux in which all boxes have $n_{4}=1$ are uniformly
weighted with $(-1)^{k}$. In other words, in this limit the gran-canonical
partition function (5.1) can be written as a weighted sum over the solid
partitions $\pi$:

$Z=\sum_{\pi}\mathsf{w}_{\pi}\,q^{|\pi|}$ (5.25)

where $|\pi|$ is the size of $\pi$, and

$\mathsf{w}_{\pi}=\begin{cases}&(-1)^{|\pi|}\quad\mbox{if~{}$Y(\pi)$~{}does~{}not~{}extend~{}in~{}the~{}$\varepsilon_{4}$
\,direction}~{},\\\
&~{}~{}~{}~{}0\quad~{}~{}\,\,\mbox{~{}\,if~{}$Y(\pi)$~{}does~{}extend~{}in~{}the~{}$\varepsilon_{4}$
\,direction}~{},\end{cases}$ (5.26)

with $Y(\pi)$ denoting the four-dimensional Young diagram associated to $\pi$.
Given the form of the weight function (5.26), it is clear that only the solid
partitions $\pi$ which are also planar with respect to the
$\varepsilon_{4}$-direction, contribute in the sum. Therefore, (5.25) actually
coincides with the partition function of the planar partitions with uniform
weight (up to an alternating sign), which are counted by the MacMahon
function, i.e.

$Z=M(-q)$ (5.27)

in agreement with (5.14) when $\mathcal{V}\mathcal{M}=1$. This argument can be
extended to more general cases along the lines discussed in [28].

##  6 ADHM construction

We have anticipated that the physical states of the open strings with at least
one end-point on a D(–1)-brane can be put in correspondence with the moduli of
instantonic configurations in eight dimensions. In this section we would like
to elaborate on this.

There are different definitions of eight-dimensional gauge instantons, which
are briefly reviewed in a similar (although not identical) brane context in
[34]. The definition appropriate to our current set-up is the one in which
these configurations correspond to Yang Mills connections whose field strength
satisfies the (anti) self-duality condition

${}^{*}\left(F\wedge F\right)=\pm\,F\wedge F~{}.$ (6.1)

When this condition holds, the quartic action $S_{4}$ for the gauge fields on
the D7 branes, given in (2.4), drastically simplifies. Indeed, in this case

$\operatorname{tr}\big{(}t_{8}\,F^{4}\big{)}=\pm\frac{1}{2}\operatorname{tr}(F\wedge
F\wedge F\wedge F)~{},$ (6.2)

and hence

$S_{4}=\begin{cases}-2\pi\mathrm{i}\,\mkern
1.5mu\overline{\mkern-1.5mu\tau\mkern-1.5mu}\mkern 1.5mu\,k~{},\\\
-2\pi\mathrm{i}\,\tau\,k~{},\end{cases}$ (6.3)

where $\tau$ is defined in (2.10) and

$k=\frac{1}{4!(2\pi)^{4}}\int\operatorname{tr}\left(F\wedge F\wedge F\wedge
F\right)$ (6.4)

is the fourth Chern number. This precisely matches the action of $k$ anti
D-instantons or $k$ D-instantons, respectively.

In analogy with the familiar case of the four-dimensional instanton, one way
to realize the conditions (6.1) is to exploit the duality properties of the
eight-dimensional Dirac matrices. Indeed, the chiral blocks $\sigma^{\mu\nu}$
(see (A.16)), satisfy

$\displaystyle\sigma^{[\mu_{1}\mu_{2}}\,\sigma^{\mu_{3}\mu_{4}]}=+\frac{1}{4!}\,\epsilon^{\mu_{1}\mu_{2}\ldots\mu_{7}\mu_{8}}\,\sigma_{\mu_{5}\mu_{6}}\,\sigma_{\mu_{7}\mu_{8}}$
(6.5)

where in the left-hand side the indices are all anti-symmetrized, while the
anti-chiral blocks $\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern 1.5mu^{\mu\nu}$ obey

