Yutaka Yoshida

Department of Physics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan

#  ’t Hooft surface operators in five dimensions and elliptic Ruijsenaars
operators

###### Contents

  1. 1 Introduction
  2. 2 Monopole surface operator in 5d $\mathcal{N}=1$ SUSY gauge theories
    1. 2.1 5d SUSY gauge theory from 10d super Yang-Mills theory
    2. 2.2 Monopole surface operators as SUSY indices on $T^{2}\times\mathbb{R}^{3}$
  3. 3 SUSY localization of surface operator
    1. 3.1 Zero locus of Q-exact term in 5d $\mathcal{N}=1$ SUSY gauge theory
    2. 3.2 Gauge fixing term and BRST transformation
    3. 3.3 Evaluation of one-loop determinants
  4. 4 Monopole bubbling effects via branes and elliptic genera
    1. 4.1 Brane construction for t’ Hooft surface operators and monopole bubbling
    2. 4.2 Elliptic genera for monopole bubbling effects
  5. 5 Deformation quantization and elliptic Ruijsenaars operators
  6. 6 Products of ’t Hooft surface operators and monopole bubbling
    1. 6.1 Surface operator $S_{(1,0,\cdots,0,-1)}$
    2. 6.2 Surface operator $S_{(1,1,0,\cdots,0,-1,-1)}$
  7. 7 Discussion
  8. A Gamma matrices
  9. B One-loop determinants in 5d $\mathcal{N}=1^{*}$ gauge theory
  10. C The differential operator $D_{10}$

##  1 Introduction

An ’t Hooft loop operator [4] in a four-dimensional (4d) gauge theory is an
example of disorder operator defined by a boundary condition of the gauge
field with a prescribed singularity along the loop. In supersymmetric (SUSY)
gauge theories, BPS analogues of disorder operators preserving a part of the
supersymmetry have interesting properties in a variety of situations.

In three dimensions, BPS monopole operators defined by singular boundary
conditions at a point in the spacetime [5, 6] play crucial roles in the study
of quantum corrections and dualities in supersymmetric gauge theories. In 3d
$\mathcal{N}=4$ non-abelian gauge theories, the moduli space of Coulomb branch
vacua receives non-perturbative corrections from ’t Hooft-Polyakov monopoles.
In general, it is difficult to exactly evaluate the non-perturbative
corrections to the hyperKähler metric of the moduli space of the Coulomb
branch vacua. An algebra consisting of the vacuum expectation values (vevs) of
the monopole operators and Coulomb branch scalars called Coulomb branch chiral
ring conjecturally gives the coordinate ring of the moduli space of Coulomb
branch vacua as an algebraic variety. Here an important point is that the
quantum corrections to a Coulomb branch chiral ring is easier to handle than
the corrections to the hyperKähler metric. In fact, the Coulomb branch chiral
rings in quiver gauge theories of ADE-type were determined in [7] with certain
assumptions; abelianization and the mirror symmetry of abelian gauge theories.
In [2], and also [3], the exact computation of vevs of monopole operators was
developed in terms of supersymmetric localization methods. Then it was found
that the algebras of monopole operators and Coulomb branch obtained by the
supersymmetric localization formula agrees with the Coulomb branch chiral
rings and their deformation quantizations in [7, 8, 9]

In four dimensions, BPS ’t Hooft loops have attracted a lot of attention from
the mathematical physics viewpoints for more than a decade. For example, there
exists the S-duality between Wilson loops and ’t Hooft loops in 4d
$\mathcal{N}=4$ super Yang-Mills theory [10, 11]. In AGT correspondence [12],
’t Hooft loops in 4d $\mathcal{N}=2$ gauge theories belonging to the class
$\mathcal{S}$ [13] conjecturally agree with Verlinde loops in Toda theories
[14, 15, 16, 17, 1]. On the $\Omega$-background, the algebra of the ’t Hooft
loops, Wilson loops, Dyonic loops and equivalently the algebra of Verlinde
loops gives a deformation quantization of Coulomb branch of 4d $\mathcal{N}=2$
gauge theory on $S^{1}\times\mathbb{R}^{3}$ and also gives a deformation
quantization of the moduli space of flat connections on a punctured Riemann
surface [18, 19, 20, 1]. In this story, supersymmetric localization method
provides a powerful method to checks the correspondence between BPS ’t Hooft
loops and Verlinde operators.

Although quantum field theories in five dimensions are non-renormalizable by
the power counting argument, some class in 5d $\mathcal{N}=1$ supersymmetric
gauge theories have non-trivial fixed points in the renormalization group flow
and make sense as quantum field theories [21, 22]. Supersymmetric localization
formulas of partition functions and supersymmetric indices on five-dimensional
manifolds give non-trivial quantitative tests of predictions for quantum
aspects of 5d $\mathcal{N}=1$ supersymmetric theories. In this paper we
introduce a BPS analogue of disorder operators in 5d $\mathcal{N}=1$
supersymmetric gauge theories on $T^{2}\times\mathbb{R}^{3}$ by imposing
boundary conditions that has a Dirac monopole singularity extending along a
two dimensional torus $T^{2}$. The BPS disorder operators which we call as BPS
’t Hooft surface operators are five-dimensional analogues of BPS ’t Hooft
loops in 4d $\mathcal{N}=2$ gauge theories on $S^{1}\times\mathbb{R}^{3}$, and
BPS monopole operators in 3d $\mathcal{N}=4$ gauge theories on
$\mathbb{R}^{3}$. We evaluate the vevs of ’t Hooft surface operators by
supersymmetric localization techniques, and study their properties.

This article is organized as follows. In Section 2, we introduce the BPS ’t
Hooft surface operators in the path integral formalism by imposing certain
boundary conditions for the fields in the five-dimensional supermultiplets. In
Section 3, we study the supersymmetric localization computation of vevs of ’t
Hooft surface operators and evaluate the classical and the one-loop
contribution to the vevs of ’t Hooft surface operators. In the path integral,
there exists non-perturbative corrections coming from the monopole bubbling
effect; the path integral over the moduli space of certain monopole solutions.
In Section 4, we evaluate monopole bubbling effect in the vacuum expectation
values (vevs) of ’t Hooft surface operators in terms of D-brane realizations
of monopole bubbling. The monopole bubbling effects contributing to the vevs
of ’t Hooft surface operators are given by elliptic genera of the low energy
world volume theories on D-branes. In Section 5, we define the products of
vevs of ’t Hooft surface operators in terms of the Moyal product and also
define the deformation quantization of vevs of the surface operators in terms
of the Weyl-Wigner transform. We find that the deformation quantization of
surface operators in 5d $\mathcal{N}=1^{*}$ gauge theory coincides with
simultaneously commuting difference operators appearing in an integrable
system, called elliptic Ruijsenaars operators. In Section 6, we study the
algebra of surface operators with respect to the Moyal product in 5d
$\mathcal{N}=1^{*}$ gauge theory. In Section 7, we discuss our results and
future problems.

