# Topological twists of massive SQCD, Part I

Johannes Aspman, Elias Furrer, Jan Manschot
School of Mathematics, Trinity College, Dublin 2, Ireland
Hamilton Mathematical Institute, Trinity College, Dublin 2

###### Abstract

We consider topological twists of four-dimensional $\mathcal{N}=2$
supersymmetric QCD with gauge group SU$(2)$ and $N_{f}\leq 3$ fundamental
hypermultiplets. The twists are labelled by a choice of background fluxes for
the flavour group, which provides an infinite family of topological partition
functions. In this Part I, we demonstrate that in the presence of such fluxes
the theories can be formulated for arbitrary gauge bundles on a compact four-
manifold. Moreover, we consider arbitrary masses for the hypermultiplets,
which introduce new intricacies for the evaluation of the low-energy path
integral on the Coulomb branch. We develop techniques for the evaluation of
these path integrals. In the forthcoming Part II, we will deal with the
explicit evaluation.

##  1 Introduction

Correlations functions of topologically twisted quantum field theories provide
many insights into non-perturbative aspects of quantum field theory as well as
the geometry of four-manifolds Witten:1988ze ; Witten:1994ev ; Witten:1994cg .
We consider topologically twisted $\mathcal{N}=2$ supersymmetric Yang-Mills
theories with additional matter multiplets on a compact four-manifold, which
were introduced in YAMRON1988325 ; anselmi1993 ; anselmi1994 ; ANSELMI1995247
; ALVAREZ1993251 ; Alvarez:1994ii . After the work by Seiberg and Witten on
the full non-perturbative solution Seiberg:1994rs ; Seiberg:1994aj ;
Witten:1994cg , these theories have received much attention in physics
Moore:1997pc ; LoNeSha ; labastida1995 ; Labastida:1995zj ; Hyun:1995mb ;
Hyun:1995hz ; laba1998 ; Dijkgraaf:1997ce ; Kanno:1998qj ; Marino:1998uy ;
mm1998 ; Moore:2017cmm ; Dedushenko:2017tdw ; Manschot:2021qqe ;
Edelstein:2000wg  and mathematics gttsche2010 ; Feehan:1997gj ; Feehan:1997rt
; Feehan:1997vp ; Feehan:2001jc ; Gorodentsev:1996cu ; Nakajima:2005fg ;
nakajima2011perverse ; marcolli_2011 ; bryan1996 ; pidstrigach1995localisation
; okonek1996recent .111For reviews, see for example Laba05 ; Marino:2000hx ;
Nakajima:2003uh ; moore2017 . Their path integrals can in many cases be
explicitly evaluated after topologically twisting. The study of the analytical
structure of partition functions as function of the parameters such as the
masses and UV couplings allows to study the limits in parameters space, as
well as relations between different field theories. Moreover, the $Q$-fixed
equations give rise to topological invariants of the underlying space-time
geometry. The path integral derivation relates distinct notions of invariants
for the same space-time geometry, and improves the understanding of such
invariants Taubes1994 ; Bershadsky:1995vm ; Gottsche:2006 ; Nakajima:2005fg ;
Harvey_1995 ; donaldson1995floer ; kim2023 ; Gadde:2013sca ;
Dedushenko:2017tdw ; Manschot:2021qqe ; Marino:1998uy .

We consider in this article topological twists of $\mathcal{N}=2$ QCD with
gauge group SU(2) and matter multiplets in the fundamental representation of
the gauge group. By including background fluxes for the flavour group, we
obtain an infinite family of topological theories Hyun:1995mb . The choice of
a background flux makes it possible to formulate topologically twists for
$\mathcal{N}=2$ SQCD for arbitrary ’t Hooft fluxes, or first Chern classes of
the gauge bundle. This is similar to the topological twist of
$\mathcal{N}=2^{*}$ SU(2) gauge theory, which requires a non-vanishing
background flux on a non-spin four-manifold Manschot:2021qqe . We moreover
develop techniques to determine correlation functions for arbitrary values of
the masses of the hypermultiplets.

The starting point of our approach is the low-energy effective field theory on
the Coulomb branch. This phase of the theory contributes for a compact four-
manifold $X$ with the topological condition that $b_{2}^{+}(X)=1$ Moore:1997pc
. In this way, the classical Donaldson invariants can be derived starting from
the Seiberg-Witten (SW) solution to $\mathcal{N}=2$ supersymmetric Yang-Mills
theory with gauge group SU(2). The Coulomb branch integral (or $u$-plane
integral) reduces to an integral over zero modes Moore:1997pc , and reads
schematically

$\Phi=\int_{\mathcal{B}}da\wedge d\bar{a}\,\rho(a)\,\Psi(a,\bar{a}),$ (1)

where $\mathcal{B}$ is the Coulomb branch with local coordinates $a$ and
$\bar{a}$, $\rho(a)$ contains the couplings to the background and
$\Psi(a,\bar{a})$ is a sum over fluxes of the unbroken U(1) gauge group. For
simplicity, we have suppressed the dependence on the metric and not included
observables here. For the pure SU(2) theory, the Coulomb branch integral can
be formulated and evaluated for arbitrary four-manifolds, without a
requirement for Kähler or toric properties.

Recently, progress has been made on evaluating these $u$-plane integrals using
a change of variables from $a$ to the running coupling $\tau$. As a result,
the integration domain becomes a fundamental domain
$\mathcal{F}\subseteq\mathbb{H}$ in the upper half-plane $\mathbb{H}$ for the
running coupling Malmendier:2008db ; Griffin:2012kw ; Malmendier:2012zz ;
Malmendier:2010ss ; Korpas:2017qdo ; Moore:2017cmm ; Korpas:2019ava ;
Korpas:2019cwg ; Manschot:2021qqe ; Aspman:2021kfp ; Korpas:2022tij . The
integral then takes the form

$\Phi=\int_{\mathcal{F}}d\tau\wedge
d\bar{\tau}\,\nu(\tau)\,\Psi(\tau,\bar{\tau}),$ (2)

where the measure factor $\nu(\tau)$ further contains the Jacobian for the
change of variables from $a$ to $\tau$. The domain $\mathcal{F}$ is a modular
fundamental domain in previous analyses, corresponding to the duality group
$\Gamma^{0}(4)$ for the pure SU(2) theory Moore:1997pc ; Korpas:2017qdo ;
Korpas:2019cwg , $\Gamma(2)$ for the $\mathcal{N}=2^{*}$ theory
Manschot:2021qqe  and similarly $\Gamma(2)$ and $\Gamma_{0}(4)$ for the
theories with two and three massless flavours Malmendier:2008db .

As mentioned above, we aim to apply this approach to $\mathcal{N}=2$
supersymmetric SU(2) theories with $N_{f}\leq 3$ hypermultiplets in the
fundamental representation. Topological correlators of these asymptotically
free theories have been considered in various papers before, in particular the
formulation of the low energy path integral in Moore:1997pc ; LoNeSha , SW
contributions for four-manifolds with $b_{2}^{+}>1$ Moore:1997pc ;
Kanno:1998qj ; Marino:1998uy ; Dedushenko:2017tdw , the $u$-plane integral for
$\mathbb{P}^{2}$ Malmendier:2008db , and the calculation of the partition
function of the AD theory within the $N_{f}=1$ theory Moore:2017cmm . Since no
background fluxes are included in these works, the ’t Hooft flux necessarily
matches the second Stiefel-Whitney class of the four-manifold,
$w_{2}(E)=w_{2}(X)$, since the twisted hypermultiplets are not well-defined
otherwise.

Extending to generic ’t Hooft fluxes, and application of the above approach
(2) to fundamental hypermultiplets with generic masses, gives rise to several
new aspects. In particular:

  1. 1.

The fundamental domain of the effective coupling constant becomes more
intricate for massive theories, and does for generic masses not correspond to
a modular fundamental domain for a subgroup of $\text{PSL}(2,\mathbb{Z})$. The
domain contains generically a set of branch points, and branch cuts starting
from these points Aspman:2021vhs ; Aspman:2020lmf ; Aspman:2021evt . These
aspects have to be dealt with appropriately.

  2. 2.

We couple the hypermultiplets to background fluxes ${\boldsymbol{k}}_{j}$ for
the flavour group to formulate the theories for arbitrary ’t Hooft fluxes.
This gives rise to additional couplings in (1) and (2),

$\prod_{j,k=1}^{N_{f}}\exp\\!\left(-2\pi i\frac{\partial^{2}F}{\partial
m_{j}\partial m_{k}}B({\boldsymbol{k}}_{j},{\boldsymbol{k}}_{k})\right),$ (3)

where $F$ is the prepotential of the massive theory, and $B(\cdot,\cdot)$ is
the quadratic form associated to the intersection form on the middle homology
$H_{2}(X,\mathbb{Z})$ of $X$. Such couplings were suggested by Shapere and
Tachikawa Shapere:2008zf , and are also essential for the formulation of the
$\mathcal{N}=2^{*}$ Yang-Mills theory on a non-spin four-manifold
Manschot:2021qqe . Similarly to Manschot:2021qqe , we also deduce a non-
holomorphic coupling to ${\boldsymbol{k}}_{j}$. Moreover, for arriving at a
single-valued integrand, we fix an ambiguity in the quadratic terms of the
prepotential. These terms have appeared earlier in the literature in the
context of singularities of the SW differential and winding numbers
Seiberg:1994aj ; Ohta_1997 .

  3. 3.

Special points on the Coulomb branch give rise to superconformal theories,
such as the Argyres-Douglas (AD) theories Argyres:1995jj ; Argyres:1995xn  and
the massless $N_{f}=4$ theory Seiberg:1994aj . Their topological partition
functions and correlators can be found by considering them in certain mass
deformations. The case of $N_{f}=1$ is analysed in moore2017 .

The paper is organised as follows. In Section 2, we present the Seiberg-Witten
solution of SU(2) $\mathcal{N}=2$ SQCD in flat space, focusing on the
fundamental domains for the effective coupling, which we illustrate in several
interesting examples. In Section 3, we formulate the topological twist by
coupling the hypermultiplets to external fluxes, such that the topological
field theory is well-defined for arbitrary ’t Hooft flux and non-spin
manifolds. The topological low-energy effective theory coupled to $N_{f}$
background fluxes is then modelled in Section 4 as a
$\mathrm{SU}(2)\times\mathrm{U}(1)^{N_{f}}$ theory, with the matter fields
corresponding to frozen U(1) factors. This allows to compute the path integral
explicitly as an integral over the $u$-plane. In Section 5, we formulate the
$u$-plane integral as an integral over the fundamental domains. We prove that
the single-valuedness under monodromies holds for a specific choice of
magnetic winding numbers. Finally, in Section 6 we demonstrate that such
integrals may be evaluated using mock modular forms, and we show that they
localise at the cusps, elliptic points and interior singularities of the
fundamental domains.

A following paper, Part II, will be dedicated to explicit analyses and
computations for specific examples. There, we will give a detailed discussion
on the contribution from the AD points for each type, as briefly mentioned in
point 3 above, as well as calculating the $u$-plane integrals for a number of
examples.

##  2 Special geometry and SW theories

In this Section, we review aspects of the non-perturbative solution for the
low energy effective theory of $\mathcal{N}=2$ SQCD with gauge group SU(2) and
$0\leq N_{f}\leq 3$ fundamental hypermultiplets Seiberg:1994rs ;
Seiberg:1994aj . See Klemm:1997gg  for a review. Throughout, we let
$\Lambda_{N_{f}}$ denote the scale of the theory with $N_{f}$ hypermultiplets
having masses $m_{j}$, $j=1,\dots,N_{f}$, and $a$ the mass of the W-boson on
the Coulomb branch.

###  2.1 Field content

The $\mathcal{N}=2$ theories we consider contain a vector multiplet and
$N_{f}\leq 3$ hypermultiplets. The fields in these multiplets form
representations of ${\rm Spin}(4)=\text{SU}(2)_{+}\times\text{SU}(2)_{-}$ and
$\text{SU}(2)_{R}$, which we denote by
$({\boldsymbol{k}},{\boldsymbol{l}},{\boldsymbol{m}})$, with
${\boldsymbol{k}},{\boldsymbol{l}}$ and ${\boldsymbol{m}}$ dimensions of the
representations.

The vector multiplet consists of a gauge field $A_{\mu}$, complex scalar field
$\phi$, and a pair of Weyl fermions $\Psi^{I}_{\alpha}$,
$\bar{\Psi}^{I}_{\dot{\alpha}}$. This multiplet transforms under the adjoint
representation of the gauge group $G$. The representation of
$\text{SU}(2)_{+}\times\text{SU}(2)_{-}\times\text{SU}(2)_{R}$ formed by the
bosonic fields is,

$({\bf 2},{\bf 2},{\bf 1})\oplus({\bf 1},{\bf 1},{\bf 1})\oplus({\bf 1},{\bf
1},{\bf 1}),$ (4)

while the representation for the fermions is

$({\bf 1},{\bf 2},{\bf 2})\oplus({\bf 2},{\bf 1},{\bf 2}).$ (5)

The hypermultiplet consist of a pair of complex scalar fields, $q$ and
$\tilde{q}$, and Weyl fermions, $\lambda_{\alpha}$,
$\bar{\lambda}_{\dot{\alpha}}$, $\chi_{\alpha}$ and
$\bar{\chi}_{\dot{\alpha}}$. We fix the gauge group $G=\text{SU}(2)$, and let
the hypermultiplets transform under the fundamental representation of this
group. With the same notation as above, the bosonic fields of this multiplet
form the representation,

$({\bf 1},{\bf 1},{\bf 2})\oplus({\bf 1},{\bf 1},{\bf 2}),$ (6)

while the fermions form the representation

$({\bf 2},{\bf 1},{\bf 1})\oplus({\bf 1},{\bf 2},{\bf 1})\oplus({\bf 2},{\bf
1},{\bf 1})\oplus({\bf 1},{\bf 2},{\bf 1}).$ (7)

###  2.2 Seiberg-Witten geometry

The Seiberg-Witten geometry underlies the Coulomb branch of $\mathcal{N}=2$
gauge theory. The Coulomb branch is the phase of the theory where
$\text{SU}(2)$ is broken to $\text{U}(1)$ by a vacuum expectation value (vev)
of the vector multiplet scalar $\phi$. The vev is semi-classically
parametrised by a complex parameter $a$,

$\phi=\left(\begin{array}[]{cc}a&0\\\ 0&-a\end{array}\right),$ (8)

up to gauge transformations. In particular, $a\to-a$ is a gauge
transformation. The gauge invariant order parameter is the Coulomb branch
expectation value of the theory in $\mathbb{R}^{4}$,

$u=\frac{1}{16\pi}\left<{\rm Tr}(\phi^{2})\right>_{\mathbb{R}^{4}}.$ (9)

The non-perturbative effective action of $\mathcal{N}=2$ SQCD is characterised
by the prepotential $F(a,{\boldsymbol{m}})$, with ${\boldsymbol{m}}$ the mass
vector ${\boldsymbol{m}}=(m_{1},\dots,m_{N_{f}})$. The semi-classical part of
$F$ reads Ohta:1996hq ; Ohta_1997 ; DHoker_1997 ; Ne

$\begin{split}&F(a,{\boldsymbol{m}})=\frac{2i}{\pi}a^{2}\left(\log(4a/\Lambda_{N_{f}})-\frac{3}{2}\right)-\frac{1}{2}\sum_{j=1}^{N_{f}}\left(n_{j}\frac{m_{j}}{\sqrt{2}}\,a+\,C\,(a^{2}+m_{j}^{2}/2)\right)\\\
&\quad-\frac{i}{4\pi}\sum_{j=1}^{N_{f}}\left(a+\tfrac{m_{j}}{\sqrt{2}}\right)^{2}\log((a+\tfrac{m_{j}}{\sqrt{2}})/\Lambda_{N_{f}})+\left(a-\tfrac{m_{j}}{\sqrt{2}}\right)^{2}\log((a-\tfrac{m_{j}}{\sqrt{2}})/\Lambda_{N_{f}})\\\
&\quad+\dots,\end{split}$ (10)

where the $\dots$ indicate further non-perturbative corrections. Here
$C=\frac{1}{2}+\frac{i}{2\pi}\log(2)-\frac{3i}{2\pi}$. The classical terms
proportional to $a^{2}$ and $m^{2}$ on the first line are chosen to facilitate
the decoupling of hypermultiplets, which will be discussed later in more
detail.

The $n_{j}\in\mathbb{Z}$ in (10) are the _magnetic winding numbers_ of the
periods $a_{D}\coloneqq\frac{\partial F}{\partial a}$ dual to $a$ Ohta:1996hq
; Ohta_1997 ; ohta1999 . These numbers seem to be only rarely discussed in the
literature beyond these references.222Nekrasov’s partition function gives a
specific choice upon expanding the function $\gamma_{\hbar}(x;\Lambda)$ in the
perturbative part Ne ; NekOk . Generally, the theory allows for $N_{f}$
electric winding numbers for $a$ and $N_{f}$ magnetic winding numbers for
$a_{D}$. These appear in the massive $N_{f}>0$ theories since the Seiberg-
Witten differentials now have poles with nonzero residues Ohta:1996hq . See
also Appendix C. The choice (10) of the prepotential corresponds to fixing the
electric winding numbers to be zero, or equivalently fixing the monodromy at
infinity to map $a\to e^{\pi i}a$. Compare for example with (Ohta:1996hq, ,
Eq. (2.17)). In Section 5, we will discuss that the single-valuedness of the
$u$-plane integral requires $n_{j}\equiv-1\mod 4$.

We introduce the period $a_{D}$ dual to $a$, and the parameters $m_{D,j}$ dual
to $m_{j}$ by

$a_{D}=\frac{\partial F}{\partial a},\qquad m_{D,j}=\sqrt{2}\frac{\partial
F}{\partial m_{j}}.$ (11)

These parameters are further combined into the $(2+2N_{f})-$dimensional vector
$\Pi$,

$\Pi=\left(\begin{array}[]{c}a_{D}\\\ a\\\ m_{D,1}\\\
\tfrac{m_{1}}{\sqrt{2}}\\\ \vdots\\\ m_{D,N_{f}}\\\
\tfrac{m_{N_{f}}}{\sqrt{2}}\end{array}\right).$ (12)

This vector forms a local system over the $u$-plane. The elements of the
vector form the symplectic form,

$\omega_{N_{f}}=da_{D}\wedge
da+\frac{1}{\sqrt{2}}\sum_{j=1}^{N_{f}}dm_{D,j}\wedge dm_{j}.$ (13)

The effective gauge coupling is related to the prepotential through

$\tau=\frac{\partial^{2}F}{\partial a^{2}}.$ (14)

We also introduce the couplings $v_{j}$ and $w_{jk}$ with $j,k\in
1,\dots,N_{f}$,333If we consider $F$ as a function of
$(a,\frac{1}{\sqrt{2}}m_{1},\dots,\frac{1}{\sqrt{2}}m_{N_{f}})$, then the dual
parameters are encoded in the Jacobian
$\boldsymbol{J}_{F}=(a_{D},{\boldsymbol{m}}_{D})$, while the couplings are the
elements of the Hessian
$\boldsymbol{H}_{F}=\begin{pmatrix}\tau&{\boldsymbol{v}}^{T}\\\
{\boldsymbol{v}}&{\boldsymbol{w}}\end{pmatrix}$.

$v_{j}=\sqrt{2}\frac{\partial^{2}F}{\partial a\partial m_{j}},\qquad
w_{jk}=2\frac{\partial^{2}F}{\partial m_{j}\partial m_{k}}.$ (15)

The derivative of the prepotential with respect to the scale $\Lambda_{N_{f}}$
provides the order parameter $u$ (9) on the Coulomb branch,

$u=\frac{4\pi i}{4-N_{f}}\Lambda_{N_{f}}\frac{\partial
F(a,{\boldsymbol{m}})}{\partial\Lambda_{N_{f}}}+\frac{1}{4-N_{f}}\sum_{j}m_{j}^{2}.$
(16)

Using the different relations introduced above together with the fact that the
prepotential satisfies the homogeneity equation Sonnenschein_1996 ;
Eguchi:1995jh ; DHoker:1996yyu

$2F=\Lambda_{N_{f}}\frac{\partial
F}{\partial\Lambda_{N_{f}}}+\sum_{j=1}^{N_{f}}m_{j}\frac{\partial F}{\partial
m_{j}}+a\frac{\partial F}{\partial a},$ (17)

we can find some non-trivial relations between the respective quantities. For
instance, from the perturbative prepotential (10), we deduce that the leading
terms of $u$ are

$u=2\,a^{2}+\mathcal{O}(a^{-2})$ (18)

The weak-coupling limit then becomes in our conventions444Note that this
differs slightly from some of the previous literature. However, it is shown in
Aspman:2021vhs  to be the unique limit consistent with the RG flow.