$\displaystyle\mkern 1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern
1.5mu^{[\mu_{1}\mu_{2}}\,\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern
1.5mu^{\mu_{3}\mu_{4}]}=-\frac{1}{4!}\,\epsilon^{\mu_{1}\mu_{2}\ldots\mu_{7}\mu_{8}}\,\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern
1.5mu_{\mu_{5}\mu_{6}}\,\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern
1.5mu_{\mu_{7}\mu_{8}}~{}.$ (6.6)

A field strength $F_{\mu\nu}$ proportional to $\sigma_{\mu\nu}$ or to
$\overline{\sigma}_{\mu\nu}$ would thus enjoy the property (6.1).

To obtain such a field strength we can follow the ADHM construction in four
dimensions (for definiteness we choose the minus sign in (6.1)). We first
introduce the ADHM matrix 151515Since the construction involves matrices of
various sizes, sometimes we find useful to indicate explicitly their sizes
with the notation $A_{[n\times n]}$ for a matrix $A$ of size $n\times n$.

$\Delta=\begin{pmatrix}\alpha_{[N\times 8k]}\\\\[5.69054pt] (B-x)_{[8k\times
8k]}\end{pmatrix}$ (6.7)

where $\alpha$ is a $N\times 8k$ matrix, while $B$ and $x$ are $8k\times 8k$
matrices given by

$\displaystyle B=(B_{\mu})_{[k\times k]}\otimes\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern 1.5mu^{\mu}~{},\qquad
x=x_{\mu}\,\mathbb{1}_{[k\times k]}\otimes\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern 1.5mu^{\mu}~{}.$ (6.8)

The matrices $\alpha$ and $B_{\mu}$ contain the bosonic moduli of the
configuration, while $x_{\mu}$ are the coordinates in eight dimensions. Then,
we define the gauge connection $A_{\mu}$, expressed as a $N\times N$ matrix,
according to

$A_{\mu}={U}^{\dagger}\partial_{\mu}U~{},$ (6.9)

where

$U=\begin{pmatrix}u_{[N\times N]}\\\ v_{[8k\times N]}\\\ \end{pmatrix}$ (6.10)

is a $(N+8k)\times N$ matrix subject to the following conditions

$\Delta^{\dagger}U=U^{\dagger}\Delta=0~{},\quad{U}^{\dagger}\,U=\mathbb{1}_{[N\times
N]}~{}.$ (6.11)

The corresponding field strength is

$\displaystyle F_{\mu\nu}$
$\displaystyle=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+\big{[}A_{\mu},A_{\nu}\big{]}=2\,\partial_{[\mu}U^{\dagger}\,\partial_{\nu]}U+2\,U^{\dagger}\partial_{[\mu}U\,{U}^{\dagger}\partial_{\nu]}U$
(6.12)
$\displaystyle=2\,\partial_{[\mu}{U}^{\dagger}\big{(}\mathbb{1}-U{U}^{\dagger}\big{)}\,\partial_{\nu]}U~{}.$

We now introduce a $8k\times 8k$ matrix $M$ such that

$\mathbb{1}-U{U}^{\dagger}=\Delta\,M\,\Delta^{\dagger}$ (6.13)

or equivalently, thanks to the conditions (6.11), such that

$\Delta^{\dagger}\Delta=M^{-1}~{}.$ (6.14)