##  2 Monopole surface operator in 5d $\mathcal{N}=1$ SUSY gauge theories

In this section we explain the decomposition of the ten-dimensional (10d)
vector multiplet on the spacetime $\mathbb{R}^{10}$ to a 5d vector multiplet
and a 5d hypermultiplet in the adjoint representation on the spacetime
$\mathbb{R}^{5}$ by the dimensional reduction. Next we define the vevs of an
’t Hooft surface operator as a supersymmetric indices on
$T^{2}\times\mathbb{R}^{3}$.

###  2.1 5d SUSY gauge theory from 10d super Yang-Mills theory

The convention of the gauge covariant derivative is $D_{M}=\partial_{M}+{\rm
i}A_{M}$. The indices $\mu,\nu,\cdots\in\\{0,1,2,3,4\\}$ express the
subscripts for the five-dimensional spacetime. The indices
$M,N,\cdots\in\\{0,1,\cdots,9\\}$ label the subscripts for the ten-dimensional
spacetime. The spacetimes $\mathbb{R}^{5}$ and $\mathbb{R}^{10}$ have the
Euclidean signature metrics $\delta_{\mu\nu}$ for $\mu,\nu=0,1,2,3,4$ and
$\delta_{MN}$ for $M,N=0,1,\cdots,9$, respectively. $x^{M}$ for
$M=0,1,\cdots,9$ denotes the coordinate of $\mathbb{R}^{10}$. The definition
and properties of the $16\times 16$ gamma matrices $\Gamma^{M}$ and
$\tilde{\Gamma}^{M}$ are summarized in Appendix A.

By the dimensional reduction, the gauge field $A_{M}$ for $M=0,1,\cdots,9$ and
the gaugino $\Psi$ in the 10d maximal super Yang-Mills theory are decomposed
to the 5d $\mathcal{N}=1$ supermultiplets as follows. In the dimensional
reduction in the directions $x^{i}$ for $i=5,6,7,8,9$, the five dimensional
gauge fields $A_{\mu}$ for $\mu=0,1,2,3,4$, an adjoint scalar $\sigma=A_{9}$
and a fermion $\lambda=\frac{1}{2}(1-\Gamma^{5678})\Psi$ form a 5d
$\mathcal{N}=1$ vector multiplet. Scalars $\Phi_{i}:=A_{i}$ for $i=5,6,7,8$
and a fermion $\psi:=\frac{1}{2}(1+\Gamma^{5678})\Psi$ form a 5d
$\mathcal{N}=1$ hypermultiplet in the adjoint representation.

The action of the 5d $\mathcal{N}=1$ super Yang-Mills theory is given by

$\displaystyle S_{\text{vec}}$
$\displaystyle=\frac{1}{g^{2}}\int_{\mathbb{R}^{5}}d^{5}x\mathrm{Tr}\Bigl{[}\frac{1}{2}F^{2}_{\mu\nu}+(D_{\mu}\sigma)^{2}-\lambda\Gamma^{\mu}D_{\mu}{\lambda}+{\rm
i}\lambda\Gamma^{9}[{\lambda},\sigma]\Bigr{]}.$ (2.1)

Here $g$ is the Yang-Mills coupling constant. A symbol $\mathrm{Tr}$ is a
trace taken over the Lie algebra $\mathfrak{g}$ of the gauge group $G$. The
action of the 5d hypermultiplet in the adjoint representation is given by

$\displaystyle S_{\text{hyp}}$
$\displaystyle=\int_{\mathbb{R}^{5}}d^{5}x\mathrm{Tr}\Bigl{[}(D_{\mu}\Phi_{i})^{2}-\frac{1}{2}[\Phi_{i},\Phi_{j}]^{2}-[\sigma,\Phi_{i}]^{2}$
$\displaystyle\quad\qquad\qquad\qquad-\psi\Gamma^{\mu}D_{\mu}\psi-{\rm
i}\psi\Gamma^{9}[\sigma,\psi]-{\rm i}\psi\Gamma^{i}[\Phi_{i},\psi]\Bigr{]}.$
(2.2)

We can introduce a fugacity (mass) $m_{\text{ad}}$ for $U(1)$ flavor symmetry
of the adjoint hypermultiplet, which breaks the $\mathcal{N}=2$ supersymmetry
to an $\mathcal{N}=1$ supersymmetry in five dimensions. In particular, the 5d
$\mathcal{N}=1$ supersymmetry obtained by the mass deformation of 5d
$\mathcal{N}=2$ supersymmetry is called 5d $\mathcal{N}=1^{*}$ supersymmetry.

To apply supersymmetric localization method, we need at least one off-shell
supercharge. For the supersymmetric gauge theories, by adding the action of
auxiliary fields $\sum_{i=1}^{7}K_{i}^{2}$ to the above actions (2.1) and
(2.2), one can keep some of the supercharges without using the equation of
motion, i.e. off-shell level [23]. To write a supersymmtry transformation, we
choose a supersymmetric variation parameter $\varepsilon$ and introduce
parameters $\nu_{j}$ for $j=1,\cdots,7$ given by

$\displaystyle\varepsilon$
$\displaystyle=\frac{1}{\sqrt{2}}({1},0^{7},1,0^{7})=\frac{1}{\sqrt{2}}({1},\underbrace{0,\cdots,0}_{7},1,\underbrace{0,\cdots,0}_{7})\,,$
(2.3) $\displaystyle\nu_{j}$ $\displaystyle=\left\\{\
\begin{array}[]{ll}\displaystyle\Gamma^{8,j+4}\varepsilon&\text{ for
}j=1,2,3,\\\ \displaystyle\Gamma^{89}\varepsilon&\text{ for }j=4,\\\
\Gamma^{8,j-4}\varepsilon&\text{ for }j=5,6,7.\end{array}\right.$ (2.7)

Then the actions (2.1) and (2.2) with the action of auxiliary fields are
invariant under the following off-shell supersymmetry transformation:

