LMU-ASC 31/11
ROM2F/2011/09

Magnetized E3-brane instantons
in F-theory

Massimo<EMAIL_ADDRESS>Andrés
<EMAIL_ADDRESS>&
Luca<EMAIL_ADDRESS>
♡I.N.F.N. Sezione di Roma “TorVergata” &
Dipartimento di Fisica, Università di Roma “TorVergata”,
Via della Ricerca Scienti ca, 00133 Roma, Italy
♢Arnold Sommerfeld Center for Theoretical Physics,
LMU München, Theresienstraße 37, D-80333 München, Germany

Abstract

We discuss E3-brane instantons in ${\cal N}=1$ F-theory compactifications to
four dimensions and clarify the structure of E3-E3 zero modes for general
world-volume fluxes. We consistently incorporate SL$(2,\mathbb{Z})$
monodromies and highlight the relation between F-theory and perturbative IIB
results. We explicitly show that world-volume fluxes can lift certain
fermionic zero-modes, whose presence would prevent the generation of non-
perturbative superpotential terms, and we discuss in detail the geometric
interpretation of the zero-mode lifting mechanism. We provide a IIB derivation
of the index for generation of superpotential terms and of its modification to
include world-volume fluxes, which reproduces and generalizes available
results. We apply our general analysis to the explicit, though very simple,
example of compactification on $\mathbb{P}^{3}$ and its orientifold weak-
coupling limit. In particular, we provide an example in which a non-rigid
divisor with fluxes contributes to the superpotential.

###### Contents

  1. 1 Introduction and summary of the results
  2. 2 A short review on F-theory vacua
    1. 2.1 Generalities on F-theory compactifications
    2. 2.2 Supersymmetric structures
    3. 2.3 Orientifold limit
    4. 2.4 Relation with the M-theory viewpoint
    5. 2.5 A working example
  3. 3 E3-instantons without fluxes: the IIB perspective
    1. 3.1 Fluxless E3-fermionic action
    2. 3.2 Zero-modes
    3. 3.3 M5-brane index from IIB perspective
    4. 3.4 Comparison with the dual M-theory result
    5. 3.5 $\chi_{\rm E3}$ as the perturbative Ext index
    6. 3.6 Implementation in the working example
  4. 4 Magnetized E3-instantons
    1. 4.1 World-volume fluxes and duality
    2. 4.2 Fermions and fluxes: the simplest case of constant $\tau$
    3. 4.3 Fermions and fluxes with generic $\tau$
    4. 4.4 Flux-modified zero modes
    5. 4.5 Index for magnetized E3-branes
    6. 4.6 On flux-induced zero-modes lifting
  5. 5 Lifting zero-modes in a one-modulus example
  6. A U(1)-bundles and holomorphic line bundles
  7. B Some useful properties of the extrinsic curvature

##  1 Introduction and summary of the results

F-theory provides a large class of non-perturbative vacua of Type IIB whereby
the complexified axio-dilaton develops a non trivial profile due to the
presence of 7-branes [1, 3, 2]. In recent times a resurgence of interest in
this class of models has been triggered by the observation that local
configurations of 7-branes can accommodate interesting aspects of particle
physics phenomenology beyond the standard model, such as gauge coupling
unification and textures in the Yukawa couplings, starting with [4, 5, 6] –
see e.g. [7, 8] for reviews and more complete lists of references.

Although it is still unclear which local F-theory solutions admit a global
embedding, non-perturbative effects generated by Euclidean D3-branes
(E3-branes) are widely recognized as crucial ingredients in IIB/F-theory model
building, see for instance [9, 10, 11, 12, 13, 14]. Most of the applications
involving M5/E3 non-perturbative effects are built on the results of [15],
which in turn relies on the study of the fermionic zero modes derived from the
standard Dirac action for a Euclidean M5-brane, dual to the E3-brane. However,
even in the absence of background fluxes, world-volume fluxes are known to
deform the standard Dirac action on D-branes [16] and M5-branes [17]. Since
the effect of world-volume fluxes on the fermionic zero-mode structure is not
taken into account in [15], understanding this and related issues remains an
open problem. This constitutes the main motivation of the present
investigation, that aims at tackling the effect of world-volume fluxes on
E3-brane instantons in general terms. Henceforth we will dub these as
magnetized E3-branes.

This is not just an academic problem. On the one hand, consistency actually
requires that for every four-cycle wrapped by an E3-brane, one must sum over
the infinite (discrete) family of possible world-volume fluxes. On the other
hand, fluxes can in fact have important physical effects. For instance, there
is evidence that they can modify the zero-mode counting [18] and thus the
nature of E3-brane instanton corrections to the effective theory. In
particular [14], fluxes can play a key role in alleviating the tension between
chirality and moduli stabilization [19].

While [15] and many other papers on F-theory are largely based on the dual
M-theory point of view, that has of course several advantages, in this paper
we will mostly work in the IIB picture. One practical motivation behind this
choice is that it allows one to use the (Wick-rotated) well-understood
effective action for D3-branes, hence avoiding the M5-brane effective action
and in particular its subtle chiral three-form flux. Furthermore, our study of
the effect of world-volume fluxes on E3-branes will have as byproduct a better
comprehension of the relation between the IIB and M-theory pictures even in
absence of fluxes – see [12, 20, 21, 13] for previous works in this direction.
An improved understanding of the IIB/M-theory dictionary can be important
since the IIB approach has the unquestionable advantage of admitting – in some
cases – local or global weak coupling descriptions, wherein one can use
perturbative string theory techniques, not available on the M-theory side.
Indeed, in the past few years, a lot of progress has been achieved in the
understanding of non-perturbative effects generated by unoriented D-brane
instantons using their open string description [22, 23]111See also [24, 25,
26, 27, 28, 29, 30] for early work and [31, 32] for recent reviews..

Although in this paper we will not consider background type IIB three-form
fluxes (or, dually, background M-theory four-form fluxes), the understanding
of E3-branes in their presence actually constitutes another motivation for our
work. Indeed, background fluxes play a crucial role in several applications
and their effect on M5/E3 brane instantons in F-theory/IIB orientifold
backgrounds has been already considered in [33, 34, 35, 36, 37, 38, 18, 39,
40], see also [41, 42] for studies on the effect of bulk fluxes on D-brane
instantons by using the string world-sheet techniques. However, the papers
[34, 35, 36, 38] work within the simplifying assumption that the world-volume
flux vanishes and [18, 39, 40] provide just partial results on its possible
effects. In this context, although legitimate in some cases, the assumption
that there is no world-volume flux constitutes a very non-generic condition,
since the world-volume Bianchi identities relate world-volume and background
fluxes. Take for instance a IIB three-form flux $H_{\it 3}$. The world-volume
Bianchi identity reads $\text{d}{\cal F}=-\iota^{*}H_{\it 3}$, where
$\iota^{*}H_{\it 3}$ denotes the pull-back of $H_{\it 3}$ onto the E3-brane
world-volume. Clearly, if $H_{\it 3}\neq 0$, then generically the world-volume
flux ${\cal F}$ will be non vanishing as well. Hence, a proper understanding
of the effect of world-volume fluxes alone on E3/M5-branes constitutes an
important step that necessarily precedes a consistent incorporation of
background fluxes.

Let us summarize our findings:

  * •

We clarify the structure of E3-E3 zero modes for general world-volume fluxes
and propose a modification of the M5-brane index [43] that reproduces and
generalizes all available results.

  * •

We consistently incorporate SL$(2,\mathbb{Z})$ non-trivial monodromies and
highlight the relation between F-theory and perturbative IIB results, that was
not so manifest even in the flux-less case [12, 20, 21].

  * •

We explicitly show that world-volume fluxes can lift certain fermionic zero-
modes. We discuss in detail the associated geometrical interpretation of the
flux-induced fermionic zero-mode lifting mechanism. We eventually apply our
general analysis to the explicit, though very simple, example with
$\mathbb{P}^{3}$ as compactification space and its orientifold weak-coupling
limit.

  * •

In particular, we discuss a concrete example that provides evidence that non-
rigid divisors can generically contribute to the superpotential. This
drastically broadens the possibilities for non-perturbatively generated
superpotential terms. For instance, once properly combined with bulk fluxes,
E3-brane instantons with world-volume fluxes can lead to significant
improvement of the moduli-lifting problem.

The plan of the paper is as follows. In Section 2, we will briefly review some
basic properties of F-theory vacua, including monodromy, S-duality and,
whenever possible, orientifold limit. We then pass to discuss in Section 3 the
fermionic parts of the E3 action and the index counting the fermionic zero-
modes and present a very useful though simple working example. Section 4 is
devoted to study the effect of turning on world-volume fluxes. In Section 5 we
present an explicit example whereby a non-rigid divisor with fluxes
contributes to the superpotential.

We have collected some useful formulae on holomorphic line bundles and
properties of the extrinsic curvature in two Appendices.

Note added: While this paper was being typed a couple of interesting papers
appeared [44, 45] that discuss instantons in F-theory and provide
complementary results to the ones in the present work.

##  2 A short review on F-theory vacua

The aim of this section is to review the structure of F-theory backgrounds. We
will emphasize the IIB viewpoint, that is somewhat less ‘standard’, being
often rephrased in terms of the dual M-theory description. The material
covered in this section is of course not new at all, but we have decided to
include it for self-completeness of the paper, in order to fix notation,
definitions and main properties of these vacua and facilitate the reading of
the subsequent sections. For more details, see for instance the recent reviews
[46, 8].

F-theory vacua are type IIB backgrounds that include full back-reaction to
7-brane induced fluxes. The 7-branes can be either D7-branes or more general
$(p,q)$ 7-branes, obtained by acting on a D7-brane – i.e. a (1,0) 7-brane – by
an SL(2,$\mathbb{Z}$) duality transformation. A D7-brane sources one unit of
Ramond-Ramond (RR) flux $F_{{\it 1}}=\text{d}C_{\it 0}$ and is then
characterized by a monodromy $\tau\rightarrow\tau+1$ of the axion-dilaton
$\tau:=C_{{\it 0}}+{\rm i}e^{-\phi}$ on a closed loop linking the D7-brane.
Analogously, a $(p,q)$ 7-brane is characterized by a more general
SL(2,$\mathbb{Z}$) monodromy of $\tau$.

Local and global tadpole constraints strongly restrict the consistent
configurations and often require the presence of mutually non-local 7-branes,
i.e. 7-branes that cannot contemporarily be seen as a set of D7-branes or an
SL(2,$\mathbb{Z}$) transform thereof. In other words, these 7-brane
configurations are intrinsically non-perturbative, in the sense that the
axion-dilaton $\tau$ generically undergoes SL(2,$\mathbb{Z})$ duality
transformations when going from one patch of the internal manifold to another.
A simple example of such a configuration is provided by the O7-planes of
perturbative string theory, more appropriately described in F-theory as a
bound states of two mutually non-local 7-branes [47, 48].

In this paper we are interested in (minimally) supersymmetric F-theory
compactification to four-dimensions. They are dual to M-theory
compactifications to three-dimensions on an elliptically fibered Calabi-Yau
four-fold, in which the complex structure of the elliptic fiber corresponds to
the axion-dilaton $\tau$. This dual description nicely geometrizes the non-
trivial features of the F-theory backgrounds and turns out to be very
convenient to study several aspects that are harder to handle within a direct
IIB framework. However, the direct type IIB viewpoint can be important for
other purposes, as we will show in this paper.

Hence, in sections 2.1, 2.2 and 2.3 we will describe some basic features of
the F-theory backgrounds within the purely IIB description, without making any
use of the dual M-theory picture. The latter will be considered only in
subsection 2.4. We will also try to make very explicit the relation between
the two descriptions. This will be important in the subsequent sections. In
section 2.5 we provide a simple example of F-theory compactification and its
weak coupling orientifold limit. This example will constitute the playground
on which we will apply the subsequent general discussions.

###  2.1 Generalities on F-theory compactifications

F-theory backgrounds are more conveniently described in the Einstein frame.
Focusing on compactifications to flat four-dimensional Minkowski space and
demanding supersymmetry, the ten-dimensional space is the direct product
$\mathbb{R}^{1,3}\times{X}$, where ${X}$ is a complex three-fold, i.e. a real
six-dimensional manifold, that is also complex and Kähler. The Einstein frame
ten-dimensional metric splits as

$\text{d}s^{2}_{10}=\text{d}x^{\mu}\text{d}x_{\mu}+\text{d}s^{2}_{{X}}$ (2.1)

Here $\text{d}s^{2}_{{X}}=g_{mn}\text{d}y^{m}\text{d}y^{n}$ is the Kähler
metric on ${{X}}$. In this paper we denote the complex structure by
$I=I^{m}{}_{n}\text{d}y^{n}\otimes\partial_{m}$ and the associated Kähler form
by $J=\frac{1}{2}J_{mn}\text{d}y^{m}\wedge\text{d}y^{n}$, where
$J_{mn}=g_{mk}I^{k}{}_{n}$. One can also include a non-trivial warping, that
can be sourced by possible D3-brane charge and three-form flux present on
${X}$, but in the present paper we will focus on the cases in which such
warping can be consistently considered constant.

As recalled at the beginning of this section, the physically non-trivial
feature characterizing the F-theory backgrounds is provided by the presence of
7-branes. In the present setting, they fill $\mathbb{R}^{1,3}$ and wrap
internal four-cycles that are required to be holomorphically embedded by
supersymmetry. Hence, by adopting the algebraic geometry terminology, the
7-branes wrap divisors of the internal space. The presence of 7-branes is
signaled by the presence of a non-trivial axion-dilaton $\tau$, that
supersymmetry requires to depend holomorphically on the internal coordinates

$\bar{\partial}\tau=0$ (2.2)

where
$\bar{\partial}:=\text{d}\bar{z}^{\bar{I}}\wedge\partial/\partial\bar{z}^{\bar{I}}$,
where $(z^{I},\bar{z}^{\bar{I}})$ are complex coordinates on ${X}$.

In general, when passing from one local patch on ${X}$ to another, the axion-
dilaton is allowed to experience non-trivial SL(2,$\mathbb{Z}$) duality
transformations

$\tau\quad\rightarrow\quad\tau^{\prime}=\frac{a\tau+b}{c\tau+d}$ (2.3)

with $a,b,c,d\in\mathbb{Z}$ and $ad-bc=1$. In particular, non-trivial
monodromies signal the presence of 7-branes. A D7-branes is associated to a
so-called $T$-monodromy

$M_{\rm D7}\equiv M_{1,0}=\left(\begin{array}[]{cc}1&1\\\
0&1\end{array}\right)\equiv T$ (2.4)

This implies that close to a D7-brane $\tau$ takes the form
$\tau\simeq\frac{1}{2\pi{\rm i}}\log(z-z_{\rm D7})$, where $z$ denotes some
local coordinate transverse to the D7-brane. Indeed, close to a D7-brane
$\tau$ must satisfy the second-order equation
$\partial\bar{\partial}\tau=-\delta^{\it 2}({\rm D7})$. More generically, the
monodromy around a $(p,q)$ 7-brane is given by

$M_{p,q}=\left(\begin{array}[]{cc}1-pq&p^{2}\\\ -q^{2}&1+pq\end{array}\right)$
(2.5)

The non-trivial axion-dilaton contributes to the energy-momentum tensor. Hence
the internal metric is not Ricci-flat and indeed supersymmetry implies the
following Einstein equations

$R^{{X}}_{{I}\bar{J}}=\nabla_{I}\bar{\nabla}_{\bar{J}}\phi$ (2.6)

It is useful to introduce the following composite one-form

$Q_{\it 1}=\frac{{\rm
i}}{2}\frac{\text{d}(\tau+\bar{\tau})}{\tau-\bar{\tau}}=\frac{1}{2}\,e^{\phi}\,F_{\it
1}=\frac{{\rm i}}{2}(\bar{\partial}\phi-\partial\phi)$ (2.7)

where, in the last equality, we have made use of (2.2). Now, $Q_{\it 1}$ can
be seen as a connection for a U(1) bundle, that we call U(1)Q. It is defined
as follows: If the background undergoes an SL(2,$\mathbb{Z})$ duality
transformation

$\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)$ (2.8)

when going from one patch to another, the corresponding U(1)Q transition
function is given by $e^{{\rm i}\arg(c\tau+d)}$. This U(1)Q bundle will play a
crucial role in the following, in that several fields can be seen to transform
as sections of the associated complex line $L_{Q}$ and powers thereof under
SL(2,$\mathbb{Z}$). Hence, $Q_{\it 1}$ can be used to construct
SL(2,$\mathbb{Z}$)-covariant derivatives, that allow one to obtain manifestly
SL(2,$\mathbb{Z}$)-covariant quantities. This will be very important in the
following discussions.

From (2.7), one can easily compute the curvature associated with $Q_{\it 1}$:

${\cal R}_{Q}=\text{d}Q={\rm i}\partial\bar{\partial}\phi$ (2.9)

Then, by (2.6) it is immediate to check that it is equal to the Ricci form
${\cal
R}_{{X}}:=R^{{X}}_{{I}\bar{J}}\,\text{d}z^{I}\wedge\text{d}\bar{z}^{\bar{J}}$:

${\cal R}_{{X}}\equiv{\cal R}_{Q}$ (2.10)

The curvature ${\cal R}_{Q}$ computes the first Chern Class of the line bundle
$L_{Q}$, $c_{1}(L_{Q})=\frac{1}{2\pi}\left[{\cal R}_{Q}\right]$, while the
Ricci form computes the first Chern Class of the tangent bundle of ${X}$,
$c_{1}({X})=\frac{1}{2\pi}\left[{\cal R}_{{X}}\right]$. Hence, one can
explicitly see that the non-triviality of the line bundle $L_{Q}$ is directly
related to the failure of the Calabi-Yau condition for ${X}$. Furthermore, by
Yau’s theorem, given a certain holomorphic axion-dilaton $\tau$ with
associated bundle $L_{Q}$ on ${X}$, the topological condition
$c_{1}(L_{Q})=c_{1}({X})$ is actually sufficient for an F-theory metric to
exist for every Kähler class $[J]\in H^{1,1}({X})$.

Notice that, by standard arguments in algebraic geometry, since ${\cal R}_{Q}$
has vanishing (2,0) and (0,2) parts, one can associate to $L_{Q}$ a
holomorphic line bundle ${\cal L}_{Q}$, whose sections $f$ satisfy holomorphic
gluing conditions $f\rightarrow(c\tau+d)f$ – see appendix A for more details.
As we will explicitly see, the possibility to trade $L_{Q}$ for ${\cal L}_{Q}$
will allow to translate several geometrical quantities in terms of purely
holomorphic data.