Plugging (6.13) into (6.12) leads to

$\displaystyle F_{\mu\nu}$
$\displaystyle=2\,\partial_{[\mu}U^{\dagger}\Delta\,M\,\Delta^{\dagger}\,\partial_{\nu]}U=2\,U^{\dagger}\,\partial_{[\mu}\Delta\,M\,\partial_{\nu]}\Delta^{\dagger}\,U$
(6.15) $\displaystyle=2\,U^{\dagger}\begin{pmatrix}0&0\\\ 0&\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern
1.5mu_{[\mu}\,M\,\sigma_{\nu]}\end{pmatrix}U=~{}2\,\mkern
1.5mu\overline{\mkern-1.5muv\mkern-1.5mu}\mkern 1.5mu\,\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern
1.5mu_{[\mu}\,M\,\sigma_{\nu]}\,v$

where in the second line, with an abuse of notation, we have denoted
$\mathbb{1}_{[k\times k]}\otimes(\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern 1.5mu_{\mu})_{[8\times
8]}$ and $\mathbb{1}_{[k\times k]}\otimes(\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern 1.5mu_{\mu})_{[8\times
8]}$ simply as $\mkern 1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern
1.5mu_{\mu}$ and $\sigma_{\nu}$.

If $N\geq 8$, it is possible to require that the $N\times N$ matrix $2\,\mkern
1.5mu\overline{\mkern-1.5muv\mkern-1.5mu}\mkern 1.5mu\,\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern
1.5mu_{[\mu}\,M\,\sigma_{\nu]}\,v$ has a single non-trivial block of size
$(8\times 8)$ proportional to $\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern 1.5mu_{\mu\nu}$. If this
is the case, then, the field strength (6.15) satisfies the relation (6.1) with
the minus sign as a consequence of the anti-self duality property (6.6). Of
course, this requirement puts stringent constraints on the form of the
matrices $M$ and $\Delta$. For $k=1$, a solution is obtained by taking

$\displaystyle M$ $\displaystyle=f\,\mathbb{1}_{[8\times
8]}~{},\quad\alpha=\begin{pmatrix}\rho\,\mathbb{1}_{[8\times 8]}\\\
0_{\,[(N-8)\times 8]}\end{pmatrix}~{},\quad\mkern
1.5mu\overline{\mkern-1.5muv\mkern-1.5mu}\mkern
1.5mu=f^{\frac{1}{2}}\,\alpha~{},$ (6.16) $\displaystyle u$
$\displaystyle=f^{\frac{1}{2}}\begin{pmatrix}-(B_{\mu}-x_{\mu})\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern
1.5mu^{\mu}&0_{[8\times(N-8)]}\\\ 0_{[(N-8)\times
8]}&0_{[(N-8)\times(N-8)]}\end{pmatrix}$

with $f=(\rho^{2}+r^{2})^{-1}$ and $r^{2}=(x_{\mu}-B_{\mu})(x^{\mu}-B^{\mu})$.
Through (6.15), this leads to the following field strength

$F_{\mu\nu}=\frac{2\rho^{2}}{(\rho^{2}+r^{2})^{2}}\begin{pmatrix}\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern
1.5mu_{\mu\nu}&0_{[8\times(N-8)]}\\\ 0_{[(N-8)\times
8]}&0_{[(N-8)\times(N-8)]}\end{pmatrix}~{}.$ (6.17)

This represents an instanton solution of size $\rho$ and center $B_{\mu}$
which corresponds to the embedding of the SO(8) octonionic field strength of
[46, 47] into U($N$) (with $N\geq 8$). One can check that the gauge connection
corresponding to (6.17) is

$A_{\mu}=\frac{(x^{\nu}-B^{\nu})}{\rho^{2}+r^{2}}\begin{pmatrix}\mkern
1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern
1.5mu_{\nu\mu}&0_{[8\times(N-8)]}\\\ 0_{[(N-8)\times
8]}&0_{[(N-8)\times(N-8)]}\end{pmatrix}~{},$ (6.18)

and that the fourth Chern number (6.4) is one [46, 47]. Generalizations to
higher $k$ are possible.