$\displaystyle{\sf Q}\cdot A_{M}$ $\displaystyle=\varepsilon\Gamma_{M}\Psi\,,$
(2.8) $\displaystyle{\sf Q}\cdot\Psi$
$\displaystyle=\frac{1}{2}\Gamma^{MN}F_{MN}\varepsilon+{\rm i}K^{i}\nu_{i}\,,$
(2.9) $\displaystyle{\sf Q}\cdot K_{i}$ $\displaystyle={\rm
i}\nu_{i}\Gamma^{M}D_{M}\Psi\,.$ (2.10)

Here the action of the off shell supercharge ${\sf Q}$ is defined by $[{\sf
Q},X]$ (resp. $\\{{\sf Q},X\\}$) for the Grassmann even fields $X=A_{M},K_{i}$
(resp. the Grassmann odd field $X=\Psi$). The square of the SUSY
transformation generates

$\displaystyle{\sf Q}^{2}\cdot A_{M}$ $\displaystyle=-2F_{\bar{z}M}\,,$ (2.11)
$\displaystyle{\sf Q}^{2}\cdot\Psi$ $\displaystyle=-2D_{\bar{z}}\Psi\,,$
(2.12) $\displaystyle{\sf Q}^{2}\cdot K_{i}$
$\displaystyle=-2D_{\bar{z}}K_{i}\,.$ (2.13)

Here $z=x^{4}+{\rm i}x^{0}$ and $\bar{z}=x^{4}-{\rm i}x^{0}$. If we replace
the representation for the fields in the adjoint hypermultiplet by a
symplectic representation $\mathcal{R}\oplus\overline{\mathcal{R}}$ of the
gauge group $G$, we obtain the supersymmetry transformation and the action for
the hypermultiplet in a symplectic representation
$\mathcal{R}\oplus\overline{\mathcal{R}}$.

###  2.2 Monopole surface operators as SUSY indices on
$T^{2}\times\mathbb{R}^{3}$

We consider a twisted compactification of 5d supersymmetric gauge theories on
the two-dimensional torus $T^{2}$ defined by

$\displaystyle T^{2}$ $\displaystyle:=\\{(x^{4},x^{0})|\,x^{4}+{\rm
i}x^{0}\equiv x^{4}+{\rm i}x^{0}+2\pi\equiv x^{4}+{\rm i}x^{0}+2\pi\tau\\}$
$\displaystyle=\\{(z,\bar{z})|\,z\equiv z+2\pi\equiv z+2\pi\tau\\}.$ (2.14)

Here $\tau=\tau_{1}+i\tau_{2}$ is the moduli of the torus $T^{2}$. A symbol
“$\equiv$” denotes the identification. We also introduce another coordinate
$0\leq s,t\leq 2\pi$ of $T^{2}$ defined by

$\displaystyle x^{4}=s+\tau_{1}t,\quad x^{0}=\tau_{2}t.$ (2.15)

We impose the following twisted boundary condition along the $x^{0}$ and
$x^{4}$ directions for the fields:

$\displaystyle X(z)\equiv X(z+2\pi),\quad X(z+2\pi\tau)\equiv e^{-2\pi{\rm
i}\epsilon({\sf J}_{3}+{\sf I}_{3})}\prod_{f}e^{-2\pi{\rm i}m_{f}{\sf
F}_{f}}X(z)$ (2.16)

where $X(z)$ denotes a field in the 5d supermultiplets. Here we have
suppressed the coordinates dependence on $x^{i}$ for $i=1,2,3$ and $\bar{z}$
to shorten the notation.

A symbol ${\sf J}_{3}$ denotes a generator of the rotation in the
$x^{1},x^{2}$-plane around the origin. When we emphasize the presence of a
fugacity $\epsilon$ called an $\Omega$-background parameter, we write a
spacetime $\mathbb{R}^{3}$ in the $x^{1,2,3}$-directions as
$\mathbb{R}^{2}_{\epsilon}\times\mathbb{R}$. A symbol ${\sf I}_{3}$ is a
generator of $\mathfrak{u}(1)\subset\mathfrak{su}(2)_{H}$, where
$\mathfrak{su}(2)_{H}$ is the Lie algebra of the R-symmetry group for the 5d
$\mathcal{N}=1$ supersymmetry algebra. ${\sf F}_{f}$’s are generators
(charges) of the Cartan subalgebra of the flavor symmetry group acting on the
hypermultiplets in the representation
$\mathcal{R}\oplus\overline{\mathcal{R}}$. $m_{f}$’s are fugacities for these
generators. If $\mathcal{R}$ is the adjoint representation, we have a single
flavor fugacity $m_{\text{ad}}:=m_{1}$.

We introduce the vev of a BPS ’t Hooft surface operator $S_{{\bm{B}}}$ for 5d
$\mathcal{N}=1$ supersymmetric gauge theory on $T^{2}\times\mathbb{R}^{3}$ as
a supersymmetric index:

$\displaystyle\langle S_{\bm{B}}\rangle$
$\displaystyle:=\mathrm{Tr}_{\mathcal{H}_{\bm{B}}}(-1)^{\sf F}e^{-2\pi{\sf
H}}e^{2\pi{\rm i}\epsilon({\sf J}_{3}+{\sf I}_{3})}\prod_{f}e^{2\pi{\rm
i}m_{f}{\sf F}_{f}}\,.$ (2.17)

To be more precise, the vev is defined by the path integral in the presence of
singular monopoles, which will mentioned later. A magnetic charge ${\bm{B}}$
is an element of the coweights lattice $\Lambda_{\text{cw}}$ of the Lie
algebra $\mathfrak{g}$. Since a Weyl group action of ${\bm{B}}$ defines a same
operator, we may assume ${\bm{B}}$ in $S_{\bm{B}}$ as a dominant coweight.
$\mathcal{H}_{\bm{B}}$ is the Hilbert space of the supersymmetric theory on
$S^{1}_{s}\times\mathbb{R}^{3}$, where $S^{1}_{s}$ is the circle in the
$s$-direction. A symbol ${\sf F}$ denotes the Fermion number operator. We take
a coordinate $t$ as the time direction and define the Hamiltonian ${\sf H}$ by
the generator of translation in the direction $t$. In the path integral
formalism, the vev of an ’t Hooft surface operator is given by

$\displaystyle\langle
S_{\bm{B}}\rangle=\int_{\text{B.C.}}{\mathcal{D}A\mathcal{D}\Psi\mathcal{D}K}\exp\left(-S_{\text{vec}}-S_{\text{hyp}}-S_{\text{b.d}}\right)\,.$
(2.18)

Here we add a boundary term $S_{\text{b.d}}$ to regularize the singularity
coming from the Dirac monopole, see (3.30). In the path integral, the twisted
boundary condition (2.16) is given by the shift of the time derivative:

$\displaystyle\partial_{t}\mapsto\partial_{t}-{\rm i}\epsilon({\sf J}_{3}+{\sf
I}_{3})-{\rm i}\sum_{f}m_{f}{\sf F}_{f}.$ (2.19)

Then the fugacities are given by background gauge field in the path integral
formalism. In the rest of this paper, we include these background gauge fields
in the definition of covariant derivative in the $t$-direction, i.e.,
$D_{t}=\partial_{t}+{\rm i}A_{t}-{\rm i}\epsilon({\sf J}_{3}+{\sf I}_{3})-{\rm
i}m_{f}{\sf F}_{f}$.