For instance, it is well known that $c_{1}({X})=-c_{1}(K_{{X}})$, where
$K_{{X}}$ is the canonical bundle on ${X}$, i.e. the holomorphic bundle of
$(3,0)$ forms on ${X}$. Hence, (2.10) implies that $c_{1}({\cal
L}_{Q})=-c_{1}(K_{{X}})$ and one can conclude that the holomorphic line bundle
${\cal L}_{Q}$ is isomorphic to the inverse of the canonical bundle on ${X}$:

${\cal L}_{Q}\simeq K^{-1}_{{X}}$ (2.11)

###  2.2 Supersymmetric structures

We are interested in minimally supersymmetric F-theory compactifications, i.e.
preserving four-supersymmetries. The supersymmetry generators are described by
the two type IIB Majorana-Weyl spinors $\epsilon_{1,2}$. They both have
positive chirality $\Gamma_{11}\epsilon_{1,2}=\epsilon_{1,2}$, where
$\Gamma_{11}=\Gamma^{\underline{0123456789}}$ is the ten-dimensional chirality
operator.222In this paper, we distinguish flat indices by underlying them. We
work in the Majorana representation in which the charge conjugation matrix is
$\Gamma^{\underline{0}}$ and all ten-dimensional gamma matrices are real (in
Minkowskian signature). Hence, one can take $\epsilon_{1,2}$ to be real. One
can group them in a bi-spinor $\epsilon$

$\epsilon=\left(\begin{array}[]{c}\epsilon_{1}\\\
\epsilon_{2}\end{array}\right)$ (2.12)

In order to describe the supersymmetry generators, one can first introduce the
following split of the ten-dimensional gamma matrices

$\Gamma^{\underline{\mu}}=\hat{\gamma}^{\underline{\mu}}\otimes\mathbbm{1}\,,\quad\Gamma^{\underline{m}}=\gamma_{5}\otimes\gamma^{\underline{m}}$
(2.13)

where $\hat{\gamma}^{\underline{\mu}}$ and $\gamma^{\underline{m}}$ are four-
and six-dimensional gamma matrices respectively, and furthermore

$\Gamma_{11}=\gamma_{5}\otimes\gamma_{7}\,,\quad\hat{\gamma}_{5}=-{\rm
i}\hat{\gamma}^{\underline{0123}}\,,\quad\gamma_{7}={\rm
i}\gamma^{\underline{123456}}$ (2.14)

with $\hat{\gamma}_{5}$ and $\gamma_{7}$ the four- and six-dimensional
chirality operators respectively.

In the Einstein frame, the supersymmetry generators in double-notation have
the structure

$\epsilon=\epsilon^{\rm R}+\epsilon^{\rm
L}=\left(\begin{array}[]{c}\epsilon^{\rm R}_{1}\\\ \epsilon^{\rm
R}_{2}\end{array}\right)+\left(\begin{array}[]{c}\epsilon^{\rm L}_{1}\\\
\epsilon^{\rm L}_{2}\end{array}\right)$ (2.15)

where

$\epsilon^{\rm R}_{1}={\rm i}\epsilon_{2}^{\rm R}=\zeta_{{\rm
R}}\otimes\eta\quad,\quad\epsilon_{1}^{\rm L}=-{\rm i}\epsilon_{2}^{\rm
L}=\zeta_{\rm L}\otimes\eta^{*}$ (2.16)

with $\zeta_{{\rm R},{\rm L}}$ and $\eta$ external and internal chiral
spinors: $\hat{\gamma}_{5}\zeta_{\rm R}=\zeta_{\rm R}$,
$\hat{\gamma}_{5}\zeta_{\rm L}=-\zeta_{\rm L}$ and $\gamma_{7}\eta=\eta$. In
Minkowskian signature, one has $\zeta_{\rm L}=\zeta^{*}_{{\rm R}}$. However,
in order to consider Euclidean D-branes, one needs to analytically continue
$\zeta_{{\rm R}}$ and $\zeta_{\rm L}$ to two independent ‘holomorphic’ 4D
spinors of opposite chirality.

Notice that the complex combination $\epsilon_{1}+{\rm i}\epsilon_{2}$
transforms with U(1)Q-charge $+1/2$ under the SL(2,$\mathbb{Z}$) duality
group. Hence, the same is true for $\eta$, that can be then considered a
section of $S_{+}\otimes L_{Q}^{1/2}$ or, more precisely, of the associated
Spin${}^{c}_{+}$ bundle over ${X}$, that reduces to $S_{+}\otimes L_{Q}^{1/2}$
only if $S_{+}$ and $L_{Q}^{1/2}$ separately exist. Indeed, consistently with
the discussion of the previous subsection, $\eta$ must satisfy the following
SL(2,$\mathbb{Z}$)-covariant Killing spinor equations

$(\nabla_{m}-\frac{{\rm i}}{2}\,Q_{m})\eta=0$ (2.17)

where $\nabla_{m}$ is the Levi-Civita spin connection. In the following we
will choose the normalization condition $\eta^{\dagger}\eta=1$.

The Kähler form $J$ on ${X}$ can be constructed from $\eta$ as

$J_{mn}={\rm i}\eta^{\dagger}\gamma_{mn}\eta\quad$ (2.18)

It is closed and in fact covariantly constant since it does not transform
under the structure group of $L_{Q}$. On the other hand, one can use $\eta$ to
construct also a holomorphic (3,0) form $\Omega$ as follows

$\Omega_{mnp}=e^{\phi/2}\eta^{T}\gamma_{mnk}\eta$ (2.19)

Notice that

$\Omega\wedge\bar{\Omega}=\frac{4{\rm i}}{3}e^{\phi}J\wedge J\wedge J$ (2.20)

This explicitly shows that the metric is not Calabi-Yau if the dilaton is not
constant.

We would like to emphasize that $\Omega$ should not be considered as a section
of the canonical bundle, since one can easily check that it transforms non-
trivially under SL(2,$\mathbb{Z}$)-duality:

$\Omega\rightarrow(c\tau+d)\Omega$ (2.21)

Hence, $\Omega$ must be rather considered as a section of $K_{Q}\otimes{\cal
L}_{Q}$. Because of the isomorphism (2.11), one can see that $\Omega$ is a
holomorphic section of a trivial line-bundle, which guarantees that $\Omega$
is globally well-defined and never vanishing.

Notice that $\Omega$ is not covariantly constant. On the other hand, according
to the general discussion of appendix A, one can associate to $\Omega$ the
$L_{Q}$-valued three-form $e^{-\phi/2}\Omega$, that is covariantly constant
under the correspondingly U(1)Q-covariant derivative:

$\nabla_{m}^{Q}(e^{-\phi/2}\Omega)\equiv(\nabla_{m}-{\rm
i}Q_{m})(e^{-\phi/2}\Omega)=0$ (2.22)

###  2.3 Orientifold limit

Backgrounds with O7-planes and D7-branes can be properly described as
particular limits of the more general F-theory backgrounds considered in the
previous sections [47, 48], in which the string coupling $g_{s}$ can be tuned
to be small. In this limit, the O7-plane can be seen as a bound state of two
mutually non-local 7-branes separated by a perturbatively invisible distance
suppressed by $e^{-1/g_{s}}$. Neglecting such a non-perturbatively resolution,
the O7-plane appears as a singular non-dynamical object characterized by an
SL(2,$\mathbb{Z}$) monodromy

$M_{\rm O7}=\left(\begin{array}[]{cc}-1&-4\\\ 0&-1\end{array}\right)$ (2.23)

When four D7-branes coincide with one O7-plane, there is a net cancellation of
the resulting backreaction, with a residual monodromy

$M_{\rm O7}M^{4}_{\rm D7}=\left(\begin{array}[]{cc}-1&0\\\
0&-1\end{array}\right)$ (2.24)

This monodromy does not act on the axion-dilaton but reverses the sign of the
fields transforming as doublets under SL(2,$\mathbb{Z}$). The standard way to
describe this effect is to construct a double cover $\tilde{X}$ and describe
the F-theory space as a $\mathbb{Z}_{2}$-quotient ${X}=\tilde{X}/\sigma$,
where $\tilde{X}$ is a double-cover of ${X}$ branched over the locus of the
O7-plane, and $\sigma$ is the orientifold involution. At the level of complex
structure, $\tilde{X}$ is a Calabi-Yau, in the sense that the holomorphic
(3,0) form $\Omega$ defined in (2.19) pulls-back to a globally defined
nowhere-vanishing section of the canonical bundle of $\tilde{X}$. When all
7-branes organize in groups of O7+4 D7, then $\tau$ is globally constant and
$\Omega$ is covariantly constant too, so that $\tilde{X}$ has Ricci-flat
Calabi-Yau metric.

One may also relax this limit where the axion-dilaton is constant by allowing
a configuration with an O7-plane and one or several D7-branes that do not lie
on it. Then, the only SL(2,$\mathbb{Z}$) monodromies will be those that shift
$\tau$ by real constants, and the sign reversing orientifold action on
doublets. In such a limit, $g_{s}$ can be kept small everywhere except for an
exponentially small region around the O7-plane. Hence, one can still proceed
within perturbative string theory as long as one is aware of this subtlety.

###  2.4 Relation with the M-theory viewpoint

The backgrounds described in the previous subsections are dual to M-theory
compactifications on elliptically fibered Calabi-Yau four-folds ${Y}$ – see
for instance [46] for a detailed discussion of this duality. In this section
we would like to provide the concrete prescription on how to construct ${Y}$
starting from the type IIB data.

Given a holomorphic axion-dilaton on a Kähler three-fold ${X}$, one defines an
elliptic fibration over it by first creating a fiber bundle over it, whose
fibers are the weighted projective space $\mathbb{CP}^{2}_{231}$ with
homogeneous coordinates $[x:y:z]$. One way to make this bundle non-trivial
over ${X}$ is by allowing the projective coordinates to transform as sections
of line bundles ${\cal L}_{x}$, ${\cal L}_{y}$ and ${\cal L}_{z}$ on ${X}$
under transitions between patches in ${X}$. In order to make the elliptically
fibered four-fold, we would like to impose fiber by fiber the Weierstrass
equation:

$y^{2}=x^{3}+f\,x\,z^{4}+g\,z^{6}$ (2.25)

that cuts out an elliptic curve inside the weighted projective space. In
particular, the complex structure of the elliptic fiber must coincide with the
axion-dilaton $\tau$ on ${X}$.

On the other hand, it is a standard fact that elliptic curves with such a
Weierstrass representation have very tractable modular properties. More
precisely, it is known that, under an SL(2,$\mathbb{Z}$) transformation, $f$
and $g$ transform as modular forms of weight 4 and 6 respectiverly, i.e.

$f\rightarrow(c\tau+d)^{4}f\quad,\quad g\rightarrow(c\tau+d)^{6}g$ (2.26)

By identifying the modular transformations of the elliptic fiber with the
S-duality group of IIB string theory, one can clearly see that according to
the definition in 2.1 of the line bundle ${\cal L}_{Q}$, $f$ and $g$ can be
regarded as holomorphic sections of ${\cal L}^{4}_{Q}$ and ${\cal L}^{6}_{Q}$:

$f\in\Gamma({\cal L}_{Q}^{4})\quad{\rm and}\quad g\in\Gamma({\cal L}_{Q}^{6})$
(2.27)

Then, by requiring consistency of (2.25), one can see that ${\cal L}_{x}={\cal
L}_{z}^{2}\otimes{\cal L}^{2}_{Q}$ and ${\cal L}_{y}={\cal
L}_{z}^{3}\otimes{\cal L}^{3}_{Q}$. At this point, one has some freedom in the
choice ${\cal L}_{z}$. One can for instance take ${\cal L}_{z}={\cal O}_{{X}}$
(i.e. the trivial line bundle on ${X}$), and then ${\cal L}_{x}={\cal
L}^{2}_{Q}$ and ${\cal L}_{y}={\cal L}^{3}_{Q}$. This choice facilitates the
description of the open patch where $z\neq 0$, by allowing one to gauge fix
$z\rightarrow 1$, and recovering the familiar Weierstrass equation. Another
useful possibility is ${\cal L}_{z}={\cal L}_{Q}^{-1}$, for which ${\cal
L}_{x}={\cal L}_{y}={\cal O}_{{X}}$. Since ${X}$ is itself holomorphically
embedded in ${Y}$ as the divisor $z=0$, this choice is useful because it
easily allows us to restrict integrals on ${Y}$ to integrals on ${X}$.

In section 2.1 we have also shown that supersymmetry constrains this line
bundle to be isomorphic to the anti-canonical bundle of the three-fold, ${\cal
L}_{Q}\simeq K_{B}^{-1}$. Via standard adjunction formulae, one can compute
the first Chern class of the F-theory elliptically fibered four-fold ${Y}$ to
be

$c_{1}({Y})=c_{1}({X})-c_{1}({\cal L}_{Q})$ (2.28)

Hence, since $c_{1}({X})=-c_{1}(K_{{X}})$, the supersymmetry condition ${\cal
L}_{Q}\simeq K_{B}^{-1}$ is equivalent to imposing the Calabi-Yau condition on
${Y}$.

Having established the basics about the M-theory perspective as well as the
IIB supergravity point of view, let us connect it to the perturbative IIB
string theory description via Sen’s limit [47, 48]. Given a choice for $f$ and
$g$, the corresponding axion-dilaton is determined by the following relation:

$j(\tau)=\frac{4\,(24\,f)^{3}}{4\,f^{3}+27\,g^{2}}\quad{\rm where}\quad
j(\tau)=e^{-2\pi{\rm i}\tau}+744+{\cal O}(e^{2\pi{\rm i}\tau})$ (2.29)

is Klein’s modular function. Let us reparametrize $f$ and $g$ as follows:

$\displaystyle f$ $\displaystyle=$ $\displaystyle-3\,h^{2}+\epsilon\,\eta$
(2.30) $\displaystyle g$ $\displaystyle=$
$\displaystyle-2\,h^{3}+\epsilon\,h\,\eta-\epsilon^{2}\,\chi/12$ (2.31)

where $h,\eta,$ and $\chi$ are sections of ${\cal L}_{Q}^{2},{\cal
L}_{Q}^{4},$ and ${\cal L}_{Q}^{6}$, respectively. Then, one finds that
$g_{s}\sim(\log\epsilon)^{-1}$ everywhere except at the locus $h=0$. A
monodromy analysis reveals that the $(p,q)$ branes have rearranged into the
following perturbative configuration:

$\displaystyle\text{ O7-plane at}:$ $\displaystyle\quad h=0$ (2.32)
$\displaystyle\text{D7-brane at}:$ $\displaystyle\quad\eta^{2}-h\,\chi=0$
(2.33)

Away from this limit, however, the total $C_{{\it 0}}$-tadpole cancelling
7-brane configuration becomes one recombined object wrapping the following
divisor:

$4\,f^{3}+27\,g^{2}=0$ (2.34)

of class ${\cal L}_{Q}^{12}$.

In the non-backreacted probe approximation, where $g_{s}\rightarrow 0$, we
expect the internal space to be Ricci flat, i.e. a Calabi-Yau three-fold.
Sen’s limit also allows one to recover the internal Calabi-Yau three-fold. It
appears as a double-cover of ${X}$ branched over the O7-plane locus $h=0$. It
is defined by tagging a new coordinate $\xi$ on ${X}$, such that $\xi$
transforms as a section of ${\cal L}_{Q}$, and imposing the equation:

$\xi^{2}-h=0$ (2.35)

Once we make the choice ${\cal L}_{Q}\equiv K_{{X}}^{-1}$, this new space
$\tilde{X}$ will be guaranteed to be Calabi-Yau.

###  2.5 A working example

In order the make the generalities of the previous section more palatable, we
will introduce a simple working example. Let the internal three-fold be
${X}=\mathbb{P}^{3}$, with homogeneous coordinates $[z_{1}:\ldots:z_{4}]$.

In general, a section of a holomorphic line bundle vanishes along a complex
codimension one holomorphic submanifold, i.e. a divisor $D$. The line bundle
can be then denoted by ${\cal O}(D)$. It turns out that its Poincaré dual
$[D]$ coincides with the first Chern class of the bundle. On $\mathbb{P}^{3}$
one can define the so-called _hyperplane bundle_ , the line bundle such that
(linear combinations of) $z_{I}$ are sections thereof. The associated divisor
is the so-called hyperplane divisor $H$ and the hyperplane bundle is indicated
with ${\cal O}_{\mathbb{P}^{3}}(1)\equiv{\cal O}_{\mathbb{P}^{3}}(H)$.
Analogously, any homogenous polynomial $P^{(n)}(z_{1},\ldots,z_{4})$ of degree
$n$ is a section of ${\cal O}_{\mathbb{P}^{3}}(n)\equiv{\cal
O}_{\mathbb{P}^{3}}(n\,H)$. Hence, one has the following useful identities

${\rm PD}(\\{P^{(n)}=0\\})=c_{1}({\cal O}_{\mathbb{P}^{3}}(n))=n\,[H]$ (2.36)

The canonical bundle of $\mathbb{P}^{3}$ is $K_{\mathbb{P}^{3}}={\cal
O}_{\mathbb{P}^{3}}(-4)$. Therefore, if $\mathbb{P}^{3}$ is chosen as F-theory
compactification space ${X}$, the supersymmetry of the background requires
that we set ${\cal L}_{Q}={\cal O}_{\mathbb{P}^{3}}(4)$. From these data, one
can easily define the corresponding F-theory Calabi-Yau four-fold ${Y}$ as a
hypersurface in the following ambient space:

$\begin{array}[]{ccccccc|c}z_{1}&z_{2}&z_{3}&z_{4}&x&y&z&\textrm{eq.}\eqref{weierstrass}\\\
\hline\cr 1&1&1&1&0&0&-4&0\\\ 0&0&0&0&2&3&1&6\end{array}\begin{array}[]{r}\\\
\\\ \end{array}$ (2.37)

That is, in terms of the general discussion of section 2.4, one has made the
choice ${\cal L}_{x}={\cal L}_{y}={\cal O}_{\mathbb{P}^{3}}$ and ${\cal
L}_{z}={\cal L}_{Q}^{-1}={\cal O}_{\mathbb{P}^{3}}(-4)$.

This space is a generalization of a weighted projective space, whereby one
takes the quotient w.r.t. two rescalings as follows:

$(z_{1},\ldots,z_{4},x,y,z)\sim(\lambda\,z_{1},\ldots,\lambda\,z_{4},\mu^{2}\,x,\mu^{3}\,y,\,\lambda^{-4}\,\mu\,z)\,,\quad\text{for}\quad\lambda,\mu\in\mathbb{C}^{*}$
(2.38)

The other necessary data to define this space are the so-called exceptional
sets that are excluded from the space, generalizing the usual exclusion of the
origin in a projective space. In the present case, there are two sets of
coordinates that are forbidden from vanishing simultaneously:

$(z_{1}\,,\ldots\,,z_{4})\neq(0,\ldots,0))\,;\quad{\rm
and}\quad(x\,,y\,,z)\neq(0,0,0)$ (2.39)

The Calabi-Yau hypersurface ${Y}$ is given by the Weierstrass equation (2.25),
with $f$ and $g$ sections of ${\cal O}_{\mathbb{P}^{3}}(16)$ and ${\cal
O}_{\mathbb{P}^{3}}(24)$ respectively, that is homogeneous of bi-degree
$(0,6)$ with respect to the two projective rescalings.

Note that one can easily construct the Calabi-Yau three-fold $\tilde{X}$ for
the corresponding perturbative string theory description of the example under
consideration in Sen’s limit. According to the general discussion of section
2.4, $\tilde{X}$ is defined by (2.35), with $h$ section of ${\cal
O}_{\mathbb{P}^{3}}(8)$, that is

$\xi^{2}=h(z_{1},\ldots,z_{4})$ (2.40)

with $h(z_{1},\ldots,z_{4})$ a degree 8 homogenous polynomial. Clearly, (2.40)
can be seen as a degree eight hypersurface on the weighted projective four-
fold $\mathbb{P}^{4}_{11114}$ with homogeneous coordinates
$[z_{1},\ldots,z_{4},\xi]$. This Calabi-Yau three-fold is denoted by
$\mathbb{P}^{4}_{11114}[8]$. On this space, the orientifold involution acts on
the target space by sending $\xi\rightarrow-\xi$.