In a supersymmetric theory, like the one we are considering, instantons
preserve a fraction of supersymmetry. A supersymmetric instanton is then
characterized by the existence of a Killing spinor $\mathfrak{e}$ such that
the supersymmetry variation of the gaugino $\Lambda$ vanishes, namely

$\delta\Lambda=\frac{1}{2}\,F_{\mu\nu}\,\gamma^{\mu\nu}\,\mathfrak{e}=0~{},$
(6.19)

which reduces to

$F_{\mu\nu}\,\sigma^{\mu\nu}\,\mathfrak{e}=0$ (6.20)

for a chiral $\mathfrak{e}$. In [32] it has been shown that to solve this
equation it is enough to require that

$\Delta^{\dagger}\Delta\,\mathfrak{e}=f^{-1}_{[k\times k]}\otimes\mathfrak{e}$
(6.21)

for some $(k\times k)$ matrix $f$. This means that the matrix $M$ introduced
in (6.14) must be such that $M\,\mathfrak{e}=f_{[k\times
k]}\otimes\mathfrak{e}$.

To make contact with the previous sections, we write the matrices $B$ and
$\alpha$ appearing in the ADHM matrix $\Delta$ in terms of the bosonic moduli
corresponding to the open strings in the D(–1)/D7 brane system. In particular
we take the eight $k\times k$ matrices $B_{\mu}$, used to construct $B$ as in
(6.8), to be given by the neutral moduli $B^{I}$ and $\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{I}$ of the D(–1)/D(–1)
sector, and take the $N\times 8k$ matrix $\alpha$ to be related to the mixed
moduli $\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu$ of the
D7/D(–1) sector according to $\alpha=\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu_{[N\times
k]}\otimes\psi^{\dagger}_{[1\times 8]}$ where $\psi$ a reference Weyl chiral
spinor [32]. Given the choices we made in the string construction, we take

$\psi=\mathfrak{e}+\mathrm{i}\,\mathfrak{e}^{\prime}$ (6.22)

with $\mathfrak{e}=\delta_{a,8}$ and $\mathfrak{e}^{\prime}=\delta_{a,7}$. The
choice on the Killing spinor $\mathfrak{e}$ reflects our choice of the
preserved supersymmetry (and hence of the BRST charge) which, in turn, is
related to how SO(8) has been broken to SO(7). The second choice on
$\mathfrak{e}^{\prime}$ reflects the choice of the complex structure with
which we have broken SO(7) to SU(4) (see Appendix A). With these positions it
is easy to verify that

$\psi\,\psi^{\dagger}\,\mathfrak{e}=-\tau^{7}\,\mathfrak{e}+\mathfrak{e}\quad\mbox{and}\quad\sigma^{\mu\nu}\,\mathfrak{e}=-(\tau_{m})^{\mu\nu}\,\tau^{m}\,\mathfrak{e}$
(6.23)

where $\tau^{m}$ are the seven octonionic matrices defined in (A.13).
Furthermore, we have

$\displaystyle\Delta^{\dagger}\Delta$
$\displaystyle=\alpha^{\dagger}\alpha+(B^{\dagger}-x^{\dagger})(B-x)$ (6.24)
$\displaystyle=w\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern
1.5mu\otimes\psi\psi^{\dagger}+\frac{1}{2}\,\big{[}B_{\mu}\,,\,B_{\nu}]\otimes\sigma^{\mu\nu}+(B_{\mu}-x_{\mu})(B^{\mu}-x^{\mu})\otimes\mathbb{1}_{[8\times
8]}~{},$

and thus projecting onto $\mathfrak{e}$ and using the relations (6.23), we
deduce that in order to satisfy (6.21) one must require that

$\frac{1}{2}\,\big{[}B_{\mu}\,,\,B_{\nu}](\tau_{m})^{\mu\nu}+w\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu\,\delta_{m}^{7}=0~{}.$
(6.25)

In complex notation, these equations read

$\big{[}B^{I},\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I}\big{]}+w\,\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern
1.5mu=0~{},\qquad\big{[}\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{I},\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{J}\big{]}+\frac{1}{2}\,\epsilon_{IJKL}\,\big{[}B^{K},B^{L}\big{]}=0~{},$
(6.26)

which are exactly the constraints reported in (4.17) and derived from the
moduli action in the D(–1)/D7-brane system.