“B.C.” in (2.18) denotes the following boundary conditions of the fields in
the path integral. At the infinitesimal neighborhood of $(x^{0},0,0,0,x^{4})$
with $\forall(x^{0},x^{4})\in T^{2}$, we impose a boundary condition admitting
a singular Dirac monopole with the magnetic charge ${\bm{B}}$:

$\displaystyle\sum_{i=1,2,3}A_{i}dx^{i}\sim\frac{\bm{B}}{2}(1-\cos\theta)d\phi,\,\,\,\sigma\sim\frac{\bm{B}}{2r},\quad\text{for
}r\to 0\,.$ (2.20)

Here $(r,\theta,\phi)$ is the polar coordinates of the space
$(x^{1},x^{2},x^{3})$. In the path integral we also sum over the boundary
conditions, where the elements of ${\bm{B}}$ in (2.20) are permuted by an
arbitrary Weyl group action of $W_{G}$.

We also have to specify the boundary conditions of the fields at a
sufficiently large $r$. At the spatial infinity, the gauge fields $A_{i}$ for
$i=0,4$ and a scalar $\sigma$ have definite values. We assume these values are
in the Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. Since gauge fields
$A_{i}$ for $i=1,2,3$ have a magnetic charge ${\bm{B}}$ near the origin
$(x^{1},x^{2},x^{3})=(0,0,0)$, a natural choice of boundary value $A_{i}$ for
$i=1,2,3$ at the spatial infinity is

$\displaystyle\sum_{i=1,2,3}A_{i}dx^{i}\sim\frac{\bm{B}}{2}(1-\cos\theta)d\phi,\quad\text{for
}r\to\infty\,.$ (2.21)

But an important point here is that we have to take into account not only the
boundary condition (2.21), but all the boundary conditions associated with
monopole bubbling (a.k.a. monopole screening ) [11]. A monopole bubbling is a
phenomenon that an ’t Hooft-Polyakov monopole screen out the charge of the
Dirac monopole ${\bm{B}}$ and reduced it to ${\bm{p}}$ with
$||{\bm{p}}||<||{\bm{B}}||$, when a smooth ’t Hooft-Polyakov monopole with a
magnetic charge ${\bm{p}}-{\bm{B}}$ in the coroot lattice
$\Lambda_{\text{cr}}$ exists at the infinitesimal neighborhood of the center
of the singular Dirac monopole with the charge
${\bm{B}}\in\Lambda_{\text{cw}}$. Here
$||{\bm{B}}||=\sqrt{\mathrm{Tr}({\bm{B}}^{2})}$.

Monopole bubbling effects in the SUSY localization formula of BPS ’t Hooft
loops for $U(N)$ and $SU(N)$ gauge theories were originally studied in [17, 1]
and were further studied in [24]. It turned out that ’t Hooft loops agree with
Verlinde loop operators by including monopole bubbling effects. Moreover
monopole bubbling effects are also necessary to reproduce the correct
properties of operator product expansions (OPEs) of BPS ’t Hooft loops [11, 1,
25, 26]. In three dimensions, monopole bubbling effects for BPS monopole
operators on $\mathbb{R}^{3}$ were studied in [2, 3]. It found that monopole
bubbling effects are necessary in order for monopole operators with higher
charges to be generated by the products of operators with smaller charges.
From these observations, we consider all the boundary conditions with monopole
bubblings for the surface operator specified by

$\displaystyle\sum_{i=1,2,3}A_{i}dx^{i}\sim\frac{\bm{p}}{2}(1-\cos\theta)d\phi,\quad\text{for
}r=\text{finite}\,,$ (2.22)

if ${\bm{p}}\in{\bm{B}}+\Lambda_{\text{cr}}$ with $||{\bm{p}}||<||{\bm{B}}||$
exists. We will study the contribution from the moduli of monopole bubbling to
the path integral in terms of brane constructions in Section 4.

##  3 SUSY localization of surface operator

In this section, we perform the path integral for the vev of surface operators
in terms of supersymmetric localization. Before we explain the technical
details of the localization computation, we first summarize the result of our
supersymmetric localization formula. We consider that the gauge group is $G$
and $N_{F}$ hypermultiplets are in a symplectic representation
$\mathcal{R}\oplus\overline{\mathcal{R}}$ of the gauge group. The localization
formula for the vev of monopole surface operator is given by

$\displaystyle\langle S_{{\bm{B}}}\rangle$
$\displaystyle=\sum_{{\bm{p}}\in{\bm{B}}+\Lambda_{\text{cr}}\atop||{\bm{p}}||\leq||{\bm{B}}||}e^{{\bm{p}}\cdot{\bm{b}}}Z^{\text{5d}}_{1\text{-loop}}({\bm{a}},{\bm{m}},{\bm{p}},\epsilon,\tau)Z_{\text{mono}}({\bm{a}},{\bm{m}},{\bm{p}},{\bm{B}},\epsilon,\tau).$
(3.1)

$Z_{\text{mono}}$’s in (3.1) are the contributions from the monopole bubbling
effects. Note that $Z_{\text{mono}}({\bm{p}},{\bm{B}})=1$ for
$||{\bm{p}}||=||{\bm{B}}||$. If ${\bm{p}}\in{\bm{B}}+\Lambda_{\text{cr}}$ with
$||{\bm{p}}||<||{\bm{B}}||$ does not exist, the localization formula is
completely determined by the one-loop computations for
$||{\bm{p}}||=||{\bm{B}}||$. The right hand side of (3.1) is given as follows.
${\bm{b}}$ is defined in (3.32). A pairing ${\bm{p}}\cdot{\bm{b}}$ of
${\bm{p}}$ and ${\bm{b}}$ is induced by the trace over $\mathfrak{g}$:

$\displaystyle{\bm{p}}\cdot{\bm{b}}=\mathrm{Tr}({\bm{p}}{\bm{b}})\,.$ (3.2)