##  3 E3-instantons without fluxes: the IIB perspective

Let us come back to our main motivation, namely the study of E3-instantons in
F-theory backgrounds. In this section we first focus on the case in which
there are no world-volume fluxes on the E3-brane, i.e.

${\cal F}:=2\pi\alpha^{\prime}F_{\rm E3}-\iota^{*}B_{\it 2}=0$ (3.1)

This is the case considered by Witten in his seminal paper [15] and by most of
the subsequent papers on this subject.

The approach of [15] starts from the M-theory viewpoint and identifies the
conditions under which an M5-brane instanton can contribute to the
superpotential. Here we would like to retrace Witten’s procedure working
directly in IIB. In this way we will set the basis for our discussion on the
inclusion of world-volume fluxes, that will be considered in section 4. As a
byproduct, we will obtain a clear physical picture on how the M-theory results
of [15] should be interpreted from the IIB viewpoint. This can be useful in
other developments based on the IIB picture, in which one can take advantage
of perturbative string theory techniques. See for instance [20, 21, 45] for
other papers exploring this perspective.

Consider an E3-brane instanton wrapping a four-cycle $D$ in the internal space
${X}$. In order to preserve supersymmetry the four-cycle must be
holomorphically embedded or, in other words, must be an effective divisor.
Hence, the on-shell E3-action is just given by the value of the complexified
Kähler modulus

$T_{\rm E3}:=\frac{2\pi}{\ell_{s}^{4}}\int_{D}\left(\frac{1}{2}\,\,J\wedge
J+{\rm i}C_{\it 4}\right)$ (3.2)

where $\ell_{s}:=2\pi\sqrt{\alpha^{\prime}}$, and then the possible correction
to the superpotential looks like

$W_{\rm np}={\cal A}(\ldots)\,e^{-T_{\rm E3}}$ (3.3)

where ${\cal A}(\ldots)$ can depend generically on other moduli in the
compactification [43, 49], see also [50].333 If in addition 7-brane U(1)
fluxes are turned on, ${\cal A}(\ldots)$ acquires a dependence on gauge-
invariant combinations of the charged massless chiral multiplets up to
anomalous U(1)’s under which $T_{\rm E3}$ shifts [24, 26, 27, 29, 28, 30].

In order to understand the precise structure of $W_{\rm np}$ one should in
principle study the complete instantonic path integral. This problem may be
attacked by either using Green-Schwarz-like effective actions, as in [51, 15,
52, 53, 54, 55], or by microscopic string theory techniques, as in [24, 26,
27, 29, 28, 30]. In particular, in order to understand whether an E3-brane can
contribute to the superpotential, one has to study the structure of the
fermionic zero-modes. Here we are going to focus on the fermions associated to
(putative) ‘open strings’ with both ends on the E3-brane, whose dynamics is
described by a Green-Schwarz-like effective action that incorporates the
effect of the non-trivial axion-dilaton in a controlled way. As we will
explicitly check, these fermions correspond to the fermions in the dual
M5-brane effective action. There could be additional chiral fermions localized
at the intersection of the E3-brane with the background 7-branes, associated
with (putative) ‘open strings’ connecting the E3 and the 7-branes. In the dual
M5-brane, they are incorporated in the chiral (self-dual) three-form and their
effect can be reabsorbed in the moduli dependence of the pre-factor ${\cal
A}(\ldots)$ in (3.3) [43].

###  3.1 Fluxless E3-fermionic action

In order to study the E3-brane fermionic zero modes, we start from the
effective action for the E3 fermions on an F-theory IIB background. By some
general arguments [56], the world-volume fermions should naturally experience
a topological twist and then be represented by world-volume forms. Here we
would like to explicitly derive the corresponding topologically twisted theory
from the general E3 action. As we will see, a new key role with respect to the
more standard topological twist on Kähler spaces will be provided by the non-
trivial axion-dilaton. Understanding the role role of the axion-dilaton in
this case will be important later, when one will introduce world-volume
fluxes.

The fermions on a D-brane in a general supergravity background are more
conveniently described in the Green-Schwarz formalism, in which the ordinary
bosonic embedding is substituted by an embedding in the ten-dimensional
superspace. Hence, the world-volume fermions are described by a pair of ten-
dimensional Majorana-Weyl spinors $\theta_{1},\theta_{2}$, that one can
combine into the two-component vector

$\Theta=\left(\begin{array}[]{c}\theta_{1}\\\ \theta_{2}\end{array}\right)$
(3.4)

The apparent mismatch between bosonic and ferminic degrees of freedom is cured
by the presence of a world-volume gauge-symmetry, usually called
$\kappa$-symmetry, that acts in the following way

$\delta_{\kappa}\Theta=(\mathbbm{1}+\Gamma_{\rm E3})\kappa$ (3.5)

where $\kappa=(\kappa_{1},\kappa_{2})$, with $\kappa_{1,2}$ two arbitrary
Majorana-Weyl spinors. In the lowest order expansion in the fermionic fields,
the operator $\Gamma_{\rm E3}$ depends just on the bosonic world-volume
degrees of freedom and one natural way to remove these redundant degrees of
freedom is to impose the $\kappa$-fixing condition

$(\mathbbm{1}+\Gamma_{\rm E3})\Theta=0$ (3.6)

In the present case, since for the moment one is assuming (3.1), $\Gamma_{\rm
E3}$ is given by

$\Gamma_{\rm E3}=\left(\begin{array}[]{cc}0&\hat{\gamma}_{\rm E3}\\\
\hat{\gamma}^{-1}_{\rm
E3}&0\end{array}\right)\qquad\qquad\text{with}\qquad\hat{\gamma}_{\rm
E3}=-\frac{{\rm i}}{4!}\,\frac{\epsilon^{a_{1}\ldots a_{4}}}{\sqrt{\det
h}}\,\Gamma_{a_{1}\ldots a_{4}}$ (3.7)

We have introduced world-volume coordinates $\sigma^{a}$, $a=1,\ldots,4$,
$h\equiv g|_{D}$ denotes the pull-back of the bulk-metric onto the E3 world-
volume $D$

$h_{ab}=g_{mn}(y)\frac{\partial y^{m}}{\partial\sigma^{a}}\frac{\partial
y^{n}}{\partial\sigma^{b}}$ (3.8)

and $\Gamma_{a}$ denotes the pull-back of the ten-dimensional gamma-matrices

The explicit $\kappa$-symmetric quadratic fermionic action for D-branes on
general bosonic backgrounds has been worked out, in the string frame, in [16].
Here one has just to apply the general result of [16] to the F-theory
backgrounds described in detailed in section 2. By taking just a little care
of the passage from string to Einstein frame444Notice that world-volume
fermions are rescaled as $\Theta\rightarrow e^{\phi/8}\Theta$ when passing
from string to Einstein frame. and imposing the $\kappa$-fixing condition
(3.6), one obtains the following Green-Schwarz action for an E3-brane wrapping
a four-cycle $D\subset B$:

$S_{\rm F}=\frac{2\pi{\rm
i}}{\ell^{4}_{s}}\int_{D}\text{d}^{4}\sigma\,\sqrt{\det
h}\,\,\overline{\Theta}\Gamma^{a}(\hat{\nabla}_{a}+\frac{{\rm
i}}{4}\,e^{\phi}F_{a}\sigma_{2})\Theta$ (3.9)

Here $\hat{\nabla}_{a}$ is the pull-back of the bulk covariant derivative and
$F_{a}$ is the pull-back of the R-R one-form $F_{\it 1}$. Furthermore
$\overline{\Theta}\equiv\Theta^{T}\Gamma^{\underline{0}}$.

So far, we have not explicitly used the supersymmetry condition on $D$. As
already mention, this boils down to the requirement that $D$ should be a
holomorphically embedded hypersurface or, in other words, an effective
divisor. More specifically, if $D$ is an effective divisor then the E3-brane
preserves the two $\epsilon_{\rm R}$ supersymmetries as defined in (2.15) and
(2.16), while if $D$ is a negative divisor, that is a anti-holomorphic
hypersurface, then the two $\epsilon_{\rm L}$ are preserved.

One way to derive these results is by imposing the usual supersymmetry
condition

$\Gamma_{\rm E3}\epsilon_{\rm R}=\epsilon_{\rm R}$ (3.10)

for an effective divisor, or alternatively $\Gamma_{\rm E3}\epsilon_{\rm
L}=\epsilon_{\rm L}$ for a negative divisor. Focusing on E3-branes preserving
(3.10), by using the decompositions (2.13) and (2.16), one can more explicitly
write (3.10) as a condition involving only the internal space:

$\gamma_{\rm E3}\eta={\rm i}\eta\quad\quad\text{($\zeta_{\rm R}$ preserved)}$
(3.11)

with

$\gamma_{\rm E3}=-\frac{{\rm i}}{4!}\,\frac{\epsilon^{a_{1}\ldots
a_{4}}}{\sqrt{\det h}}\,\gamma_{a_{1}\ldots a_{4}}$ (3.12)

We are now in a position to introduce a more natural parametrization of the
world-volume fermion $\Theta$, that explicitly uses supersymmetry and
automatically solves the $\kappa$-fixing condition (3.6). Indeed, one can
decompose the two components $\theta_{1}$ and $\theta_{2}$ as follows. Split
$\theta_{1}=\theta_{1}^{\rm R}+\theta_{1}^{\rm L}$ and
$\theta_{2}=\theta_{2}^{\rm R}+\theta_{2}^{\rm L}$ and set:

$\displaystyle\left\\{\begin{array}[]{l}\theta_{1}^{\rm
R}=\frac{1}{2}\big{(}\tilde{\lambda}\otimes\eta+\tilde{\psi}_{a}\otimes\gamma^{a}\eta^{*}+\frac{1}{2}\tilde{\rho}_{ab}\otimes\gamma^{ab}\eta\big{)}\\\
\theta_{2}^{\rm R}=\frac{{\rm
i}}{2}\big{(}\tilde{\lambda}\otimes\eta-\tilde{\psi}_{a}\otimes\gamma^{a}\eta^{*}+\frac{1}{2}\tilde{\rho}_{ab}\otimes\gamma^{ab}\eta\big{)}\par\end{array}\right.$
(3.15) $\displaystyle\left\\{\begin{array}[]{l}\theta_{1}^{\rm
L}=\frac{1}{2}\big{(}\lambda\otimes\eta^{*}+\psi_{a}\otimes\gamma^{a}\eta+\frac{1}{2}\rho_{ab}\otimes\gamma^{ab}\eta^{*}\big{)}\\\
\theta_{2}^{\rm L}=\frac{{\rm
i}}{2}\big{(}\lambda\otimes\eta^{*}-\psi_{a}\otimes\gamma^{a}\eta+\frac{1}{2}\rho_{ab}\otimes\gamma^{ab}\eta^{*}\big{)}\end{array}\right.$
(3.18)

Hence, one has traded $\Theta$ for a new sets of world-volume fermionic
fields. They are not spinors on $D$ but rather forms, hence explicitly
realizing the expected topological twist [56], with purely holomorphic or
anti-holomorphic indices. More explicitly, if $s^{i}$
($\bar{s}^{\bar{\imath}}$) are (anti-)holomorphic world-volume coordinates,
one can write

$\tilde{\psi}=\tilde{\psi}_{i}\,\text{d}s^{i}\,,\quad\psi=\tilde{\psi}_{\bar{\imath}}\,\text{d}\bar{s}^{\bar{\imath}}\,,\quad\tilde{\rho}=\frac{1}{2}\rho_{\bar{\imath}\bar{\jmath}}\,\text{d}\bar{s}^{\bar{\imath}}\wedge\text{d}\bar{s}^{\bar{\jmath}}\,,\quad\rho=\frac{1}{2}\rho_{ij}\,\text{d}s^{i}\wedge\text{d}s^{j}$
(3.19)

On the other hand, the new world-volume fermions keep their spinor nature
related to four-dimensional external flat space. Namely, if $S_{\pm}$ denote
the (anti-)chiral spin bundles associated to the four flat directions,
$\tilde{\lambda},\tilde{\psi}$ and $\tilde{\rho}$ have spinorial index in
$S_{+}$, while $\lambda,\psi$ and $\rho$ have spinorial index in $S_{-}$.

Finally, since the E3-brane can cover different background patches related by
a possible SL(2,$\mathbb{Z}$)-duality transformations, it is important to
understand how the world-volume fermions transform under the duality. By
consistency with the superspace formulation, the Green-Schwarz fermion (3.4)
must transform as as the bulk supersymmetry generator $\epsilon$. Namely, the
combinations $\theta_{1}\pm{\rm i}\theta_{2}$ must transform with U(1)Q
charges $\pm 1/2$ – see for instance [57]. Since the internal spinor $\eta$
transforms itself with U(1)Q charges $\pm 1/2$, it is easy to recover that
$\lambda,\psi$ and $\rho$ are neutral under the SL(2,$\mathbb{Z}$)-duality,
while $\tilde{\lambda}$, $\tilde{\psi}$ and $\tilde{\rho}$ transform with
U(1)Q charges $-1,+1$ and $-1$ respectively. One can then more synthetically
summarize these properties by saying that

$\displaystyle\begin{array}[]{c|c|c}\text{l.h.\
fermions}&\text{U(1)${}_{Q}$-charge}&\text{associated bundle}\\\
\hline\cr\lambda^{\alpha}&0&S_{-}\otimes\Lambda^{0,0}\\\
\psi^{\alpha}&0&S_{-}\otimes\Lambda^{0,1}\\\
\rho^{\alpha}&0&S_{-}\otimes\Lambda^{2,0}\end{array}$ (3.24) (3.25)
$\displaystyle\begin{array}[]{c|c|c}\text{r.h.\
fermions}&\text{U(1)${}_{Q}$-charge}&\text{associated bundle}\\\
\hline\cr\tilde{\lambda}^{\dot{\alpha}}&-1&S_{+}\otimes\Lambda^{0,0}\otimes
L_{Q}^{-1}\\\ \tilde{\psi}^{\dot{\alpha}}&+1&S_{+}\otimes\Lambda^{1,0}\otimes
L_{Q}\\\ \tilde{\rho}^{\dot{\alpha}}&-1&S_{+}\otimes\Lambda^{0,2}\otimes
L_{Q}^{-1}\end{array}$ (3.30)

where the restriction to $D$ of the bundles defined on $X$ is understood.

By plugging the expansion (3.15) into the action (3.9), after some
manipulations, one gets the following effective action for the new fermionic
fields

$S_{F}=\frac{4\pi{\rm
i}}{\ell_{s}^{4}}\int_{D}(\psi\wedge*\partial\lambda-\tilde{\psi}\wedge*\bar{\partial}_{Q}\tilde{\lambda}-\rho\wedge*\bar{\partial}\psi+\tilde{\rho}\wedge*\partial_{Q}\tilde{\psi})$
(3.31)

where $*$ is the Hodge-star computed using the induced metric $h$ and
$\partial_{Q}$ and $\bar{\partial}_{Q}$ are U(1)Q-covariant Dolbeault
differentials, that is $\partial_{Q}=\partial\mp iQ^{1,0}$ and
$\bar{\partial}_{Q}=\bar{\partial}\mp iQ^{0,1}$ on fields with U(1)Q-charge
$\pm 1$.

A couple of comments are in order. First of all, the axion-dilaton enter in
exactly the right and minimal way to render the fermionic Lagrangian
manifestly invariant under SL(2,$\mathbb{Z}$)-duality transformations. The
preservation of this important property will be a guiding principle when we
will introduce a non-vanishing world-volume flux ${\cal F}\neq 0$. Second, the
action does not depend on the components of background tensorial fields that
are transversal to the E3-brane. As we will see, this will not any longer be
true when ${\cal F}\neq 0$.

###  3.2 Zero-modes

From the action (3.31) one can easily obtain the following fermionic equations
of motion

$\displaystyle\partial\lambda=0\quad$ ,
$\displaystyle\quad\bar{\partial}_{Q}\tilde{\lambda}=0$ (3.32)
$\displaystyle\bar{\partial}^{\dagger}\psi=0\quad$ ,
$\displaystyle\quad\partial_{Q}^{\dagger}\tilde{\psi}=0$ (3.33)
$\displaystyle\partial^{\dagger}\rho=0\quad$ ,
$\displaystyle\quad\bar{\partial}_{Q}^{\dagger}\tilde{\rho}=0$ (3.34)
$\displaystyle\bar{\partial}\psi=0\quad$ ,
$\displaystyle\quad\partial_{Q}\tilde{\psi}=0$

where $\bar{\partial}_{Q}^{\dagger}$ and $\partial_{Q}^{\dagger}$ are the
adjoint Dolbeault operator obtained by using the usual hermitian inner product
between forms: $(\chi_{1},\chi_{2})=\int\chi_{1}\wedge*\bar{\chi}_{2}$.

Hence, the zero modes are given by the (twisted) harmonic representatives of
the following cohomology groups

$\displaystyle\begin{array}[]{c|c}\text{l.h.\ zero modes}&\text{cohomology
group}\\\ \hline\cr\lambda^{\alpha}_{\rm z.m.}&H_{\partial}^{0,0}(D)\simeq
H^{0}(D,\bar{\cal O}_{D})\\\ \psi^{\alpha}_{\rm
z.m.}&H_{\bar{\partial}}^{0,1}(D)\simeq H^{1}(D,{\cal O}_{D})\\\
\rho^{\alpha}_{\rm z.m.}&H_{\partial}^{2,0}(D)\simeq H^{2}(D,\bar{\cal
O}_{D})\end{array}$ (3.39) (3.40)
$\displaystyle\begin{array}[]{c|c}\text{r.h.\ zero modes}&\text{cohomology
group}\\\ \hline\cr\tilde{\lambda}^{\dot{\alpha}}_{\rm
z.m.}&H_{\bar{\partial}}^{0,0}(D,{L}_{Q}^{-1})\simeq H^{0}(D,{\cal
L}_{Q}^{-1})\\\ \tilde{\psi}^{\dot{\alpha}}_{\rm
z.m.}&H_{\partial}^{1,0}(D,{L}_{Q})\simeq H^{1}(D,\bar{\cal L}^{-1}_{Q})\\\
\tilde{\rho}^{\dot{\alpha}}_{\rm
z.m.}&H_{\bar{\partial}}^{0,2}(D,{L}_{Q}^{-1})\simeq
H^{2}(D,\mathcal{L}_{Q}^{-1})\end{array}$ (3.45)

We have already used the fact that, as discussed in section 2.1 and more in
detail in appendix A, the $U(1)_{Q}$ connection $Q$ defines the holomorphic
line bundle $\mathcal{L}_{Q}=K_{{X}}^{-1}$, whose holomorphic sections
correspond to the $\bar{\partial}_{Q}$-closed section of $L_{Q}$. This allows
to give an interpretation of the fermionic zero modes in terms of sheaf
cohomology groups $H^{i}(D,{\cal O}_{D}),H^{i}(D,{\cal L}^{-1}_{Q})$ and their
complex conjugated, as explicitly indicated in (3.39).