We observe that for $k=1$ the first constraint leads to $w=\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu=0$. This implies that
the ADHM data reduce to those in (6.16) for $\rho=0$, thus describing a point-
like instanton configuration with vanishing size. It would be interesting to
explore what happens if one introduces a non-commutative deformation parameter
in such a way that the right-hand side of the real constraint in (6.26)
differs from zero, thus allowing to have configurations with non-zero size,
and more generally to investigate the features of the solutions of the ADHM
constraints in the general case. We leave these issues to future
investigations.

##  7 Conclusions

In this paper we have studied a D(–1)/D7-brane system in Type IIB string
theory that describes the non-perturbative sector of a U($N$) gauge theory in
eight dimensions, and provided an explicit analysis of the massless open
string states and of their vertex operators in the various sectors. The new
ingredient with respect to previous work is the introduction of a constant
magnetic flux on the world-volume of the D7-branes which allows for the
existence of bosonic moduli in the mixed sectors. After discussing the moduli
action, we have computed the instanton partition function using localization
and confirmed the results of [27] obtaining a closed form expression in terms
of the MacMahon function.

It is important to remark that this instanton partition function and the
corresponding prepotential given in (5.16) involve only the fields of the
overall U(1) factor inside U($N$). Furthermore, the prepotential is cubic in
the $\varepsilon_{I}$ parameters used in the localization process, showing
that the gauge effective action receives non-perturbative corrections only in
curved space. To see this in an explicit way, let us first turn off the
parameters $m_{u}$ so that the quantity $\mathcal{M}$ in (5.13) simply reduces
to $\operatorname{tr}\phi$, where $\phi$ is the scalar component of the vector
superfield (see (2.8)). Then let us note that

$(\varepsilon_{1}+\varepsilon_{2})(\varepsilon_{1}+\varepsilon_{3})(\varepsilon_{2}+\varepsilon_{3})\,\sim\,t_{8}^{\mu_{1}\ldots\mu_{8}}\,\mathcal{W}_{\mu_{1}\mu_{2}}\,\mathcal{W}_{\mu_{3}\mu_{4}}\,\mathcal{W}_{\mu_{5}\mu_{6}}\,\mathcal{I}_{\mu_{7}\mu_{8}}$
(7.1)

where $\mathcal{W}_{\mu\nu}\,\equiv\,\mathcal{W}_{\mu\nu z}$ is the
graviphoton field strength defined in (4.27a). The tensor
$\mathcal{I}_{\mu\nu}$, instead, describes the background magnetic flux on the
D7-branes and is obtained from (2.12) by setting $f_{I}=-1$ for all
$I=1,\ldots 4$ 161616According to (2.14), this corresponds to the choice of
the twist parameters $\theta_{I}=1/4$ that we have made in (4.1).. Finally,
$t_{8}$ is the tensor that typically appears in higher derivative actions,
like for instance (2.4) 171717Given the particular form of
$\mathcal{W}_{\mu\nu}$ and $\mathcal{I}_{\mu\nu}$, one has
$t_{8}^{\mu_{1}\ldots\mu_{8}}\,\mathcal{W}_{\mu_{1}\mu_{2}}\,\mathcal{W}_{\mu_{3}\mu_{4}}\,\mathcal{W}_{\mu_{5}\mu_{6}}\,\mathcal{I}_{\mu_{7}\mu_{8}}\sim
2\,\operatorname{tr}\big{(}\mathcal{W}\,\mathcal{W}\,\mathcal{W}\,\mathcal{I}\big{)}-\frac{1}{2}\,\operatorname{tr}\big{(}\mathcal{W}\,\mathcal{W}\big{)}\,\operatorname{tr}\big{(}\mathcal{W}\,\mathcal{I}\big{)}$
where the trace is over the space-time indices, from which the result (7.1)
easily follows.. Suppressing all indices for simplicity, the prepotential
(5.16) is therefore