$\left\\{\begin{array}[]{l}\tau\to+i\infty,\\\ a\to-i\infty,\\\
u\to-\infty.\end{array}\right.$ (19)

The Seiberg-Witten (SW) solution provides a family of elliptic curves
parametrised by the order parameter $u$ and the masses $m_{i}$, whose complex
structure corresponds to the running coupling
$\tau=\frac{\theta}{\pi}+\frac{8\pi i}{g^{2}}$. For the theories of interest
in this paper, the curves $\mathcal{S}_{N_{f}}$ are given by Seiberg:1994aj

$\displaystyle N_{f}=0:\quad y^{2}$
$\displaystyle=x^{3}-ux^{2}+\frac{1}{4}\Lambda_{0}^{4}x,$ (20) $\displaystyle
N_{f}=1:\quad y^{2}$
$\displaystyle=x^{2}(x-u)+\frac{1}{4}m\Lambda_{1}^{3}x-\frac{1}{64}\Lambda_{1}^{6},$
$\displaystyle N_{f}=2:\quad y^{2}$
$\displaystyle=(x^{2}-\frac{1}{64}\Lambda_{2}^{4})(x-u)+\frac{1}{4}m_{1}m_{2}\Lambda_{2}^{2}x-\frac{1}{64}(m_{1}^{2}+m_{2}^{2})\Lambda_{2}^{4},$
$\displaystyle N_{f}=3:\quad y^{2}$
$\displaystyle=x^{2}(x-u)-\frac{1}{64}\Lambda_{3}^{2}(x-u)^{2}-\frac{1}{64}(m_{1}^{2}+m_{2}^{2}+m_{3}^{2})\Lambda_{3}^{2}(x-u)$
$\displaystyle\quad+\frac{1}{4}m_{1}m_{2}m_{3}\Lambda_{3}x-\frac{1}{64}(m_{1}^{2}m_{2}^{2}+m_{2}^{2}m_{3}^{2}+m_{1}^{2}m_{3}^{2})\Lambda_{3}^{2}.$

The family of SW curves (20) are Jacobian rational elliptic surfaces with
singular fibres Malmendier:2008yj ; Caorsi:2018ahl ; Closset:2021lhd . These
surfaces are well studied in the mathematical literature shioda1972 ;
schuett2009 ; maier2006 . Rational in this context means that $g_{2}$ and
$g_{3}$ are polynomials in $u$ of degree at most $4$ and $6$, respectively
miranda1995 . In Appendix B, we summarise the class $\mathcal{S}$
representation of the SW curves Gaiotto:2009hg .

The curve for $N_{f}$ hypermultiplets reduces to the curve for $N_{f}-1$
hypermultiplets upon decoupling a hypermultiplet in the double scaling limit
Eguchi1999

$m_{N_{f}}\to\infty,\quad\Lambda_{N_{f}}\to 0,\quad
m_{N_{f}}\Lambda_{N_{f}}^{4-N_{f}}=\Lambda_{N_{f}-1}^{4-(N_{f}-1)}.$ (21)

The curves (20) provide the exact results for the vevs of the scalar $a$ and
its dual $a_{D}$ as period integrals. We have explicitly,

$a=\int_{A}\lambda,\qquad a_{D}=\int_{B}\lambda,$ (22)

with $\lambda$ the SW differential, for which we refer to Seiberg:1994aj ;
Ohta_1997 ; Ohta:1996hq .

###  2.3 Characteristic functions on the Coulomb branch

In this subsection, we discuss various characteristic functions on the Coulomb
branch, which are instrumental for determining the $u$-plane integral. They
are:

  * •

the physical discriminant $\Delta_{N_{f}}$,

  * •

the order parameter $u$,

  * •

the period $da/du$, and

  * •

the derivatives $du/d\tau$ and $da/d\tau$.

The physical discriminant $\Delta_{N_{f}}$
We define the physical discriminant $\Delta_{N_{f}}$ as the monic polynomial

$\Delta_{N_{f}}=\prod_{j=1}^{N_{f}+2}(u-u_{j}),$ (23)

where $u_{j}$ for $j=1,\dots,N_{f}+2$ are the singular points of the effective
theory. We let $j=1,\dots,N_{f}$ label the singular points where one of the
matter hypermultiplets becomes massless; and $j=N_{f}+1,N_{f}+2$ denote the
strong coupling singularities where a monopole and a dyon, respectively,
becomes massless.

The discriminant is a polynomial of degree $2+N_{f}$ in $u$. It can also be
determined directly from the SW curve Aspman:2021vhs ,

$\Delta_{N_{f}}=(-1)^{N_{f}}\Lambda_{N_{f}}^{2N_{f}-8}(g_{2}^{3}-27g_{3}^{2}),$
(24)

where $g_{2}$ and $g_{3}$ are the Weierstraß invariants of the SW curve. Up to
the sign and scale, the physical discriminant $\Delta_{N_{f}}$ equals the
mathematical discriminant $g_{2}^{3}-27g_{3}^{2}$ for these theories.

As the masses are tuned, some of the singularities on the Coulomb branch can
collide. If we consider $\Delta_{N_{f}}$ as a polynomial in $u$, its
discriminant $D(\Delta_{N_{f}})$ vanishes if and only if two roots coincide.
It is straightforward to show that for $N_{f}\leq 3$,

$D(\Delta_{N_{f}})=\left(D^{\text{AD}}_{N_{f}}\right)^{3}\prod_{i<j}(m_{i}-m_{j})^{2}(m_{i}+m_{j})^{2}.$
(25)

This factorises the locus in mass space where singularities collide into two
orthogonal components: The first component is the Argyres-Douglas (AD) locus
given by the polynomial equation $D^{\text{AD}}_{N_{f}}=0$, where mutually
_non-local_ singularities collide Argyres:1995xn ; Aspman:2021vhs ;
EGUCHI1996430 . The other component is characterised by the equations
$m_{i}=\pm m_{j}$, and one can check that this gives rise to mutually _local_
singularities colliding. Here, the flavour symmetry gets enhanced and a Higgs
branch opens up Seiberg:1994aj . This extends the analysis of Aspman:2021vhs .

Given a mass configuration ${\boldsymbol{m}}=(m_{1},\dots,m_{N_{f}})$, we can
denote by $k_{l}$ the weight (or multiplicity) of the $l$-th singularity, and
by ${\boldsymbol{k}}({\boldsymbol{m}})=(k_{1},k_{2},\dots)$ the vector of
those weights. Since the Coulomb branch
$\mathcal{B}_{N_{f}}({\boldsymbol{m}})$ contains $2+N_{f}$ singularities aside
from weak coupling $u=\infty$, it is clear that
${\boldsymbol{k}}({\boldsymbol{m}})$ provides a partition of $2+N_{f}$. This
in turn partitions the mass space $\mathbb{C}^{N_{f}}\ni{\boldsymbol{m}}$ into
finitely many regions where ${\boldsymbol{k}}({\boldsymbol{m}})$ is locally
constant. As an example, in Fig. 1 we plot the contours of (25) for $N_{f}=2$
in the real ${\boldsymbol{m}}=(m_{1},m_{2})$ plane.

Figure 1: Partitioning of the real $\boldsymbol{m}=(m_{1},m_{2})$ plane in
$N_{f}=2$, in units of $\Lambda_{2}=1$. On the AD component the Coulomb branch
$\mathcal{B}_{2}(\boldsymbol{m})$ contains an AD point of Kodaira type II
(blue) or III (green). On the other component (orange), mutually local
singularities collide. If $\boldsymbol{m}$ is varied along a continuous path
that does not cross the partitioning $0=D(\Delta_{N_{f}})$, the weight vector
$\boldsymbol{k}(\boldsymbol{m})$ is constant.

The possible singularity structures of the rational elliptic curves (20) are
classified in Persson’s list of allowed configurations of singular fibres
Persson:1990 ; Miranda:1990 . From Kodaira’s classification, it follows that
any solution to (25) gives rise to a singularity on the Coulomb branch of
Kodaira type $I_{k}$, $II$, $III$, or $IV$. As described in Aspman:2021vhs ;
Closset:2021lhd , the solutions to $0=D^{\text{AD}}_{N_{f}}$ give rise to AD
points of Kodaira type $II$, $III$ and $IV$. The second component
$0=\prod_{i<j}(m_{i}-m_{j})(m_{i}+m_{j})$ can be studied in more detail. These
are $2(N_{f}-1)$ independent equations. Whenever one of the factors vanishes,
the SW surface contains an $I_{k}$ singularity with $k\geq 2$. For $N_{f}=2$,
the only possibility is $I_{2}$, while for $N_{f}=3$ singularities of type
$I_{2}$, $I_{3}$ and $I_{4}$ are possible. The point in the Coulomb branch
$\mathcal{B}_{N_{f}}$ corresponding to an $I_{k}$ singularity with $k\geq 2$
intersects with a Higgs branch of quaternionic dimension $k-1\geq 1$
Seiberg:1994aj . Further merging these $I_{k}$ singularities with a mutually
non-local singularity does not affect the Higgs branch, such that the points
with a $III$ or $IV$ singularity also intersect with a Higgs branch of
quaternionic dimension one or two, respectively, while the points with AD
theories of type $II$ do not intersect with any Higgs branch.
Order parameter $u$
The order parameter $u$, introduced in (9), is invariant under monodromies on
the Coulomb branch. By bringing the SW curve to Weierstraß form, $u$ can be
expressed in terms of the running coupling $\tau\in\mathbb{H}$ (14)
Nahm:1996di ; Brandhuber:1996ng ; Huang:2009md ; Aspman:2021vhs ;
Aspman:2021evt . As in Aspman:2021vhs , we restrict the domain for
$\tau\in\mathbb{H}$ to $\mathcal{F}({\boldsymbol{m}})\subseteq\mathbb{H}$,
such that $u$ is 1-to-1. In specific cases, $u$ can be expressed in terms of
modular forms. We follow the conventions of (Aspman:2021vhs, , Section 2.2) in
picking the solutions. We discuss the order parameters in more detail in the
following Subsection 2.4.
The period $da/du$
The period $\frac{da}{du}$ can be related to the complex structure $\tau$ of
the curve through Brandhuber:1996ng

$\frac{da}{du}=\frac{1}{6}\sqrt{\frac{g_{2}}{g_{3}}\frac{E_{6}}{E_{4}}},$ (26)

where the Weierstraß invariants $g_{2}$, $g_{3}$ are polynomials in $u$, and
$E_{4}=E_{4}(\tau)$, $E_{6}=E_{6}(\tau)$ are the Eisenstein series of modular
weights four and six respectively, defined in Eq. (154). The square root in
(26) leads to an ambiguity in the sign of $da/du$. This can be resolved by
taking a branch of the square root, for example at weak coupling. On the other
hand, $da/du$ is not a single-valued function on the $u$-plane. It changes by
a sign under monodromies at weak coupling, and transforms with a modular
weight under monodromies around strong coupling singularities.
The derivatives $du/d\tau$ and $da/d\tau$
The derivative $du/d\tau$ is related to the period $\frac{da}{du}$ through
Matone:1995rx ; Aspman:2021vhs

$\frac{du}{d\tau}=-\frac{16\pi
i}{4-N_{f}}\frac{\Delta_{N_{f}}}{P^{\text{M}}_{N_{f}}}\left(\frac{da}{du}\right)^{2},$
(27)

where $P^{\text{M}}_{N_{f}}$ is a polynomial in $u$. For $N_{f}\leq 3$, it
reads

$P^{\text{M}}_{N_{f}}=\frac{6}{4-N_{f}}(-1)^{N_{f}}\Lambda_{N_{f}}^{2N_{f}-8}(2g_{2}g_{3}^{\prime}-3g_{2}^{\prime}g_{3}),$
(28)

where the prime denotes differentiation with respect to $u$. It appears
naturally in differential equations associated with elliptic surfaces (see
e.g. stiller1981 ).

When $u$ is known in terms of modular forms, it is also straightforward to
determine $\frac{du}{d\tau}$ by differentiating modular forms Aspman:2021vhs .
For generic masses ${\boldsymbol{m}}$, we can compute the $q$-series of $u$
and thus $\frac{du}{d\tau}$ to any order. We will not explicitly need
$du/d\tau$ for evaluating the $u$-plane integral, but through

$\frac{da}{d\tau}=\frac{da}{du}\frac{du}{d\tau}$ (29)

we can use it to evaluate $da/d\tau$. This derivative appears as the Jacobian
for the change of integration variables from the periods $a$ to the couplings
$\tau$.

###  2.4 Fundamental domains

The recent progress in computing $u$-plane integrals has been enabled by
mapping the $u$-plane to a modular fundamental domain, on which the $u$-plane
integrand can be related to mock modular forms and thus be efficiently
evaluated Malmendier:2008db ; Griffin:2012kw ; Malmendier:2012zz ;
Malmendier:2010ss ; Korpas:2017qdo ; Moore:2017cmm ; Korpas:2019ava ;
Korpas:2019cwg ; Manschot:2021qqe ; Aspman:2021kfp ; Korpas:2022tij . It has
been known since the 1990s that the $u$-planes for $\mathcal{N}=2$ SQCD with
$N_{f}=0,2,3$ massless hypermultiplets are modular and correspond to
fundamental domains for congruence subgroups of $\text{PSL}(2,\mathbb{Z})$
Nahm:1996di . On the other hand, the generic mass case including the peculiar
role of massless $N_{f}=1$ have remained elusive.

The order parameter $u$ for a given mass
${\boldsymbol{m}}=(m_{1},\dots,m_{N_{f}})$ can be considered as a function

$u:\mathbb{H}\to\mathcal{B}_{N_{f}}({\boldsymbol{m}}),$ (30)

where $\mathcal{B}_{N_{f}}({\boldsymbol{m}})$ is the Coulomb branch of the
theory with $N_{f}$ hypermultiplets of mass ${\boldsymbol{m}}$. In
Aspman:2021vhs , it was found that for $N_{f}\geq 1$ and generic masses the
duality group does not act on $\tau$ by fractional linear transformations.
This prevents the preimage $u^{-1}(\mathcal{B}_{N_{f}}({\boldsymbol{m}}))$
from being a modular fundamental domain for a subgroup of
$\text{PSL}(2,\mathbb{Z})$. Instead, we can define a fundamental domain
$\mathcal{F}_{N_{f}}({\boldsymbol{m}})$ through the equivalence relation
$\tau\sim\tau^{\prime}\Leftrightarrow u(\tau)=u(\tau^{\prime})$, such that
$u:\mathcal{F}_{N_{f}}({\boldsymbol{m}})\to\mathcal{B}_{N_{f}}({\boldsymbol{m}})$
is bijective. In Aspman:2021vhs ; Aspman:2021evt , such fundamental domains
for the effective gauge coupling have been found explicitly for
$\mathcal{N}=2$ SQCD with $N_{f}\leq 4$ generic masses. They decompose as

$\mathcal{F}_{N_{f}}({\boldsymbol{m}})=\bigcup_{j=1}^{n}\alpha_{j}\mathcal{F},$
(31)

where $\mathcal{F}=\text{PSL}(2,\mathbb{Z})\backslash\mathbb{H}$ is the key-
hole fundamental domain of $\text{PSL}(2,\mathbb{Z})$, and
$\alpha_{j}\in\text{PSL}(2,\mathbb{Z})$ are $n\leq 6$ maps that are locally
constant as functions of the masses ${\boldsymbol{m}}$. More precisely,
$\mathcal{F}_{N_{f}}({\boldsymbol{m}})$ is constant on any of the finitely
many connected components of
$\\{{\boldsymbol{m}}\in\mathbb{C}^{N_{f}}\,|\,D(\Delta_{N_{f}})=0\\}$, where
$D(\Delta_{N_{f}})$ is given in (25). We also call $n$ the index of the domain
$\mathcal{F}_{N_{f}}({\boldsymbol{m}})$.

The geometry of $\mathcal{F}_{N_{f}}({\boldsymbol{m}})$ agrees with that of
the Coulomb branch $\mathcal{B}_{N_{f}}({\boldsymbol{m}})$. Namely,
$\mathcal{B}_{N_{f}}({\boldsymbol{m}})$ is the complex plane with $2+N_{f}$
singular points removed. In (31), those singularities are reflected in the
cusps $\alpha_{j}(\infty)$, where the number of $\alpha_{j}$ with the same
$\alpha_{j}(\infty)$ (giving the _width_ of the cusp) agrees with the
multiplicity of the singularity $u(\alpha_{j}(\infty))$.

The domains
$\mathcal{F}({\boldsymbol{m}})\coloneqq\mathcal{F}_{N_{f}}({\boldsymbol{m}})$
are endowed with further data. For generic masses ${\boldsymbol{m}}$, there
are $N_{f}$ pairs of branch points on $\mathcal{F}({\boldsymbol{m}})$ that are
connected by branch cuts. The branch points are determined by the non-
removable singularities of the rational function
$\Delta_{N_{f}}(u)/P^{\text{M}}_{N_{f}}(u)$, as it appears in Matone’s
relation (27). They correspond to the nontrivial zeros of the discriminant of
the sextic equation associated with the SW curve Aspman:2021vhs . As the
masses are varied, the branch points move on continuous paths in the
fundamental domain.

When two branch points collide, there appears an elliptic point, which is a
further dedicated point on the domain $\mathcal{F}({\boldsymbol{m}})$. These
elliptic points correspond to the superconformal Argyres-Douglas (AD) points
Argyres:1995jj ; Argyres:1995xn . By Kodaira’s classification of singular
fibres, they are always in the $\text{PSL}(2,\mathbb{Z})$ orbit of $i$ or
$e^{\pi i/3}$. Their presence is also responsible for the index $n$ of the
domain to be reduced. The possible values are $n=2,3,4$ if
$\mathcal{F}({\boldsymbol{m}})$ contains such an elliptic point, and $n=6$
otherwise.

The $N_{f}=4$ theory furthermore contains a UV coupling
$\tau_{\text{\tiny{UV}}}$. Since $u(\tau)\to\infty$ for
$\tau\to\tau_{\text{\tiny{UV}}}$, it can be viewed as a puncture of the
fundamental domain, and has to be excluded from the integration domain.

Let us review the explicit construction of $\mathcal{F}({\boldsymbol{m}})$ in
a few important examples. Many more examples can be found in Aspman:2021vhs ;
Aspman:2021evt . Here, we are mainly considering the masses to be small
compared to the period $a$, such that the mass singularities discussed in
previous sections can be considered as strong coupling singularities, or in
other words as cusps on the real line of the fundamental domains. When the
masses are increased it is more natural to think of the mass singularities as
weakly coupled.

The pure SU(2) $N_{f}=0$ theory
It is well-known that the pure SU(2) Coulomb branch corresponds to the
fundamental domain for the congruence subgroup $\Gamma^{0}(4)$ of
$\text{PSL}(2,\mathbb{Z})$. The order parameter can be explicitly determined
in terms of a modular function for the duality group matone1996 ,

$\displaystyle\frac{u}{\Lambda_{0}^{2}}=$
$\displaystyle-\frac{1}{2}\frac{\vartheta_{2}^{4}+\vartheta_{3}^{4}}{\vartheta_{2}^{2}\vartheta_{3}^{2}},$
(32)

where the Jacobi theta functions $\vartheta_{i}$ are defined in Appendix A.1.
A fundamental domain for the pure SU(2) theory is given in Fig. 2. Since the
domain is modular, the branch points are absent.

Figure 2: Fundamental domain of $\Gamma^{0}(4)$. This is the duality group of
the pure SU(2) theory. The two cusps on the real line correspond to the strong
coupling singularities of the gauge theory, while the cusp at $\tau=i\infty$
corresponds to weak coupling. Boundaries of the same colour are identified.

The massless $N_{f}=1$ theory
From the massless $N_{f}=1$ SW curve, one directly finds Nahm:1996di

$\displaystyle\frac{u}{\Lambda_{1}^{2}}=$
$\displaystyle-\frac{3}{2^{\frac{7}{3}}}\frac{E_{4}^{\frac{1}{2}}}{(E_{4}^{\frac{3}{2}}-E_{6})^{\frac{1}{3}}},$
(33)

where $E_{4}$ and $E_{6}$ are the Eisenstein series defined in (154). This
function also appears as an order parameter in pure SU(3) SW theory
Aspman:2020lmf  as well as in the description of certain elliptically fibred
Calabi-Yau spaces Klemm:2012sx . Due to the presence of the square root, this
function is not modular. The massless $N_{f}=1$ curve has three distinct
strongly coupled singularities, which become the three cusps of the
fundamental domain aside from the weakly coupled region $u\to\infty$. The
fundamental domain is plotted in Fig. 3. Note that, this is not the
fundamental domain of any subgroup of $\text{PSL}(2,\mathbb{Z})$.