$Z^{\text{5d}}_{1\text{-loop}}({\bm{a}},{\bm{m}},{\bm{p}},\epsilon,\tau)$ is
the one-loop determinant of the Q-exact action around the saddle point
specified by a magnetic charge ${\bm{p}}$, and factorizes to the one-loop
determinant of the 5d $\mathcal{N}=1$ vector multiplet
$Z^{\text{5d.vec}}_{1\text{-loop}}$ and the one of the 5d $\mathcal{N}=1$
hypermultiplet $Z^{\text{5d.hyp}}_{1\text{-loop}}$:

$\displaystyle
Z^{\text{5d}}_{1\text{-loop}}({\bm{a}},{\bm{m}},{\bm{p}},\epsilon,\tau)=Z^{\text{5d.vec}}_{1\text{-loop}}({\bm{a}},{\bm{p}},\epsilon,\tau)Z^{\text{5d.hyp}}_{1\text{-loop}}({\bm{a}},{\bm{m}},{\bm{p}},\epsilon,\tau),$
(3.3)

where

$\displaystyle Z^{\text{5d.vec}}_{1\mathchar
45\relax\text{loop}}({\bm{a}},\bm{p},\epsilon,\tau)=\left[\prod_{\alpha\in\mathrm{rt}(\mathfrak{g})}\prod_{k=0}^{|\alpha\cdot\bm{p}|-1}\frac{\vartheta_{1}\left(\alpha\cdot{\bm{a}}+\left(\frac{|\alpha\cdot\bm{p}|}{2}-k\right)\epsilon;\tau\right)}{\eta(\tau)}\right]^{-\frac{1}{2}},$
(3.4) $\displaystyle Z^{\text{5d.hyp}}_{1\mathchar
45\relax\text{loop}}({\bm{a}},{\bm{m}},{\bm{p}},\epsilon,\tau)=\left[\prod_{{w}\in\Delta(\mathcal{R})}\prod_{f=1}^{N_{F}}\prod_{k=0}^{|{w}\cdot\bm{p}|-1}\frac{\vartheta_{1}\left({w}\cdot{\bm{a}}-{m_{f}}+\left(\frac{|{w}\cdot\bm{p}|-1}{2}-k\right)\epsilon;\tau\right)}{\eta(\tau)}\right]^{\frac{1}{2}},$
(3.5)

$\mathrm{rt}(\mathfrak{g})$ is the set of roots of $\mathfrak{g}$ and
$\Delta(\mathcal{R})$ is the set of weights of a representation $\mathcal{R}$.

A theta function $\vartheta_{1}\left(u;\tau\right)$ and eta function
$\eta(\tau)$ are defined by

$\displaystyle\vartheta_{1}(u)=\vartheta_{1}\left(u;\tau\right)$
$\displaystyle:=2e^{\frac{{\rm i}\pi}{4}\tau}\sin(\pi
u)\prod_{i=1}^{\infty}(1-e^{2\pi{\rm i}n\tau})(1-e^{2\pi{\rm
i}n\tau}e^{2\pi{\rm i}u})(1-e^{2\pi{\rm i}n\tau}e^{-2\pi{\rm i}u})\,,$ (3.6)
$\displaystyle\eta(\tau)$ $\displaystyle:=e^{\frac{\pi{\rm
i}\tau}{12}}\prod_{n=1}^{\infty}(1-e^{2\pi{\rm i}n\tau})\,.$ (3.7)

To shorten expressions, we introduce the following notation for the theta
function:

$\displaystyle\vartheta_{1}(\pm x+y)$ $\displaystyle=\prod_{\alpha=\pm
1}\vartheta_{1}(\alpha x+y),\quad\vartheta_{1}(\pm x\pm
y+w)=\prod_{\alpha,\beta=\pm 1}\vartheta_{1}(\alpha x+\beta y+w).$ (3.8)

$Z_{\text{mono}}({\bm{p}},{\bm{B}})$ in (3.1) is interpreted as a contribution
from the path integral over the moduli space of Bogomolny equation (3.21) with
a monopole bubbling explained in Section 4. We gives explicit computations of
$\langle S_{\bm{B}}\rangle$ for small magnetic charges in Section 6.

###  3.1 Zero locus of Q-exact term in 5d $\mathcal{N}=1$ SUSY gauge theory

To apply supersymmetric localization procedure, we introduce the Faddeev–Popov
ghosts $c,\bar{c},$ the Nakanishi-Lautrup B-field $\bar{b}$, and a BRST charge
${\sf Q}_{B}$. We will explain the definition of the BRST charge ${\sf Q}_{B}$
and the gauge fixing term in the next subsection. We add one-prameter family
of Q-exact action ${\rm t}\widehat{\sf Q}\cdot V$ to the original action, and
take a limit ${{\rm t}\to\infty}$:

$\displaystyle\langle S_{\bm{B}}\rangle=\lim_{{\rm
t}\to\infty}\int\mathcal{D}A\mathcal{D}\Psi\mathcal{D}K\mathcal{D}c\mathcal{D}\bar{c}\mathcal{D}\bar{b}\exp\left({-S_{\text{vec}}-S_{\text{hyp}}-S_{\text{b.d}}-{\rm
t}\widehat{\sf Q}\cdot V}\right)\,,$ (3.9)

with

$\displaystyle\widehat{\sf Q}:={\sf Q}+{\sf Q}_{B}\,.$ (3.10)

Then the principle of supersymmetric localization [27] tells us that the path
integral in (2.18) is exactly evaluated in terms of the one-loop integral
around $\widehat{\sf Q}\cdot V=0$ and integrals over certain moduli spaces of
equations associated with $\widehat{\sf Q}\cdot V=0$ in (3.9).

In this section we focus on the following matter part in the Q-exact action
$\widehat{\sf Q}\cdot V$ :

$\displaystyle{\sf Q}\cdot(\Psi,\overline{{\sf Q}\cdot\Psi})\,,$ (3.11)

and study the zero locus (saddle point locus) of (3.11). Here $\overline{{\sf
Q}\cdot\Psi}$ is the complex conjugate of ${\sf Q}\cdot\Psi$.
$\left(A,B\right)=\int_{T^{2}\times\mathbb{R}^{3}}AB$, where the spinor
indices and Lie algebra indices in five dimensional fields $A$ and $B$ are
contracted. To write down explicitly the saddle equation ${\sf Q}\cdot\Psi=0$
in (3.11), it is convenient to decompose the fermion $\Psi$ with the sixteen
components into Grassmann odd functions $\Psi_{M}$ for $M=1,\cdots,9$ and
$\Upsilon^{i}$ for $i=1,\cdots,7$ as

$\displaystyle\Psi$
$\displaystyle=\sum_{M=1}^{9}\Psi_{M}\tilde{\Gamma}^{M}{\varepsilon}+{\rm
i}\sum_{i=1}^{7}\Upsilon^{i}\nu_{i}\,.$ (3.12)

Here $\Psi_{M}$ and $\Upsilon^{i}$ are defined by

$\displaystyle\Psi_{M}$ $\displaystyle=\varepsilon\Gamma^{M}\Psi,\quad{\rm
i}\Upsilon^{i}=\bar{\nu}_{i}\Psi\,,$ (3.13)

with $\bar{\nu}_{i}=-\nu_{i}$. Then, we have the supersymmetry transformation
of $\Psi_{M}$ and $\Upsilon_{i}$ as follows.