For later notational convenience, let us define the associated Hodge numbers
as follows:

$\displaystyle h^{i}(D)$ $\displaystyle\equiv$ $\displaystyle{\rm
dim}\,H^{i}(D,{\cal O}_{D})={\rm dim}\,H^{i}(D,\bar{\cal O}_{D})$ (3.46)
$\displaystyle h^{i}_{Q}(D)$ $\displaystyle\equiv$ $\displaystyle{\rm
dim}\,H^{i}(D,\mathcal{L}_{Q}^{-1})={\rm dim}\,H^{i}(D,\bar{\cal L}_{Q}^{-1})$
(3.47)

Hence, since each fermion has two four-dimensional spinorial indices, the
left-handed zero modes are counted by $h^{i}(D)$, while the right-handed zero-
modes are counted by $h^{i}_{Q}(D)$.

The structure of this fermionic zero-mode spectrum can be interpreted as
follows.

First, consider the case in which the restriction of the line bundle ${\cal
L}_{Q}$ onto the E3-brane, ${\cal L}_{Q}|_{\rm E3}$, is trivial. Since, as
recalled in section 2.4, the seven-brane divisor $D_{7\text{-brane}}$ is
associated to a section of ${\cal L}^{12}_{Q}$, whose restriction on $D$ is
also trivial, one is requiring that the intersection two-cycle $D\cap
D_{\text{7-brane}}$ is homologically trivial. In this case, the E3-brane
spectrum is transparent to the background 7-branes and is similar to the
spectrum for an E3-brane on a Calabi-Yau three-fold. Namely, the right- and
left-handed zero-modes are in one-to-one correspondence and have a precise
physical interpretation. First, since $h^{0}(D)=1$ for a connected $D$, there
are two left-handed universal zero-modes $\lambda^{\alpha}_{\rm z.m.}$, often
denoted as $\theta^{\alpha}$ in the literature, that can be seen as goldstini
associated to the supersymmetries $\zeta^{\alpha}_{\rm L}$ broken by the
E3-brane. Analogously, in the present case, one has $h^{0}_{Q}(D)=1$ too, that
implies that there are two additional right-handed zero-modes
$\tilde{\lambda}^{\dot{\alpha}}$, often denoted as $\bar{\tau}^{\dot{\alpha}}$
in the literature – see for instance [31] – associated to the hidden
supersymmetry broken by the 7-branes. The other zero-modes have a clear
geometrical interpretation. $h^{1}(D)=h^{1}_{Q}(D)$ counts left- and right-
handed zero-modes that are supersymmetric partners of the Wilson lines, while
$h^{2}(D)=h^{2}_{Q}(D)$ counts left- and right-handed zero-modes that are
supersymmetric partners to the geometric deformations. The latter statements
uses the fact that the restriction to $D$ of the three-fold canonical bundle
$K_{Q}\simeq{\cal L}_{Q}^{-1}$ is trivial too. This implies that
$h^{2}(D)=h^{2}_{Q}(D)=\dim H^{0}(D,N_{D})$ through the standard adjunction
formula and Serre duality.

Let us now assume that ${\cal L}_{Q}|_{\rm E3}$, is non-trivial, in the sense
that the two-cycle $D\cap D_{\text{7-brane}}$ is now homologically non-
trivial. In this case, generically, there is a mismatch between left- and
right-handed zero modes. For instance, one obviously still has the two
universal zero-modes $\lambda^{\alpha}_{\rm z.m.}\sim\theta^{\alpha}$, but
there could be a different number $2h^{0}_{Q}(D)$ of almost-universal zero
modes $\tilde{\lambda}_{\rm
z.m.}^{\dot{\alpha}}\sim\bar{\tau}^{\dot{\alpha}}$. More precisely,
$h^{0}_{Q}(D)$ counts the number of holomorphic sections of $K_{{X}}|_{\rm
E3}$. Requiring $h^{0}_{Q}(D)=0$ implies that $K_{{X}}|_{\rm E3}$ must be a
negative line-bundle on the E3-brane.

Finally, it can be useful to consider the weak coupling orientifold limit of
the above results, in which the axion-dilaton can be approximated as constant
and each O7-plane, covered by four D7-branes, generate a monodromy (2.24).
This monodromy acts on the world-volume fermions
$\tilde{\lambda},\tilde{\psi},\tilde{\rho}$ by reversing their sign, while it
leaves $\lambda,\psi,\rho$ invariant. By uplifting ${X}$ to the double cover
Calabi-Yau $\tilde{X}$, one can see that the $\lambda,\psi,\rho$ are even
under the orientifold involution, while
$\tilde{\lambda},\tilde{\psi},\tilde{\rho}$ are odd. Then the fermionic zero
modes are even/odd harmonic representatives of (complex conjugated of the)
even/odd cohomology groups $H^{0,i}_{\pm}(\tilde{D})$, where $\tilde{D}$ is
the double cover of $D$:

$\begin{array}[]{c|c}\text{l.h.\ zero modes}&\text{cohomology group}\\\
\hline\cr\lambda^{\alpha}_{\rm z.m.}&H_{+}^{0,0}(\tilde{D})\\\
\psi^{\alpha}_{\rm z.m.}&H_{+}^{0,1}(\tilde{D})\\\ \rho^{\alpha}_{\rm
z.m.}&H_{+}^{2,0}(\tilde{D})\end{array}\qquad\begin{array}[]{c|c}\text{r.h.\
zero modes}&\text{cohomology group}\\\
\hline\cr\tilde{\lambda}^{\dot{\alpha}}_{\rm z.m.}&H_{-}^{0,0}(\tilde{D})\\\
\tilde{\psi}^{\dot{\alpha}}_{\rm z.m.}&H_{-}^{1,0}(\tilde{D})\\\
\tilde{\rho}^{\dot{\alpha}}_{\rm z.m.}&H_{-}^{0,2}(\tilde{D})\end{array}$
(3.48)

This spectrum is counted by

$h^{i}_{\pm}(D):=\dim H^{0,i}_{\pm}(D)$ (3.49)

and agrees with the results obtained in the literature based on specific
orientifolds models. In particular, in the case in which
$h^{1}_{\pm}(D)=h^{2}_{\pm}(D)=0$ (that in particular implies that the divisor
is rigid), the non-universal zero modes
$\tilde{\lambda}_{\text{z.m.}}^{\dot{\alpha}}\sim\bar{\tau}^{\dot{\alpha}}$
are absent only if $h^{0,0}_{-}=0$, that is possible only if the E3-brane
coincides with its orientifold image, i.e. only if it is a so-called O(1)
E3-brane [27, 28, 29].

###  3.3 M5-brane index from IIB perspective

In [15], Witten provided a necessary condition for an E3-brane to contribute
to the superpotential in terms of the arithmetic genus of the divisor wrapped
by the dual M5-brane instanton. Here we would like to revisit Witten’s
argument from the IIB perspective. Of course, the two approaches must be
equivalent, as we will discuss in the the following section.

Witten’s argument is based on the observation that the M5-brane configuration
is symmetric under a U(1) rotation along the normal bundle to the divisor
wrapped by the M5-brane. In the IIB picture, one has to consider a rotation
along the two directions transverse to the divisor $D\subset{X}$. In
particular, one has to consider the action of this rotation on the world-
volume fermionic fields.

Let us denote with $({\underline{5}},\underline{6})$ the two (flat) directions
normal to the E3-brane. Rotations in these two directions can be seen as
R-symmetry tranformations. We denote this symmetry by U(1)R. The ten-
dimensional spinorial action of the U(1)R rotation is generated by the matrix
$R=\frac{1}{2}\mathbbm{1}\otimes\gamma_{\underline{56}}$, where we have used
the gamma-matrices decomposition (2.13). By using (2.14) and (3.12), one can
alternatively write $R=-\frac{1}{2}\mathbbm{1}\otimes(\gamma_{\rm
E3}\gamma_{7})$. This generator acts in the same way on both components
$\theta_{1}$ and $\theta_{2}$ of (3.4). By looking at the expansion (3.15) and
using (3.11), it is easy to see that the world-volume fields transform with
charges

$\begin{array}[]{c|c}\text{l.h.\ fermions}&\text{U(1)${}_{R}$ charge}\\\
\hline\cr\lambda^{\alpha}&-1/2\\\ \psi^{\alpha}&+1/2\\\
\rho^{\alpha}&-1/2\end{array}\quad\quad\begin{array}[]{c|c}\text{r.h.\
fermions}&\text{U(1)${}_{R}$ charge}\\\
\hline\cr\tilde{\lambda}^{\dot{\alpha}}&+1/2\\\
\tilde{\psi}^{\dot{\alpha}}&-1/2\\\
\tilde{\rho}^{\dot{\alpha}}&+1/2\end{array}$ (3.50)

Taking into account the two-component spinorial structure of the world-volume
zero-modes, the path integral integration measure produces a violation of the
U(1)R symmetry given by

$\displaystyle\chi_{\rm E3}$ $\displaystyle=$
$\displaystyle(h^{0}(D)-h^{1}(D)+h^{2}(D))-\big{[}h^{0}_{Q}(D)-h^{1}_{Q}(D)+h^{2}_{Q}(D)\big{]}$
(3.51) $\displaystyle\equiv$ $\displaystyle\chi(D,{\cal O}_{D})-\chi(D,{\cal
L}^{-1}_{Q})$ (3.52)

of the U(1)R symmetry. As already written in (3.51), $\chi_{\rm E3}$ can be
seen as the difference of the holomorphic Euler characteristics

$\displaystyle\chi(D,{\cal O}_{D})$ $\displaystyle:=$ $\displaystyle
h^{0}(D)-h^{1}(D)+h^{2}(D)$ (3.53) $\displaystyle\chi(D,{\cal L}^{-1}_{Q})$
$\displaystyle:=$ $\displaystyle h^{0}_{Q}(D)-h^{1}_{Q}(D)+h^{2}_{Q}(D)$
(3.54)

The U(1)R symmetry should not be anomalous, that implies that the U(1)R
violation generated by the fermion zero-modes must be compensated by a shift
$T_{\rm E3}\rightarrow T_{\rm E3}-{\rm i}\chi_{\rm E3}$ in the term
$e^{-T_{\rm E3}}$ appearing in the non-perturbative contribution. In
particular, a superpotential is generated only if in the complete path-
integral there are exactly two zero modes, that implies that a necessary
condition for having a non-trivial superpotential is that

$\chi_{\rm E3}\equiv\chi(D,{\cal O}_{D})-\chi(D,{\cal L}^{-1}_{Q})=1$ (3.55)

In order to evaluate $\chi_{\rm E3}$, it can be useful to express it in terms
of characteristic classes by using the Hirzebruch-Riemann-Roch index theorem:

$\chi(D,{\cal O}_{D})=\int_{D}{\rm
Td}(D)\,,\quad\chi(D,\mathcal{L}_{Q}^{-1})=\int_{D}{\rm ch}({\cal
L}^{-1}_{Q})\wedge{\rm Td}(D)$ (3.56)

By using the isomorphism ${\cal L}_{Q}\simeq K_{{X}}^{-1}$ and the adjunction
formulas, the identities (3.56) can be useful for separately computing
$\chi(D,{\cal O}_{D})$ and $\chi(D,{\cal L}^{-1}_{Q})$. For instance, the
combination that appears in $\chi_{\rm E3}$ simplifies to

$\chi_{\rm E3}=-\tfrac{1}{2}\,\int_{{X}}c_{1}({\cal L}_{Q})\wedge[D]\wedge[D]$
(3.57)

where $[D]$ is the Poincaré dual to $D$ in ${X}$.

By recalling the topology of the total divisor wrapped by the background
7-branes, it is immediate to see from (3.57) that

$\chi_{\rm E3}=-\frac{1}{24}n$ (3.58)

where $n$ is the total number of intersections of the 7-branes with the
divisor self-intersection $D\cap D$, with the appropriate multiplicities. In
particular, the tadpole conditions ensure that $n$ is a multiple of $24$.

Finally, notice that in the weak-coupling orientifold limit $\chi_{\rm E3}$
reduces to the equivariant index known as the holomorphic Lefschetz number
$L(\tilde{D})$ discussed in [20]

$\chi_{\rm
E3}=L(D)=h^{0}_{+}(\tilde{D})-\big{[}h^{0}_{-}(\tilde{D})+h^{1}_{+}(\tilde{D})\big{]}+\big{[}h^{1}_{-}(\tilde{D})+h^{2}_{+}(\tilde{D})\big{]}-h^{2}_{-}(\tilde{D})$
(3.59)

that is computed in the double cover Calabi-Yau $\tilde{X}$, with $\tilde{D}$
being the double cover of the divisor $D$.

In the orientifold limit, the formula (3.58) for $\chi_{\rm E3}$ can be
expressed just in terms of intersections with the O7-planes. Let us continue
working on $X=\tilde{X}/\mathbb{Z}_{2}$. Taking into account that each
O7-plane is the bound-state of two mutually nonlocal 7-branes, one can write

$n=n_{\rm D7}+2n_{\rm O7}$ (3.60)

On the other hand, the tadpole condition restricted on $D\cap D$ gives the
constraint

$n_{\rm D7}=4n_{\rm O7}$ (3.61)

Hence, $n=6n_{\rm O7}$ and one can write the index $\chi_{\rm E3}$ just as

$\displaystyle\chi_{\rm E3}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\,n_{\rm O7}=-\frac{1}{4}\,\int_{{X}}[D_{\rm
O7}]\wedge[D]\wedge[D]$ (3.62) $\displaystyle=$
$\displaystyle-\frac{1}{4}\,\int_{\tilde{X}}[D_{\rm
O7}]\wedge[\tilde{D}]\wedge[\tilde{D}]$ (3.63)

where in the second line we have rewritten the result in the double cover
Calabi-Yau $\tilde{X}$.

More directly, in the weak coupling description reviewed in section 2.4, the
total divisor $D_{\rm O7}$ wrapped by the O7-planes is defined by (2.32),
where $h$ is a section of ${\cal L}^{2}_{Q}$. Hence $[D_{\rm O7}]=c_{1}({\cal
L}_{Q}^{2})=2c_{1}({\cal L}_{Q})$ and then (3.57) immediately implies the
first line of (3.62).

###  3.4 Comparison with the dual M-theory result

In the past sections we have learned that the twisted theory on an E3-brane
allows us to define the zero-modes as elements of sheaf cohomologies of the
form $H^{i}(D,{\cal O}_{D})$ and $H^{i}(D,{\cal L}^{-1}_{Q})$. In this
section, we will briefly establish the link between these objects and the
zero-modes of the dual M5-brane considered by Witten in [15]. Most of this
analysis is based on the papers [60, 61, 62], see also [63].

Witten’s analysis translates the fermionic zero-modes on the M5-instanton in
terms of its worldvolume $(0,i)$-forms. Given an E3-brane on a divisor
$D\subset{X}$, its dual M5-brane is simply the restriction onto $D$ of the
elliptic fibration over ${X}$. Hence, if $\pi$ is the projection of the
elliptic fiber $\pi:{Y}\rightarrow{X}$, then the dual M5 wraps a divisor of
the Calabi-Yau four-fold ${Y}$ is defined as

$\hat{D}\equiv\pi^{-1}(D)$ (3.64)

Since the fibration is holomorphic, the cohomologies of $D$ and $\hat{D}$ are
related by the so-called Leray spectral sequence (see [58], and [59] for some
material), that eventually implies that the Euler characteristics be related
as follows:

$\chi(\hat{D},{\cal O}_{\hat{D}})=\chi_{E3}\equiv\chi(D,{\cal
O}_{D})-\chi(D,K_{{X}})$ (3.65)

Therefore, since $K_{{X}}={\cal L}^{-1}_{Q}$, the new IIB index (3.51)
computes exactly the index introduced by Witten in [43] from the M-theory
vantage point.

In order to learn about the zero-modes individually, however, the results of
Kollár [62] are crucial, because they show that the Leray spectral sequence
simplifies drastically555I.e. it degenerates at ‘page’ $E_{2}$. and one
actually has the following relations:

$\displaystyle\text{dim }H^{0}(\hat{D},{\cal O}_{\hat{D}})$ $\displaystyle=$
$\displaystyle\text{dim }H^{0}(D,{\cal O}_{D})$ $\displaystyle\text{dim
}H^{1}(\hat{D},{\cal O}_{\hat{D}})$ $\displaystyle=$ $\displaystyle\text{dim
}H^{0}(D,K_{X})+\text{dim }H^{1}(D,{\cal O}_{D})$ (3.66)
$\displaystyle\text{dim }H^{2}(\hat{D},{\cal O}_{\hat{D}})$ $\displaystyle=$
$\displaystyle\text{dim }H^{1}(D,K_{X})+\text{dim }H^{2}(D,{\cal O}_{D})$
$\displaystyle\text{dim }H^{3}(\hat{D},{\cal O}_{\hat{D}})$ $\displaystyle=$
$\displaystyle\text{dim }H^{2}(D,K_{X})$

that originally appeared in [60, 61]. Clearly, the cohomology groups in the
r.h.s. of (3.4) coincide with the (complex conjugated) cohomologies appearing
in (3.39), that were found from the direct IIB zero mode spectrum.

The relations (3.4) were only understood as cohomological identities with no
particular meaning. The present treatment of the E3-instanton gives them
physical significance and provides an interpretation of the M5 Hodge numbers
in terms of more palatable quantities from the IIB perspective, that can be
summarized by the diagram in Fig. 1. Therefore, given an instanton in a IIB
setup, one does not necessarily need to construct an M/F-theory lift of it, in
order to determine whether a non-perturbative superpotential will be
generated.666It would be interesting to use the IIB/M-theory relation to
explore the geometrization of non-perturbative effects proposed in [64, 65,
66, 67] from the point of view of E3/M5-brane instantons.

$h^{2}_{Q}({\rm E3})$ $h^{2}({\rm E3})$ $h^{1}_{Q}({\rm E3})$ $h^{1}({\rm
E3})$ $h^{0}_{Q}({\rm E3})$ $h^{0}({\rm E3})$ geom. mod. tw. geom. mod. tw.
Wilson lines Wilson lines $\bar{\tau}^{\dot{\alpha}}$ $\theta^{\alpha}$
$h^{3}({\rm M5})$ $h^{2}({\rm M5})$ $h^{1}({\rm M5})$ $h^{0}({\rm M5})$ Figure
1: Schematic description of the reorganization of the E3 fermionic zero modes
in terms of the corresponding zero modes on the dual M5-brane. Here
$h^{i}({\rm E3})\equiv h^{i}(D)$, $h^{i}_{Q}({\rm E3})\equiv h^{i}_{Q}(D)=\dim
H^{i}(D,K_{X})$ and $h^{i}({\rm M5})\equiv\dim H^{i}(\hat{D},{\cal
O}_{\hat{D}})\equiv\dim H^{0,i}_{\bar{\partial}}(\hat{D})$.

###  3.5 $\chi_{\rm E3}$ as the perturbative Ext index

In this section, we will show that the new index $\chi_{\rm E3}$, despite
being defined in a non-perturbative context, can be understood in terms of
objects that appear naturally in string theory, namely Ext groups.