Figure 3: Fundamental domain for massless $N_{f}=1$.

When a positive mass $m$ is turned on, the branch points move in the
fundamental domain emerging from those of the massless case. They move as a
function of the mass as shown in Aspman:2021vhs , and depicted in Fig. 4. This
further incorporates the hypermultiplet decoupling on the level of the
fundamental domain.

(a)

(b)

(c)

(d)

Figure 4: Choice of branch cuts (zigzag lines) for varying (real) mass in
$N_{f}=1$. Lines with the same color are identified. Starting with a small
mass in Figure (a), we cut along the arcs of radius 1 around $\tau=1$ and
$\tau=2$. At the AD mass we can use the identifications of the different
boundaries to reorganise the domain in Figure (b) to the one of Figure (c).
When we increase the mass further the cuts of Figure (c) move upwards as in
Figure (d) eventually reaching infinity and disappearing, leaving us with the
domain in Fig. 2 of the pure theory.

The equal mass $N_{f}=2$ theory
For $N_{f}=2$ with $m_{1}=m_{2}=m$, the order parameter can be determined
explicitly Aspman:2021vhs ,

$\frac{u}{\Lambda_{2}^{2}}=-\frac{\vartheta_{4}^{8}+\vartheta_{2}^{4}\vartheta_{3}^{4}+(\vartheta_{2}^{4}+\vartheta_{3}^{4})\sqrt{16\frac{m^{2}}{\Lambda_{2}^{2}}\vartheta_{2}^{4}\vartheta_{3}^{4}+\vartheta_{4}^{8}}}{8\vartheta_{2}^{4}\vartheta_{3}^{4}}.$
(34)

The curve has a double singularity at $u_{*}=m^{2}+\frac{\Lambda_{2}^{2}}{8}$
and two simple singularities at $u_{\pm}=-\frac{\Lambda_{2}^{2}}{8}\pm
m\Lambda_{2}$. When $m$ is varied, it can be determined how the branch point
$\tau_{\text{bp}}$ moves in the fundamental domain. When $m$ is not zero and
not equal to $m_{\text{AD}}=\tfrac{\Lambda_{2}}{2}$, the singularities
$u_{*}$, $u_{+}$ and $u_{-}$ are distinct and the fundamental domain
$\mathcal{F}(m,m)$ is given in Fig. 5. For any such mass, there is a pair of
branch points in $\mathcal{F}(m,m)$, which are connected by a branch cut. Near
a branch point $\tau_{\rm bp}$, $u$ reads

$u(\tau)=u_{\rm bp}+c(\tau_{\rm bp})\,(\tau-\tau_{{\rm bp}})^{1/2},$ (35)

for some constant $c(\tau_{\rm bp})$.

Figure 5: Fundamental domain $\mathcal{F}_{2}(m,m)$ of the massive
$\boldsymbol{m}=(m,m)$ $N_{f}=2$ theory. The dashed lines correspond to the
paths of the branch points from zero to infinite mass. For given positive mass
$m$, the two branch points are identified under $TST^{-1}$, such that there is
only one branch point $u_{\text{bp}}=2m^{2}-\tfrac{\Lambda_{2}^{2}}{8}$ on the
$u$-plane.

The massless limit in $N_{f}=2$
In the limit where $m_{1}=m_{2}=m\to 0$, one finds from (34) that

$\frac{u}{\Lambda_{2}^{2}}=-\frac{1}{8}\frac{\vartheta_{3}^{4}+\vartheta_{4}^{4}}{\vartheta_{2}^{4}}$
(36)

This expression is a modular function for the principal congruence subgroup
$\Gamma(2)$ of $\text{PSL}(2,\mathbb{Z})$. The holomorphy of (36) is explained
by the fact that the pair of branch points for $m\neq 0$ collides in the limit
$m\to 0$, which annihilates the branch cut. This is because the mutually local
singularities $u_{\pm}\to-\frac{\Lambda_{2}^{2}}{8}$ collide, and thus the
singularities corresponding to $\tau\to 0$ and $\tau\to 2$ in Fig. 5 are
identified. The fundamental domain of the massless theory is thus the modular
fundamental domain for the duality group $\Gamma(2)$, as depicted in Fig. 6.

Figure 6: Fundamental domain for $\Gamma(2)$, the duality group of the
massless $N_{f}=2$ theory.

The AD limit in $N_{f}=2$
When $m\to m_{\text{AD}}=\tfrac{\Lambda_{2}}{2}$, the singularities $u_{*}$
and $u_{+}$ collide and form the superconformal AD point $(A_{1},A_{3})$ of
Kodaira type III. As studied in detail in Aspman:2021vhs ; Closset:2021lhd ;
Magureanu:2022qym , this AD point forms an elliptic point of the fundamental
domain. Since three mutually non-local singularities collide for this mass,
the index of the domain $\mathcal{F}(m_{\text{AD}},m_{\text{AD}})$ is reduced
by 3, giving a domain of index $6-3=3$. By taking the limit from (34), one
finds that the square root resolves by virtue of the Jacobi identity (151).
This restores holomorphy and even modularity. One finds that the order
parameter

$\frac{u}{\Lambda_{2}^{2}}=-\frac{1}{64}\left(\frac{16\vartheta_{4}^{8}}{\vartheta_{2}^{4}\vartheta_{3}^{4}}+40\right)$
(37)

is a modular function for the congruence subgroup $\Gamma_{0}(2)$ of
$\text{PSL}(2,\mathbb{Z})$ Aspman:2021vhs . Its elliptic fixed point
$\tau_{\text{AD}}=1+i$ is precisely the AD point. We depict the corresponding
fundamental domain in Fig. 7.

Figure 7: Fundamental domain of the $N_{f}=2$ theory with equal AD mass
$m_{1}=m_{2}=m_{\text{AD}}=\frac{\Lambda_{2}}{2}$. It is the modular
fundamental domain for the congruence subgroup $\Gamma_{0}(2)$ of
$\text{PSL}(2,\mathbb{Z})$. The AD point $\tau_{\text{AD}}=1+i$ is the
elliptic fixed point of $\Gamma_{0}(2)$. The three singularities collided in
$u_{\text{AD}}$ results in a reduction of the index from 6 to 3.

The $N_{f}=4$ theory
In $N_{f}=4$, the fundamental domain for the effective coupling $\tau$ can be
found analogously to the asymptotically free cases. One notable new feature is
the additional singularity of the theory when $\tau$ approaches the UV
coupling $\tau_{\text{\tiny{UV}}}$. For example, with the mass configuration
${\boldsymbol{m}}=(m,m,0,0)$ one finds Aspman:2021evt

$u(\tau,\tau_{\text{\tiny{UV}}})=-\frac{m^{2}}{3}\vartheta_{3}(\tau_{\text{\tiny{UV}}})^{4}\frac{\lambda(\tau_{\text{\tiny{UV}}})^{2}+2\left(\lambda(\tau)-1\right)\lambda(\tau_{\text{\tiny{UV}}})-\lambda(\tau)}{\lambda(\tau_{\text{\tiny{UV}}})-\lambda(\tau)},$
(38)

where $\lambda=\frac{\vartheta_{2}^{4}}{\vartheta_{3}^{4}}$. This function is
a meromorphic $\Gamma(2)$ modular form of weight $2$ in
$\tau_{\text{\tiny{UV}}}$ for fixed $\tau$, and of weight $0$ in $\tau$ for
fixed $\tau_{\text{\tiny{UV}}}$. Furthermore, it has mixed weight $(0,2)$
under $\text{PSL}(2,\mathbb{Z})$ acting on $\tau$ and
$\tau_{\text{\tiny{UV}}}$ simultaneously. As such, it is an example of a
_bimodular_ form for the triple
$(\Gamma(2),\Gamma(2);\text{PSL}(2,\mathbb{Z}))$ Aspman:2021evt ;
Manschot:2021qqe . Since $u(\tau,\tau_{\text{\tiny{UV}}})$ has a pole as
$\tau\to\tau_{\text{\tiny{UV}}}$, the fundamental domain
$\mathcal{F}_{4}({\boldsymbol{m}})$ has a puncture at
$\tau_{\text{\tiny{UV}}}$. This is depicted in Fig. 8.

Figure 8: Fundamental domain of the $N_{f}=4$ theory with mass
$\boldsymbol{m}=(m,m,0,0)$. The six singularities on the Coulomb branch
$\mathcal{B}_{4}$ are described by the three cusps, each of width $2$. The
$I_{0}^{*}$ singularity corresponding to $u=\infty$ sits at
$\tau=\tau_{\text{\tiny{UV}}}$.

###  2.5 Monodromies

This section determines the monodromies around the $N_{f}+2$ monodromies. We
leave the winding numbers $n_{j}$, $j=1,\dots,N_{f}$, for $a_{D}$ generic.
Starting with the monodromy around infinity, $a\to e^{\pi i}a$, we deduce from
the (10) that the vector $\Pi$ transforms as $\Pi\to{\bf M}_{\infty}\,\Pi$,
with ${\bf M}_{\infty}$ given by

${\bf M}_{\infty}=\begin{pmatrix}-1&4-N_{f}&0&-n_{1}&\cdots&0&-n_{N_{f}}\\\
0&-1&0&0&\cdots&0&0\\\ 0&n_{1}&1&1&\cdots&0&0\\\ 0&0&0&1&\cdots&0&0\\\
\vdots&\vdots&&&\ddots&&\\\ 0&n_{N_{f}}&0&0&\cdots&1&1\\\
0&0&0&0&\cdots&0&1\end{pmatrix}.$ (39)

The monodromy matrix ${\bf M}_{\infty}$ is in
$\text{SL}(2+2N_{f},\mathbb{Z})$, while it acts on the couplings by a
symplectic transformation, i.e. it preserves the symplectic form (13). This
can be checked by requiring that any monodromy ${\bf M}_{\infty}$ satisfies
${\bf M}^{T}{\bf J}{\bf M}={\bf J}$, with

${\bf J}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}^{\oplus N_{f}+1}.$ (40)

The action on the couplings $\tau$ (14), $v_{j}$ and $w_{jk}$ (15) is thus

${\bf M}_{\infty}:\begin{cases}\tau\to\tau+N_{f}-4,\\\ v_{j}\to-
v_{j}-n_{j},\\\ w_{jk}\to w_{jk}+\delta_{jk},\end{cases}$ (41)

with $\delta_{jk}$ the Kronecker delta.

If we assume that the mass $m_{j}$ is large, we can also deduce the
monodromies around $a=\tfrac{m_{j}}{\sqrt{2}}$, $j=1,\dots,N_{f}$ from the
perturbative prepotential (10). For $a$ encircling $\tfrac{m_{1}}{\sqrt{2}}$
counterclockwise, $\Pi\to{\bf M}_{1}\Pi$, we find for the monodromy matrix
${\bf M}_{1}$,

${\bf M}_{1}=\begin{pmatrix}1&1&0&-1&\cdots&0&0\\\ 0&1&0&0&\cdots&0&0\\\
0&-1&1&1&\cdots&0&0\\\ 0&0&0&1&\cdots&0&0\\\ \vdots&\vdots&&&\ddots&&\\\
0&0&0&0&\cdots&1&0\\\ 0&0&0&0&\cdots&0&1\end{pmatrix},$ (42)

while the ${\bf M}_{j}$ for other values of $j$ are given by permutations. Its
action on the couplings is

${\bf M}_{j}:\begin{cases}\tau\to\tau+1,\\\ v_{k}\to v_{k}-\delta_{jk},\\\
w_{kl}\to w_{kl}+\delta_{kl}\delta_{jl}.\end{cases}$ (43)

To visualise the monodromies with large ${\rm Im}(\tau)$ for $N_{f}=1$, we
refer to Fig. 4(d). In this regime of the masses, there is one monodromy with
periodicity 3 and one with periodicity 1.

Besides the monodromies ${\bf M}_{\infty}$ and ${\bf M}_{j}$, there are
monodromies ${\bf M}_{m}$ and ${\bf M}_{d}$ around the points where a monopole
and a dyon becomes massless, respectively. By requiring that the electro-
magnetic charges of the massless particles are $(n_{m},n_{e})=(1,0)$ and
$(1,-2)$, respectively, we can fix the upper left blocks of the monodromies.
We fix the remaining entries by assuming that the masses remains invariant,
$m_{j}\to m_{j}$, and that the other periods only change by a multiple of the
vanishing cycle at the corresponding cusp, together with the requirement that

${\bf M}_{\infty}={\bf M}_{m}{\bf M}_{d}\prod_{j=1}^{N_{f}}{\bf M}_{j}.$ (44)

For $N_{f}=1$ and $n_{1}=n$, this gives for ${\bf M}_{m}$,

${\bf M}_{m}=\begin{pmatrix}1&0&0&0\\\ -1&1&0&-(n+1)/2\\\
(n+1)/2&0&1&(n+1)^{2}/4\\\ 0&0&0&1\end{pmatrix}.$ (45)

This acts on the couplings as

${\bf M}_{m}:\begin{cases}\tau\to\frac{\tau}{-\tau+1},\\\
v\to\frac{v+(n+1)\tau/2}{-\tau+1},\\\ w\to
w+\frac{(v+(n+1)/2)^{2}}{-\tau+1}.\end{cases}$ (46)

The monodromy ${\bf M}_{d}$ around the dyon singularity for $N_{f}=1$ is

${\bf M}_{d}=\begin{pmatrix}-1&4&0&-n-1\\\ -1&3&0&-(n+1)/2\\\
(n+1)/2&-n-1&1&(n+1)^{2}/4\\\ 0&0&0&1\end{pmatrix},$ (47)

This acts on the couplings as

${\bf M}_{d}:\begin{cases}\tau\to\frac{-\tau+4}{-\tau+3},\\\
v\to\frac{v+(n+1)\tau-n-1}{-\tau+3},\\\ w\to
w+\frac{(v+(n+1)/2)^{2}}{-\tau+3}.\end{cases}$ (48)

We can note that all the above monodromy matrices leave the symplectic form
(13) invariant and are independent of the masses.

For a small mass $m$, the fourth “hypermultiplet” cusp of the fundamental
domain for $N_{f}=1$ lies naturally near the real axis, $\tau\to 1$. See for
example Fig. 4(a). Having determined ${\bf M}_{\infty}$, ${\bf M}_{m}$ and
${\bf M}_{d}$, we can easily determine the monodromy ${\bf\tilde{M}}_{1}$ in
this regime as

${\bf\tilde{M}}_{1}={\bf M}_{m}^{-1}{\bf M}_{\infty}{\bf
M}_{d}^{-1}=\begin{pmatrix}0&1&0&(1-n)/2\\\ -1&2&0&(1-n)/2\\\
(n-1)/2&(1-n)/2&1&(n-1)^{2}/4\\\ 0&0&0&1\end{pmatrix}$ (49)

Thus for $n=-1$, the massless particle has charge $\pm(-1,1,0,1)$.

We get similar monodromies for $N_{f}=2,3$. The action on the running
couplings $\tau$ are the same for all $N_{f}$, by construction. The
transformations of $v_{j}$ and $w_{jk}$ also take the same form for all
$N_{f}$ and can be summarised as

$\displaystyle{\bf
M}_{m}:\begin{cases}v_{j}&\to\frac{v_{j}+(n_{j}+1)\tau/2}{-\tau+1},\\\
w_{jk}&\to
w_{jk}+\frac{(v_{j}+(n_{j}+1)/2)(v_{k}+(n_{k}+1)/2)}{-\tau+1},\end{cases}$
(50) $\displaystyle{\bf
M}_{d}:\begin{cases}v_{j}&\to\frac{v_{j}+(n_{j}+1)\tau-n_{j}-1}{-\tau+3},\\\
w_{jk}&\to
w_{jk}+\frac{(v_{j}+(n_{j}+1)/2)(v_{k}+(n_{k}+1)/2)}{-\tau+3}.\end{cases}$

##  3 The UV theory on a four-manifold

We review various aspects of the formulation of the UV theory on a compact
smooth four-manifold.

###  3.1 Aspects of four-manifolds

We let $X$ be a smooth, compact, oriented Riemannian four-manifold, with Euler
number $\chi=\chi(X)$ and signature $\sigma=\sigma(X)=b_{2}^{+}-b_{2}^{-}$.
The $u$-plane integral is non-vanishing only for four-manifolds $X$ with
$b_{2}^{+}\leq 1$. In this article, we consider manifolds with $b_{2}^{+}=1$.
Such four-manifolds admit a linear complex structure $\mathcal{J}$ on the
tangent space $TX_{p}$ at each point $p$ of $X$. The complex structure varies
smoothly on $X$, such that $TX$ is a complex bundle. We introduce furthermore
the canonical class $K_{X}=-c_{1}(TX)$ of $X$, with $c_{1}(TX)$ the first
Chern class of $TX$. For a manifold $X$ with $(b_{1},b_{2}^{+})=(0,1)$, we
have that

$K_{X}^{2}=8+\sigma(X).$ (51)

The middle cohomology $H^{2}(X,\mathbb{Z})$ of $X$ gives rise to the uni-
modular lattice $L$. More precisely, we identify $L$ with the natural
embedding of $H^{2}(X,\mathbb{Z})$ in $H^{2}(X,\mathbb{Z})\otimes\mathbb{R}$,
which mods out the torsion of $H^{2}(X,\mathbb{Z})$. A characteristic element
$K\in L$ is an element which satisfies
${\boldsymbol{l}}^{2}+B(K,{\boldsymbol{l}})\in 2\mathbb{Z}$ for all
${\boldsymbol{l}}\in L$. The Riemann-Roch theorem demonstrates that the
canonical class $K_{X}$ of $X$ is a characteristic element of $L$. The Wu
formula furthermore shows that any characteristic vector $K$ of $L$ is a lift
of $w_{2}(X)$.

The quadratic form $Q$ of the lattice $L$ for a 4-manifold with
$(b_{1},b_{2}^{+})=(0,1)$ can be brought to a simple standard form depending
on whether $Q$ is even or odd Donaldson90 . This divides such manifolds into
two classes, for which the evaluation of their $u$-plane integrals needs to be
done separately Korpas:2019cwg ; Aspman:2023ate . The period point $J\in
H^{2}(X,\mathbb{R})$ is defined as the unique class in the forward light cone
of $H^{2}(X,\mathbb{R})$ that satisfies $J=*J$ and $J^{2}=1$.

All four-manifolds without torsion and even intersection form admit a Spin
structure. More generally, for any oriented four-manifold one can define a
$\text{Spin}^{c}$-structure. The group $\text{Spin}^{c}(4)$ can be defined as
pairs of unitary $2\times 2$ matrices with coinciding determinant,

$\text{Spin}^{c}(4)=\\{(u_{1},u_{2})\in\mathrm{U}(2)\times\mathrm{U}(2)|\det{u_{1}}=\det
u_{2}\\}.$ (52)

There exists a short exact sequence

$1\longrightarrow U(1)\longrightarrow\text{Spin}^{c}(4)\longrightarrow
SO(4)\longrightarrow 1.$ (53)

A $\text{Spin}^{c}$-structure $\mathfrak{s}$ on a four-manifold $X$ is then a
reduction of the structure group of the tangent bundle on $X$, i.e. SO(4), to
the group $\text{Spin}^{c}(4)$. The different $\text{Spin}^{c}$-structures
correspond to the inequivalent ways of choosing transition functions of the
tangent bundle such that the cocycle condition is satisfied. The
$\text{Spin}^{c}$-structure defines two rank two hermitian vector bundles
$W^{\pm}$. We let $c(\mathfrak{s})$ be the first Chern class of the
determinant bundles, $c(\mathfrak{s})\coloneqq c_{1}(\det W^{\pm})\in
H^{2}(X,\mathbb{Z})$.

If $\mathfrak{s}$ is the canonical $\text{Spin}^{c}$ structure associated to
an almost complex structure on $X$, then $c(\mathfrak{s})^{2}=2\chi+3\sigma$.
More generally,

$c_{1}(\mathfrak{s})^{2}\equiv\sigma\mod 8.$ (54)

###  3.2 Topological twisting with background fluxes

We discuss in this section topological twisting of theories with fundamental
hypermultiplets including background fluxes. The discussion is parallel to the
case of $\mathcal{N}=2^{*}$ Manschot:2021qqe , where the hypermultiplet is in
the adjoint representation of the gauge group.