$\displaystyle{\sf Q}\cdot\Psi_{M}$ $\displaystyle=\varepsilon\Gamma^{M}{\sf
Q}\cdot\Psi={\rm i}F_{M0}+F_{M4}\,.$ (3.14)

Here we used the following properties;
$\Gamma^{M}\Gamma^{NL}=\Gamma^{[M}\Gamma^{NL]}+2\delta^{M[N}\Gamma^{L]}$,
$\varepsilon\Gamma^{[M}\Gamma^{NL]}\varepsilon=0$ and
$(\varepsilon\Gamma^{0}\varepsilon,\varepsilon\Gamma^{1}\varepsilon,\cdots,\varepsilon\Gamma^{9}\varepsilon)=({\rm
i},0^{3},1,0^{5})$.

$\displaystyle{\rm i}{\sf Q}\cdot\Upsilon_{i}$
$\displaystyle=\bar{\nu}_{i}{\sf Q}\cdot\Psi$
$\displaystyle=\frac{1}{2}\left(\sum_{j,k=1}^{3}F_{jk}\bar{\nu}_{i}\Gamma^{jk}\varepsilon+2\sum_{j=1}^{3}\sum_{k=5}^{8}F_{jk}\bar{\nu}_{i}\Gamma^{jk}\varepsilon+2\sum_{j=1}^{3}F_{j9}\bar{\nu}_{i}\Gamma^{j9}\varepsilon+2\sum_{j=5}^{8}F_{j9}\bar{\nu}_{i}\Gamma^{j9}\varepsilon\right)$
$\displaystyle\qquad+{\rm i}K^{i}\,.$ (3.15)

Here we used $\bar{\nu}_{i}\nu_{j}=-\delta_{ij}$ and
$\bar{\nu}_{i}\Gamma^{M4}\varepsilon=0$ for $i=1\cdots,7$ and
$M=0,1,\cdots,9$. Then

$\displaystyle{\rm i}{\sf Q}\cdot\Upsilon_{i=1,2,3}$
$\displaystyle=-\frac{1}{2}\sum_{j,k=1}^{3}\epsilon_{ijk}F_{jk}+D_{i}\sigma-{\rm
i}K^{i}.$ (3.16) $\displaystyle{\rm i}{\sf Q}\cdot\Upsilon_{i=4,5,6,7}$
$\displaystyle=\sum_{j=1}^{3}\sum_{k=5}^{8}D_{j}\Phi_{k}\bar{\nu}_{i}\Gamma^{jk}\varepsilon+\sum_{j=5}^{8}{\rm
i}[\Phi_{j},\sigma]\bar{\nu}_{i}\Gamma^{j9}\varepsilon-{\rm i}K^{i}$ (3.17)

Here we used

$\displaystyle\bar{\nu}_{i}\Gamma^{jk}\varepsilon=0\quad\text{for}\quad
i\in\\{5,6,7\\},\,\,j,k\in\\{1,2,3\\}$ (3.18)
$\displaystyle\bar{\nu}_{i}\Gamma^{j9}\varepsilon=0\quad\text{for}\quad
i\in\\{4,5,6,7\\},\,\,j\in\\{1,2,3\\}.$ (3.19)

From (3.14) and (3.16), we find that the saddle point equation ${\sf
Q}\cdot\Psi=0$ for the bosonic fields in the 5d vector multiplet are
decomposed to

$\displaystyle F_{04}$ $\displaystyle=0\,,$ (3.20) $\displaystyle D_{i}\sigma$
$\displaystyle=\frac{1}{2}\epsilon_{ijk}F_{jk}\,\,\,(i,j,k=1,2,3)\,,$ (3.21)
$\displaystyle K_{i}$ $\displaystyle=0\,.$ (3.22)

From (3.17), the saddle point equation ${\sf Q}\cdot\Psi=0$ for the 5d
hypermultiplet (3.17) is written as

$\displaystyle\sum_{i=1}^{3}\sigma^{i}D_{i}q+[\sigma,q]=0\,,$ (3.23)

where $\sigma^{i},i=1,2,3$ are the Pauli matrices, and $q=(q_{1},q_{2})^{T}$
is defined by

$\displaystyle q_{1}:=\Phi_{5}-{\rm i}\Phi_{6}+{\rm i}\Phi_{7}+\Phi_{8},\quad
q_{2}:={\rm i}\Phi_{5}-\Phi_{6}-\Phi_{7}-{\rm i}\Phi_{8}\,.$ (3.24)

The invariance under the flavor symmetry rotation require the saddle point
value of $\Phi_{i}$ for $i=5,\cdots,9$ is zero.

We evaluate the saddle point value of the super Yang-Mills action and the
boundary term. $F_{04}=0$ means that the saddle point configuration for the
gauge fields $A_{i}$ for $i=0,4$ are flat connections which are fixed by the
value at $r\to\infty$. Let $\bar{A}_{i}$, $i=0,4$ be the boundary values
$A_{i}|_{r=\infty}=\text{constant}$ for $i=0,4$ and define ${\bm{a}}$ as a
holomorphic combination of the constant gauge fields:

$\displaystyle{\bm{a}}:=\bar{A}_{0}+{\rm
i}\bar{A}_{4}\in\mathfrak{h}\otimes{\mathbb{C}}\,.$ (3.25)

Next let $(\bar{A}_{i},\bar{\sigma})$ for $i=1,2,3$ be a solution of the
Bogomolny equation (3.21). From the boundary condition of the path integral
(2.20), $(\bar{A}_{i},\bar{\sigma})$ behaves as