$\mathcal{F}\,\sim\,t_{8}\,\mathcal{W}^{3}\,\mathcal{I}\,\operatorname{tr}\phi\,\log
M(-q)~{}.$ (7.2)

To obtain the effective action, we promote $\phi$ to the full scalar
superfield $\Phi(x,\theta)$ given in (2.8) and the graviphoton field strength
$\mathcal{W}_{\mu\nu}$ to the full graviphoton superfield [31]

$W_{\mu\nu}(x,\theta)=\mathcal{W}_{\mu\nu}(x)+\theta\,\chi_{\mu\nu}(x)+\frac{1}{2}\,\theta\sigma^{\lambda\rho}\,\theta\,R_{\mu\nu\lambda\rho}(x)+\ldots$
(7.3)

where $\chi_{\mu\nu}(x)$ and $R_{\mu\nu\rho\sigma}(x)$ are respectively, the
self-dual parts of the gravitino field strength and of the Riemann tensor. In
this way we obtain a prepotential $\mathcal{F}(\Phi,W)$ that is linear in
$\Phi$ and cubic in $W$. The eight-dimensional effective action is then given
by

$\int d^{8}x\,d^{8}\theta\,\mathcal{F}(\Phi,W)+\mathrm{c.c.}$ (7.4)

After integration over the $\theta$’s, our results predict an all-instanton
formula for higher derivative couplings in Type II B supergravity, which are
schematically of the form
$t_{8}\,t_{8}\,R^{3}\operatorname{tr}F\,F_{\mathrm{class}}$, in presence of
D7-branes carrying a non-trivial world-volume flux
$F_{\mathrm{class}}\sim\mathcal{I}$ along a U(1) subgroup of U($N$). It would
be very interesting to further investigate the properties and the implications
of these non-perturbative gravitational couplings, and also to explore the
structure of the instanton partition function of our system when the
graviphoton parameters $\varepsilon_{I}$ do not add up to zero.

Acknowledgments We would like to thank Igor Pesando for helpful discussions.
This research is partly supported by the INFN Iniziativa Specifica ST&FI
“String Theory & Fundamental Interactions”.

##  Appendix A Notations and conventions

In this appendix we collect our notations for the various symmetry groups of
the brane system and for the spinors and Dirac matrices.

### From ten to eight dimensions

The symmetry of the ten-dimensional flat background in which the superstring
theory is defined is broken by the presence of the seven-branes as

$\displaystyle\mathrm{SO}(10)\to\mathrm{SO}(8)\otimes\mathrm{SO}(2)~{},$ (A.1)

with $\mathrm{SO}(8)$ rotating the first eight coordinates $x^{\mu}$. Thus,
the vector representation decomposes as

$\displaystyle\mathbf{10}\to(\mathbf{8}_{v},0)\oplus(\mathbf{1},+1)\oplus(\mathbf{1},-1)~{},$
(A.2)

where each couple $(\mathbf{R},p)$ indicates a representation $\mathbf{R}$ of
$\mathrm{SO}(8)$ and a charge $p$ under $\mathrm{U}(1)\sim\mathrm{SO}(2)$. The
$\mathrm{U}(1)^{4}$ Cartan subgroup of $\mathrm{SO}(8)$ corresponds to the
rotations in the $(x^{2I-1},x^{2I})$ planes, with $I=1,\ldots 4$, and here we
will denote its four parameters as $\varepsilon_{I}$.