We let $(E\to X,\nabla)$ be a principal
$\text{SU}(2)/\mathbb{Z}_{2}\cong\text{SO}(3)$-bundle with connection
$\nabla$. The second Stiefel-Whitney class $w_{2}(E)\in
H^{2}(X,\mathbb{Z}_{2})$ measures the obstruction to lift $E$ to an SU(2)
bundle, which will exist locally but not globally if $w_{2}(E)\neq 0$. We
denote a lift of $w_{2}(E)$ to the middle cohomology lattice $L$ by
$\bar{w}_{2}(E)\in L$, and define the ’t Hooft flux
${\boldsymbol{\mu}}=\bar{w}_{2}(E)/2\in L/2$. The instanton number of the
principal bundle is defined as $k=-\frac{1}{4}\int_{X}p_{1}(E)$ and satisfies
$k\in-{\boldsymbol{\mu}}^{2}+\mathbb{Z}$, where $p_{1}$ is the first
Pontryagin class.

To formulate the theories with $N_{f}$ fundamental hypermultiplets on a
compact four-manifold, we perform a topological twist. Coupling the four-
dimensional $\mathcal{N}=2$ $\text{SU}(2)$ theory to background fields means
choosing two sets of data:

  * •

A principal $\text{SU}(2)_{R}$ R-symmetry bundle, with connection
$\nabla_{R}$,

  * •

and a principal bundle $\mathcal{L}$ with connection for global symmetries
(the flavour symmetries) Manschot:2021qqe .

The relevant twist for the $\mathcal{N}=2$ supersymmetry algebra in four
dimensions is the Donaldson-Witten twist. This twist is the local
identification of the SU(2)+ with the diagonal subgroup of the
SU(2)${}_{+}\times$SU(2)R factor of the spin lift of the local spin group
$\text{Spin}(4)\cong\text{SU}(2)_{+}\times\text{SU}(2)_{-}$ Witten:1988ze .
Alternatively, one can view the fields as sections of a non-trivial R-symmetry
bundle, isomorphic to the spin bundle $S^{+}$. Application of this to the
representations of the vector multiplet (4) and (7) gives:

bosons: $\displaystyle({\bf 2},{\bf 2})\oplus({\bf 1},{\bf 1})\oplus({\bf
1},{\bf 1}),$ (55) fermions: $\displaystyle({\bf 2},{\bf 2})\oplus({\bf
3},{\bf 1})\oplus({\bf 1},{\bf 1}).$

Thus the bosons remain unchanged, a vector and a complex scalar, while the
fermions reorganise to a vector, self-dual two-form and real scalar, which we
denote as $\psi$, $\chi$ and $\eta$, respectively. We note that none of these
fields are spinors, and can thus be considered on a non-spin four-manifold.
The original supersymmetry generators also transform in the representations
for the fermions above. Thus the theory contains a scalar fermionic
supercharge
$\mathcal{Q}=\epsilon^{\dot{A}\dot{B}}\overline{\mathcal{Q}}_{\dot{A}\dot{B}}$,
whose cohomology provides the operators in the topological theory
Witten:1988ze .

For the fields of a hypermultiplet, (6) and (7), one finds

bosons: $\displaystyle({\bf 1},{\bf 2})\oplus({\bf 1},{\bf 2}),$ (56)
fermions: $\displaystyle({\bf 2},{\bf 1})\oplus({\bf 1},{\bf 2})\oplus({\bf
2},{\bf 1})\oplus({\bf 1},{\bf 2}).$

Thus hypermultiplet bosons become spinors, i.e. sections of the spin bundle
$S^{+}$, while the fermions are sections of $S^{+}$ and $S^{-}$. Thus the
twisted hypermultiplets can in this case only be formulated on four-manifolds
which are spin, i.e. $w_{2}(X)=0$ Hyun:1995mb ; Moore:1997pc .

However, if the hypermultiplets are charged under a gauge field or flux, the
product of these bundles with $S^{\pm}$ may be a Spinc bundle, $W^{+}$ or
$W^{-}$ Hyun:1995mb ; Laba05 ; Manschot:2021qqe . The latter are defined for
arbitrary four-manifolds. For example, an almost complex structure on $X$
determines two canonical Spinc bundles $W^{\pm}\simeq S^{\pm}\otimes
K^{-1/2}_{X}$ with $K_{X}$ the canonical class determined by the almost
complex structure. Since the hypermultiplets are in the fundamental, two-
dimensional representation of SU(2), the topologically twisted hypermultiplets
are well-defined on a non-spin four-manifold if ${\boldsymbol{\mu}}=-K_{X}/2$
Moore:1997pc .

Let us state this also in terms of the gauge bundle $E$. To this end, we label
the two components of the fundamental, two-dimensional representation of SU(2)
by $\pm$. The two components are sections of a line bundle
$\mathcal{L}_{E}^{\pm 1/2}$ with $c_{1}(\mathcal{L}_{E})=\bar{w}_{2}(E)$. Of
course, the square root $\mathcal{L}_{E}^{1/2}$ only exists if $w_{2}(E)\in
2L$. On the other hand, the physical requirement is that
$S^{+}\otimes\mathcal{L}_{E}^{1/2}$ is well defined, or
$\bar{w}_{2}(X)+\bar{w}_{2}(E)\in 2L$. Therefore, the obstructions can cancel
each other for a suitable choice of $w_{2}(E)$. Thus the topological twisted
theory is not well-defined for an arbitrary choice of ’t Hooft flux
${\boldsymbol{\mu}}\coloneqq\frac{1}{2}\bar{w}_{2}(E)$; but rather
${\boldsymbol{\mu}}$ has to satisfy
${\boldsymbol{\mu}}=\frac{1}{2}\bar{w}_{2}(X)\mod L$ Moore:1997pc , or

$\bar{w}_{2}(X)=\bar{w}_{2}(E)\mod 2L.$ (57)

To consider more general ’t Hooft fluxes ${\boldsymbol{\mu}}$ or equivalently
$w_{2}(E)$, we can couple the $j$’th hypermultiplet to a background flux or
line bundle $\mathcal{L}_{j}$, with $\mathcal{L}_{j}$ possibly different for
each $j$. We let $\mathcal{E}_{j}=\mathcal{L}_{E}\otimes\mathcal{L}_{j}$. Then
the requirement that $S^{\pm}\otimes\mathcal{E}_{j}^{\pm 1/2}$ is globally
well-defined is that

$c_{1}(\mathcal{E}_{j})\in\bar{w}_{2}(X)+2L,$ (58)

which can be satisfied for any $\bar{w}_{2}(E)$ for a suitable choice of
$\mathcal{L}_{j}$. Thus we can formulate the $u$-plane integral for arbitrary
$\bar{w}_{2}(E)$, if we require that the background fluxes satisfy

$c_{1}(\mathcal{L}_{j})\in\bar{w}_{2}(X)+\bar{w}_{2}(E)+2L,$ (59)

for each $j$. This is consistent with (57) for $c_{1}(\mathcal{L}_{j})=0$.

The Chern classes $c_{1}(\mathcal{L}_{j})$ can also be seen as the splitting
classes of the $\text{Spin}(2N_{f})$ principal bundle $\mathcal{L}$. The Chern
class of $\mathcal{L}$ reads

$c(\mathcal{L})=\sum_{l=0}^{2}c_{l}(\mathcal{L})=\prod_{j=1}^{N_{f}}(1+c_{1}(\mathcal{L}_{j})).$
(60)

The scalar generators of the equivariant cohomology of $\text{Spin}(2N_{f})$
are the masses $m_{j}$, which generate the $N_{f}$-dimensional Cartan
subalgebra of $\text{Spin}(2N_{f})$. The gauge bundle $E_{k}$ is also
$\text{Spin}(2N_{f})$ equivariant. For generic masses, the flavour group is
U(1)${}^{N_{f}}$, and is enhanced for special loci of the masses, for example
to $\text{U}(N_{f})$ for equal masses Seiberg:1994aj .

The $\mathcal{Q}$-fixed equations are the non-Abelian monopole equations with
$N_{f}$ matter fields in the fundamental representation. For generic gauge
group $G$ and with representation $R$, these equations read Labastida:1995zj

$\begin{split}&\left(F^{a}_{\dot{\alpha}\dot{\beta}}\right)^{+}+\frac{i}{2}\sum_{j=1}^{N_{f}}\bar{M}^{j}_{(\dot{\alpha}}T^{a}M^{j}_{\dot{\beta})}=0,\\\
&\not{D}M^{j}=\sum_{\mu}\sigma^{\mu}\,D_{\mu}M^{j}=0,\end{split}$ (61)

where $T^{a}$ is a generator of the Lie algebra in the representation $R$.
Including the sum over matrix elements, we have

$M^{j}_{(\alpha}T^{a}M^{j}_{\beta)}=\sum_{k,l}(M^{j})^{k}_{(\alpha}(T^{a})^{kl}(M^{j})^{l}_{\beta)}.$
(62)

We denote the moduli space of solutions to (61) by
$\mathcal{M}^{Q,N_{f}}_{k,\mathcal{L}_{j}}$, and leave the dependence on the
’t Hooft flux ${\boldsymbol{\mu}}$ and the metric $J$ implicit. For $N_{f}=4$
on $X=\mathbb{CP}^{2}$, such moduli spaces are studied in Gorodentsev:1996cu .

The moduli spaces $\mathcal{M}^{Q,N_{f}}_{k,\mathcal{L}_{j}}$ is non-compact
for vanishing masses bryan1996 ; Moore:1997dj ; Dedushenko:2017tdw . This is
improved upon turning on masses and localizing with respect to the
$\text{U}(1)^{N_{f}}$ flavour symmetry, $M^{j}_{\alpha}\to
e^{i\,\varphi_{j}}M^{j}_{\alpha}$, which leave invariant the
$\mathcal{Q}$-fixed equations (61). There are two components:

  * •

the instanton component, with $F^{+}=0$ and $M^{j}=0$, $j=1,\dots,N_{f}$. The
moduli space for this component is denoted $\mathcal{M}^{\rm i}_{k}$. Since
the hypermultiplet fields vanish, this component is associated to the Coulomb
branch.

  * •

the abelian or monopole component, for which a U$(1)$ subgroup of the flavour
group acts as pure gauge. Here the connection is reducible, and a U(1)
subgroup of the SU(2) gauge group is preserved. For generic masses, there are
$N_{f}$ such components, where $M^{\ell}$ is upper or lower triangular for
some $\ell$, and $M^{j}=0$ for all $j\neq\ell$. The moduli space of this
component is denoted $\mathcal{M}^{{\rm a},j}_{k}$, $j=1,\dots,N_{f}$. Since
some of the hypermultiplet fields are non-vanishing, this component is
associated to the Higgs branch Moore:1997dj ; Marino:1998uy .

The instanton component $\mathcal{M}^{\rm i}_{k}$ is non-compact due to point-
like instantons. This can be cured using the Uhlenbeck compactification or
algebraic-geometric compactifications. We assume that the physical path
integral chooses a specific compactification, whose details are however not
manifest at the level of the low energy effective field theory other than that
the compactification must be in agreement with the correlation functions.

The topological twist for $\mathcal{N}=2$ supersymmetric QCD can be further
made dependent on a cocycle $\zeta_{\alpha\beta\gamma}^{\text{gauge}}$
representing the ’t Hooft flux, and $\zeta_{\alpha\beta\gamma}^{s}$ a cocycle
representing $w_{2}(X)$ (the cocycles are the U(1) valued functions measuring
the obstruction of the cocycle condition for transition functions)
Manschot:2021qqe . Without additional line bundles,
$\bar{w}_{2}(X)=\bar{w}_{2}(E)$ is equivalent to the cocycle
$\zeta_{\alpha\beta\gamma}^{\text{gauge}}\zeta_{\alpha\beta\gamma}^{s}$ being
trivialisable. We leave it for future work to explore whether the invariants
depend on the choice of trivialisation.

###  3.3 Correlation functions and moduli spaces

The $\mathcal{Q}$-fixed equations (61) include a Dirac equation for each
hypermultiplet $j=1,\dots,N_{f}$ in the fundamental representation. The
corresponding index bundle $W_{k}^{j}$ defines an element of the $K$-group of
$\mathcal{M}^{\rm i}_{k}$. Its virtual rank $\text{rk}(W^{j}_{k})$ is the
formal difference of two infinite dimensions. It is given by an index theorem
and reads

$\text{rk}(W^{j}_{k})=-k+\frac{1}{4}(c_{1}(\mathcal{L}_{j})^{2}-\sigma)\in\mathbb{Z},$
(63)

where $c_{1}(\mathcal{L}_{j})$ is the first Chern class of the bundle
$\mathcal{L}_{j}$. Note that the rhs is not an integer for an arbitrary
$c_{1}(\mathcal{L}_{j})\in H^{2}(X,\mathbb{Z})$. To verify that the rhs is
integral for the $c_{1}(\mathcal{L}_{j})$’s satisfying (59), we rewrite
$\text{rk}(W^{j}_{k})$ as

$\text{rk}(W^{j}_{k})=-(k+{\boldsymbol{\mu}}^{2})-c_{1}(\mathcal{L}_{j})\cdot{\boldsymbol{\mu}}+\frac{1}{4}\left((c_{1}(\mathcal{L}_{j})+2{\boldsymbol{\mu}})^{2}-\sigma\right).$
(64)

Then the first term on the rhs is an integer since
$k\in-\frac{1}{4}w_{2}(E)^{2}+\mathbb{Z}$ for an $\text{SO}(3)$ bundle. The
second term is an integer because
$c_{1}(\mathcal{L}_{j})\cdot{\boldsymbol{\mu}}=(\bar{w}_{2}(X)-2{\boldsymbol{\mu}})\cdot{\boldsymbol{\mu}}\mod\mathbb{Z}\in\mathbb{Z}$,
and the third term is an integer using (54) and the fact that
$c_{1}(\mathcal{L}_{j})+2{\boldsymbol{\mu}}$ equals the characteristic class
of a $\text{Spin}^{c}$-structure $\mathfrak{s}_{j}$ by (59),

$c_{1}(\mathcal{L}_{j})+2{\boldsymbol{\mu}}=c(\mathfrak{s}_{j}),$ (65)

for each $j$.

The mass $m_{j}$ is the equivariant parameter of the U(1) flavour symmetry
associated to the $j$’th hypermultiplet. The equivariant Chern class of
$W^{j}_{k}$ reads in terms of the splitting class $x_{l}$,

$c(W_{k}^{j})=\prod_{l=0}^{-\text{rk}(W^{j}_{k})}(x_{l}+m_{j})=m_{j}^{-\text{rk}(W^{j}_{k})}\sum_{l}\frac{c_{l}(W_{k}^{j})}{m_{j}^{l}}.$
(66)

We abbreviate $c_{l}(W_{k}^{j})$ to $c_{l,j}$, and let
$c(W_{k})=\prod_{j=1}^{N_{f}}c(W_{k}^{j})$.

The moduli space $\mathcal{M}_{k,{\boldsymbol{\mu}},\mathcal{L}_{j}}^{Q}$ for
$N_{f}$ hypermultiplets corresponds to the vanishing locus of the obstructions
for the existence of $N_{f}$ zero modes of the Dirac operator. As a result,
the virtual complex dimension of the moduli space
$\mathcal{M}_{k,{\boldsymbol{\mu}},\mathcal{L}_{j}}^{Q}$ is that of the
instanton moduli space plus the sum of (typically negative) ranks of the index
bundles $W_{k}^{j}$,
$\text{vdim}(\mathcal{M}^{Q,N_{f}}_{k,\mathcal{L}_{j}})=\text{vdim}(\mathcal{M}^{Q}_{k})_{N_{f}=0}+\sum_{j=1}^{N_{f}}\text{rk}(W_{k}^{j})$
Hyun:1995hz ; bryan1996 ; Feehan:2001jc ; Manschot:2021qqe . This gives

$\text{vdim}(\mathcal{M}_{k,\mathcal{L}_{j}}^{Q,N_{f}})=(4-N_{f})k+\frac{1}{4}\left(-3\chi-(3+N_{f})\sigma+\sum_{j=1}^{N_{f}}c_{1}(\mathcal{L}_{j})^{2}\right).$
(67)

It is argued in LoNeSha  that the inclusion of massive matter amounts to
inserting an integral of the equivariant Euler class of the Dirac index bundle
over the moduli space. Therefore, the correlation functions are the generating
functions for the intersection numbers of the standard Donaldson observables
and the Poincaré duals to the Chern classes of the various vector bundles.

The correlation functions on $X$ in the theory with $N_{f}$ massive
fundamental hypermultiplets are conjectured to be

$\langle\mathcal{O}_{1}\dots\mathcal{O}_{p}\rangle=\sum_{k}\Lambda_{N_{f}}^{{\rm
vdim}(\mathcal{M}^{Q,N_{f}}_{k,\mathcal{L}_{j}})}\,\int_{\mathcal{M}_{k,\mathcal{L}_{j}}^{Q,N_{f}}}c(W_{k})\,\omega_{1}\wedge\dots\wedge\omega_{p},$
(68)

where $\omega_{i}=\mu(\mathcal{O}_{i})$ are the Donaldson classes associated
to the physical observable $\mathcal{O}_{i}$, and $c(M)$ is the Euler class of
the matter bundle Mathai:1986tc ; Atiyah:1990tm ; losev1998 ; Cordes:1994fc .
Localising to the fixed point locus in
$\mathcal{M}^{Q,N_{f}}_{k,\mathcal{L}_{j}}$ with respect to $U(1)^{N_{f}}$
gives

$\begin{split}&\langle\mathcal{O}_{1}\dots\mathcal{O}_{p}\rangle=\sum_{k}\Lambda_{N_{f}}^{{\rm
vdim}(\mathcal{M}^{Q,N_{f}}_{k,\mathcal{L}_{j}})}\\\
&\qquad\qquad\times\int_{\mathcal{M}^{\rm i}_{k}\cup\mathcal{M}^{\rm
a}_{k}}\left(\prod_{j=1}^{N_{f}}m_{j}^{-\text{rk}(W^{j}_{k})}\sum_{l}\frac{c_{l,j}}{m_{j}^{l}}\right)\,\omega_{1}\wedge\dots\wedge\omega_{p}.\end{split}$
(69)

where the integral is over the union $\mathcal{M}^{\rm
i}_{k}\cup\mathcal{M}^{\rm a}_{k}$ of the instanton component
$\mathcal{M}^{\rm i}_{k}$ (LoNeSha, , Eq. (5.13)) and the monopole component
$\mathcal{M}^{\rm a}_{k}$ Manschot:2021qqe ; Vafa:1994tf . The equation
together with the dimension of the moduli spaces (67) demonstrate a selection
rule for observables together with powers of $\Lambda_{N_{f}}$ and $m_{j}$.

In the decoupling limit $m_{N_{f}}\to\infty$, $\Lambda_{N_{f}}\to 0$ (21), the
only contribution for $j=N_{f}$ is from $l=0$, $c_{0,N_{f}}=1$. The powers of
$m_{j}$ and $\Lambda_{N_{f}}$ work out such that the correlation functions
reduce to those of the theory with $N_{f}-1$ hypermultiplets Marino:1998uy

$\left(\frac{\Lambda_{N_{f}}}{\Lambda_{N_{f}-1}}\right)^{-\alpha}\langle\mathcal{O}_{1}\dots\mathcal{O}_{p}\rangle_{N_{f}}\to\langle\mathcal{O}_{1}\dots\mathcal{O}_{p}\rangle_{N_{f}-1},$
(70)

We deduce from (69) that

$\begin{split}\alpha&={\rm
vdim}(\mathcal{M}^{Q,N_{f}}_{k})+(4-N_{f})\,\text{rk}(W_{k}^{N_{f}})\\\
&=\frac{1}{4}\Big{(}-3\chi-7\sigma+(5-N_{f})\,c_{1}(\mathcal{L}_{N_{f}})^{2}+\sum_{j=1}^{N_{f}-1}c_{1}(\mathcal{L}_{j})^{2}\Big{)}.\end{split}$
(71)

The overall factor can be accounted for by an overall renormalization in the
decoupling limit.

The correlation function has a smooth massless limit $m_{j}\to 0$, for which
only terms with the top Chern classes contribute. These are given by

$\langle\mathcal{O}_{1}\dots\mathcal{O}_{p}\rangle=\sum_{k}\prod_{j=1}^{N_{f}}\Lambda_{N_{f}}^{{\rm
vdim}(\mathcal{M}^{Q,N_{f}}_{k,\mathcal{L}_{j}})}\int_{\mathcal{M}^{\rm
i}_{k}\cup\mathcal{M}^{\rm
a}_{k}}c_{l_{j},j}\,\omega_{1}\wedge\dots\wedge\omega_{p}$ (72)

with $l_{j}=-{\rm rk}(W^{j}_{k})$ for each $c_{l,j}$. For a non-vanishing
result, the degree of $\omega_{1}\dots\omega_{p}$ must equal ${\rm
vdim}(\mathcal{M}^{Q,N_{f}}_{k,\mathcal{L}_{j}})$.