$\displaystyle\sum_{i=1,2,3}\bar{A}_{i}dx^{i}$
$\displaystyle\sim\frac{\bm{B}}{2}(1-\cos\theta)d\phi,\quad\bar{\sigma}\sim\frac{{\bm{B}}}{2r},\quad\text{for
}r\to 0\,,$ (3.26) $\displaystyle\sum_{i=1,2,3}A_{i}dx^{i}$
$\displaystyle\sim\frac{\bm{p}}{2}(1-\cos\theta)d\phi,\quad\bar{\sigma}\sim\frac{{\bm{p}}}{2r}+\sigma_{0}\quad\text{for
}r=\text{finite}\,.$ (3.27)

Here we assume that $\sigma_{0}$ is a Cartan valued constant. We substitute a
saddle point value $(\bar{A}_{i=0,\cdots,4},\bar{\sigma})$ into the 5d super
Yang-Mills action. Then saddle point value of the 5d super Yang-Mills action
is given

$\displaystyle S_{\text{vec}}|_{\text{saddle}}$
$\displaystyle=\frac{1}{g^{2}}\int_{T^{2}\times(\mathbb{R}^{3}\backslash
B^{3}_{R})}d^{5}x\mathrm{Tr}\Bigl{[}\frac{1}{2}F^{\mu\nu}F_{\mu\nu}+D^{\mu}\sigma
D_{\mu}\sigma-\lambda\Gamma^{\mu}D_{\mu}{\lambda}+{\rm
i}\lambda\Gamma^{9}[{\lambda},\Phi_{9}]\Bigr{]}\Big{|}_{\text{saddle}}$
$\displaystyle=\frac{2\pi\text{vol}(T^{2})}{g^{2}R}\mathrm{Tr}({\bm{p}}^{2})\,.$
(3.28)

where

$\displaystyle
B^{3}_{R}:=\\{(x^{1},x^{2},x^{3})|\,\,|x^{1}|^{2}+|x^{2}|^{2}+|x^{3}|^{2}\leq
R^{2}\\}\,.$ (3.29)

In (3.28), we removed the three-dimensional ball $B^{3}_{R}$ from the
integration region to regularize the divergence coming from the center of the
Dirac monopole. The divergence coming from $R\to 0$ is compensated by the
saddle point value of the boundary term $S_{\text{b.d}}$ [1] as follows.

$\displaystyle S_{\text{b.d}}|_{\text{saddle}}$
$\displaystyle=\frac{2}{g^{2}}\int_{T^{2}\times\partial
B^{3}_{R}}\text{dvol}(T^{2})\mathrm{Tr}({\sigma}F_{A})\Big{|}_{\text{saddle}}$
$\displaystyle=-\frac{1}{g^{2}}\text{vol}(T^{2})\mathrm{Tr}\left(2\pi\frac{\bm{p}^{2}}{R}+4\pi^{2}R{\sigma}_{0}{\bm{p}}\right)\,.$
(3.30)

Here $\text{dvol}(T^{2})$ is the volume form of $T^{2}$, and $F_{A}$ is the
restriction of the field strength
$\frac{1}{2}\sum_{i,j=1}^{3}F_{ij}dx^{i}\wedge dx^{j}$ on the boundary
$\partial B^{3}_{R}$. The the saddle point value of the sum of the super Yang-
Mills action and the boundary term has a finite value:

$\displaystyle(S_{\text{vec}}+S_{\text{b.d}})|_{\text{saddle}}$
$\displaystyle=-\frac{4\pi^{2}R}{g^{2}}\text{vol}(T^{2})\mathrm{Tr}\left({\bm{p}}{\sigma}_{0}\right)=:{{\bm{p}}\cdot{\bm{b}}}\,,$
(3.31)

with

$\displaystyle{\bm{b}}=-\frac{4\pi^{2}R}{g^{2}}\text{vol}(T^{2}){\sigma}_{0}.$
(3.32)

On the other hand, the saddle point values of the fields in the hypermultiplet
are zero, which means that the saddle point values of the action of
hypermultiplet is zero.

###  3.2 Gauge fixing term and BRST transformation

To evaluate the one-loop determinant, we introduce the action for the ghosts
and the gauge fixing term:

$\displaystyle S_{\text{g.f}}$
$\displaystyle=\int_{T^{2}\times\mathbb{R}^{3}}d^{5}x{\sf
Q}_{B}\cdot\mathrm{Tr}\left(\bar{c}\left(\sum_{i=1,2,3,9}\bar{D}_{M}\tilde{A}_{M}+\frac{\xi}{2}\bar{b}\right)\right)$
$\displaystyle=\int_{T^{2}\times\mathbb{R}^{3}}d^{5}x\mathrm{Tr}\left(-{\rm
i}\bar{c}\bar{D}_{M}D_{M}c+\bar{b}\left({\rm
i}\bar{D}^{M}\tilde{A}_{M}+\frac{\xi}{2}\bar{b}\right)\right)$ (3.33)

Here the $\mathfrak{g}$-valued ghost fields $c,\bar{c}$ are Grassmann odd, and
the $\mathfrak{g}$-valued B-field $\bar{b}$ is a Grassmann even.
$\bar{D}_{M}=\partial_{M}+{\rm i}\bar{A}_{M}$, where $\bar{A}_{M}$ denotes the
saddle point values of the gauge fields and the vector multiplet scalar given
by (3.25), (3.26), (3.27), and $\bar{A}_{i=5,6,7,8}=0$. $\tilde{A}_{M}$ is the
fluctuation of around $\bar{A}_{M}$ mentioned in the beginning of Section 3.3.
We define the BRST transformation by

$\displaystyle{\sf Q}_{B}\cdot A_{M}$ $\displaystyle=D_{M}c,\quad{\sf
Q}_{B}\cdot\Psi=-{\rm i}[c,\Psi],\quad{\sf Q}_{B}\cdot K_{i}=-{\rm
i}[c,K_{i}],$ $\displaystyle{\sf Q}_{B}\cdot c$ $\displaystyle=-\frac{\rm
i}{2}[c,c],\quad{\sf Q}_{B}\cdot\bar{c}=\bar{b},\quad{\sf
Q}_{B}\cdot\bar{b}=0.$ (3.34)

Note that the BRST charge is nilpotent $\\{{\sf Q}_{B},{\sf Q}_{B}\\}=0$. The
supersymmetry transformation for $(c,\bar{c},\bar{b})$ is defined by

$\displaystyle{\sf Q}\cdot c$ $\displaystyle=-{\rm
i}\tilde{A}_{0}-\tilde{A}_{4}\,,$ $\displaystyle{\sf Q}\cdot\bar{c}$
$\displaystyle=0\,,$ $\displaystyle{\sf Q}\cdot\bar{b}$ $\displaystyle=-({\rm
i}D_{0}+D_{4})\bar{c}\,.$ (3.35)

We define the Q-exact term by

$\displaystyle\widehat{\sf Q}\cdot V:=\widehat{\sf
Q}\left[(\Psi,\overline{\widehat{\sf Q}\cdot\Psi})+V_{\text{g.f}}\right]$
(3.36)

with $\widehat{\sf Q}={\sf Q}+{\sf Q}_{B}$.