Under the breaking (A.1), the 32-components of the ten dimensional spinors
(which can be chiral or anti-chiral) decompose into $S^{A}S^{\pm}$, where
$S^{A}$ are the 16 components of $\mathrm{SO}(8)$ spinors (chiral or anti-
chiral) and $S^{\pm}$ have charge $\pm 1/2$ with respect to $\mathrm{U}(1)$.
More precisely, we have

$\displaystyle\mathbf{16}_{s}$
$\displaystyle\to\big{(}\mathbf{8}_{s},+1/2\big{)}\oplus\big{(}\mathbf{8}_{c},-1/2\big{)}~{},$
$\displaystyle\mathbf{16}_{c}$
$\displaystyle\to\big{(}\mathbf{8}_{s},-1/2\big{)}\oplus\big{(}\mathbf{8}_{c},+1/2\big{)}~{}.$
(A.3)

The ten-dimensional Dirac matrices take the form

$\displaystyle\Gamma^{\mu}=\gamma^{\mu}\otimes\mathbb{1}~{},~{}~{}~{}\Gamma^{9}=\gamma\otimes\sigma_{1}~{},~{}~{}~{}\Gamma^{10}=\gamma\otimes\sigma_{2}~{},$
(A.4)

where $\gamma^{\mu}$ with $\mu=1,2\ldots 8$ are the eight-dimensional Dirac
matrices, $\gamma$ is the eight-dimensional chirality matrix and the sigmas
are the usual Pauli matrices. The ten-dimensional charge conjugation matrix
$\mathcal{C}$ satisfying
$\mathcal{C}\,\Gamma^{M}\left(\mathcal{C}^{-1}\right)^{T}=-\left(\Gamma^{M}\right)^{T}$
for $M=1,\ldots,10$ is written as

$\displaystyle\mathcal{C}=C\otimes\sigma_{2}~{},$ (A.5)

where $C$ is the eight-dimensional charge conjugation which is such that
$C\,\gamma^{\mu}\left(C^{-1}\right)^{T}=-\left(\gamma^{\mu}\right)^{T}$.

### Eight-dimensional spinors

For the eight-dimensional spinors we can use either the “spin field” basis or
the Majorana-Weyl basis, which we are going to describe.

####  $\bullet$ The spin field basis:

In the “spin field” basis the spinor components are labeled by (twice) their
SO(8) weight vectors and are

$\displaystyle S^{A}$
$\displaystyle=\Big{(}S^{++++}\,,\,S^{-+++}\,,\,S^{+-++}\,,\,S^{--++}\,,\,S^{++-+}\,,\,S^{-+-+}\,,\,S^{+--+}\,,\,S^{---+}\,,$
(A.6)
$\displaystyle~{}~{}~{}~{}~{}~{}S^{+++-}\,,\,S^{-++-}\,,\,S^{+-+-}\,,\,S^{--+-}\,,\,S^{++--}\,,\,S^{-+--}\,,\,S^{+---}\,,\,S^{----}\Big{)}~{}.$

In this basis the Dirac matrices are

$\displaystyle\gamma^{1}$
$\displaystyle=\sigma_{1}\otimes\mathbb{1}\otimes\mathbb{1}\otimes\mathbb{1}~{},~{}~{}~{}~{}~{}~{}\gamma^{2}=\sigma_{2}\otimes\mathbb{1}\otimes\mathbb{1}\otimes\mathbb{1}~{},$
(A.7) $\displaystyle\gamma^{3}$
$\displaystyle=\sigma_{3}\otimes\sigma_{1}\otimes\mathbb{1}\otimes\mathbb{1}~{},~{}~{}~{}~{}~{}\gamma^{4}=\sigma_{3}\otimes\sigma_{2}\otimes\mathbb{1}\otimes\mathbb{1}~{},$
$\displaystyle\gamma^{5}$
$\displaystyle=\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{1}\otimes\mathbb{1}~{},~{}~{}~{}\,\gamma^{6}=\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{2}\otimes\mathbb{1}~{},$
$\displaystyle\gamma^{7}$
$\displaystyle=\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{1}~{},~{}~{}\,\gamma^{8}=\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{2}~{}.$

The chirality matrix is

$\gamma=\gamma^{1}\gamma^{2}\gamma^{3}\gamma^{4}\gamma^{5}\gamma^{6}\gamma^{7}\gamma^{8}=\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{3}~{},$
(A.8)

while the charge conjugation matrix reads

$C=\sigma_{2}\otimes\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{1}~{}.$ (A.9)

The chirality matrix is diagonal, and the eight components in (A.6) with an
even number of minuses are chiral, i.e. they span the $\mathbf{8}_{s}$
representation of SO(8), while the eight ones with an odd number of minuses
are anti-chiral, i.e. they span the $\mathbf{8}_{c}$ representation.