By comparing the form degrees (or ghost numbers) of operators with the virtual
dimension of the moduli space, one derives selection rules for correlation
functions of point and surface operators. Since $u$ has ghost number 4, the
fugacity $p$ naturally has ghost number $-4$. Similarly, we associate ghost
number $-2$ to the surface ${\boldsymbol{x}}$. The mass $m_{j}$ finally has
ghost number $2$. Thus correlation functions will evaluate to sums of
monomials of the form
$p^{s}{\boldsymbol{x}}^{t}\prod_{j=1}^{N_{f}}m_{j}^{r_{j}}$, with the
selection rule

$-4s-2t+2\sum_{j=1}^{N_{f}}r_{j}=-\text{vdim}_{\mathbb{R}}(\mathcal{M}_{k,\mathcal{L}_{j}}^{Q,N_{f}}).$
(73)

##  4 The effective theory on a four-manifold

We consider in this Section the low energy effective field theory on a four-
manifold. We derive the semi-classical action of the theory coupled to
background $\text{U}(1)$ fields. As in previous cases Witten:1995gf ;
Shapere:2008zf ; Manschot:2021qqe ; Moore:1997pc , the final expression takes
the form of a Siegel-Narain theta series multiplied by a measure factor.

###  4.1 Hypermultiplets and background fields

The effective theory coupled to $N_{f}$ background fluxes can be modelled as
that of a theory with gauge group $\text{SU}(2)\times\text{U}(1)^{N_{f}}$,
where the fields of the U(1) factors have been frozen in a special way
Nelson:1993nf ; Manschot:2021qqe . To derive the precise form, we recall the
low-energy effective Lagrangian for the $r$ multiplets
$(\phi^{J},\eta^{J},\chi^{J},\psi^{J},F^{J})$ of the topologically twisted
$\text{U}(1)^{r}$ SYM theory Marino:1998bm . Since the $u$-plane integral
reduces to an integral over zero-modes Moore:1997pc , it suffices to only
include the zero-modes in the Lagrangian. For simply connected four-manifolds,
there is no contribution from the one-form fields $\psi^{J}$. The Lagrangian
is then given in terms of the prepotential $F(\\{a^{J}\\})$ and its
derivatives to the vevs $\langle\phi^{J}\rangle=a^{J}$, as

$\begin{split}\mathcal{L}&=\frac{i}{16\pi}(\bar{\tau}_{JK}F^{J}_{+}\wedge
F_{+}^{K}+\tau_{JK}F^{J}_{-}\wedge F^{K}_{-})-\frac{1}{8\pi}y_{JK}D^{J}\wedge
D^{K}\\\
&\qquad+\frac{i\sqrt{2}}{16\pi}\bar{\mathcal{F}}_{JKL}\eta^{J}\chi^{K}\wedge(D+F_{+})^{L},\end{split}$
(74)

with $y_{JK}=\text{Im}(\tau_{JK})$,
$\tau_{JK}=\partial_{J}\partial_{K}F(\\{a^{J}\\})$ and
$\mathcal{F}_{JKL}=\partial_{J}\partial_{K}\partial_{L}F(\\{a^{J}\\})$. It is
left invariant by the BRST operator $Q$, which acts on the zero modes as

$\displaystyle=\psi^{J}=0,\qquad$ $\displaystyle[Q,\psi^{J}]=4\sqrt{2}da^{J},$
(75) $\displaystyle[Q,a^{J}]$ $\displaystyle=0,\qquad$
$\displaystyle[Q,\bar{a}^{J}]=\sqrt{2}i\eta^{J},$ $\displaystyle[Q,\eta^{J}]$
$\displaystyle=0,\qquad$ $\displaystyle[Q,\chi^{J}]=i(F_{+}-D_{+})^{J},$
$\displaystyle[Q,D^{J}]$ $\displaystyle=(d\psi^{J})_{+}=0.$

Using this operator, we can write $\mathcal{L}$ as the sum of a topological,
holomorphic term and a $Q$-exact term,

$\mathcal{L}=\frac{i}{16\pi}\tau_{JK}F^{J}\wedge F^{K}+\\{Q,W\\},$ (76)

with

$W=-\frac{i}{8\pi}y_{JK}\chi^{J}(F_{+}+D)^{K}.$ (77)

The low-energy theory of $\text{SU}(2)$ gauge theory with $N_{f}$
hypermultiplets coupled to $N_{f}$ background fluxes can then be modelled by
the above rank $r$ description with $r=N_{f}+1$. We identify $F(\\{a^{J}\\})$
with $F(a,{\boldsymbol{m}})$. We let the indices $J,K$ run from 0 to $N_{f}$
and identify the index 0 with the unbroken U(1) of the SU(2) gauge group and
the indices $j,k,l=1,\dots,N_{f}$ with that of the frozen
$\text{U}(1)^{N_{f}}$ factors. We further set $\phi^{0}\coloneqq\phi$ for any
field $\phi$. We will proceed by using lower indices for $j,k,l$, except where
the summation convention is explicitly used, to avoid confusion with powers of
the fields.

The masses of the hypermultiplets are the vevs of the frozen scalar fields of
the corresponding vector multiplets,
$\tfrac{m_{j}}{\sqrt{2}}=\langle\phi_{j}\rangle=a_{j}$ Nelson:1993nf . We set
$[F_{j}]=4\pi{\boldsymbol{k}}_{j}$ with

${\boldsymbol{k}}_{j}=c_{1}(\mathcal{L}_{j})/2\in L/2.$ (78)

To make the BRST variations of the fields from the frozen U(1) factors vanish,
we set $\eta_{j}=\chi_{j}=0$, as well as $D^{j}=F_{+}^{j}$. With these
identifications, the Lagrangian becomes

$\displaystyle\mathcal{L}=$ $\displaystyle\frac{i}{16\pi}\tau_{JK}F^{J}\wedge
F^{K}+\frac{1}{8\pi}y_{00}F_{+}\wedge F_{+}-\frac{1}{8\pi}y_{00}D\wedge D$
(79)
$\displaystyle+\frac{i\sqrt{2}}{16\pi}\bar{\mathcal{F}}_{000}\eta\chi\wedge(D+F_{+})+\frac{i\sqrt{2}}{8\pi}\bar{\mathcal{F}}_{00j}\eta\chi\wedge
F_{+}^{j}$ $\displaystyle+\frac{1}{4\pi}y_{0j}(F_{+}-D)\wedge F_{+}^{j}.$

Integrating over $D$, $\eta$ and $\chi$ in the standard way Marino:1998bm ;
Moore:1997pc ; Manschot:2021qqe , we end up with

$\displaystyle\int dDd\eta d\chi\,e^{-\int_{X}\mathcal{L}}$ (80)
$\displaystyle=\frac{\partial}{\partial\bar{a}}\left(i\sqrt{y_{00}}\,B\left(F+\frac{y_{0j}}{y_{00}}F^{j},J\right)\right)e^{-\int_{X}\mathcal{L}_{0}},$

where

$\displaystyle\mathcal{L}_{0}$
$\displaystyle=\frac{i}{16\pi}\tau_{JK}F^{J}\wedge
F^{K}+\frac{1}{8\pi}y_{00}F_{+}\wedge F_{+}+\frac{y_{0j}}{4\pi}F_{+}\wedge
F_{+}^{j}+\frac{1}{8\pi}\frac{y_{0j}y_{0k}}{y_{00}}F^{j}_{+}\wedge F^{k}_{+}$
(81) $\displaystyle=\frac{i}{16\pi}(\bar{\tau}F_{+}\wedge F_{+}+\tau
F_{-}\wedge F_{-})+\frac{i}{8\pi}(v_{j}F_{-}\wedge
F_{-}^{j}+\bar{v}_{j}F_{+}\wedge F^{j}_{+})$
$\displaystyle\quad+\frac{i}{16\pi}w_{jk}F^{j}\wedge
F^{k}+\frac{y}{8\pi}\text{Im}(v_{j})\text{Im}(v_{k})F_{+}^{j}\wedge
F_{+}^{k},$

and we identified $\tau\coloneqq\tau_{00}$, $y={\rm Im}(\tau)=y_{00}$,
$v_{j}\coloneqq\tau_{0j}$ and $w_{jk}\coloneqq\tau_{jk}$. Thus the coupling
$w_{jk}$ is holomorphic, but the coupling $v_{j}$ is non-holomorphic. This is
similar to the couplings for $\mathcal{N}=2^{*}$ Manschot:2021qqe .

###  4.2 Sum over fluxes

The path integral includes a sum over fluxes ${\boldsymbol{k}}=[F]/4\pi\in
L/2$. After summing the exponentiated action (80) over the fluxes
${\boldsymbol{k}}$ and multiplying by $\frac{d\bar{a}}{d\bar{\tau}}$, we find
that this takes the form

$\sum_{{\boldsymbol{k}}\in L+{\boldsymbol{\mu}}}\int dDd\eta
d\chi\,e^{-\int_{X}\mathcal{L}}=\left(\prod_{j,k=1}^{N_{f}}C_{jk}^{B({\boldsymbol{k}}_{j},{\boldsymbol{k}}_{k})}\right)\,\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,\bar{\tau},{\boldsymbol{z}},\bar{\boldsymbol{z}}).$
(82)

The couplings $C_{jk}$ are given in terms of $w_{jk}$ (15) by

$C_{jk}=e^{-\pi iw_{jk}},$ (83)

for $j,k=1,\dots,N_{f}$. Such couplings were first put forward in
Shapere:2008zf , and were also crucial in Manschot:2021qqe .

The term $\Psi_{{\boldsymbol{\mu}}}^{J}$ is an example of a Siegel-Narain
theta function. It reads explicitly

$\displaystyle\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,\bar{\tau},{\boldsymbol{z}},\bar{\boldsymbol{z}})$
$\displaystyle=e^{-2\pi y{\boldsymbol{b}}^{2}_{+}}\sum_{{\boldsymbol{k}}\in
L+{\boldsymbol{\mu}}}\partial_{\bar{\tau}}\left(4\pi
i\sqrt{y}B({\boldsymbol{k}}+{\boldsymbol{b}},J)\right)$ (84)
$\displaystyle\times(-1)^{B({\boldsymbol{k}},K)}q^{-{\boldsymbol{k}}_{-}^{2}/2}\bar{q}^{{\boldsymbol{k}}_{+}^{2}/2}e^{-2\pi
iB({\boldsymbol{z}},{\boldsymbol{k}}_{-})-2\pi
iB(\bar{\boldsymbol{z}},{\boldsymbol{k}}_{+})},$

and discussed in more detail in Appendix A.2. The elliptic variable reads in
terms of $v_{j}$ and ${\boldsymbol{k}}_{j}$,

${\boldsymbol{z}}=\sum_{j=1}^{N_{f}}v_{j}{\boldsymbol{k}}_{j},\quad\text{and}\quad{\boldsymbol{b}}=\frac{\text{Im}({\boldsymbol{z}})}{y},$
(85)

thus inducing a non-holomorphic dependence on $v_{j}$. Furthermore, $K$
appearing in the fourth root of unity $(-1)^{B({\boldsymbol{k}},K)}$ is a
characteristic vector of $L$. Note that $\Psi_{\boldsymbol{\mu}}^{J}$ changes
by the sign $(-1)^{B({\boldsymbol{\mu}},K-K^{\prime})}$ upon replacing $K$ by
a different characteristic vector $K^{\prime}$ Moore:1997pc ; Cordova:2018acb
; Manschot:2021qqe .

For $N_{f}=0$, this phase can be understood as arising from integrating out
massive fermionic modes Witten:1995gf . It also appears naturally in
decoupling the adjoint hypermultiplet in the analogous function for
$\mathcal{N}=2^{*}$ Manschot:2021qqe . For $N_{f}>0$, the constant part of the
couplings $v_{j}$ (15) effectively contribute to the phase, such that the
total phase reads

$e^{\pi iB({\boldsymbol{k}},K)}\prod_{j=1}^{N_{f}}e^{\pi
i\,n_{j}\,B({\boldsymbol{k}}_{j},{\boldsymbol{k}})},$ (86)

with $n_{j}$ the magnetic winding numbers. For arbitrary $n_{j}\in\mathbb{Z}$,
the phase is an eighth root of unity. It would be interesting to understand
this phase from integrating out massive modes.

We deduce from (86) that the summand of $\Psi^{J}_{\boldsymbol{\mu}}$ changes
by a phase

$e^{\pi i(n_{j}^{\prime}-n_{j})B({\boldsymbol{k}}_{j},{\boldsymbol{k}})}$ (87)

if the winding numbers $n_{j}$ are replaced by $n^{\prime}_{j}$. Since
${\boldsymbol{k}}_{j}\in K/2-{\boldsymbol{\mu}}\mod L$ (see (59)) and
${\boldsymbol{k}}\in L+{\boldsymbol{\mu}}$, this phase is 1 if
$n_{j}^{\prime}-n_{j}=0\mod 4$. We can therefore restrict to
$n_{j}\in\mathbb{Z}_{4}$. For specific choices of ${\boldsymbol{\mu}}$ and
${\boldsymbol{k}}_{j}$, the $n_{j}$ can lie in a subgroup of $\mathbb{Z}_{4}$.

The modular transformations of $\Psi^{J}_{\boldsymbol{\mu}}$ are discussed in
Appendix A.2, which are crucial input for single-valuedness of the $u$-plane
integrand. We will demonstrate in Section 5.2 that the $u$-plane integrand is
single-valued if we impose further constraints on the winding numbers $n_{j}$.

Finally, if the theory is considered on a curved background, topological
couplings arise in the effective field theory Witten:1995gf . These terms
couple to the Euler characteristic and the signature of the four-manifold $X$,
respectively denoted $A$ and $B$. These take the form Witten:1995gf ;
Moore:1997pc ,

$A=\alpha\left(\frac{du}{da}\right)^{1/2},\qquad B=\beta\Delta_{N_{f}}^{1/8}.$
(88)

Here, $\Delta_{N_{f}}$ is the physical discriminant incorporating the
singularities of the effective theory, while $\frac{du}{da}$ is the
(reciprocal of) the periods of the SW curves as introduced in Section 2. Both
can be determined directly from the SW curve, as described in Section 2.3. The
prefactors $\alpha$ and $\beta$ are independent of $u$, but can be functions
of other moduli such as the masses ${\boldsymbol{m}}$, the dynamical scale
$\Lambda_{N_{f}}$ or the UV coupling $\tau_{\text{\tiny{UV}}}$. However, it
turns out that for the theories with fundamental matter they are independent
of the masses and only depend on the scale Marino:1998uy ; Manschot:2019pog .
They satisfy several constraints from holomorphy, RG flow, homogeneity and
dimensional analysis, and can in principle be fixed for any Lagrangian theory
from a computation in the $\Omega$-background Moore:2017cmm ; Marino:1998uy ;
Manschot:2019pog ; Closset:2021lhd .

###  4.3 Observables and contact terms

The observables in the topologically twisted theories are the point observable
or 0-observable $u$, as well as $d$-observables supported on a $d$-dimensional
submanifold of $X$. The $d$-observables are only non-vanishing if the
submanifold corresponds to a non-trivial homology class. For $b_{1}=0$, the
$d$-observables with $d$ odd therefore do not contribute.

To introduce the surface observable, let ${\boldsymbol{x}}\in
H_{2}(X,\mathbb{Q})$. Then the surface observable reads in terms of the UV
fields,

$I({\boldsymbol{x}})=\frac{1}{4\pi^{2}}\int_{{\boldsymbol{x}}}{\rm
Tr}\left[\psi\wedge\psi-\frac{1}{\sqrt{2}}\phi\,F\right].$ (89)

In the effective infrared theory, this operator becomes,

$\tilde{I}({\boldsymbol{x}})=\frac{i}{\sqrt{2}\pi}\int_{\boldsymbol{x}}\frac{1}{32}\frac{d^{2}u}{da^{2}}\,\psi\wedge\psi-\frac{\sqrt{2}}{4}\frac{du}{da}(F_{-}+D).$
(90)

Generating functions of correlation functions are obtained by inserting

$e^{p\,u/\Lambda_{N_{f}}^{2}+\tilde{I}({\boldsymbol{x}})/\Lambda_{N_{f}}}$
(91)

in the path integral. The surface observable leads to a change in the argument
of the sum over fluxes (84),

${\boldsymbol{z}}\to{\boldsymbol{z}}+\frac{{\boldsymbol{x}}}{2\pi\,\Lambda_{N_{f}}}\,\frac{du}{da},\qquad\bar{\boldsymbol{z}}\to\bar{\boldsymbol{z}}.$
(92)

and to analytically continue ${\boldsymbol{b}}$ (85) to the complex number by
setting ${\boldsymbol{b}}=({\boldsymbol{z}}-\bar{\boldsymbol{z}})/(2iy)$.

The inclusion of the surface observable also gives rise to a contact term
Witten:1994ev ; Moore:1997pc ; losev1998 , which in particular ensures that
the $u$-plane integrand is single-valued. For $0\leq N_{f}\leq 3$, the contact
term is $\exp({\boldsymbol{x}}^{2}\,G_{N_{f}})$ with mm1998 ; LoNeSha ;
Mari_o_1999

$G_{N_{f}}=-\frac{1}{24\,\Lambda_{N_{f}}^{2}}E_{2}\left(\frac{du}{da}\right)^{2}+\frac{1}{3\,\Lambda_{N_{f}}^{2}}\left(u+\frac{\Lambda_{3}^{2}}{64}\delta_{N_{f},3}\right),$
(93)

while for $N_{f}=4$ it is given by mm1998 ; Labastida:1998sk

$\displaystyle G_{N_{f}=4}$
$\displaystyle=-\frac{1}{24\,\Lambda_{4}^{2}}E_{2}\left(\frac{du}{da}\right)^{2}+\frac{u}{3\,\Lambda_{4}^{2}}E_{2}(\tau_{\text{\tiny{UV}}})+\frac{1}{18\,\Lambda_{4}^{2}}\left\llbracket
m_{1}^{2}\right\rrbracket E_{4}(\tau_{\text{\tiny{UV}}}).$ (94)

This expression (93) is valid for the theories with $N_{f}$ arbitrary
hypermultiplet masses. The reason for it is the following mm1998 ;
Marino:1998bm ; Marino:1998ru : $G$ is guaranteed to be
$\bar{\mathcal{Q}}$-closed and hence locally holomorphic. First, notice that
$\frac{\partial F}{\partial\tau_{0}}=\frac{u}{4}$, where
$\Lambda_{N_{f}}^{4-N_{f}}\eqqcolon e^{\pi i\tau_{0}}$ for the asymptotically
free theories ($N_{f}\leq 3$) and $\tau_{0}=\tau_{\text{\tiny{UV}}}$ for
$N_{f}=4$. The real part of the exponential prefactor of
$\Psi_{\boldsymbol{\mu}}^{J}$ can be added to $G$ to give a monodromy-
invariant contribution $\hat{G}$ which multiplies the intersection
${\boldsymbol{x}}^{2}$. From the action of a duality transformation on
$\hat{G}$ it can be inferred that

$G_{N_{f}}=-\frac{4i}{\pi\Lambda_{N_{f}}^{2}}\frac{\partial^{2}F}{\partial\tau_{0}^{2}}.$
(95)

The expressions (93) follow by direct computation. A more general scheme to
fix the contact terms is proposed in LoNeSha . Contact terms can also be
derived from the corresponding Whitham hierarchies TAKASAKI_1999 ; Mari_o_1999
. In the presence of surface observables, there are additional mixed contact
terms $\frac{\partial^{2}\mathcal{F}}{\partial\tau_{0}\partial m}$ for the
external fluxes $\\{{\boldsymbol{k}}_{j}\\}$ as encountered in
Manschot:2021qqe  for the $\mathcal{N}=2^{*}$ theory.

##  5 The $u$-plane integral

In this section, we set up the $u$-plane integral schematically given in (2),
and demonstrate that it is well-defined on the integration domain for any
${\boldsymbol{\mu}}$ with appropriate background fluxes. The case
${\boldsymbol{\mu}}=\bar{w}_{2}(X)/2$ and ${\boldsymbol{k}}_{j}=0$ was
analyzed in Moore:1997pc .