A divisor $D\subset{X}$ is associated to a holomorphic sheaf ${\cal E}$ with
support on $D$, defined by the short exact sequence

$0\rightarrow{\cal O}_{\rm{X}}(-D)\stackrel{{\scriptstyle\cdot
P_{D}}}{{\longrightarrow}}{\cal
O}_{\rm{X}}\stackrel{{\scriptstyle|_{P_{D}=0}}}{{\longrightarrow}}{\cal
E}_{D}\rightarrow 0$ (3.67)

The first object corresponds to the negative bundle whose sections have degree
opposite to $P_{D}$, where $P_{D}$ is the section of ${\cal O}_{\rm{X}}(D)$
that vanishes on $D$. The middle map corresponds to multiplication by $P_{D}$,
and the map into ${\cal E}_{D}$ corresponds to setting $P_{D}=0$, i.e.
restricting onto $D$. In mathematics, this is called a _resolution_ of ${\cal
E}_{D}$ by line-bundles (locally free sheaves).

One can now compute the groups ${\rm Ext}^{i}({\cal E}_{D},{\cal E}_{D})$ –
see for instance [68, 69] for an introduction to Ext-groups and to their
applications to D-brane physics. Using the so called local-to-global spectral
sequence one gets

$\displaystyle\dim{\rm Ext}^{0}({\cal E}_{D},{\cal E}_{D})$ $\displaystyle=$
$\displaystyle\dim H^{0}(D,{\cal O}_{D})$ $\displaystyle\dim{\rm
Ext}^{1}({\cal E}_{D},{\cal E}_{D})$ $\displaystyle=$ $\displaystyle\dim
H^{1}(D,{\cal O}_{D})+\dim H^{0}(D,N_{D})$ (3.68) $\displaystyle=$
$\displaystyle\dim H^{1}(D,{\cal O}_{D})+\dim H^{2}(D,K_{{X}})$ (3.69)
$\displaystyle\dim{\rm Ext}^{2}({\cal E}_{D},{\cal E}_{D})$ $\displaystyle=$
$\displaystyle\dim H^{2}(D,{\cal O}_{D})+\dim H^{1}(D,N_{D})$ $\displaystyle=$
$\displaystyle\dim H^{0}(D,K_{{X}})+\dim H^{1}(D,K_{{X}})$
$\displaystyle\dim{\rm Ext}^{3}({\cal E}_{D},{\cal E}_{D})$ $\displaystyle=$
$\displaystyle\dim H^{2}(D,N_{D})=\dim H^{0}(D,K_{{X}})$

where $N_{D}$ is the holomorphic normal bundle to $D$.

By recalling that $K_{{X}}\simeq{\cal L}^{-1}_{Q}$, one can see that the the
${\rm Ext}^{i}({\cal E}_{D},{\cal E}_{D})$ classes coincide with the E3
fermionic zero-mode spectrum (3.39). Furthermore, it is easy to see that
$\chi_{\rm E3}$ reproduces the Ext-index777The Hirzebruch-Riemann-Roch theorem
for Ext-groups reads $\int_{{X}}{\rm ch}({\cal E})\,{\rm ch}({\cal
F})^{*}\,{\rm Td}({X})$, that can be used to further check the equivalence
with $\chi_{\rm E3}$.

$\chi_{\rm E3}\equiv{\rm Ind}_{\rm Ext}({\cal E}_{D},{\cal
E}_{D}):=\sum_{i}(-)^{i}\dim{\rm Ext}^{i}({\cal E}_{D},{\cal E}_{D})$ (3.70)

In general, in the perturbative framework in which there are no
SL$(2,\mathbb{Z})$-monodromies and ${X}$ is a Calabi-Yau, given two
holomorphic sheaves ${\cal E}$ and ${\cal F}$ on ${X}$ associated to two
D-branes, the Ext groups count the perturbative spectrum of open strings
stretching between the two D-branes [70]. However, if ${X}$ is a Calabi-Yau
then Serre-duality (i.e. CPT conjugation) implies that $\dim{\rm
Ext}^{i}({\cal E},{\cal F})=\dim{\rm Ext}^{i}({\cal F},{\cal G})$ and then
${\rm Ind}_{\rm Ext}({\cal E}_{D},{\cal E}_{D})\equiv 0$. In the present case,
one gets in general a non-vanishing (3.70) because $K_{{X}}\simeq{\cal
L}_{Q}^{-1}$ is non-trivial, due to the fact that one is actually considering
non-perturbative backgrounds.

In any case, it is somewhat surprising from a physical perspective that the
Ext groups and the associated index, that are usually applied to perturbative
D-brane physics, capture the spectrum of the E3-brane in non-perturbative
backgrounds. It is not completely clear to us whether this just an accidental
correspondence or it hides a deeper motivation.

###  3.6 Implementation in the working example

Let us now put the results of this section into practice by using the example
with ${X}=\mathbb{P}^{3}$ that was introduced in 2.5. We will first show them
from the general IIB point of view on ${X}$, then in Sen’s limit from the
double-cover point of view, and finally in terms of the dual M5-instanton.

Since this is a one-modulus three-fold , all divisors are Poincaré dual to
some multiple of the hyperplane class $[H]\in H^{2}({X},\mathbb{Z})$. Let use
then choose as divisor wrapped by the E3-brane an hypersurface defined by

$D:\\{P^{(n)}(z_{1},\ldots,z_{4})=0\\}$ (3.71)

for any degree $n$ polynomial $P^{(n)}(z_{1},\ldots,z_{4})$. This fixes the
cohomology class of its Poincaré dual to be

$[D]\simeq n[H]$ (3.72)

According to the discussion of section (3.2), in order to compute the spectrum
of E3 fermionic zero-modes, one needs to compute $H^{i}(D,{\cal O}_{D})$ and
$H^{i}(D,{\cal L}^{-1}_{Q})$.

Let us first focus on the cohomology groups $H^{i}(D,{\cal O}_{D})$, that
counts the zero modes associated to the left-handed world-volume fields
$\lambda,\psi,\rho$, see (3.39). In order to compute them, one can use the
short exact sequence:

$0\rightarrow{\cal O}_{\mathbb{P}^{3}}(-n)\rightarrow{\cal
O}_{\mathbb{P}^{3}}\rightarrow{\cal O}_{D}\rightarrow 0$ (3.73)

where the first map is multiplication by the section of ${\cal
O}_{\mathbb{P}^{3}}(n)$ that vanishes on $D$ and the second map is just
restriction.

This induces a long exact sequence of cohomologies. However, it is known that
$H^{i}(\mathbb{P}^{3},{\cal O}_{\mathbb{P}^{3}}(k))=0$ for $i\neq 0,3$ and any
$k$. Hence the long sequence breaks into several short ones and one eventually
gets

$\displaystyle\text{dim }H^{0}(D,{\cal O}_{D})$ $\displaystyle=$
$\displaystyle 1$ (3.74) $\displaystyle\text{dim }H^{1}(D,{\cal O}_{D})$
$\displaystyle=$ $\displaystyle 0$ (3.75) $\displaystyle\text{dim
}H^{2}(D,{\cal O}_{D})$ $\displaystyle=$ $\displaystyle\text{dim
}H^{0}(D,{\cal O}_{\mathbb{P}^{3}}(n-4))$ (3.76)

The dimension of $H^{2}(D,{\cal O}_{D})$ can be more explicitly computed by
recalling that, in general, one can count the sections of a line bundle ${\cal
O}_{\mathbb{P}^{3}}(d)$ by counting monomials of degree $d$, that yields:

$\text{dim }H^{0}(\mathbb{P}^{3},{\cal O}(d))=\binom{d+3}{3}$ (3.77)

Hence, one obtains the following spectrum of left-handed fermions on the
E3-brane

$\begin{array}[]{c|c}\text{l.h.\ fermions}&\text{\\#\ zero modes}\\\
\hline\cr\lambda^{\alpha}&2\times 1\\\ \psi^{\alpha}&0\\\
\rho^{\alpha}&2\times\binom{n-1}{3}\end{array}\begin{array}[]{r}\\\ \\\
\end{array}$ (3.78)

As expected, for every $n$, there are the two universal zero modes
$\lambda^{\alpha}_{\rm z.m.}\sim\theta^{\alpha}$. On the other hand, if $n\geq
4$, there are additional $\frac{2(n-1)!}{3!(n-4)!}$ zero-modes
$\rho^{\alpha}_{\rm z.m.}$.

One can now turn to the right-handed fermions
$\tilde{\lambda},\tilde{\psi},\tilde{\rho}$, whose zero modes are counted by
$H^{i}(D,{\cal L}_{Q}^{-1})\equiv H^{i}(D,{\cal O}_{\mathbb{P}^{3}}(-4))$, see
(3.39). In order to compute their dimensions, one can follow the same strategy
as above, using now the short exact sequence

$0\rightarrow{\cal O}_{\mathbb{P}^{3}}(-n)\otimes
K_{\mathbb{P}^{3}}\rightarrow K_{\mathbb{P}^{3}}\rightarrow
K_{\mathbb{P}^{3}}|_{D}\rightarrow 0$ (3.79)

The result is

$\displaystyle\text{dim }H^{0}(D,K_{\mathbb{P}^{3}})$ $\displaystyle=$
$\displaystyle 0$ (3.80) $\displaystyle\text{dim }H^{1}(D,K_{\mathbb{P}^{3}})$
$\displaystyle=$ $\displaystyle 0$ (3.81) $\displaystyle\text{dim
}H^{2}(D,K_{\mathbb{P}^{3}})$ $\displaystyle=$ $\displaystyle\text{dim
}H^{0}(\mathbb{P}^{3},{\cal O}_{\mathbb{P}^{3}}(n))-1$ (3.82)

By using the general formula (3.77), for the right-handed fermionic spectrum
one then obtains

$\begin{array}[]{c|c}\text{r.h.\ fermions}&\text{\\#\ zero modes}\\\
\hline\cr\tilde{\lambda}^{\dot{\alpha}}&0\\\ \tilde{\psi}^{\dot{\alpha}}&0\\\
\tilde{\rho}^{\dot{\alpha}}&2\times\big{[}\binom{n+3}{3}-1\big{]}\end{array}\begin{array}[]{r}\\\
\\\ \end{array}$ (3.83)

One can then see that the right-handed sector contributes with
$\frac{2(n+3)!}{n!3!}-2$ zero modes $\tilde{\rho}^{\dot{\alpha}}_{\rm z.m.}$.

In summary, in addition to the two universal zero modes $\lambda^{\alpha}_{\rm
z.m.}\sim\theta^{\alpha}$ there is always a certain number of zero modes
$\tilde{\rho}^{\dot{\alpha}}_{\rm z.m.}$ and, for $n\geq 4$, additional zero
modes $\rho^{\alpha}_{\rm z.m.}$. The zero modes
$\tilde{\rho}^{\dot{\alpha}}_{\rm z.m.}$ have a clear geometrical
interpretation, as $\text{dim}\,H^{2}(D,K_{\mathbb{P}^{3}})={\rm
dim}\,H^{0}(D,N_{D})$ by Serre duality and then they can be seen as the
supersymmetric partner of the geometrical deformations of the divisor $D$.
Since $N_{D}\simeq{\cal O}_{\mathbb{P}^{3}}(n)|_{D}$, they can be directly
counted as follows:

$\displaystyle\text{dim }H^{2}(D,K_{\mathbb{P}^{3}})$ $\displaystyle=$
$\displaystyle\text{dim }H^{0}({\cal O}_{\mathbb{P}^{3}}(n))$ $\displaystyle=$
$\displaystyle\text{\\# monomials of degree n in }z_{1},\ldots z_{4}\text{
minus one}$ $\displaystyle=$ $\displaystyle\binom{n+3}{3}-1$ (3.85)

where the minus one corresponds subtracting the defining polynomial $P^{(n)}$,
since we only count deformations. On the other hand, $\rho^{\alpha}_{\rm
z.m.}$ are associated to some kind of twisted geometrical deformations, whose
interpretation will become more transparent in the weak coupling orientifold
limit.

Now, the index $\chi_{\rm E3}$ can be immediately computed from (3.51) to be

$\chi_{\rm E3}=-2\,n^{2}$ (3.86)

that is always different from $1$. Alternatively, one can easily get the same
result from (3.57), by using (3.72), together with the identity $c_{1}({\cal
L}_{Q})=-c_{1}(K_{\mathbb{P}^{3}})=4[H]$ and the fact that
$\int_{\mathbb{P}^{3}}[H]\wedge[H]\wedge[H]=1$.

Let us now revisit these results from the orientifold point of view, as was
done in [20]. The divisor $D\subset\mathbb{P}^{3}$ has a double-cover divisor
$\tilde{D}$ in the double-cover Calabi-Yau three-fold
$\subset\tilde{X}=\mathbb{P}^{4}_{11114}[8]$ discussed in section 2.5. Now,
the canonical bundle of the three-fold is trivial, but the relevant
information is encoded in the $\mathbb{Z}_{2}$-equivariant structure of the
orientifold involution $\xi\rightarrow-\xi$.

The condition (3.72) translates into $[\tilde{D}]=n\,[H]$ in $\tilde{X}$,
where we continue to use the symbol $H$ to denote the double cover of $H$,
hoping that this will not cause confusion. Its Poicaré dual $[H]$ in
$\tilde{X}$ can be thought of as the pullback of the hyperplane class $[H]$ in
$\mathbb{P}^{3}$ under the $\mathbb{Z}_{2}$ projection and we use the notation
${\cal O}_{\tilde{X}}(n)\equiv{\cal O}_{\tilde{X}}(n\,H)$. The most general
effective divisor $\tilde{D}$ of this kind is given by the intersection of
(2.40) and the vanishing locus

$\tilde{P}^{(n)}(z_{1},\ldots,z_{4},\xi)=0$ (3.87)

of a homogeneous degree $n$ polynomial
$\tilde{P}^{(n)}(z_{1},\ldots,z_{4},\xi)$, where $\xi$ itself has degree four,
that is furthermore even in $\xi$ in order to respect the orientifold
involution $\xi\rightarrow-\xi$. However, we restrict to configurations that
are irreducible double covers of single connected holomorphic hypersurfaces
$D$ in ${X}\equiv\mathbb{P}^{3}$, with (3.87) just given by (3.71), i.e.
independent of $\xi$. This means that the divisor $\tilde{D}$ is transversely
invariant under the orientifold action, i.e. is a so-called O(1) instanton.

In order to understand the fermionic zero-mode spectrum, one needs to compute
the orientifold-even and -odd cohomology groups $H^{i}_{\pm}(\tilde{D},{\cal
O}_{\tilde{D}})$.

First, considering only connected transversely invariant divisors, one finds

$\text{dim }H^{0,0}_{+}(\tilde{D})=1\,,\quad\text{dim
}H^{0,0}_{-}(\tilde{D})=0$ (3.88)

This matches (3.78) and (3.83), cf. with (3.48).

Furthermore, it is useful to note that the line bundle ${\cal
O}_{\tilde{X}}(1)$ is very ample. Then, the Lefschetz hyperplane theorem
guarantees that

$H^{1}(\tilde{D},{\cal O}_{\tilde{D}})=0\quad\Leftrightarrow\quad
H_{\pm}^{0,1}(\tilde{D})=0$ (3.89)

By inspecting (3.48), one can see that this is again in agreement with (3.78)
and (3.83).

Finally, by Serre duality, one knows that

$\text{dim }H^{0,2}_{\pm}(\tilde{D})=\text{dim
}H^{0}_{\mp}(\tilde{D},N_{\tilde{D}})$ (3.90)

where $N_{\tilde{D}}$ is the normal bundle to $\tilde{D}$ in $\tilde{X}$. Note
that, since the holomorphic $(3,0)$ form is orientifold-odd, Serre duality
exchanges the two subgroups of the cohomology with opposite $\mathbb{Z}_{2}$
parity. Therefore, $H^{0,2}_{-}(\tilde{D})$ corresponds to divisor
deformations that respect the orientifold involution, whereas
$H^{0,2}_{+}(\tilde{D})$ corresponds to those that are odd under it. In fact,
one can directly count the even and odd sections. Orientifold _odd_ sections
of the normal bundle correspond to deformations of the divisor’s polynomial
$\tilde{P}^{(n)}(z_{1},\ldots,z_{4},\xi)$ that are linear in $\xi$. The even
ones do not contain $\xi$. Notice that higher powers of $\xi$ can be
eliminated via the hypersurface equation (2.40).

It is then easy to see that the counting of $H^{0,2}_{-}(\tilde{D})$ is
identical to the counting in (3.6) and gives the same result. This is of
course what one was to reproduce: the number of zero modes $\tilde{\rho}_{\rm
z.m.}^{\dot{\alpha}}$ in (3.83).

On the other hand, the odd sections, that are linear in $\xi$, are counted by

$\displaystyle\dim H^{0,2}_{+}(\tilde{D})$ $\displaystyle=$
$\displaystyle\text{dim }H^{0}_{-}(\tilde{D},N_{\tilde{D}})$ $\displaystyle=$
$\displaystyle\text{\\# monomials of degree $n-4$ in }z_{1},\ldots z_{4}$
$\displaystyle=$ $\displaystyle\binom{n-4+3}{3}$ (3.92)

This indeed coincides with the dimension of $H^{2}(D,{\cal O}_{D})$ and gives
a clearer interpretation of the zero modes $\rho^{\alpha}_{\rm z.m.}$ found in
(3.78).

Finally, according to the discussion in section 3.4 one can easily lift all
this information to the dual M5-brane. By using (3.4), one can relate the
cohomologies of $D\subset{X}$ to the cohomologies of the M5-instanton divisor
$\hat{D}\equiv\pi^{-1}(D)\subset{Y}$, where $\pi:{Y}\rightarrow{X}$ is the
projection from the elliptically fibered Calabi-Yau four-fold to its base
${X}$, as follows:

$\displaystyle\text{dim }H^{0}(\hat{D},{\cal O}_{\hat{D}})$ $\displaystyle=$
$\displaystyle\text{dim }H^{0}(D,{\cal O}_{D})=1$ $\displaystyle\text{dim
}H^{1}(\hat{D},{\cal O}_{\hat{D}})$ $\displaystyle=$ $\displaystyle\text{dim
}H^{0}(D,K_{X})+\text{dim }H^{1}(D,{\cal O}_{D})=0$ (3.93)
$\displaystyle\text{dim }H^{2}(\hat{D},{\cal O}_{\hat{D}})$ $\displaystyle=$
$\displaystyle\text{dim }H^{1}(D,K_{X})+\text{dim }H^{2}(D,{\cal
O}_{D})=\binom{n-1}{3}$ $\displaystyle\text{dim }H^{3}(\hat{D},{\cal
O}_{\hat{D}})$ $\displaystyle=$ $\displaystyle\text{dim
}H^{2}(D,K_{X})=\binom{n+3}{3}-1$

The relevant index for the M5, i.e. the holomorphic characteristic
$\chi(\hat{D},{\cal O})$ [15], does obviously coincide with (3.86).

##  4 Magnetized E3-instantons

From now on, we would like to relax the simplifying assumption (3.1) and study
how a non-trivial world-volume flux

${\cal F}:=2\pi\alpha^{\prime}F_{\rm E3}-\iota^{*}B_{\it 2}\neq 0$ (4.1)

can change the story. The supersymmetry of the E3-brane imposes that it still
wrap an effective divisor $D$ and that the world-volume flux ${\cal F}$
satisfy the anti-self-duality condition

${\cal F}=-*{\cal F}$ (4.2)

that can be alternatively expressed by the pair of conditions

$\displaystyle{\cal F}^{0,2}$ $\displaystyle=$ $\displaystyle
0\qquad(\text{holomorphy})$ (4.3) $\displaystyle J\wedge{\cal F}$
$\displaystyle=$ $\displaystyle 0\qquad(\text{primitivity})$ (4.4)

This conditions can be derived from standard arguments, paying particular
attention to the analytic continuation necessary to extend the fermions to
Euclidean signature, as in section 3.1.888See for instance appendix D of [64]
for a detailed general discussion, that includes the setting considered in
this paper as a particular subcase.