####  $\bullet$ The Majorana-Weyl basis:

For $\mathrm{SO}(8)$ it is also possible to use a Majorana-Weyl basis. This
can be achieved introducing the chiral combinations

$\displaystyle\mathcal{S}^{\alpha}=\frac{1}{\sqrt{2}}\begin{pmatrix}\mathrm{i}\,S^{+--+}-\mathrm{i}\,S^{-++-}\\\\[2.84526pt]
-S^{+--+}-S^{-++-}\\\\[2.84526pt]
-\mathrm{i}\,S^{+-+-}-\mathrm{i}\,S^{-+-+}\\\\[2.84526pt]
-S^{+-+-}+S^{-+-+}\\\\[2.84526pt]
-\mathrm{i}\,S^{++--}+\mathrm{i}\,S^{--++}\\\\[2.84526pt]
-S^{++--}-S^{--++}\\\\[2.84526pt]
-\mathrm{i}\,S^{++++}-\mathrm{i}\,S^{----}\\\\[2.84526pt]
S^{++++}-S^{----}\end{pmatrix}~{},$ (A.10)

and the anti-chiral ones

$\displaystyle\mathcal{S}^{\dot{\alpha}}=\frac{1}{\sqrt{2}}\begin{pmatrix}\mathrm{i}\,S^{+---}-\mathrm{i}\,S^{-+++}\\\\[2.84526pt]
-S^{+---}-S^{-+++}\\\\[2.84526pt]
-\mathrm{i}\,S^{+-++}-\mathrm{i}\,S^{-+--}\\\\[2.84526pt]
-S^{+-++}+S^{-+--}\\\\[2.84526pt]
-\mathrm{i}\,S^{++-+}+\mathrm{i}\,S^{--+-}\\\\[2.84526pt]
-S^{++-+}-S^{--+-}\\\\[2.84526pt]
-\mathrm{i}\,S^{+++-}-\mathrm{i}\,S^{---+}\\\\[2.84526pt]
S^{+++-}-S^{---+}\end{pmatrix}~{}.$ (A.11)

In the basis $(\mathcal{S}^{\alpha},\mathcal{S}^{\dot{\alpha}})$, the
chirality matrix is $\mathbb{1}_{8}\otimes\sigma_{3}$, the charge conjugation
is the identity and all Dirac matrices are purely imaginary. They take the
form 181818With an abuse of notation we denote these matrices again by
$\gamma^{\mu}$, even if they do not coincide with the ones in (A.7); rather
they are equivalent to the latter under the change of basis in (A.10) and
(A.11).

$\displaystyle{\gamma}^{m}=\tau^{m}\otimes\sigma_{1}~{},~{}~{}~{}\gamma^{8}=\mathbb{1}_{8}\otimes\sigma_{2}~{},$
(A.12)

where $\tau^{m}$, with $m=1,\ldots,7$, are ($-\mathrm{i}$ times) seven-
dimensional Dirac matrices. These latter have elements

$\displaystyle\left(\tau^{m}\right)_{\alpha\beta}$
$\displaystyle=-\mathrm{i}\left(\delta^{m}_{\alpha}\,\delta^{8}_{\beta}-\delta^{m}_{\beta}\,\delta^{8}_{\alpha}+c_{m\alpha\beta}\right)~{},$
(A.13)

where the completely anti-symmetric tensor $c_{mnp}$ describes the octonionic
structure constants:

$\displaystyle c_{127}=c_{163}=c_{154}=c_{253}=c_{246}=c_{347}=c_{567}=1~{},$
(A.14)