###  5.1 Definition of the integrand

As discussed in the previous sections, the $u$-plane integral on a closed
four-manifold $X$ with $(b_{1},b_{2}^{+})=(0,1)$ depends on many parameters.
We summarise:

  * •

The scale $\Lambda_{N_{f}}$ and masses
${\boldsymbol{m}}=(m_{1},\dots,m_{N_{f}})$ of the theory. See Section 2.

  * •

The magnetic winding numbers $n_{j}$, $j=1,\dots,N_{f}$. See Section 2.2.

  * •

The four-manifold $X$, in particular its signature $\sigma=\sigma(X)$, Euler
characteristic $\chi=\chi(X)$, period point $J$ and intersection form $Q$. See
Section 3.1.

  * •

The ’t Hooft flux ${\boldsymbol{\mu}}$, and the external fluxes
$\\{{\boldsymbol{k}}_{j}\\}=({\boldsymbol{k}}_{1},\dots,{\boldsymbol{k}}_{N_{f}})$.
See Section 3.3.

  * •

The fugacities for the point and surface observables $p$ and
${\boldsymbol{x}}$. See Section 4.3.

In terms of these parameters, the $u$-plane path integral reduces to the
following finite dimensional integral over
$\mathcal{F}_{N_{f}}({\boldsymbol{m}})$,

$\boxed{\begin{aligned}
&\Phi_{{\boldsymbol{\mu}},\\{{\boldsymbol{k}}_{j}\\}}^{J}(p,{\boldsymbol{x}},{\boldsymbol{m}},\Lambda_{N_{f}})=\\\
&\qquad\int_{\mathcal{F}_{N_{f}}({\boldsymbol{m}})}d\tau\wedge
d\bar{\tau}\,\nu(\tau;\\{{\boldsymbol{k}}_{j}\\})\,\Psi_{\boldsymbol{\mu}}^{J}(\tau,\bar{\tau},{\boldsymbol{z}},\bar{\boldsymbol{z}})\,e^{2pu/\Lambda_{N_{f}}^{2}+{\boldsymbol{x}}^{2}G_{N_{f}}}.\end{aligned}}$
(96)

We summarise the different elements on the rhs:

  * •

The integration domain $\mathcal{F}_{N_{f}}({\boldsymbol{m}})$ in (96) is
crucially the fundamental domain of the effective gauge coupling. As discussed
in Section 2.4, this domain requires new aspects compared to integration
domains for earlier discussions of $u$-plane integrals. The evaluation of
integrals over $\mathcal{F}_{N_{f}}({\boldsymbol{m}})$ will be discussed in
more detail in Section 6.

  * •

$\nu$ is the “measure factor” Witten:1995gf ; Moore:1997pc ; Marino:1998uy ;
LoNeSha ; Manschot:2021qqe

$\nu(\tau;\\{{\boldsymbol{k}}_{j}\\})=\mathcal{K}_{N_{f}}\,\frac{da}{d\tau}\,A^{\chi}B^{\sigma}\prod_{i,j=1}^{N_{f}}C_{ij}^{B({\boldsymbol{k}}_{i},{\boldsymbol{k}}_{j})}.$
(97)

It combines the topological couplings (88) and the couplings to the background
fluxes (83) with the Jacobian $\frac{da}{d\tau}$ of the change of variables
from $a$ to $\tau$.

$\mathcal{K}_{N_{f}}$ is an overall normalisation factor. For $N_{f}=0$, it is
fixed by matching to known Donaldson invariants. Since $\chi+\sigma=4$, there
is an ambiguity Moore:2017cmm

$(\mathcal{K}_{N_{f}},\alpha,\beta)\sim(\zeta^{-4}\mathcal{K}_{N_{f}},\zeta\alpha,\zeta\beta),$
(98)

with $\alpha$ and $\beta$ the $u$-independent prefactors in (88).

  * •

The function $\Psi^{J}_{\boldsymbol{\mu}}$ arises from the sum over $U(1)$
fluxes. It is a Siegel-Narain theta function (84) and discussed in detail in
Section 4.2. The elliptic parameter ${\boldsymbol{z}}$ of the Siegel-Narain
theta function reads

$\begin{split}{\boldsymbol{z}}&=\frac{{\boldsymbol{x}}}{2\pi\,\Lambda_{N_{f}}}\frac{du}{da}+\sum_{j=1}^{N_{f}}v_{j}{\boldsymbol{k}}_{j},\\\
\bar{\boldsymbol{z}}&=\sum_{j=1}^{N_{f}}\bar{v}_{j}{\boldsymbol{k}}_{j}.\end{split}$
(99)

  * •

Finally, $G_{N_{f}}$ is the contact term, discussed in more detail in Section
4.3.

While the path integral set up in Section 4 integrates the exponentiated
action over the local coordinates $a$ and $\bar{a}$, in (96) we have changed
variables to $\tau$ and $\bar{\tau}$. This change of variables
$(a,\bar{a})\to(\tau,\bar{\tau})$ is valid as long as the Jacobian is
nonsingular in the integration region. Since the coordinates $a$ and $\bar{a}$
are holomorphic and anti-holomorphic respectively, the Jacobian is diagonal
and the functional determinant accordingly reads
$\frac{da}{d\tau}\frac{d\bar{a}}{d\bar{\tau}}$. We thus need to show that
$\frac{da}{d\tau}$ is not singular away from isolated points in
$\mathcal{F}_{N_{f}}({\boldsymbol{m}})$, which in (96) we remove implicitly
from the integration domain.

Using $\frac{da}{du}=\frac{da}{du}\frac{du}{d\tau}$, we can study the singular
points in detail. First, it is shown in Aspman:2021vhs  that the singularities
of $\frac{du}{d\tau}$ are in one-to-one correspondence with the branch points.
In fact, both $\frac{du}{d\tau}=0$ and $\frac{du}{d\tau}=\infty$ are realised
as branch points in $\mathcal{N}=2$ SQCD. In the following Section 6, we
remove a small circle in $\mathcal{F}_{N_{f}}({\boldsymbol{m}})$ around the
branch points, and show that they do not give an extra contribution.
Furthermore, the solutions to $\frac{du}{da}=0$ are shown to be the Argyres-
Douglas (AD) points. We exclude them from the integration region, and study
their contribution also in Section 6. Finally, we know that
$\eta^{24}\propto\left(\frac{da}{du}\right)^{12}\Delta_{N_{f}}$ Aspman:2021vhs
, with $\eta$ the Dedekind eta-function as defined in (152). Since $\eta\neq
0$ and $\Delta_{N_{f}}$ does not have poles, we find that $\frac{da}{du}$
never vanishes. This agrees with the fact that $\frac{da}{du}$ is the period
of a holomorphic differential and therefore is never zero.

We conclude that the functional determinant is singular in $\mathbb{H}$
precisely at the branch points and AD points, however with the proper
exclusion of those as done in the following Section, it is non-singular and
the change of variables is well-defined. This furthermore conveniently solves
the problem that there is no natural integration region in $(a,\bar{a})$ space
Moore:1997pc .

###  5.2 Monodromy transformations of the integrand

We continue by explicitly verifying that the $u$-plane integral is single-
valued around the singular points of the moduli space. We find that this puts
a constraint on the magnetic winding numbers $n_{j}$, in addition to the
constraints on the background fluxes ${\boldsymbol{k}}_{j}$ discussed in
Section 3.3.

#### Monodromy around infinity

Let us determine how the $u$-plane integrand transforms under the monodromy
around infinity. As a function of the effective coupling $\tau$, the measure
factor (97) is proportional to
$\frac{da}{d\tau}\left(\frac{du}{da}\right)^{\frac{\chi}{2}}\Delta^{\frac{\sigma}{8}}$
times the product over the couplings $C_{ij}$. We take the monodromy at
infinity to be oriented as $u\to e^{2\pi i}u$ and $a\to e^{\pi i}a$, as in
Section 2.5. Then this path also encircles all singularities $u_{j}$, which
are the roots of the physical discriminant,
$\Delta=\prod_{j=1}^{N_{f}+2}(u-u_{j})$. We thus have that $\Delta\to e^{2\pi
i(N_{f}+2)}\Delta$, and hence

$\Delta^{\frac{\sigma}{8}}\to e^{\pi
i(N_{f}+2)\sigma/4}\Delta^{\frac{\sigma}{8}}$ (100)

Next, since $u\to e^{2\pi i}u$ and $a\to e^{\pi i}a$ we find $\frac{du}{da}\to
e^{\pi i}\frac{du}{da}$, and therefore

$\left(\frac{du}{da}\right)^{\frac{\chi}{2}}\to e^{\pi
i\chi/2}\left(\frac{du}{da}\right)^{\frac{\chi}{2}}.$ (101)

For $\frac{da}{d\tau}$ we have that $a\to e^{\pi i}a$, while $d\tau\to d\tau$,
and thus

$\frac{da}{d\tau}\to-\frac{da}{d\tau}.$ (102)

From (41) we recall that $w_{ij}\to w_{ij}+\delta_{ij}$, such that with the
definition (83) we find $C_{ij}\to e^{-\pi i\delta_{ij}}C_{ij}$. The couplings
$C_{ij}$ transform in the measure factor as

$\prod_{i,j=1}^{N_{f}}C_{ij}^{B({\boldsymbol{k}}_{i},{\boldsymbol{k}}_{j})}\to
e^{-\pi
i\sum_{j}{\boldsymbol{k}}_{j}^{2}}\prod_{i,j=1}^{N_{f}}C_{ij}^{B({\boldsymbol{k}}_{i},{\boldsymbol{k}}_{j})}.$
(103)

Combining (100), (101), (102), (103), and using $\chi=4-\sigma$, we obtain

$\nu\to-e^{\pi iN_{f}\sigma/4}e^{-\pi
i\sum_{j}{\boldsymbol{k}}_{j}^{2}}\,\nu.$ (104)

This phase for ${\boldsymbol{k}}_{j}=0$ can be checked directly by taking
$q$-expansions from the SW curves, for generic masses.

From (41) we recall that under the monodromy around infinity $v_{j}\to-
v_{j}-n_{j}$, and thus

${\boldsymbol{z}}\to-{\boldsymbol{z}}-\sum_{j=1}^{N_{f}}n_{j}{\boldsymbol{k}}_{j}.$
(105)

For the sum over fluxes we can now deduce using (159) that

$\begin{split}&\Psi_{\boldsymbol{\mu}}^{J}\Big{(}\tau+N_{f}-4,-{\boldsymbol{z}}-\sum_{j=1}^{N_{f}}n_{j}{\boldsymbol{k}}_{j}\Big{)}\\\
&\quad=e^{\pi i(N_{f}-4)({\boldsymbol{\mu}}^{2}-{\boldsymbol{\mu}}\cdot
K)}\,\Psi^{J}_{\boldsymbol{\mu}}\Big{(}\tau,-{\boldsymbol{z}}-\sum_{j=1}^{N_{f}}n_{j}{\boldsymbol{k}}_{j}+(N_{f}-4)({\boldsymbol{\mu}}-\tfrac{K}{2})\Big{)},\end{split}$
(106)

where we suppressed the dependence on the anti-holomorphic parts. Recall from
(59) that

$c_{1}(\mathcal{L}_{j})\equiv K-2{\boldsymbol{\mu}}\mod 2L.$ (107)

and as such we can express ${\boldsymbol{k}}_{j}=c_{1}(\mathcal{L}_{j})/2$ as

${\boldsymbol{k}}_{j}=\frac{K}{2}-{\boldsymbol{\mu}}+{\boldsymbol{\ell}}_{j},\qquad{\boldsymbol{\ell}}_{j}\in
L.$ (108)

We have then

${\boldsymbol{k}}^{2}_{j}=\frac{\sigma}{4}-K\cdot{\boldsymbol{\mu}}+{\boldsymbol{\mu}}^{2}-2{\boldsymbol{\mu}}\cdot{\boldsymbol{\ell}}_{j}\mod
2\mathbb{Z},$ (109)

where we used that $K$ is a characteristic vector of $L$, and
$K^{2}=\sigma\mod 8$. Using (159) and substitution of (108) in this
expression, (106) equals

$e^{\pi i(N_{f}-4)({\boldsymbol{\mu}}^{2}-{\boldsymbol{\mu}}\cdot
K)}\,\Psi^{J}_{\boldsymbol{\mu}}\Big{(}\tau,-{\boldsymbol{z}}-\sum_{j=1}^{N_{f}}n_{j}{\boldsymbol{\ell}}_{j}+(N_{f}-4+\sum_{j}n_{j})({\boldsymbol{\mu}}-\tfrac{K}{2})\Big{)}.$
(110)

Our aim is to write this as a phase times
$\Psi^{J}_{\boldsymbol{\mu}}(\tau,{\boldsymbol{z}})$. The constraints on the
winding numbers should be independent of ${\boldsymbol{\mu}}$ and
${\boldsymbol{k}}_{j}$, since the prepotential is. From (162), we therefore
get the first constraint

$\sum_{j}n_{j}=N_{f}\mod 2.$ (111)

Using identity (162),
$2{\boldsymbol{\mu}}^{2}-K\cdot{\boldsymbol{\mu}}\in\mathbb{Z}$ and
$4({\boldsymbol{\mu}}-\tfrac{K}{2})\in 2L$, this simplifies to

$\begin{split}&e^{\pi
i(N_{f}-4)({\boldsymbol{\mu}}^{2}-{\boldsymbol{\mu}}\cdot K)+2\pi
i{\boldsymbol{\mu}}\cdot\sum_{j}n_{j}{\boldsymbol{\ell}}_{j}-2\pi
i(N_{f}+\sum_{j}n_{j})({\boldsymbol{\mu}}^{2}-{\boldsymbol{\mu}}\cdot
K/2)}\Psi^{J}_{\boldsymbol{\mu}}(\tau,-{\boldsymbol{z}})\\\ &=-e^{\pi
iN_{f}({\boldsymbol{\mu}}^{2}-{\boldsymbol{\mu}}\cdot K)+2\pi
i{\boldsymbol{\mu}}\cdot\sum_{j}n_{j}{\boldsymbol{\ell}}_{j}-2\pi
i(N_{f}+\sum_{j}n_{j})({\boldsymbol{\mu}}^{2}-{\boldsymbol{\mu}}\cdot
K/2)}\,\Psi^{J}_{\boldsymbol{\mu}}(\tau,{\boldsymbol{z}})\\\ &=-e^{-\pi
iN_{f}{\boldsymbol{\mu}}^{2}+2\pi
i{\boldsymbol{\mu}}\cdot\sum_{j}n_{j}{\boldsymbol{\ell}}_{j}-2\pi
i\sum_{j}n_{j}({\boldsymbol{\mu}}^{2}-{\boldsymbol{\mu}}\cdot
K/2)}\,\Psi^{J}_{\boldsymbol{\mu}}(\tau,{\boldsymbol{z}}).\end{split}$ (112)

Finally using (109), we can express the phase in terms of
${\boldsymbol{k}}_{j}$,

${\bf
M}_{\infty}:\qquad\Psi^{J}_{\boldsymbol{\mu}}(\tau,{\boldsymbol{z}})\to-e^{-\pi
iN_{f}{\boldsymbol{\mu}}^{2}-\pi
i\sum_{j}n_{j}({\boldsymbol{k}}_{j}^{2}+{\boldsymbol{\mu}}^{2}-\sigma/4)}\,\Psi_{\boldsymbol{\mu}}^{J}(\tau,{\boldsymbol{z}}).$
(113)

By multiplying (104) with (113), we find

$\nu(\tau;\\{{\boldsymbol{k}}_{j}\\})\,\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,{\boldsymbol{z}})\to
e^{-\pi i\sum_{j}(n_{j}+1){\boldsymbol{k}}_{j}^{2}+\frac{\pi
i}{4}\sum_{j}(\sigma-4{\boldsymbol{\mu}}^{2})(n_{j}+1)}\,\nu(\tau;\\{{\boldsymbol{k}}_{j}\\})\,\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,{\boldsymbol{z}}).$
(114)

Combining (65) with (54), we have that
$4({\boldsymbol{k}}_{j}+{\boldsymbol{\mu}})^{2}\equiv\sigma\mod 8$ for every
$j=1,\dots,N_{f}$. We insert this into the second exponential of (114), such
that

${\bf
M}_{\infty}:\qquad\nu(\tau;\\{{\boldsymbol{k}}_{j}\\})\,\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,{\boldsymbol{z}})\to
e^{2\pi
i{\boldsymbol{\mu}}\sum_{j}(n_{j}+1){\boldsymbol{k}}_{j}}\,\nu(\tau;\\{{\boldsymbol{k}}_{j}\\})\,\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,{\boldsymbol{z}}),$
(115)

and the $u$-plane integrand is invariant under $T^{N_{f}-4}$ if and only if
${\boldsymbol{\mu}}\sum_{j}(n_{j}+1){\boldsymbol{k}}_{j}\in\mathbb{Z}$. Using
(108) and the fact that $K$ is a characteristic vector of $L$, we find

$n_{j}=1\mod 2$ (116)

for all $j=1,\dots,N_{f}$, which implies the above constraint (111).

#### Monodromy ${\bf M}_{j}$

Let us determine how the integrand transforms under the monodromy $M_{j}$
around the mass singularity $m_{j}/\sqrt{2}$. Since the mass singularity
corresponds to a singularity $u_{j}$ on the $u$-plane, we have that
$(u-u_{j})\to e^{2\pi i}(u-u_{j})$. This implies that
$\Delta=(u-u_{j})\prod_{i\neq j}^{2+N_{f}}(u-u_{i})\to e^{2\pi i}\Delta$, such
that $\Delta^{\frac{\sigma}{8}}\to e^{\pi
i\sigma/4}\Delta^{\frac{\sigma}{8}}$. The transformation of $\frac{da}{du}$
can be determined from (26): While $u\to u_{j}$, both $g_{2}$ and $g_{3}$
remain finite and nonzero (otherwise $u_{j}$ would be an AD point). This
implies that $\frac{g_{3}}{g_{2}}$ contains no factors of $(u-u_{j})$, and
thus $\frac{du}{da}\to\frac{du}{da}$. Similarly, we have that
$\frac{da}{d\tau}\to\frac{da}{d\tau}$. From (43) we finally have that
$w_{ik}\to w_{ik}+\delta_{ij}\delta_{ik}$. We combine

${\bf M}_{j}:\quad\nu\to e^{\pi i\sigma/4}e^{-\pi
i{\boldsymbol{k}}_{j}^{2}}\nu.$ (117)

For the monodromy around the mass singularity $m_{j}/\sqrt{2}$, we find for
$\Psi_{\boldsymbol{\mu}}^{J}$ with (159)

$\begin{split}&\Psi_{{\boldsymbol{\mu}}}^{J}(\tau+1,{\boldsymbol{z}}-{\boldsymbol{k}}_{j})=e^{\pi
i({\boldsymbol{\mu}}^{2}-{\boldsymbol{\mu}}\cdot
K)}\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,{\boldsymbol{z}}-{\boldsymbol{\ell}}_{j})\\\
&=e^{\pi i({\boldsymbol{\mu}}^{2}-{\boldsymbol{\mu}}\cdot K)+2\pi
iB({\boldsymbol{\mu}},{\boldsymbol{\ell}}_{j})}\,\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,{\boldsymbol{z}})\\\
&=e^{-\pi i\sigma/4+\pi
i{\boldsymbol{k}}_{j}^{2}}\,\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,{\boldsymbol{z}}).\end{split}$
(118)

The phases thus cancel precisely,

${\bf
M}_{j}:\quad\nu(\tau+1)\,\Psi_{{\boldsymbol{\mu}}}^{J}(\tau+1,{\boldsymbol{z}}-{\boldsymbol{k}}_{j})=\nu(\tau)\,\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,{\boldsymbol{z}}),$
(119)

without any constraints.