###  4.1 World-volume fluxes and duality

As it is clear form (4.1), the world-volume flux ${\cal F}$ receives two kinds
of contributions. One comes from the pull-back of the bulk NS-NS $B_{\it 2}$
field, and the other from the world-volume flux $F_{\rm E3}$. Both transform
non-trivially under the SL(2,$\mathbb{Z}$)-duality transformation.

Let us first recall that $B_{\it 2}$ pairs up with the R-R two form $C_{\it
2}$ to transform as a doublet under SL(2,$\mathbb{Z}$)-duality

$\left(\begin{array}[]{c}C_{\it 2}\\\ B_{\it
2}\end{array}\right)\rightarrow\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\left(\begin{array}[]{c}C_{\it 2}\\\ B_{\it
2}\end{array}\right)$ (4.5)

As we are going to discuss, this transformation is consistent with the
transformation of the world-volume flux $F_{\rm E3}$.

Let us then turn to the purely world-volume part $F_{\rm E3}$. First introduce
the dual field-strength $F^{D}_{\rm E3}$ defined as

$F^{D}_{\rm E3}=-2\pi{\rm i}\,\frac{\delta S_{\rm E3}}{\delta F_{\rm E3}}$
(4.6)

In this section, as already stated, we work in Euclidean signature. Thus most
equations should be considered as the analytical continuation of equations in
Minkwskian signature, wherein their meaning would be more transparent. In
(4.6) the functional derivative is defined by $\delta S_{\rm
E3}=\int_{D}\frac{\delta S_{\rm E3}}{\delta F_{\rm E3}}\wedge\delta F_{\rm
E3}$ and $S_{\rm E3}$ is the (Wick rotatated) general off-shell action

$S_{\rm E3}=\frac{1}{(2\pi)^{3}\alpha^{\prime
2}}\int_{D}\text{d}^{4}\sigma\sqrt{\det(h+e^{-\phi/2}{\cal F})}\,+\frac{{\rm
i}}{(2\pi)^{3}\alpha^{\prime 2}}\int_{D}\Big{(}C_{\it 4}+C_{\it 2}\wedge{\cal
F}+\frac{1}{2}C_{\it 0}{\cal F}\wedge{\cal F}\Big{)}$ (4.7)

This action, and actually its full supersymmetric extension, is invariant
under SL(2,$\mathbb{Z}$)-duality, under which $F^{D}_{\rm E3}$ and $F_{\rm
E3}$ transform as a doublet – see for instance the explicit discussion in
[57]. Hence, it is useful to introduce

${\cal F}^{D}:=2\pi\alpha^{\prime}F^{D}_{\rm E3}-\iota^{*}C_{\it 2}$ (4.8)

that pairs with ${\cal F}$ into a doublet under SL(2,$\mathbb{Z}$)-duality:

$\left(\begin{array}[]{c}{\cal F}^{D}\\\ {\cal
F}\end{array}\right)\rightarrow\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\left(\begin{array}[]{c}{\cal F}^{D}\\\ {\cal
F}\end{array}\right)$ (4.9)

It is then convenient to consider the combinations

$\displaystyle{\cal F}^{-}$ $\displaystyle:=$ $\displaystyle{\rm
i}e^{\phi}(\bar{\tau}{\cal F}-{\cal F}^{D})$ (4.10) $\displaystyle{\cal
F}^{+}$ $\displaystyle:=$ $\displaystyle-{\rm i}e^{\phi}(\tau{\cal F}-{\cal
F}^{D})$ (4.11)

that transform as sections of ${\cal L}_{Q}$ and $\bar{\cal L}_{Q}$
respectively under SL(2,$\mathbb{Z}$)-duality.

Now, the key observation is that in the case we are interested in, namely
E3-branes wrapping a divisor $D$ with anti-self-dual flux (4.2), one has

${\cal F}^{-}=2{\cal F}\,,\qquad{\cal F}^{+}=0\qquad\qquad\text{(for $*{\cal
F}=-{\cal F}$)}$ (4.12)

A quick check of this is obtained by using the fact that the supersymmetric
E3-brane is calibrated, in generalized sense of [71]. For the purpose of
evaluating the on-shell value of $F^{D}_{\rm E3}$ from (4.6), this allows to
use the simplified action obtained by substituting the DBI term in (4.7) with
the value provided by the generalized calibration ${\rm Re\hskip
1.00006pt}(e^{{\rm i}J})$, that is

$S_{\rm E3}=\frac{1}{(2\pi)^{3}\alpha^{\prime
2}}\int_{D}\Big{[}\frac{1}{2}J\wedge J+{\rm i}\Big{(}\frac{1}{2}\,\tau\,{\cal
F}\wedge{\cal F}+C_{\it 4}+C_{\it 2}\wedge{\cal F}\Big{)}\Big{]}+\ldots$
(4.13)

where the $\ldots$ on the right are contributions at least quadratic in terms
that vanish on the supersymmetric configurations [72] and then they do not
effectively contribute to the evaluation of the on-shell $F^{D}_{\rm E3}$.

From (4.13) and (4.6) one readily gets $2\pi\alpha^{\prime}F^{D}_{\rm
E3}=\tau{\cal F}+C_{\it 2}$, that indeed implies (4.12). Equations (4.12) have
the consequence that, in the case we are interested in, one should consider
${\cal F}$ as taking values in ${\cal L}_{Q}$, that is

$e^{-\phi/2}{\cal F}\in\Lambda^{1,1}\otimes L_{Q}$ (4.14)

Furthermore, since $\bar{\partial}{\cal F}=0$, then
$\bar{\partial}_{Q}(e^{-\phi/2}{\cal F})=0$ and ${\cal F}$ identifies the
following cohomology classes

$[e^{-\phi/2}{\cal F}]\in
H^{1,1}_{\bar{\partial}}(D,L_{Q})\quad\Leftrightarrow\quad[{\cal F}]\in
H^{1}(D,T^{*}_{D}\otimes{\cal L}_{Q})$ (4.15)

where $T^{*}_{D}$ is the holomorphic cotangent bundle on $D$.

At this point one has to face a little subtlety. Take the Minkowskian Bianchi
identity $\text{d}{\cal F}=0$ (if $H_{\it 3}=0$) and assume that it is valid
in the Euclidean case we are interested in. Being ${\cal F}$ purely $(1,1)$,
this naturally splits into $\bar{\partial}{\cal F}=0$ and $\partial{\cal
F}=0$. Now, as we saw above, $\bar{\partial}{\cal F}=0$ is
SL$(2,\mathbb{Z})$-covariant, being equivalent to
$\bar{\partial}_{Q}(e^{-\phi/2}{\cal F})=0$. On the other hand, $\partial{\cal
F}=0$ is not SL$(2,\mathbb{Z})$-covariant. This effect is accompanied by a
failure of the expected field equation $\text{d}{\cal
F}^{D}=\text{d}(\tau{\cal F})=0$. Again, this can be split into two parts:
$\bar{\partial}{\cal F}^{D}=0$ and $\partial{\cal F}^{D}=0$. Since ${\cal
F}^{D}=\tau{\cal F}$, the first is indeed automatically implied by the Bianchi
identity $\bar{\partial}{\cal F}=0$ and then it is
SL$(2,\mathbb{Z})$-covariant by itself. On the other hand, the second is
incompatible with the SL$(2,\mathbb{Z})$ non-covariant Bianchi Identity
$\partial{\cal F}=0$, and furthermore it is not SL$(2,\mathbb{Z})$-covariant
by itself. Hence, by invoking the principle of SL$(2,\mathbb{Z})$-covariance,
it is natural to consider the possibility that $\partial{\cal F}=0$ and
$\partial(\tau{\cal F})=0$ are both incorrect and should be substituted by a
single SL$(2,\mathbb{Z})$-covariant equation, that should reduce to
$\partial{\cal F}=0$ when $\tau$ is constant. The most natural choice is

$\partial_{Q}(e^{-\phi/2}{\cal F})=0$ (4.16)

We may interpret this possibility as follows. In Minkowski signature Bianchi
identities and equations of motion are usually put on different footings and
only the second ones are influenced by the background axion-dilaton. On the
other hand, switching to Euclidean signature, the supersymmetric anti-self-
duality condition mixes Bianchi identities and equations of motion into an
anti-holomorphic ($\bar{\partial}$) and holomorphic ($\partial$) part. Now it
would be the anti-holomorphic part that is insensitive to the non-trivial
$\tau$, while the holomorphic one would be modified by it.999This unbalancing
between holomorphic and anti-holomorphic sectors generated by the non-trivial
$\tau$, that is an ubiquitous ingredient in the present discussion, has a
simpler manifestation if one considers E($-1$)-instantons. Indeed, the action
would simply be given by $\tau(z_{\rm E(-1)})$, and so it would automatically
be extremized wrt the anti-holomorphic variables ($\bar{z}_{\rm E(-1)}$) but
not wrt the holomorphic ones ($z_{\rm E(-1)}$).  In fact, we will never
explicitly need (4.16), with the only exception of section 4.6, where one will
be naturally led to consider it.

The present situation is reminiscent of the fate of YM instantons in the
presence of non-trivial VEV’s for charged scalars. Instanton solutions no
longer satisfy the classical coupled field equations and they only extremize
the Euclidean action if some ‘constraint’ is added by including additional
terms in the action, whence the name ‘constrained instantons’. Although one
can feel uneasy by expanding around a configuration that is not a solution of
the ‘naive’ field equations, the success of supersymmetric instanton
calculus101010See e.g.[73] for a recent review., that relies on elegant and
sound localization techniques, should be taken as an encouraging analogy for
the validity of our analysis. At any rate, whenever a perturbative orientifold
limit with constant $\tau$ is possible, these subtleties seems to disappear.
Furthermore, even for weakly coupled orientifold limits with non-constant
$\tau$, one could check the validity of the analysis by directly using open-
string techniques.

###  4.2 Fermions and fluxes: the simplest case of constant $\tau$

We now come back to our main problem, the study of the effect of a non-trivial
world-volume flux (4.1) on the fermionic zero-modes spectrum. In this
subsection we restrict to the simpler subcase in which the axion-dilaton is
(or is approximated to be) constant and the internal metric is Ricci-flat.
Clearly the weak-coupling orientifold backgrounds fit into this class.

What one needs to do is to repeat the steps of section 3.1, so one can proceed
quite speedily, mutatis mutandis for taking into account the non-trivial flux.
First of all, the operator $\Gamma_{\rm E3}$ gets modified. Namely, by using
the gamma-matrix decomposition (2.13), in (3.7) one should take
$\hat{\gamma}_{\rm E3}=\mathbbm{1}\otimes\gamma_{\rm E3}({\cal F})$, with

$\gamma_{\rm E3}({\cal F})=\frac{-{\rm i}\,\epsilon^{a_{1}\ldots
a_{4}}}{\sqrt{\det(h+e^{-\phi/2}{\cal
F})}}\Big{(}\frac{1}{4!}\gamma_{a_{1}\ldots
a_{4}}+\frac{1}{4}\,e^{-\phi/2}{\cal
F}_{a_{1}a_{2}}\gamma_{a_{3}a_{4}}+\frac{1}{8}\,e^{-\phi}{\cal
F}_{a_{1}a_{2}}{\cal F}_{a_{3}a_{4}}\Big{)}$ (4.17)

We continue imposing $\kappa$-fixing condition in the form (3.6) and the
supersymmetry condition still reads (3.12).

The quadratic action for the $\kappa$-fixed Green-Schwarz fermions is now
given by [16]

$S_{\rm F}=\frac{2\pi{\rm
i}}{\ell^{4}_{s}}\int_{D}\text{d}^{4}\sigma\,\sqrt{\det
M}\,\,\overline{\Theta}({\cal M}^{-1})^{ab}\Gamma_{a}\hat{\nabla}_{b}\Theta$
(4.18)

where $M_{ab}:=h_{ab}+e^{-\phi/2}{\cal F}_{ab}$ and ${\cal
M}_{ab}:=h_{ab}\mathbbm{1}+e^{-\phi/2}{\cal F}_{ab}\,\sigma_{3}$.

One also needs to decompose $\Theta$ in terms of new (topologically twisted)
world-volume fields. Because of the modification of $\Gamma_{\rm E3}$, the
decomposition (3.15) does not fulfill the $\kappa$-fixing condition (3.6)
anymore and must be deformed into a new one. A new decomposition that we will
find convenient is

$\displaystyle\left\\{\begin{array}[]{l}\theta_{1}^{\rm
R}=\frac{1}{2}\Big{(}\tilde{\lambda}\otimes\eta+\sqrt{\frac{\det h}{\det
M}}M^{a}{}_{b}\tilde{\psi}_{a}\otimes\gamma^{b}\eta^{*}+\frac{1}{2}\tilde{\rho}_{ab}\otimes\gamma^{ab}\eta\Big{)}\\\
\theta_{2}^{\rm R}=\frac{{\rm
i}}{2}\Big{(}\tilde{\lambda}\otimes\eta-\sqrt{\frac{\det h}{\det
M}}(M^{T})^{a}{}_{b}\tilde{\psi}_{a}\otimes\gamma^{a}\eta^{*}+\frac{1}{2}\tilde{\rho}_{ab}\otimes\gamma^{ab}\eta\Big{)}\end{array}\right.$
(4.21) $\displaystyle\left\\{\begin{array}[]{l}\theta_{1}^{\rm
L}=\frac{1}{2}\Big{(}\lambda\otimes\eta^{*}+\sqrt{\frac{\det h}{\det
M}}M^{a}{}_{b}\psi_{a}\otimes\gamma^{b}\eta+\frac{1}{2}\rho_{ab}\otimes\gamma^{ab}\eta^{*}\Big{)}\\\
\theta_{2}^{\rm L}=\frac{{\rm
i}}{2}\Big{(}\lambda\otimes\eta^{*}-\sqrt{\frac{\det h}{\det
M}}(M^{T})^{a}{}_{b}\psi_{a}\otimes\gamma^{a}\eta+\frac{1}{2}\rho_{ab}\otimes\gamma^{ab}\eta^{*}\Big{)}\end{array}\right.$
(4.24)

The decomposition (4.21) clearly reduces to (3.15) in the limit ${\cal
F}\rightarrow 0$, in which $M\rightarrow h$.

By pugging (4.21) into the action (4.18), after some manipulations, one gets

$\displaystyle S_{\rm F}$ $\displaystyle=$ $\displaystyle\frac{4\pi{\rm
i}}{\ell_{s}^{4}}\int_{D}\big{(}\psi\wedge*\partial\lambda-\tilde{\psi}\wedge*\bar{\partial}\tilde{\lambda}-\rho\wedge*\bar{\partial}\psi+\tilde{\rho}\wedge*\partial\tilde{\psi}\big{)}$
(4.26) $\displaystyle\quad\qquad+\frac{4\pi{\rm
i}}{\ell_{s}^{4}}\int_{D}\sqrt{\det h}\,\big{(}\rho\cdot{\cal S}_{\cal
F}\cdot\rho-\tilde{\rho}\cdot\tilde{\cal S}_{\cal F}\cdot\tilde{\rho}\big{)}$

We have introduced the short-handed notation

$\rho\cdot{\cal S}_{\cal F}\cdot\rho:=\frac{1}{4}\rho^{ab}({\cal S}_{{\cal
F}})_{ab}{}^{cd}\rho_{cd}\,,\quad\tilde{\rho}\cdot\tilde{\cal S}_{\cal
F}\cdot\tilde{\rho}:=\frac{1}{4}\tilde{\rho}^{ab}(\tilde{\cal S}_{{\cal
F}})_{ab}{}^{cd}\tilde{\rho}_{cd}$ (4.27)

where ${\cal S}_{\cal F}$ and $\tilde{\cal S}_{{\cal F}}$ are tensors that are
anti-symmetric and purely (anti-)holomorphic in $(ab)$ and $(cd)$, explicitly
given in (anti-)holomorphic indices by

$\displaystyle({\cal S}_{\cal F})_{\bar{\imath}\bar{\jmath}}{}^{uv}$
$\displaystyle=$ $\displaystyle-e^{-\phi}{\cal
K}^{m}{}_{\bar{t}[\bar{\imath}}{\cal
F}_{\bar{\jmath}]}{}^{\bar{t}}\bar{\Omega}_{m}{}^{uv}$ (4.28)
$\displaystyle(\tilde{\cal S}_{\cal F})_{ij}{}^{\bar{u}\bar{v}}$
$\displaystyle=$ $\displaystyle-e^{-\phi}{\cal K}^{m}{}_{t[i}{\cal
F}_{j]}{}^{t}\Omega_{m}{}^{\bar{u}\bar{v}}\ $ (4.29)

The first line in (4.26) is just (3.31) in the case of constant axion-dilaton,
$Q_{\it 1}=0$. Hence, one can see that the new decomposition (4.21) allows us
to get the same kinetic terms for the topologically twisted world-volume
fields. The possibility to reach this result is not a priory obvious at all.

The second line of (4.26) provides a new mass-like term for $\rho$ and
$\tilde{\rho}$, induced by the world-volume flux through the appearance of the
extrinsic curvature ${\cal K}^{m}{}_{ab}$. The definition and some useful
properties of the extrinsic curvature are reviewed in appendix B.

In the case of an orientifold background, the action of the E3-brane wrapping
the double-cover divisor $\tilde{D}$ in the double cover Calabi-Yau
$\tilde{X}$ take the very same form as (4.26). Both $\Omega$ and ${\cal F}$
are odd under the orientifold involution and then $S_{\cal F}$ and
$\tilde{S}_{\cal F}$ stay invariant. Furthermore, since the monodromy (2.24)
acts on $\Theta$ as $-{\rm i}\sigma_{2}$ and on $\eta$ as $\eta\rightarrow{\rm
i}\eta$, it is easy to see from (4.21) that $\lambda$, $\psi$ and $\rho$ are
even under the O7-involution, while $\tilde{\lambda}$, $\tilde{\psi}$ and
$\tilde{\rho}$ are odd. Hence the action (4.26) is manifestly invariant under
the orientifold involution, as required.

We stress that the above transformation rules under the orientifold involution
for the topologically twisted world-volume fermions are the same as those
found in the flux-less case discussed in section 3.2. This is probably one of
the reasons why our choice (4.21) leads to the particularly simple action
(4.26). Crucially, as we will see in the next subsection, this observation
continue to hold for the complete SL(2,$\mathbb{Z}$)-duality group.

###  4.3 Fermions and fluxes with generic $\tau$

We can now address the generalization of the above results to the case in
which $\tau$ has a more general non-constant (and holomorphic) profile. In
principle, one could repeat the steps of the previous subsection, starting
back again from the general action in [16] and simply allowing for a non-
constant $\tau$. Although straightforward, this procedure turns out to be
technically quite intricate. Then, here we follow an alternative strategy,
that allows to obtain the desired result without much effort.