#### Monodromy ${\bf M}_{m}$

For the monopole singularity in $N_{f}=1$ we find that

$\displaystyle\Psi_{{\boldsymbol{\mu}}}^{J}(\tfrac{\tau}{-\tau+1},\tfrac{v{\boldsymbol{k}}_{1}+(n+1)/2\,\tau{\boldsymbol{k}}_{1}}{-\tau+1})=e^{\pi
i(n+1)\frac{v}{-\tau+1}{\boldsymbol{k}}_{1}^{2}+\pi
i\frac{(n+1)^{2}}{4}\frac{\tau}{-\tau+1}{\boldsymbol{k}}_{1}^{2}}\,(-1)^{(n+1){\boldsymbol{k}}_{1}\cdot
K/2}$ (120)
$\displaystyle\qquad\qquad\times\Psi_{{\boldsymbol{\mu}}+(n+1){\boldsymbol{k}}_{1}/2}^{J}(\tfrac{\tau}{-\tau+1},\tfrac{v{\boldsymbol{k}}_{1}}{-\tau+1}),$

where we have used (163). Then using (160), we arrive at

$\displaystyle\Psi_{{\boldsymbol{\mu}}}^{J}(\tfrac{\tau}{-\tau+1},\tfrac{v{\boldsymbol{k}}_{1}+(n+1)/2\,\tau{\boldsymbol{k}}_{1}}{-\tau+1})=(-\tau+1)^{b_{2}/2}(-\bar{\tau}+1)^{2}\,e^{\pi
i(n+1){\boldsymbol{k}}_{1}\cdot
K/2-(n+1)^{2}{\boldsymbol{k}}_{1}^{2}/4}\,e^{-\pi i\sigma/4}$ (121)
$\displaystyle\times\exp\\!\left[\pi
i\frac{(v+(n+1)/2)^{2}}{-\tau+1}{\boldsymbol{k}}_{1}^{2}\right]\Psi_{{\boldsymbol{\mu}}+(n+1){\boldsymbol{k}}_{1}/2}^{J}(\tau,{\boldsymbol{z}}).$

Since $\Psi_{\boldsymbol{\mu}}^{J}$ is required to transform to itself up to
an overall factor, we must demand that $(n+1){\boldsymbol{k}}_{1}/2\in L$.
Therefore for ${\boldsymbol{k}}_{1}\in L/2$, we find the requirement that
$n=-1\in\mathbb{Z}_{4}$. This simplifies the transformations considerably, and
we find

$\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,{\boldsymbol{z}})\to(-\tau+1)^{b_{2}/2}(-\bar{\tau}+1)^{2}e^{-\pi
i\sigma/4}e^{\pi
i{\boldsymbol{k}}_{1}^{2}\frac{v^{2}}{-\tau+1}}\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,{\boldsymbol{z}}).$
(122)

The ${\boldsymbol{k}}_{j}$-independent part of the measure factor transforms
precisely as under ${\bf M}_{j}$ (see (117)), as the same argument holds.
However, due to the transformation $\tau\to\frac{\tau}{-\tau+1}$, the measure
also picks up its modular weight $\frac{\sigma}{2}+1$. From (46) we
furthermore find the transformation of $C_{11}$, such that

$\nu(\tau,{\boldsymbol{k}}_{1})\to e^{\pi i\sigma/4}e^{-\pi
i{\boldsymbol{k}}_{1}^{2}\frac{v^{2}}{-\tau+1}}(-\tau+1)^{\frac{\sigma}{2}+1}\nu(\tau,{\boldsymbol{k}}_{1}),$
(123)

where we have already used $n=-1\in\mathbb{Z}_{4}$. If we multiply (122) and
(123) with $d\tau\wedge d\bar{\tau}$ (which has modular weight $(-2,-2)$),
then

${\bf M}_{m}:\quad d\tau\wedge
d\bar{\tau}\nu(\tau,{\boldsymbol{k}}_{1})\,\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,{\boldsymbol{z}})\to
d\tau\wedge
d\bar{\tau}\nu(\tau,{\boldsymbol{k}}_{1})\,\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,{\boldsymbol{z}}),$
(124)

where we have used $\sigma+b_{2}=2$. Thus, the $u$-plane integrand is also
invariant under ${\bf M}_{m}$.

For $N_{f}>1$ we find the same condition, namely that $n_{j}=-1\mod 4$ for all
$j$.

#### Monodromy ${\bf M}_{d}$

Given the relation (44), it is not necessary to explicitly check single-
valuedness of the integrand under this monodromy, as it is a product of the
above monodromies.
To conclude this section, let us stress the constraints for the winding number
$n_{j}$, such that the $u$-plane integral is invariant under all monodromies
in $N_{f}\leq 3$. To this end, we need to satisfy the constraints $n_{j}=1\mod
2$ (116) from ${\bf M}_{\infty}$, and $n_{j}=-1\mod 4$ for ${\bf M}_{m}$.
Since the latter is the stronger constraint, we require

$n_{j}=-1\mod 4,$ (125)

for all $j=1,\dots,N_{f}$.

##  6 Integration over fundamental domains

As discussed in Sections 2 and 5, $u$-plane integrals for massive
$\mathcal{N}=2$ theories with fundamental hypermultiplets include new aspects.
This section discusses how to evaluate such integrals (96). More concretely,
we aim to define and evaluate integrals of the form

$\mathcal{I}_{f}=\int_{\mathcal{F}({\boldsymbol{m}})}d\tau\wedge
d\bar{\tau}\,y^{-s}\,f(\tau,\bar{\tau}),$ (126)

with $s\leq 1$. The domain $\mathcal{F}({\boldsymbol{m}})$ is the fundamental
domain for the effective coupling constant as discussed in Section 2.4, and
$f$ a non-holomorphic function of weight $(2-s,2-s)$ arising from the
topologically twisted Yang-Mills theory. For $\mathcal{F}({\boldsymbol{m}})$ a
fundamental domain of a congruence subgroup, such integrals (126) have been
studied in the context of theta lifts of weakly holomorphic modular forms and
harmonic Maass forms Borcherds:1996uda ; bringmann2017regularized ;
bruinier2004two  as well as one-loop amplitudes in string theory Lerche:1988np
; Dixon:1990pc ; Harvey:1995fq .

We assume that the integrand $y^{-s}\,f(\tau,\bar{\tau})$ can be expressed as

$\partial_{\bar{\tau}}\widehat{h}(\tau,\bar{\tau})=y^{-s}\,f(\tau,\bar{\tau}),$
(127)

for a suitable function $\widehat{h}(\tau,\bar{\tau})$ using mock modular
forms. This was indeed the case in Korpas:2017qdo ; Korpas:2019ava ;
Manschot:2021qqe , and will be demonstrated for massive $\mathcal{N}=2$
theories with fundamental hypermultiplets. The integral $\mathcal{I}_{f}$ then
reads

$\mathcal{I}_{f}=-\int_{\partial\mathcal{F}({\boldsymbol{m}})}d\tau\,\widehat{h}(\tau,\bar{\tau}),$
(128)

with $\partial\mathcal{F}({\boldsymbol{m}})$ the boundary of
$\mathcal{F}({\boldsymbol{m}})$. We will carry this out evaluation in Part II
Aspman:2023ate .

There are a number of aspects to be addressed in order to evaluate integrals
over $\mathcal{F}({\boldsymbol{m}})$:

  1. 1.

Identifications of boundary components of $\mathcal{F}({\boldsymbol{m}})$ due
to monodromies on the $u$-plane.

  2. 2.

Contributions from the cusps, that is $\tau\to i\infty$ or
$\tau\to\gamma(i\infty)\in\mathbb{Q}$ for an element
$\gamma\in\text{PSL}(2,\mathbb{Z})$.

  3. 3.

Contributions from a singular point in the interior of
$\mathcal{F}({\boldsymbol{m}})$.

  4. 4.

Contributions from an elliptic point $p\in\mathbb{H}$ of
$\text{PSL}(2,\mathbb{Z})$.

  5. 5.

Branch points and branch cuts.

We will discuss these aspects 1.–5. in the following.
1\. Identifications
The modular transformation induced by monodromies identify components of the
boundary of the fundamental domain $\partial\mathcal{F}({\boldsymbol{m}})$
pairwise. Their contributions to the integral (128) vanish, which is, for
example, familiar from deriving valence formulas for modular forms
(Bruinier08, , Fig. 2). See Fig. 4 for an example.
2\. Cusps
Contributions near the cusps require a regularisation Moore:1997pc ;
Korpas:2019ava . Such regularisations have been developed in the context of
string amplitudes Lerche:1988np ; Dixon:1990pc ; Harvey:1995fq  and analytic
number theory Petersson1950 ; Bruinier08 ; 1603.03056 .

Let us first consider the cusp $\tau\to i\infty$. To regularise the
divergence, one introduces a cut-off $\text{Im}\,\tau=Y\gg 1$, and takes the
limit $Y\to\infty$ after evaluation. We require that $f$ near $i\infty$ has a
Fourier expansion of the form555Also if $f$ does not satisfy this requirement,
the integral can be regularised as explained in Korpas:2019ava ; 1603.03056 .
We do not need this regularisation for the correlators in this paper.

$f(\tau,\bar{\tau})=\sum_{m\gg-\infty,n\geq 0}c(m,n)\,q^{m}\,\bar{q}^{n}.$
(129)

Then the function $\widehat{h}$ has the form,

$\widehat{h}(\tau,\bar{\tau})=h(\tau)+2^{s}\int^{i\infty}_{-\bar{\tau}}\frac{f(\tau,-v)}{(-i(v+\tau))^{s}}dv,$
(130)

where $h(\tau)$ is a weakly holomorphic $q$-series, with expansion

$h(\tau)=\sum_{m\gg-\infty}d(m)\,q^{m}.$ (131)

The cusp $\tau\to i\infty$ then contributes

$\left[\mathcal{I}_{f}\right]_{\infty}=w_{\infty}\,d(0),$ (132)

with $d(0)$ the constant term of $h(\tau)$ (131), and $w_{\infty}$ the width
of the cusp $\mathcal{F}({\boldsymbol{m}})$ at $i\infty$. For $N_{f}\leq 3$,
$w_{\infty}$ is $4-N_{f}$ Aspman:2021vhs .

The other cusps can be treated in a similar fashion using modular
transformations. We label the $n_{c}$ cusps in $\mathcal{F}({\boldsymbol{m}})$
by $j=1,\dots,n_{c}$. If the cusp is on the horizontal axis at
$-\frac{d_{j}}{c_{j}}\in\mathbb{Q}$ with relative prime
$(c_{j},d_{j})\in\mathbb{Z}^{2}$, we can map the cusp to $i\infty$ by a
modular transformation

$\gamma_{j}=\begin{pmatrix}a_{j}&b_{j}\\\ c_{j}&d_{j}\end{pmatrix}.$ (133)

We let $\tau_{j}=\gamma_{j}\tau$. Then the holomorphic part $h_{j}(\tau_{j})$
of $(c_{j}\tau+d_{j})^{-2}\,\widehat{h}(\gamma\tau_{j},\gamma\bar{\tau}_{j})$
can be expanded for $\tau$ near $-\frac{d_{j}}{c_{j}}$ as

$h_{j}(\tau_{j})=\sum d_{j}(n)\,q_{j}^{n},\qquad q_{j}=e^{2\pi i\tau_{j}}.$
(134)

As a result, the cusp $j$ contributes

$\left[\mathcal{I}_{f}\right]_{j}=w_{j}\,d_{j}(0).$ (135)

3\. Singular points in the interior of $\mathcal{F}({\boldsymbol{m}})$
The integrand can be singular at a point $\tau_{\rm s}$ in the interior of
$\mathcal{F}({\boldsymbol{m}})$. Such singularities appear typically for
deformations of superconformal theories, such as the $\mathcal{N}=2^{*}$
theory and the $N_{f}=4$ theory, where the UV coupling
$\tau_{\text{\tiny{UV}}}$ gives rise to such a singularity Manschot:2021qqe ;
Aspman:2021evt . See Fig. 8 for an example. We require that the expansion of
$f$ near such a singularity reads,

$f(\tau,\bar{\tau})=\sum_{m\gg-\infty,n\geq 0}c_{s}(m,n)\,(\tau-\tau_{\rm
s})^{m}\,(\bar{\tau}-\bar{\tau}_{\rm s})^{n}.$ (136)

Then, the anti-derivative $\widehat{h}(\tau,\bar{\tau})$ has similar
expansion,

$\widehat{h}(\tau,\bar{\tau})=\sum_{m\gg-\infty,n\geq
0}d_{s}(m,n)\,(\tau-\tau_{\rm s})^{m}\,(\bar{\tau}-\bar{\tau}_{\rm s})^{n}.$
(137)

The contour integral for a small contour around $\tau_{\rm s}$,

$C_{\varepsilon}(\tau_{\rm s})=\left\\{\tau=\tau_{\rm
s}+\varepsilon\,e^{i\varphi},\varphi\in[0,2\pi)\right\\},$ (138)

is bounded for such a function. Moreover, in the limit $\varepsilon\to 0$, the
contour integral is finite. We define the “residue” of a non-holomorphic
function $g(\tau,\bar{\tau})$

$\begin{split}\underset{\tau=\tau_{\rm
s}}{\text{nRes}}\left[g(\tau,\bar{\tau})\right]&=\frac{1}{2\pi
i}\lim_{\varepsilon\to 0}\oint_{C_{\varepsilon}(\tau_{\rm
s})}g(\tau,\bar{\tau})\,d\tau.\end{split}$ (139)

For the expansion (136) this evaluates to

$\left[\mathcal{I}_{f}\right]_{s}=2\pi i\,\underset{\tau=\tau_{\rm
s}}{\text{nRes}}\left[\widehat{h}(\tau,\bar{\tau})\right]=d_{s}(-1,0),$ (140)

with $d_{s}(-1,0)$ the coefficient in the expansion (137).
4\. Elliptic points
For $\mathcal{N}=2$ QCD, AD points are the elliptic points of the duality
group, and lie on the boundary of $\mathcal{F}({\boldsymbol{m}})$. See Fig. 7
for an example. The elliptic points are $\alpha=e^{\pi i/3}$ and $i$, and
their images under $\text{PSL}(2,\mathbb{Z})$. Contour integrals around such
points can be regularised using a cut-off $\varepsilon$. We assume that the
anti-derivative $\widehat{h}$ has the following expansion near an elliptic
point $\tau_{e}$,

$\widehat{h}(\tau,\bar{\tau})=\sum_{m\gg-\infty,n\geq 0}d_{\rm
e}(m,n)\,(\tau-\tau_{\rm e})^{m}\,(\bar{\tau}-\bar{\tau}_{\rm e})^{n}.$ (141)

As a result, the boundary arc around $\tau_{\rm AD}$ in $\mathbb{H}$ is a
fraction of $2\pi$, which needs to be properly accounted for. These
neighbourhoods have an angle $\frac{2\pi}{k_{e}}$, with $k_{e}=2$ for
$\tau_{e}=i$, and $k_{e}=6$ for $\tau_{e}=\alpha$ Diamond . Furthermore, it is
important how many images of $\mathcal{F}$ in $\mathcal{F}({\boldsymbol{m}})$
coincide at the elliptic point. We denote this number by $n_{e}$. For
$\mathcal{N}=2$ SQCD, we found examples with $n_{e}=2$ and $4$ for
$\tau_{e}\sim\alpha$, while for $\tau_{e}\sim i$, $n_{e}=1$ Aspman:2021vhs .
The contribution from an elliptic point is then,

$\left[\mathcal{I}_{f}\right]_{e}=2\pi i\frac{n_{\text{e}}}{k_{\rm
e}}\,\underset{\tau=\tau_{\rm
e}}{\text{nRes}}\left[\widehat{h}(\tau,\bar{\tau})\right]=\frac{n_{\text{e}}}{k_{\rm
e}}\,d_{e}(-1,0),$ (142)

5\. Branch points and cuts
Branch points and cuts are a new aspect compared to previous analyses (see for
instance Fig. 4 and 5). We will demonstrate that their contribution vanishes
for the integrands of interest.

We assume that the integrand $f$ satisfies

$\widehat{h}(\tau,\bar{\tau})=(\tau-\tau_{{\rm bp}})^{n}\,g(\tau,\bar{\tau}),$
(143)

with $n\in\mathbb{Z}/2$ and $n\geq-1/2$, $g(\tau,\bar{\tau})$ being a real
analytic function near $\tau_{\rm bp}$. This assumption is satisfied for the
twisted Yang-Mills theories Aspman:2023ate . To treat this type of
singularity, we remove a $\delta$ neighbourhood and analyse the $\delta\to 0$
limit. Let $C_{\delta}$ be the contour

$C_{\delta}=\\{\tau_{\rm bp}+\delta\,e^{i\theta}\,|\,\theta\in(0,2\pi)\\}$
(144)

around $\tau_{\rm bp}$ with radius $\delta>0$. Therefore, on the contour
$|y^{-s}f|$ is bounded by

$|\widehat{h}|\leq\delta^{n}\,K$ (145)

for some $K>0$. The integral around the branch point therefore vanishes in the
limit,

$\begin{split}\mathcal{I}^{\rm bp}_{f}&=\lim_{\delta\to
0}\,\int_{C_{\delta}}\widehat{h}\,|d\tau|\leq\lim_{\delta\to
0}\,\int_{0}^{2\pi}\delta^{n}\,K\,\delta\,d\theta\\\ &=\lim_{\delta\to
0}\,2\pi K\delta^{n+1}=0.\end{split}$ (146)

The branch points necessarily give rise to branch cuts. For the purpose of
integration, we remove a neighbourhood with distance $r$ from the cut, and
take the limit $r\to 0$ after determining the integral. Since the value of the
integrand is finite near the branch cut, the contribution to the integral
vanishes.
Summary
Combining all the contributions discussed above, we find

$\mathcal{I}_{f}=\sum_{j=1}^{n}w_{j}\,d_{j}(0)+\sum_{s}d_{s}(-1,0)+\sum_{e}\,\frac{n_{\text{e}}}{k_{\rm
e}}\,d_{e}(-1,0).$ (147)

This formula generalises Moore:1997pc  for the pure $N_{f}=0$ theory on a
smooth four-manifold $X$ that admits a metric of positive scalar curvature,
(Korpas:2019cwg, , Equation (5.10)) for the pure theory on generic $X$,
(Manschot:2021qqe, , Equation (4.88)) for the $\mathcal{N}=2^{*}$ theory on
$X$, and Malmendier:2008db  for the massless $N_{f}=2$ and $N_{f}=3$ theories
on $X=\mathbb{CP}^{2}$.

###### Acknowledgements.

We are happy to thank Cyril Closset, Horia Magureanu, Greg Moore and Samson
Shatashvili for correspondence and discussions. JM thanks Greg Moore for
collaboration on related projects. JA is supported by the Government of
Ireland Postgraduate Scholarship Programme GOIPG/2020/910 of the Irish
Research Council. EF is supported by the TCD Provost’s PhD Project Award. JM
is supported by the Laureate Award 15175 “Modularity in Quantum Field Theory
and Gravity” of the Irish Research Council.

##  Appendix A Modular forms

In this Appendix, we collect some properties of modular forms for subgroups of
$\text{PSL}(2,\mathbb{Z})$. For further reading, see Bruinier08 ; ono2004 ;
Zagier92 ; Diamond ; schultz2015 ; koblitz1993 .

###  A.1 Theta functions and Eisenstein series

We make use of modular forms for the congruence subgroups $\Gamma_{0}(n)$ and
$\Gamma^{0}(n)$ of $\text{PSL}(2,\mathbb{Z})$. These subgroups are defined as

$\displaystyle\Gamma_{0}(n)=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\text{SL}(2,\mathbb{Z})\big{|}\,c\equiv 0\;\mod
n\right\\},$ (148) $\displaystyle\Gamma^{0}(n)=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\text{SL}(2,\mathbb{Z})\big{|}\,b\equiv 0\;\mod
n\right\\},$

and are related by conjugation with the matrix $\text{diag}(n,1)$. We
furthermore define the _principal congruence subgroup_ $\Gamma(n)$ as the
subgroup of $\text{SL}(2,\mathbb{Z})\ni A$ with $A\equiv\mathbbm{1}\mod n$. A
subgroup $\Gamma$ of $\text{SL}(2,\mathbb{Z})$ is called a congruence subgroup
if it contains $\Gamma(n)$ for some $n\in\mathbb{N}$.

The above introduced congruence subgroups host a number of interesting modular
forms. The Jacobi theta functions $\vartheta_{j}:\mathbb{H}\to\mathbb{C}$,
$j=2,3,4$, are defined as

$\begin{split}\vartheta_{2}(\tau)=\sum_{r\in\mathbb{Z}+\frac{1}{2}}q^{r^{2}/2},\quad\vartheta_{3}(\tau)=\sum_{n\in\mathbb{Z}}q^{n^{2}/2},\quad\vartheta_{4}(\tau)=\sum_{n\in\mathbb{Z}}(-1)^{n}q^{n^{2}/2},\end{split}$
(149)

with $q=e^{2\pi i\tau}$. These functions transform under
$T,S\in\text{SL}(2,\mathbb{Z})$ as

$\begin{split}S:\quad&\begin{array}[]{l}\vartheta_{2}(-1/\tau)=\sqrt{-i\tau}\vartheta_{4}(\tau),\\\
\vartheta_{3}(-1/\tau)=\sqrt{-i\tau}\vartheta_{3}(\tau),\\\
\vartheta_{4}(-1/\tau)=\sqrt{-i\tau}\vartheta_{2}(\tau),\end{array}\\\
T:\quad&\begin{array}[]{l}\vartheta_{2}(\tau+1)=e^{\frac{\pi
i}{4}}\vartheta_{2}(\tau),\\\ \vartheta_{3}(\tau+1)=\vartheta_{4}(\tau),\\\
\vartheta_{4}(\tau+1)=\vartheta_{3}(\tau).\end{array}\end{split}$ (150)

They furthermore satisfy the Jacobi abstruse identity

$\vartheta_{2}^{4}+\vartheta_{4}^{4}=\vartheta_{3}^{4}.$ (151)

The Dedekind eta function $\eta:\mathbb{H}\to\mathbb{C}$ is defined as the
infinite product

$\eta(\tau)=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n}),\quad q=e^{2\pi
i\tau}.$ (152)

It transforms under the generators of $\text{SL}(2,\mathbb{Z})$ as

$\displaystyle S:$ $\displaystyle\eta(-1/\tau)=\sqrt{-i\tau}\,\eta(\tau),$
(153) $\displaystyle T:$ $\displaystyle\eta(\tau+1)=e^{\frac{\pi
i}{12}}\,\eta(\tau),$

and relates to the Jacobi theta series as
$\eta^{3}=\frac{1}{2}\vartheta_{2}\vartheta_{3}\vartheta_{4}$.