The important ingredient is that the action should be invariant under
SL(2,$\mathbb{Z}$) duality. This has been demonstrated in full generality in
[57] and in this section we would like to explore the consequences of this
general result for the specific problem at hand. In the flux-less case of
section 3 we found that the topologically twisted world-volume fermions
transform nicely under the SL(2,$\mathbb{Z}$) duality and in section 4.2 we
have already seen that, for constant $\tau$, this property is preserved for
the $\mathbb{Z}_{2}$ subgroup corresponding to the orientifold involution.
What we would like to argue here is that, even for non-constant $\tau$, the
decomposition (4.21) provides topologically twisted world-volume fermions that
continue to transform under the SL(2,$\mathbb{Z}$) duality in exactly the same
simple way that was found in the flux-less case.

The understanding of the transformation properties of the topologically
twisted world-volume fermions is not as straightforward as in the fluxless
case for the following reason. As already mentioned, in general the
combinations $\theta_{1}\pm{\rm i}\theta_{2}$ of the Majorana-Weyl fermions
appearing in (3.4) transform with U(1)Q charge $\pm 1/2$ under
SL(2,$\mathbb{Z}$) duality. If one tries to naively apply this transformation
to (4.21), taking into account that $\eta$ and ${\cal F}$ have U(1)Q charge
$+1/2$ and $+1$, one seems to encounters an horrible non-linear propagation of
this U(1)Q action on the topologically twisted world-volume fermions, that
originates exactly in the (non-linear) presence of the non-vanishing ${\cal
F}$ in (4.21). However, one has to be careful since, in presence of world-
volume fluxes, this U(1)Q action generically breaks the $\kappa$-fixing
condition (3.6). Let us try to be very explicit on this point.

In bi-spinor formalism, the U(1)Q action is given by

$\Theta\rightarrow\Theta^{\prime}=e^{-\frac{{\rm
i}}{2}\alpha\,\sigma_{2}}\Theta$ (4.30)

where $\alpha=\arg(c\tau+d)$. On the other hand, the infinitesimal
$\kappa$-symmetry gauge transformation around a bosonic background can be
written as

$\delta_{\kappa}\Theta=P^{+}_{\kappa}({\cal F})\kappa$ (4.31)

for an arbitrary spinor $\kappa$, where use of the projectors has been made

$P^{\pm}_{\kappa}({\cal F})=\frac{1}{2}\big{(}\mathbbm{1}\pm\Gamma_{\rm
E3}\big{)}$ (4.32)

that, as explicitly indicated, depend on ${\cal F}$. In this paper we have
chosen to gauge-fix (4.31) by imposing that

$P^{+}_{\kappa}({\cal F})\Theta=0$ (4.33)

Now, it should be evident that the SL(2,$\mathbb{Z}$) action (4.30)
generically breaks (4.33), that must then be re-established by acting with a
compensating $\kappa$-symmetry transformation.

In order to find the $\kappa$-fixing preserving transformation, it is
convenient to work with an infinitesimal SL(2,$\mathbb{R}$)-trasformation of
the form (4.30), that should still be preserved at the supergravity level.
Hence

$\delta_{S}\Theta=-\frac{{\rm i}}{2}\,\alpha\,\sigma_{2}\Theta$ (4.34)

Now, the deformed bispinor $\Theta^{\prime}\simeq\Theta+\delta_{S}\Theta$
should be corrected by a $\kappa$-symmetry transformation in such a way that
it preserves the $\kappa$-fixing condition $P^{+}_{\kappa}({\cal
F}^{\prime})\Theta^{\prime}=0$, where $e^{-\phi^{\prime}/2}{\cal
F}^{\prime}=e^{{\rm i}\alpha}e^{-\phi/2}{\cal F}\simeq(1+{\rm
i}\alpha)e^{-\phi/2}{\cal F}$. Hence, by defining
$\delta_{S}P^{-}_{\kappa}({\cal F})=P^{-}_{\kappa}({\cal
F}^{\prime})-P^{-}_{\kappa}({\cal F})$, the $\kappa$-fixing preserving first
order deformation is given by

$\hat{\delta}_{S}\Theta=P^{-}_{\kappa}({\cal F})\,\delta_{S}\Theta+\delta
P^{-}_{\kappa}({\cal F})\,\Theta$ (4.35)

At this point, all one has to do is to plug the decomposition (4.21) into
(4.35), taking into account that $\eta$ transform with U(1)Q charge $+1/2$ and
extract the resulting transformations for the world-volume fermions. After
some tedious algebra, one eventually finds that all nasty terms drop out and
one is left with a simple linear action of the topologically twisted world-
volume fermions

$\displaystyle\hat{\delta}_{S}\lambda=\hat{\delta}_{S}\psi=\hat{\delta}_{S}\rho=0$
(4.36) $\displaystyle\hat{\delta}_{S}\tilde{\lambda}=-{\rm
i}\alpha\tilde{\lambda}\,,\quad\hat{\delta}_{S}\tilde{\psi}={\rm
i}\alpha\tilde{\psi}\,,\quad\hat{\delta}_{S}\tilde{\rho}=-{\rm
i}\alpha\tilde{\rho}$ (4.37)

Hence, on the twisted world-volume fermions the apparently complicated action
of the infinitesimal SL(2,$\mathbb{R}$) duality simplifies drastically and
reduces to a simple linear action. In particular, this can be readily
exponentiated into a simple finite SL(2,$\mathbb{R}$), and hence
SL(2,$\mathbb{Z}$), action of the form

$\lambda,\psi,\rho\rightarrow\lambda,\psi,\rho\quad,\quad\tilde{\lambda}\rightarrow
e^{-{\rm i}\alpha}\tilde{\lambda}\quad,\quad\tilde{\psi}\rightarrow e^{{\rm
i}\alpha}\tilde{\psi}\quad,\quad\tilde{\rho}\rightarrow e^{-{\rm
i}\alpha}\tilde{\rho}$ (4.38)

Hence, even in presence of fluxes, the topologically twisted world-volume
fermions continue to transform under SL(2,$\mathbb{Z}$)-duality with the same
U(1)Q charges as in the fluxless case. In particular, their properties are
still summarized by (3.24). One can then see the naturalness of the
decomposition (4.21), even in the case of non-constant $\tau$.

Furthermore, notice that the tensors ${\cal S}_{\cal F}$ and $\tilde{S}_{\cal
F}$ transform nicely under SL(2,$\mathbb{Z}$)-transformation, namely with
charge $0$ and $+2$. In particular, they can be seen as operators

${\cal S}_{\cal F}:\Lambda^{2,0}\rightarrow\Lambda^{0,2}\,,\quad\tilde{\cal
S}_{\cal F}:\Lambda^{0,2}\otimes L_{Q}^{-1}\rightarrow\Lambda^{2,0}\otimes
L_{Q}$ (4.39)

One can now make the following observation. Take $\tau$-constant. With the
above rules, it is clear that the action (4.26) is invariant under a global
SL(2,$\mathbb{Z}$)-duality. This action must extend to the case of a more
general F-theory background with non-constant $\tau$ in such a way as to be
invariant under the SL(2,$\mathbb{Z}$)-transformations that characterize it.
Furthermore this generalization must reduce to (3.31) in the case of constant
flux. All these ingredients mashed together practically uniquely fix the
complete effective action to be the simple ‘covariantization’ of (4.26)
obtained by replacing the ordinary derivatives on U(1)Q-charged fields with
the U(1)Q-covariant derivatives.

Hence, we conclude that, in presence of a non-trivial world-volume flux, the
fermionic E3-brane action is given by

$\displaystyle S_{\rm F}$ $\displaystyle=$ $\displaystyle\frac{4\pi{\rm
i}}{\ell_{s}^{4}}\int_{D}\big{(}\psi\wedge*\partial\lambda-\tilde{\psi}\wedge*\bar{\partial}_{Q}\tilde{\lambda}-\rho\wedge*\bar{\partial}\psi+\tilde{\rho}\wedge*\partial_{Q}\tilde{\psi}\big{)}$
(4.41) $\displaystyle\quad\qquad+\frac{4\pi{\rm
i}}{\ell_{s}^{4}}\int_{D}\sqrt{\det h}\,\big{(}\rho\cdot{\cal S}_{\cal
F}\cdot\rho-\tilde{\rho}\cdot\tilde{\cal S}_{\cal F}\cdot\tilde{\rho}\big{)}$

where $\partial_{Q}:=\partial-{\rm i}qQ^{1,0}$,
$\bar{\partial}_{Q}:=\bar{\partial}-{\rm i}qQ^{0,1}$, with $q$ being the
U(1)Q-charge given in (3.24).

###  4.4 Flux-modified zero modes

The action (3.31) leads to the following flux-modified fermionic equations

$\displaystyle\partial\lambda=0\quad$ ,
$\displaystyle\quad\bar{\partial}_{Q}\tilde{\lambda}=0$ (4.42)
$\displaystyle\bar{\partial}^{\dagger}\psi=0\quad$ ,
$\displaystyle\quad\partial_{Q}^{\dagger}\tilde{\psi}=0$ (4.43)
$\displaystyle\partial^{\dagger}\rho=0\quad$ ,
$\displaystyle\quad\bar{\partial}_{Q}^{\dagger}\tilde{\rho}=0$ (4.44)
$\displaystyle\bar{\partial}\psi={\cal S}_{\cal F}\cdot\rho\quad$ ,
$\displaystyle\quad\partial_{Q}\tilde{\psi}=\tilde{\cal S}_{\cal
F}\cdot\tilde{\rho}$ (4.45)

Clearly, comparing these equations with the flux-less ones (3.32), one can see
that the only effect of the flux is to slightly mix $\psi$ and $\rho$, and
analogously for $\tilde{\psi}$ and $\tilde{\rho}$. However, in order to count
the zero modes associated to $\psi$ and $\tilde{\psi}$, one can consistently
set $\rho=\tilde{\rho}=0$. So one obtains that $\psi_{\rm z.m.}$ and
$\tilde{\psi}_{\rm z.m.}$ are still given by harmonic representatives of the
cohomology groups $H^{0,1}_{\bar{\partial}}(D)$ and
$H^{1,0}_{\partial}(D,L_{Q})$ respectively. Nothing changes for $\lambda_{\rm
z.m.}$ and $\tilde{\lambda}_{\rm z.m.}$ as well, that are still given by the
harmonic representatives of $H^{0,0}_{\partial}(D)$ and
$H^{0,0}_{\bar{\partial}}(D,L^{-1}_{Q})$.

The story changes for $\rho$ and $\tilde{\rho}$. On the one hand, the third
line of (4.42) implies that $\rho_{\rm z.m.}$ and $\tilde{\rho}_{\rm z.m.}$
must still be a harmonic representatives of $H^{2,0}_{\partial}(D)$ and
$H^{0,2}_{\bar{\partial}}(D,L^{-1}_{Q})$ respectively, as in the flux-less
case. However, for $\rho_{\rm z.m.}$ and $\tilde{\rho}_{\rm z.m.}$ to be true
zero-modes, the last line in (4.42) requires ${\cal S}_{\cal F}\cdot\rho_{\rm
z.m.}$ to be $\bar{\partial}$-exact and $\tilde{\cal S}_{\cal
F}\cdot\tilde{\rho}_{\rm z.m.}$ must be $\partial_{Q}$-exact.

One can rewrite this condition as follows. First, restricting the action of
${\cal S}_{\cal F}$ and $\tilde{S}_{\cal F}$ onto harmonic forms, one can see
them as maps between cohomology classes

$\displaystyle{\cal S}_{\cal F}$ $\displaystyle:$ $\displaystyle
H^{2,0}_{\partial}(D)\rightarrow H^{0,2}_{\bar{\partial}}(D)$ (4.46)
$\displaystyle\tilde{\cal S}_{\cal F}$ $\displaystyle:$ $\displaystyle
H^{0,2}_{\bar{\partial}}(D,L_{Q}^{-1})\rightarrow H^{2,0}_{\partial}(D,L_{Q})$
(4.47)

Hence, the condition for harmonic $\rho_{\rm z.m.}$ and $\tilde{\rho}_{\rm
z.m.}$ to be true zero-modes is that

$\displaystyle[{\cal S}_{\cal F}\cdot\rho]=0$ $\displaystyle\in$
$\displaystyle H_{\bar{\partial}}^{0,2}(D)$ (4.48) $\displaystyle[\tilde{\cal
S}_{\cal F}\cdot\tilde{\rho}]=0$ $\displaystyle\in$ $\displaystyle
H_{\partial}^{2,0}(D,L_{Q})$ (4.49)

Once (4.48) is guaranteed, the last equation in (4.42) can be integrated by
using the standard Hodge decomposition

$\displaystyle\psi$ $\displaystyle=$ $\displaystyle\psi_{\rm
z.m.}+\bar{\partial}^{\dagger}\Delta_{\bar{\partial}}^{-1}({\cal S}_{\cal
F}\cdot\rho_{\rm z.m.})$ (4.50) $\displaystyle\tilde{\psi}$ $\displaystyle=$
$\displaystyle\tilde{\psi}_{\rm
z.m.}+\partial^{\dagger}_{Q}\Delta^{-1}_{\partial_{Q}}(\tilde{\cal S}_{\cal
F}\cdot\tilde{\rho}_{\rm z.m.})$ (4.51)

Here $\psi_{\rm z.m.}$ and $\tilde{\psi}_{\rm z.m.}$ are the harmonic forms
introduced above, that count as independent zero modes. On the other hand, one
can see that if $\rho_{\rm z.m.}$ ($\tilde{\rho}_{\rm z.m.}$) is non-vanishing
and satisfies the condition (4.48), then $\psi$ ($\tilde{\psi}$) acquires an
additional corresponding non-vanishing but uniquely determined profile.

One could re-express the above results in terms of sheaf cohomologies. Namely,
one can see ${\cal S}_{\cal F}$ and $\tilde{S}_{\cal F}$ as maps

$\displaystyle{\cal S}_{\cal F}$ $\displaystyle:$ $\displaystyle
H^{2}(D,\bar{\cal O})\rightarrow H^{2}(D,{\cal O})$ (4.52)
$\displaystyle\tilde{\cal S}_{\cal F}$ $\displaystyle:$ $\displaystyle
H^{2}(D,{\cal L}_{Q}^{-1})\rightarrow H^{2}(D,\bar{\cal L}^{-1}_{Q})$ (4.53)

Then, in summary, one finds that the in presence of fluxes the fermionic zero
modes are associated with

$\displaystyle\begin{array}[]{c|c}\text{l.h.\ zero modes}&\text{vector
space}\\\ \hline\cr\lambda^{\alpha}_{\rm z.m.}&H^{0}(D,\bar{\cal O}_{D})\\\
\psi^{\alpha}_{\rm z.m.}&H^{1}(D,{\cal O}_{D})\\\ \rho^{\alpha}_{\rm
z.m.}&\ker{\cal S}_{\cal F}\subset H^{2}(D,\bar{\cal
O}_{D})\end{array}\qquad\begin{array}[]{c|c}\text{r.h.\ zero
modes}&\text{vector space}\\\ \hline\cr\tilde{\lambda}^{\dot{\alpha}}_{\rm
z.m.}&H^{0}(D,{\cal L}_{Q}^{-1})\\\ \tilde{\psi}^{\dot{\alpha}}_{\rm
z.m.}&H^{1}(D,\bar{\cal L}^{-1}_{Q})\\\ \tilde{\rho}^{\dot{\alpha}}_{\rm
z.m.}&\ker\tilde{\cal S}_{\cal F}\subset H^{2}(D,{\cal
L}_{Q}^{-1})\end{array}$ (4.62)

These results clearly show that a world-volume flux can potentiallly lift part
of the zero-modes of $\rho$ and $\tilde{\rho}$. According to the discussion of
section 3.2, these zero-modes are the ($Q$-twisted) supersymmetric partners of
the bosonic zero modes describing the infinitesimal deformations of the
divisor. In fact, the above flux-induced fermionic moduli lifting has a
geometrical counterpart – see for instance [70, 68, 74]. We will come back to
this point in subsection 4.6

On the other hand, one can also see very explicitly that world-volume fluxes
cannot lift the $h^{0}(D)$, $h^{1}(D)$, $h^{0}_{Q}(D)$ and $h^{1}_{Q}(D)$ zero
modes. This observation fits in well with the results scattered in the
literature, mostly based on specific examples – see for instance [31] and
references therein.

###  4.5 Index for magnetized E3-branes

In section 3.3 we have provided a direct IIB derivation of the index in [15].
The derivation is based on the U(1)R symmetry correponding to a rotation of
the complex coordinate orthogonal to the divisor. Notice that, even in
presence of fluxes, the U(1)R charges of the topologically twisted world-
volume fermions are still given by (3.50). Hence by looking at the quadratic
fermionic action (4.41) it is immediate to realize that, contrary to the naive
expectation, the presence of world-volume flux explicitly breaks the U(1)R
symmetry. The breaking has its origin in the appearance of the extrinsic
curvature. However, even though the U(1)R symmetry acting only on the word-
volume fermions is broken, the extrinsic curvature appears contracted with
three-form $\Omega$ or its complex conjugated. As it is evident from (2.19),
$\Omega$ transforms with U(1)R charge $-1$. Hence, from the world-volume
perspective, $\Omega$ can be thought of as a spurion restoring the U(1)R that,
regarded as a local Lorentz symmetry, should not be broken in the complete
theory.

Such spurionic couplings give a mass to the $\rho$ and $\tilde{\rho}$ zero
modes that lie in the kernel of the operators ${\cal S}_{\cal F}$ and
$\tilde{\cal S}_{\cal F}$ defined above. Hence, in the path-integral, the
integration of these massive would-be zero-modes will pull-down such spurionic
mass terms that, combined with the factor $e^{-T_{\rm E3}}$, will modify the
anomaly argument.

Hence by defining, in addition to (3.46), the flux-modified Hodge numbers

$h^{2}({\cal F}):={\rm dim}\,[\ker{\cal S}_{\cal F}\subset H^{2}(D,\bar{\cal
O})]\,,\quad h^{2}_{Q}({\cal F}):=\dim[\ker\tilde{\cal S}_{\cal F}\subset
H^{2}(D,{\cal L}_{Q}^{-1})]$ (4.63)

one is led to consider the modified index

$\displaystyle\chi_{\rm E3}({\cal F})$ $\displaystyle:=$ $\displaystyle
h^{0}-(h^{0}_{Q}+h^{1})+[h^{1}_{Q}+h^{2}({\cal F})]-h^{2}_{Q}({\cal F})$
(4.64) $\displaystyle\equiv$ $\displaystyle\chi^{0}_{\rm E3}-(\dim{\rm
Im\hskip 1.00006pt}{\cal S}_{\cal F}-\dim{\rm Im\hskip 1.00006pt}\tilde{\cal
S}_{\cal F})$ (4.65)

where we have defined $\chi^{0}_{\rm E3}:=\chi_{\rm E3}({\cal F}=0)$.

Clearly $\chi_{\rm E3}({\cal F})$ counts the amount of surviving zero-modes
weighted by the sign of their U(1)R charge. The natural generalization of
Witten’s criterion, necessary but by no means sufficient, is then given by

$\chi_{\rm E3}({\cal F})=1$ (4.66)

So far, we have only considered the quadratic fermionic action. Hence, it
could appear that in principle there could be other U(1)R violating higher
order terms that could spoil (4.66). However, standard non-rinormalization
arguments imply that, up to the overall factor $e^{-T_{\rm E3}}$, the
resulting superpotential can be computed in the limit $T_{\rm
E3}\rightarrow\infty$, in which the metric of the bulk three-fold $X$ is
infinitely scaled up. In this limit the higher order terms vanish and hence we
do not expect them to modify the above arguments. Under this reasonable
assumption, (4.66) appears as the most natural necessary condition to be
considered.