#### Eisenstein series

We let $\tau\in\mathbb{H}$ and define $q=e^{2\pi i\tau}$. Then the Eisenstein
series $E_{k}:\mathbb{H}\to\mathbb{C}$ for even $k\geq 2$ are defined as the
$q$-series

$E_{k}(\tau)=1-\frac{2k}{B_{k}}\sum_{n=1}^{\infty}\sigma_{k-1}(n)\,q^{n},$
(154)

where $B_{k}$ are the Bernoulli numbers and $\sigma_{k}(n)=\sum_{d|n}d^{k}$ is
the divisor sum. For $k\geq 4$ even, $E_{k}$ is a modular form of weight $k$
for $\operatorname{SL}(2,\mathbb{Z})$. Any modular form for
$\text{SL}(2,\mathbb{Z})$ can be related to the Jacobi theta functions (149)
by

$E_{4}=\frac{1}{2}(\vartheta_{2}^{8}+\vartheta_{3}^{8}+\vartheta_{4}^{8}),\qquad
E_{6}=\frac{1}{2}(\vartheta_{2}^{4}+\vartheta_{3}^{4})(\vartheta_{3}^{4}+\vartheta_{4}^{4})(\vartheta_{4}^{4}-\vartheta_{2}^{4}).$
(155)

With our normalisation (154), the $j$-invariant can be written as

$j=1728\frac{E_{4}^{3}}{E_{4}^{3}-E_{6}^{2}}=256\frac{(\vartheta_{3}^{8}-\vartheta_{3}^{4}\vartheta_{4}^{4}+\vartheta_{4}^{8})^{3}}{\vartheta_{2}^{8}\vartheta_{3}^{8}\vartheta_{4}^{8}}.$
(156)

###  A.2 Siegel-Narain theta function

Let $L$ be an $n$-dimensional uni-modular lattice with signature $(1,n-1)$.
For the application to the $u$-plane integral, $n=b_{2}(X)$. Let $K$ be a
characteristic vector of $L$. Its defining property is
${\boldsymbol{l}}^{2}={\boldsymbol{l}}\cdot K\mod 2$ for every
${\boldsymbol{l}}\in L$. Furthermore, we have that ${\boldsymbol{\mu}}\in
L/2$.

We consider the Siegel-Narain theta function
$\Psi_{{\boldsymbol{\mu}}}^{J}:\mathbb{H}\times\mathbb{C}\to\mathbb{C}$
defined in the main text in (84). We repeat it here for convenience,

$\displaystyle\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,\bar{\tau},{\boldsymbol{z}},\bar{\boldsymbol{z}})$
$\displaystyle=e^{-2\pi y{\boldsymbol{b}}^{2}_{+}}\sum_{{\boldsymbol{k}}\in
L+{\boldsymbol{\mu}}}\partial_{\bar{\tau}}\left(4\pi
i\sqrt{y}B({\boldsymbol{k}}+{\boldsymbol{b}},J)\right)$ (157)
$\displaystyle\times(-1)^{B({\boldsymbol{k}},K)}q^{-{\boldsymbol{k}}_{-}^{2}/2}\bar{q}^{{\boldsymbol{k}}_{+}^{2}/2}e^{-2\pi
iB({\boldsymbol{z}},{\boldsymbol{k}}_{-})-2\pi
iB(\bar{\boldsymbol{z}},{\boldsymbol{k}}_{+})},$

where $J$ is a normalized positive vector in $L\otimes\mathbb{R}$,
${\boldsymbol{k}}_{+}=B({\boldsymbol{k}},J)\,J$,
${\boldsymbol{k}}_{-}={\boldsymbol{k}}-{\boldsymbol{k}}_{+}$ and
${\boldsymbol{b}}=\text{Im}({\boldsymbol{z}})/y$. The transformations under
the generators $S$ and $T$ of $\text{PSL}(2,\mathbb{Z})$ are most easily
determined if we shift ${\boldsymbol{\mu}}\to{\boldsymbol{\mu}}+K/2$. One
finds Korpas:2017qdo ; Korpas:2019cwg

$\begin{split}&S:\qquad\Psi^{J}_{{\boldsymbol{\mu}}+K/2}(-1/\tau,-1/\bar{\tau},{\boldsymbol{z}}/\tau,\bar{\boldsymbol{z}}/\bar{\tau})=-i(-i\tau)^{n/2}(i\bar{\tau})^{2}\\\
&\hskip 85.35826pt\times\,e^{-\pi i{\boldsymbol{z}}^{2}/\tau+\pi
iK^{2}/2}\,(-1)^{B({\boldsymbol{\mu}},K)}\,\Psi_{K/2}^{J}(\tau,\bar{\tau},{\boldsymbol{z}}-{\boldsymbol{\mu}},\bar{\boldsymbol{z}}-{\boldsymbol{\mu}}),\\\
&T:\qquad\Psi^{J}_{{\boldsymbol{\mu}}+K/2}(\tau+1,\bar{\tau}+1,{\boldsymbol{z}},\bar{\boldsymbol{z}})=\\\
&\hskip 85.35826pte^{\pi
i({\boldsymbol{\mu}}^{2}-K^{2}/4)}\,\Psi^{J}_{{\boldsymbol{\mu}}+K/2}(\tau,\bar{\tau},{\boldsymbol{z}}+{\boldsymbol{\mu}},\bar{\boldsymbol{z}}+{\boldsymbol{\mu}}).\end{split}$
(158)

Using these transformations, one finds for the periodicity in $\tau$,

$\begin{split}\Psi_{{\boldsymbol{\mu}}}^{J}(\tau+1,\bar{\tau}+1,{\boldsymbol{z}},\bar{\boldsymbol{z}})&=e^{\pi
i({\boldsymbol{\mu}}^{2}-B({\boldsymbol{\mu}},K))}\Psi_{{\boldsymbol{\mu}}}^{J}(\tau,\bar{\tau},{\boldsymbol{z}}+{\boldsymbol{\mu}}-K/2,\bar{\boldsymbol{z}}+{\boldsymbol{\mu}}-K/2)\\\
\end{split}$ (159)

and for $S^{-1}T^{-k}S=\left(\begin{smallmatrix}1&0\\\
k&1\end{smallmatrix}\right)$,

$\Psi^{J}_{\boldsymbol{\mu}}\\!\left(\tfrac{\tau}{k\tau+1},\tfrac{\bar{\tau}}{k\bar{\tau}+1},\tfrac{{\boldsymbol{z}}}{k\tau+1},\tfrac{\bar{\boldsymbol{z}}}{k\bar{\tau}+1}\right)=(k\tau+1)^{\frac{n}{2}}(k\bar{\tau}+1)^{2}e^{-\frac{\pi
ik{\boldsymbol{z}}^{2}}{k\tau+1}}e^{\frac{\pi
i}{4}kK^{2}}\Psi^{J}_{{\boldsymbol{\mu}}}(\tau,\bar{\tau},{\boldsymbol{z}},\bar{\boldsymbol{z}}).$
(160)

We furthermore list the following transformations for ${\boldsymbol{z}}$:

  * •

For the reflection ${\boldsymbol{z}}\to-{\boldsymbol{z}}$,

$\Psi^{J}_{\boldsymbol{\mu}}(\tau,\bar{\tau},-{\boldsymbol{z}},-\bar{\boldsymbol{z}})=-e^{2\pi
iB({\boldsymbol{\mu}},K)}\,\Psi^{J}_{\boldsymbol{\mu}}(\tau,\bar{\tau},{\boldsymbol{z}},\bar{\boldsymbol{z}}).$
(161)

  * •

For shifting ${\boldsymbol{z}}\to{\boldsymbol{z}}+{\boldsymbol{\nu}}$ with
${\boldsymbol{\nu}}\in L$,

$\Psi^{J}_{\boldsymbol{\mu}}(\tau,\bar{\tau},{\boldsymbol{z}}+{\boldsymbol{\nu}},\bar{\boldsymbol{z}}+{\boldsymbol{\nu}})=e^{-2\pi
iB({\boldsymbol{\nu}},{\boldsymbol{\mu}})}\,\Psi^{J}_{{\boldsymbol{\mu}}}(\tau,\bar{\tau},{\boldsymbol{z}},\bar{\boldsymbol{z}}).$
(162)

  * •

For shifting ${\boldsymbol{z}}\to{\boldsymbol{z}}+{\boldsymbol{\nu}}\tau$ with
${\boldsymbol{\nu}}\in L\otimes\mathbb{R}$,

$\Psi^{J}_{\boldsymbol{\mu}}(\tau,{\boldsymbol{z}}+{\boldsymbol{\nu}}\tau)=e^{2\pi
iB({\boldsymbol{z}},{\boldsymbol{\nu}})}q^{{\boldsymbol{\nu}}^{2}/2}(-1)^{-B({\boldsymbol{\nu}},K)}\,\Psi^{J}_{{\boldsymbol{\mu}}+{\boldsymbol{\nu}}}(\tau,\bar{\tau},{\boldsymbol{z}},\bar{\boldsymbol{z}}).$
(163)

We can restrict to ${\boldsymbol{\nu}}\in L/2$, if the characteristic
${{\boldsymbol{\mu}}+{\boldsymbol{\nu}}}$ is required to be in $L/2$.

##  Appendix B Class $\mathcal{S}$ representation

A different parametrisation of the SW curves (20) is the class $\mathcal{S}$
representation. This representation gives the SW differential in a canonical
form

$\lambda^{2}=p_{N_{f}}(z,u,\Lambda_{N_{f}},{\boldsymbol{m}})dz^{2},$ (164)

where the Laurent polynomials $p_{N_{f}}$ read (Gaiotto:2009hg, , Section 10)

$\displaystyle p_{0}$
$\displaystyle=\frac{\Lambda_{0}^{2}}{z^{3}}+\frac{2u}{z^{2}}+\frac{\Lambda_{0}^{2}}{z},$
(165) $\displaystyle p_{1}$
$\displaystyle=\frac{\Lambda_{1}^{2}}{z^{3}}+\frac{3u}{z^{2}}+\frac{2\Lambda_{1}m}{z}+\Lambda_{1}^{2},$
$\displaystyle p_{2}$
$\displaystyle=\frac{\Lambda_{2}^{2}}{z^{4}}+\frac{2\Lambda_{2}m_{1}}{z^{3}}+\frac{4u}{z^{2}}+\frac{2\Lambda_{2}m_{2}}{z}+\Lambda_{2}^{2}.$

The corresponding elliptic curves can be found as $x^{2}=z^{4}\,p_{N_{f}}$,
which is quartic for $N_{f}=0,1$ and cubic for $N_{f}=2$. By comparing
invariants of the SW curves (20) with those of the class $\mathcal{S}$ curve
(164), one finds the dictionary

$\displaystyle N_{f}=0:$ $\displaystyle u_{\mathcal{S}}=u_{\text{SW}},$ (166)
$\displaystyle N_{f}=1:$ $\displaystyle
u_{\mathcal{S}}=\frac{4}{3}u_{\text{SW}},\quad
m_{\mathcal{S}}=2m_{\text{SW}},$ $\displaystyle N_{f}=2:$ $\displaystyle
u_{\mathcal{S}}=4u_{\text{SW}},\quad m_{\mathcal{S},i}=4m_{\text{SW},i}.$

These relations merely amount to a rescaling of the parametrisation of the
Coulomb branch, and in particular leave its geometry invariant. For this
reason, we proceed above with using the SW curves (20).

##  Appendix C Winding numbers

In this Appendix, we discuss the winding numbers appearing in (10) in some
more detail. As mentioned in Section 2.2, generally the theory admits $N_{f}$
electric winding numbers and $N_{f}$ magnetic winding numbers. While in the
main text we set the electric winding numbers to zero, in this Appendix we
keep the possibility open for them to be nonzero.

The $\mathcal{N}=2$ supersymmetry algebra requires that the central charge $Z$
is a linear combination of conserved charges. The U(1) conserved charges
$S_{i}$ of the hypermultiplets must appear in $Z$ as follows Seiberg:1994aj ,

$Z=n_{m}a_{D}+n_{e}a+\sum_{j=1}^{N_{f}}S_{j}\frac{m_{j}}{\sqrt{2}}.$ (167)

We note that $Z$ is an inner product of periods
$(a_{D},a,\frac{1}{\sqrt{2}}{\boldsymbol{m}})$ and conserved charges
$(n_{m},n_{e},\boldsymbol{S})$. The periods are given by contour integrals

$a=\int_{\gamma_{1}}\lambda,\qquad a_{D}=\int_{\gamma_{2}}\lambda,$ (168)

where $\gamma_{1},\gamma_{2}$ generate $H_{1}(E,\mathbb{Z})$, and $\lambda$ is
a meromorphic 1-form, the SW differential. In massive SQCD, $\lambda$ has
poles with nonzero residues. If the contours $\gamma_{i}$ are deformed across
poles of $\lambda$, then $a$ and $a_{D}$ pick up contributions from the
residues. By invariance of (167), these jumps are linear combinations of the
masses,

$\sum_{j=1}^{N_{f}}\Delta S_{j}\frac{m_{j}}{\sqrt{2}},$ (169)

i.e. the global charges of the hypermultiplets are shifted as $S_{j}\to
S_{j}+\Delta S_{j}$. From the residue theorem, the shift in the periods gives

$2\pi i\sum_{i=1}^{N_{f}}n_{i}\text{Res}(\lambda,x_{i}),$ (170)

where $x_{i}$ are the $N_{f}$ poles of $\lambda$, and $n_{i}\in\mathbb{Z}$ are
the winding numbers of the contour deformation. Since both $a_{D}$ and $a$ are
given by contour integrals of the same differential, a deformation of either
$\gamma_{1}$ or $\gamma_{2}$ or both results in such a shift.

_Example._ In order to see how $n_{i}$ and $\Delta S_{j}$ are related, let us
study $N_{f}=2$ as an example. In $N_{f}=2$, the $S_{j}$ are integers for
fundamental particles and half-integers for monopoles Seiberg:1994aj . The two
poles of $\lambda$ are $x^{\pm}=\pm\frac{1}{8}\Lambda_{2}^{2}$, with residue
(Seiberg:1994aj, , (15.4))

$2\pi i\text{Res}(\lambda,x^{\pm})=\pm\frac{m_{1}+m_{2}}{2\sqrt{2}}.$ (171)

Then,

$\sum_{j=1}^{2}\Delta S_{j}\frac{m_{j}}{\sqrt{2}}=2\pi
i\sum_{i=1}^{2}n_{i}\text{Res}(\lambda,x_{i})=\sum_{j=1}^{2}\frac{n_{1}-n_{2}}{2}\frac{m_{j}}{\sqrt{2}},$
(172)

such that $\Delta S_{j}=\frac{n_{1}-n_{2}}{2}\in\mathbb{Z}/2$ for both
$j=1,2$. Constraints on the $n_{i}$ entail constraints on the change $\Delta
S_{j}$. For instance, if both $n_{1}$ and $n_{2}$ are odd, or if both are
even, then $\Delta S_{1}$ and $\Delta S_{2}$ are integers. In the main text of
this paper, we find such a constraint.

For generic $N_{f}\leq 3$, by construction of the SW curves the residues of
$\lambda$ are linear combinations of the masses

$2\pi
i\text{Res}(\lambda,x_{i})=\sum_{j=1}^{N_{f}}l_{ij}\frac{m_{j}}{\sqrt{2}},$
(173)

with $i=1,\dots,N_{f}$, and $l_{ij}\in\mathbb{Z}/2$ (Seiberg:1994aj, ,
(17.1)). Then we can compute

$\sum_{j=1}^{N_{f}}\Delta S_{j}\frac{m_{j}}{\sqrt{2}}=2\pi
i\sum_{i=1}^{N_{f}}n_{i}\text{Res}(\lambda,x_{i})=\sum_{i=1}^{N_{f}}\sum_{j=1}^{N_{f}}n_{i}l_{ij}\frac{m_{j}}{\sqrt{2}}=\sum_{j=1}^{N_{f}}\frac{m_{j}}{\sqrt{2}}\sum_{i=1}^{N_{f}}l_{ij}n_{i}.$
(174)

We find that the change in the abelian global charges $\Delta S_{j}$ is given
by a half-integral linear combination of winding numbers,

$\Delta S_{j}=\sum_{i=1}^{N_{f}}l_{ij}n_{i}.$ (175)

In order to relate the winding numbers $n_{i}$ to those for the periods
$a_{i}$ and $a_{D,i}$ as found in Ohta_1997 , we can expand

${\bf M}_{\infty}^{\text{Ohta}}:\quad a_{D}\to-
a_{D}+(4-N_{f})a-\frac{1}{\sqrt{2}}\left(\frac{n^{a_{D}}_{N_{f}}}{n^{a}_{N_{f}}}-\frac{4-N_{f}}{2}\right)\sum_{j=1}^{N_{f}}n_{j}^{a}m_{j},$
(176)

with $n_{j}^{a}$ being the coefficients of $-\frac{1}{2\sqrt{2}}m_{j}$ in the
constant term of $a$, and similarly $n_{j}^{a_{D}}$ those for $a_{D}$
(Ohta_1997, , (5.2))666In Ohta_1997 , the $n_{j}^{a}$ are called $n_{j}$ and
$n_{j}^{a_{D}}$ are called $n_{j}^{\prime}$.. We are aiming to compare this to
(39),

${\bf M}_{\infty}:\quad a_{D}\to-
a_{D}+(4-N_{f})a-\frac{1}{\sqrt{2}}\sum_{j=1}^{N_{f}}n_{j}m_{j}.$ (177)

Since in (176) we divide and multiply by $n_{j}^{a}$, we cannot directly
relate the two expressions by setting $n_{j}^{a}=0$. However, if we relate
first

$n_{j}^{a}=\frac{n^{a}_{N_{f}}}{n^{a_{D}}_{N_{f}}}n_{j}^{a_{D}},\qquad
j=1,\dots N_{f}-1,$ (178)

then we find

$\displaystyle{\bf M}_{\infty}^{\text{Ohta}}:\quad a_{D}\to$ $\displaystyle-
a_{D}+(4-N_{f})a-\frac{1}{\sqrt{2}}\sum_{j=1}^{N_{f}}\left(n_{j}^{a_{D}}-\frac{4-N_{f}}{2}n_{j}^{a}\right)m_{j}.$
(179)

In this case, it is well-defined to set $n_{j}^{a}=0$, such that the action of
${\bf M}_{\infty}^{\text{Ohta}}$ and ${\bf M}_{\infty}$ coincide for
$n_{j}=n_{j}^{a_{D}}$, as anticipated.

The $N_{f}-1$ conditions (178) reproduce the constraints on the winding
numbers in (Ohta_1997, , (5.3)): For $N_{f}=1$, the condition is empty. For
$N_{f}=2$, it agrees precisely with Ohta. For $N_{f}=3$ finally, the
$N_{f}-1=2$ equations (178) are equivalent to those found in Ohta_1997 . It
seems however that our result $n_{j}^{a}=0,n_{j}^{a_{D}}=-1$ (125) is merely
one particular of the infinitely many solutions to the geometric constraint
(178).777One other simple solution is $n_{j}^{a}=n$ and
$n_{j}^{a_{D}}=\tilde{n}$, with arbitrary $n,\tilde{n}\in\mathbb{Z}$. Thus it
appears that the formulation of the theory on a compact four-manifold
introduces further constraints. It would be interesting to understand whether
it is possible to introduce non-vanishing electric winding numbers in the
$u$-plane integral, and whether this leads to different correlation functions.

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