Finally, in section 3.5 we have pointed out the identity (3.70) between
$\chi_{\rm E3}$ in the absence of fluxes and the Ext-index. It is not clear to
us whether this correspondence can be in some way extended to the case in
which world-volume fluxes are turned on. One of the reasons is that, in an
F-theory background, the flux cannot be easily associated to standard
perturbative data underlying the definition of Ext-groups.

It is conceivable that one might be able to calculate the index (4.66) in the
orientifold weak coupling limit via some generalization of the
$\mathbb{Z}_{2}$-equivariant Hirzebruch-Riemann-Roch index theorems exploited
in [75], but for Ext-groups.

###  4.6 On flux-induced zero-modes lifting

We would like to understand better the mechanism that regulates the flux-
induced moduli-lifting. As already mentioned, this mechanism has a geometric
counterpart.

The infinitesimal deformations of the divisor are generated by the holomorphic
sections of the normal bundle $H^{0}(D,N_{D})$. Even though by definition
these deformations preserve the holomorphy of the embedding, it is important
to realize that part of them can also produce a deformation of the complex
structure induced on $D$ by the bulk. This effect originates from the non-
holomorphic split of the short exact sequence

$0\rightarrow T_{D}\rightarrow T_{X}|_{D}\rightarrow N_{D}\rightarrow 0$
(4.67)

Namely, not every holomorphic section of $N_{D}$ can be uplifted to a
holomorphic section of $T_{X}|_{D}$. Take the long exact sequence

$\ldots\rightarrow H^{0}(D,T_{X}|_{D})\stackrel{{\scriptstyle
m}}{{\rightarrow}}H^{0}(D,N_{D})\stackrel{{\scriptstyle\delta}}{{\rightarrow}}H^{1}(D,T_{D})\rightarrow\ldots$
(4.68)

Since Im$\,m={\rm Ker}\,\delta$, one can see that the holomorphic sections of
$N_{D}$ which cannot be uplifted to holomorphic sections of $T_{X}|_{D}$ are
the ones whose $\delta$-image in $H^{1}(D,T_{D})$ does not vanish.

In appendix B it is shown how the extrinsic curvature provides an explicit
realization of the $\delta$-map. Namely, working in terms of the associated
smooth complex bundles, given a covariantly holomorphic section $V$ of $N_{D}$
(regarded as a U(1) line bundle), one finds that a representative of
$H^{1}(D,T_{D})\simeq H^{0,1}_{\bar{\partial}}(D,T_{D})$ is provided by
$\Upsilon_{m}V^{m}$, where $\Upsilon_{m}$ is defined in (B.12).

Before proceeding, notice that one can analogously consider the twisted short
exact sequence

$0\rightarrow T_{D}\otimes{\cal L}_{Q}|_{D}\rightarrow(T_{X}\otimes{\cal
L}_{Q})|_{D}\rightarrow N_{D}\otimes{\cal L}_{Q}|_{D}\rightarrow 0$ (4.69)

and repeat all the arguments above. Take a covariantly holomorphic section $V$
of $N_{D}\otimes L_{Q}$, which corresponds to a holomorphic section of
$N_{D}\otimes L_{Q}$. Then the obstruction for the latter to become a
holomorphic section of $(T_{X}\otimes{\cal L}_{Q})|_{D}$ is given by a non-
vanishing cohomology class of $\Upsilon_{m}V^{m}$ in
$H^{0,1}_{\bar{\partial}}(D,T_{D}\otimes L_{Q})\simeq
H^{1}(D,T_{D}\otimes{\cal L}_{Q})$.

The above arguments are clearly valid for the complex-conjugated anti-
holomorphic quantities or if one replaces ${\cal L}_{Q}$ and $L_{Q}$ by any
other line bundles on ${X}$.

Take now the world-volume fermions $\rho$ and $\tilde{\rho}$. We have seen
that, for them to be zero-modes, they must define harmonic representatives of
$H_{\partial}^{2,0}(D)$ and $H_{\bar{\partial}}^{0,2}(D,L^{-1}_{Q})$
respectively. By using $\Omega$, one can associate with them the vectors $V\in
H_{\bar{\partial}}^{0,0}(D,N_{D}\otimes L^{-1}_{Q})$ and $\tilde{V}\in
H_{\partial}^{0,0}(D,\bar{N}_{D})$ given by

$V^{m}=\frac{1}{2}e^{-\phi/2}\bar{\Omega}^{mab}\rho_{ab}\quad,\quad\tilde{V}^{m}=\frac{1}{2}e^{-\phi/2}\Omega^{mab}\tilde{\rho}_{ab}$
(4.70)

In holomorphic indices, one then has

$(\Upsilon\cdot V)^{i}{}_{\bar{\jmath}}=-\frac{1}{2}e^{-\phi/2}{\cal
K}^{mi}{}_{\bar{\jmath}}\bar{\Omega}_{m}{}^{uv}\rho_{uv}\,,\quad(\Upsilon\cdot\tilde{V})^{\bar{\imath}}{}_{j}=-\frac{1}{2}e^{-\phi/2}{\cal
K}^{m\bar{\imath}}{}_{j}\Omega_{m}{}^{\bar{u}\bar{v}}\tilde{\rho}_{\bar{u}\bar{v}}$
(4.71)

so that $\Upsilon\cdot V\in H^{0,1}_{\bar{\partial}}(D,T_{D}\otimes
L^{-1}_{Q})$ and $\Upsilon\cdot\tilde{V}\in H^{1,0}_{\partial}(D,\bar{T}_{D})$
are exactly the combinations which appear in the fermionic equations of motion
(4.42).

We can now turn our attention onto the world-volume flux. Supersymmetry
demands it to be $(1,1)$. Clearly, any deformation of the complex structure of
$D$ generated by $v\in H^{0,1}_{\bar{\partial}}(D,T_{D})\simeq H^{1}(D,T_{D})$
can break supersymmetry since it could generate a non-vanishing

$\delta_{v}{\cal F}^{0,2}\equiv v\cdot{\cal F}:=v^{i}{}_{\bar{\jmath}}{\cal
F}_{i\bar{u}}\text{d}\bar{s}^{\bar{\jmath}}\wedge\text{d}s^{\bar{u}}$ (4.72)

More precisely, supersymmetry is actually broken only if $\delta_{v}{\cal
F}^{0,2}$ cannot be reabsorbed by deforming the world-volume gauge-fields,
that is only if $\delta_{v}{\cal F}^{0,2}$ is cohomologically non-trivial.

In the fermionic equations of motion, one can write

$\displaystyle{\cal S}_{\cal F}\cdot\rho$ $\displaystyle\equiv$ $\displaystyle
e^{-\phi/2}(\Upsilon\cdot V)\cdot{\cal F}\equiv e^{-\phi/2}\delta_{V}{\cal
F}^{0,2}$ (4.73) $\displaystyle\tilde{\cal S}_{\cal F}\cdot\tilde{\rho}$
$\displaystyle\equiv$ $\displaystyle
e^{-\phi/2}(\Upsilon\cdot\tilde{V})\cdot{\cal F}\equiv
e^{-\phi/2}\delta_{\tilde{V}}{\cal F}^{2,0}$ (4.74)

Hence ${\cal S}_{\cal F}\cdot\rho$ and $\tilde{\cal S}_{\cal
F}\cdot\tilde{\rho}$ exactly account for the generation of $(0,2)$ and $(2,0)$
components produced by the (twisted) complex structure deformations associated
to $\Upsilon\cdot V$ and $\Upsilon\cdot\tilde{V}$ respectively. From this
point of view, the moduli lifting conditions (4.48) are exactly demanding that
the good zero modes should not be associated to (twisted) geometrical
deformations which generate cohomologically non-trivial components
$\delta_{V}{\cal F}^{0,2}$ or $\delta_{\tilde{V}}{\cal F}^{2,0}$, i.e.
deformations which break supersymmetry.

The question is now: are there general circumstances under which zero-mode
lifting is for sure impossible? First of all, this happens if $\Upsilon\cdot
V$ and $\Upsilon\cdot\tilde{V}$ are all trivial in cohomology, that is, the
short exact sequence (4.67) splits holomorphically. Indeed, in this case one
can write

$\Upsilon\cdot
V=\bar{\partial}_{Q}U\,,\quad\Upsilon\cdot\tilde{V}=\partial\tilde{U}$ (4.75)

with $U\in T_{D}\otimes L^{-1}_{Q}$ and $\tilde{U}\in\bar{T}_{D}$. Then,
assuming (4.16), one can deduce that

${\cal S}_{\cal F}\cdot\rho=\bar{\partial}(e^{-\phi/2}U\cdot{\cal
F})\,,\quad\tilde{\cal S}_{\cal
F}\cdot\tilde{\rho}=\partial_{Q}(e^{-\phi/2}\tilde{U}\cdot{\cal F})$ (4.76)

and one can see that the conditions (4.48) are always satisfied. If on the
other hand (4.16) is not assumed, then one cannot reach any definitive
conclusion about $\tilde{\rho}$.

The other special possibility is that the cohomological non-triviality of
${\cal F}$ is inherited from the bulk. Remember that $e^{-\phi/2}{\cal F}$
defines a cohomology class in $H^{1,1}_{\bar{\partial}}(D,L_{Q})$ and,
assuming (4.16), also in $H^{1,1}_{\partial}(D,L_{Q})$. On the other hand,
through the associated pull-back, the embedding $\iota:D\rightarrow{X}$ allows
to see part of these cohomology groups as inherited from the bulk. In other
words, ${\rm Im}\,\iota^{*}$ contains the classes which can be seen as the
pull-back of a non-trivial class in ${X}$.

Now, if $e^{-\phi/2}{\cal F}\in{\rm Im}\,\iota^{*}$ in $\partial_{Q}$ and
$\bar{\partial}_{Q}$ cohomology, then it cannot generate any zero-mode
lifting. Indeed, one can always write $\Upsilon\cdot V=\bar{\partial}_{Q}{\cal
U}$ and $\Upsilon\cdot\tilde{V}=\partial\tilde{\cal U}$, where ${\cal U}$ and
$\tilde{\cal U}$ are smooth sections of the (anti-)holomorphic bundles
$T_{X}|_{D}\otimes L^{-1}_{Q}$ and $\bar{T}_{X}|_{D}$ respectively. Hence, if
${\cal F}$ is the pull-back of a bulk form $\hat{\cal F}$, one can write

${\cal S}_{\cal F}\cdot\rho=\bar{\partial}[\iota^{*}(e^{-\phi/2}{\cal
U}\cdot\hat{\cal F})]\,,\quad\tilde{\cal S}_{\cal
F}\cdot\tilde{\rho}=\partial_{Q}[\iota^{*}(e^{-\phi/2}\tilde{\cal
U}\cdot\hat{\cal F})]$ (4.77)

which implies that the $\rho$ and $\tilde{\rho}$ zero modes cannot be lifted
by the flux. Again, we stress that we need (4.16) to arrive to such a
conceivable conclusion on $\tilde{\rho}$.

Finally, notice that the subtleties related to (4.16) disappear if one only
considers F-theory vacua admitting a weak coupling orientifold limit and works
on the double cover Calabi-Yau three-fold $\tilde{X}$. In this case ${\cal F}$
defines an odd cohomology class in $H^{1,1}_{-}(\tilde{D})$, where $\tilde{D}$
is the double cover of $D$. The above arguments run smoothly in this case and
then one can conclude that there is no flux-induced lifting of the
$\rho^{\alpha}_{\rm z.m.}$ and $\tilde{\rho}^{\dot{\alpha}}_{\rm z.m.}$ zero
modes when either the short exact sequence (4.67) splits holomorphically or
${\cal F}$ can be written as the pull-back of a two-form in
$H^{1,1}_{-}(\tilde{X})$.

##  5 Lifting zero-modes in a one-modulus example

In this section we will use the model developed in sections 2.5 and 3.6. The
setting is simply ${X}=\mathbb{P}^{3}$, and the Calabi-Yau double-cover three-
fold $\tilde{X}$ is the octic hypersurface $\mathbb{P}^{4}_{11114}[8]$ defined
by (2.40).

This provides a very simple example that shows that even in a one-modulus
case, non-perturbative superpotentials can be generated. We choose as divisor
wrapped by the E3-brane the hyperplane divisor, $D\simeq H$. This corresponds
to the choice $n=1$ in the discussion of section 3.6. Hence, from the general
results (3.78) and (3.83), we see that in absence of world-volume fluxes the
fermionic spectrum for this specific case is

$\begin{array}[]{c|c}\text{l.h.\ fermions}&\text{\\#\ zero m.\ (${\cal
F}=0$)}\\\ \hline\cr\lambda^{\alpha}&2\times 1\\\ \psi^{\alpha}&0\\\
\rho^{\alpha}&0\end{array}\quad\quad\begin{array}[]{c|c}\text{r.h.\
fermions}&\text{\\#\ zero m.\ (${\cal F}=0$)}\\\
\hline\cr\tilde{\lambda}^{\dot{\alpha}}&0\\\ \tilde{\psi}^{\dot{\alpha}}&0\\\
\tilde{\rho}^{\dot{\alpha}}&2\times 3\end{array}$ (5.1)

Hence, in addition to the two universal zero modes $\lambda_{\rm
z.m.}^{\alpha}\sim\theta^{\alpha}$, there are $2\times 3$ (the factor $2$
counts the two Weyl indices) zero modes $\tilde{\rho}_{\rm
z.m.}^{\dot{\alpha}}$. In absence of fluxes, one has $\chi_{E3}=-2$ and then,
a priori, such an instanton cannot contribute to a superpotential.

We will now show how the addition of world-volume fluxes on $D$ can lead to
the removal of the six zero modes $\tilde{\rho}_{\rm z.m.}^{\dot{\alpha}}$, so
that the divisor can contribute to the superpotential. Actually, as it should
be clear from the general discussion of section 4.1, describing a general
world-volume flux ${\cal F}$ is difficult, as it can undergo non-trivial
monodromies, and hence cannot be treated as curvatures of ordinary line
bundles or as ordinary two-forms. This difficulty is analogous to the
difficulty in describing worldvolume self-dual 3-form fluxes on the dual
M5-instanton explicitly in a concrete situation.

On the other hand, via the weak coupling orientifold limit, these difficulties
are greatly alleviated, basically because they can be addressed in a
perturbative IIB string theory picture. In that case, the purely worldvolume
fluxes $F_{\rm E3}$ on the double-cover divisor $\tilde{D}$ are closed,
orientifold-odd two-forms of $(1,1)$-type, i.e. elements of
$H^{1,1}_{-}(\tilde{D})$. Hence, via Poincaré duality they can be understood
as divisor classes on $\tilde{D}$, i.e. holomorphic curves:111111There is a
subtlety, related to the Freed-Witten quantization condition, that reads
${\cal F}+\iota^{*}B_{\it 2}+\frac{1}{2}c_{1}(K_{\tilde{D}})\in
H^{2}(\tilde{D},\mathbb{Z})$ [76, 77]. In our case
$c_{1}(K_{\tilde{D}})=\iota^{*}[H]$. Since $[H]\in
H^{2}_{+}(\tilde{X};\mathbb{Z})$, one must turn on an even $[B^{+}_{\it
2}]=\frac{1}{2}[H]$ to cancel it, which is consistent with the orientifold
action because of the periodic identification $[B^{+}_{\it 2}]\sim[B^{+}_{\it
2}]+[\omega_{\it 2}]$ for any $[\omega_{\it 2}]\in
H^{2}(\tilde{X},\mathbb{Z})$, and then in particular for $[\omega_{\it
2}]=-[H]$.

$\frac{F_{\rm
E3}}{2\pi}=\tfrac{1}{2}[\tilde{D}]+\sum_{i}n_{i}\big{(}[C_{i}]-[C_{i}^{\prime}]\big{)}\,,\quad
n_{i}\in\mathbb{Z}$ (5.2)

where the $[C_{i}]$ and $[C_{i}^{\prime}]$ are the Poincaré duals of
holomorphic curves $C_{i}\subset\tilde{D}$ and their orientifold images,
respectively. We suppress pullback symbols for simplicity. In order for the
flux to survive the orientifold projection $\sigma^{*}({\cal F}_{E3})=-{\cal
F}_{E3}$, where $\sigma$ is the involution, we turn on a B-field $B_{\it
2}=\tfrac{1}{2}\,[H]$, such that

${\cal
F}=(2\pi\ell_{s})^{2}\sum_{i}n_{i}\big{(}[C_{i}]-[C_{i}^{\prime}]\big{)}\,,\quad
n_{i}\in\mathbb{Z}$ (5.3)

Furthermore, the primitivity condition ${\cal F}\wedge J=0$ translates into
the integrated condition

$\sum_{i}n_{i}\int_{C_{i}}J=0$ (5.4)

Such curves can be explicitly constructed and hence provide with a very hands-
on way of introducing fluxes. This approach was pioneered in the context of
black hole microstate counting in [78, 79, 80], and further developed and
exploited in [81, 82, 83].

The key point is that, as explained in section 4.6, the flux-induced lifting
of the zero modes $\tilde{\rho}^{\dot{\alpha}}_{\rm z.m.}$ have a clear
geometric counterpart in terms of lifting of geometric moduli. In particular,
only world-volume fluxes that cannot be seen as the pull-back of some $(1,1)$
flux on the bulk can contribute. Hence, only the purely world-volume flux
$F_{\rm E3}$ really matters. Then, instead of directly calculating the action
of the operator $\tilde{\cal S}_{\cal F}$, one can Poincaré-dualize the
problem and look for world-volume fluxes (5.2) that ‘rigidify’ the divisor
$D$. Geometrically, this will boil down to requiring that the holomorphic
curves $C_{i}$ cannot be deformed together with $D$ while preserving their
holomorphy .

Let us first discuss the geometric deformations more in detail. In the
orientifold limit, they are associated to the $H^{2}_{-}(\tilde{D})$
cohomology. They correspond via Serre duality to $H^{0}_{+}(H,{\cal
O}_{\tilde{X}}(1))$, i.e. to the sections of the normal bundle
$N_{\tilde{D}}\simeq{\cal O}_{\tilde{X}}(1)|_{\tilde{D}}$ that are _even_
under the orientifold involution. These can be seen quite directly by writing
down the equation for a generic hyperplane divisor $H$ in
$\tilde{X}=\mathbb{P}^{4}_{11114}[8]$. Let the coordinates of the three-fold
be as defined earlier, $[z_{1}:\ldots:z_{4}:\xi]$, where $\xi\rightarrow-\xi$
under the involution, and $\xi$ has projective weight four. Then the E3
divisor $\tilde{D}$ is given by:

$P_{D}=a_{1}\,z_{1}+\ldots+a_{4}\,z_{4}=0$ (5.5)

The vanishing locus of this equation is invariant under rescalings, hence the
moduli space of this divisor is a $\mathbb{P}^{3}$. Hence, $\text{dim
}H_{+}^{0}(H,{\cal O}_{\tilde{X}}(1))=3$, that indeed gives the $2\times 3$
zero modes $\tilde{\rho}^{\dot{\alpha}}$. Notice however that, because $\xi$