###  3.3 Evaluation of one-loop determinants

In the localization computation of the path integral in (3.9), We decompose
the fields to $X=\bar{X}+\tilde{X}/\sqrt{\sf t}$, where $\bar{X}$ denotes the
fields satisfy $\widehat{\sf Q}\cdot V=0$, and $\tilde{X}$ is called as the
fluctuation fields around the saddle point value. In the limit ${\sf
t}\to\infty$, the quadratic part of the fluctuation fields in the Q-exact
action only contributes to the path integral, and the higher order interaction
terms in the Q-exact action are negligible, i.e., the one-loop computation for
the fluctuation fields gives the exact answer. Then we may take the first
order approximation of $\widehat{\sf Q}\cdot\tilde{X}$ in the Q-exact action
with respect to the fluctuation fields.

In order to evaluate the one-loop determinant in terms of the index theorem of
the transversally elliptic operators, let us define collections of the
fluctuation fields ${\bm{X}}_{i}$, $i=0,1$ by

$\displaystyle{\bm{X}}_{0}$
$\displaystyle=(X_{0,1},X_{0,2},\cdots,X_{0,9})=(\tilde{A}_{1},\tilde{A}_{2},\cdots,\tilde{A}_{9})\,,$
$\displaystyle{\bm{X}}_{1}$
$\displaystyle=(X_{1,1},X_{1,2},\cdots,X_{1,9})=(\Upsilon_{1},\cdots,\Upsilon_{7},c,\bar{c})\,.$
(3.37)

To make the expression concise, we omitted $\tilde{}$ for the fluctuation
fields $\Upsilon_{i}$ and the ghosts. Note that the saddle point configuration
for the ghost and B-field is trivial. The square of $\widehat{\sf
Q}$-transformation generates

$\displaystyle\widehat{\sf Q}^{2}$ $\displaystyle=-\frac{\rm
i}{\tau_{2}}\left(\partial_{t}-\tau\partial_{s}+{\rm i}{\bm{a}}-{\rm
i}\epsilon({\sf J}_{3}+{\sf I}_{3})-{\rm i}\sum_{f}m_{f}{\sf F}_{f}\right)\,.$
(3.38)

At the quadratic order of ${\bm{X}}_{i}$, $V$ can be written as

$\displaystyle V$ $\displaystyle=(\Psi,\overline{\widehat{\sf
Q}\cdot\Psi})+V_{\text{g.f}}$
$\displaystyle=\left(\left(\begin{array}[]{cc}\widehat{\sf
Q}\cdot{\bm{X}}_{0},&{\bm{X}}_{1}\end{array}\right),\left(\begin{array}[]{cc}D_{00}&D_{01}\\\
D_{10}&D_{11}\end{array}\right)\left(\begin{array}[]{c}{\bm{X}}_{0}\\\
\widehat{\sf Q}\cdot{\bm{X}}_{1}\end{array}\right)\right)\,,$ (3.44)

where $D_{ij}$ for $i,j=0,1$ are linear differential operators which depend on
the saddle point field configuration, but are independent of the fluctuations
field ${\bm{X}}_{i}$. Then the Q-exact action $\widehat{\sf Q}\cdot V$ at the
quadratic order is given by

$\displaystyle\widehat{\sf Q}\cdot V$
$\displaystyle=\left(\left(\begin{array}[]{cc}{\bm{X}}_{0}&\widehat{\sf
Q}\cdot{\bm{X}}_{1}\end{array}\right),\left(\begin{array}[]{cc}-\widehat{\sf
Q}^{2}&0\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}D_{00}&D_{01}\\\
D_{10}&D_{11}\end{array}\right)\left(\begin{array}[]{c}{\bm{X}}_{0}\\\
\widehat{\sf Q}\cdot{\bm{X}}_{1}\end{array}\right)\right)$ (3.52)
$\displaystyle+\left(\left(\begin{array}[]{cc}\widehat{\sf
Q}\cdot{\bm{X}}_{0}&{\bm{X}}_{1}\end{array}\right),\left(\begin{array}[]{cc}D_{00}&D_{01}\\\
D_{10}&D_{11}\end{array}\right)\left(\begin{array}[]{cc}-1&0\\\
0&-\widehat{\sf Q}^{2}\end{array}\right)\left(\begin{array}[]{c}\widehat{\sf
Q}\cdot{\bm{X}}_{0}\\\ {\bm{X}}_{1}\end{array}\right)\right)\,.$ (3.60)

Following the argument in [27], one can show that the one-loop determinant is
reduced to the ratio of the functional determinants over the spaces
$\mathrm{Det}_{\mathrm{coker}(D_{10})}$ and
$\mathrm{Det}_{\mathrm{ker}(D_{10})}$:

$\displaystyle Z^{\text{5d}}_{1\text{-loop}}$
$\displaystyle=\left[\frac{\mathrm{Det}\left(\left(\begin{array}[]{cc}D_{00}&D_{01}\\\
D_{10}&D_{11}\end{array}\right)\left(\begin{array}[]{cc}-1&0\\\
0&-\widehat{\sf
Q}^{2}\end{array}\right)\right)}{\mathrm{Det}\left(\left(\begin{array}[]{cc}-\widehat{\sf
Q}^{2}&0\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}D_{00}&D_{01}\\\
D_{10}&D_{11}\end{array}\right)\right)}\right]^{\frac{1}{2}}=\left[\frac{\mathrm{Det}_{\mathrm{coker}(D_{10})}\widehat{\sf
Q}^{2}}{\mathrm{Det}_{\mathrm{ker}(D_{10})}\widehat{\sf
Q}^{2}}\right]^{\frac{1}{2}}\,.$ (3.69)

Here $\mathrm{ker}(D_{10})$ is the space of the kernel of the differential
operator $D_{10}$, and $\mathrm{coker}(D_{10})$ is the space of the cokernel
of the differential operator $D_{10}$. Since the differential operator
$D_{10}$ is decomposed to $D_{10}=D_{\text{vec}}\oplus D_{\text{hyp}}$, where
$D_{\text{vec}}$ acting on the fields in the vector multiplet and
$D_{\text{hyp}}$ acting on the ones in the hypermultiplet, the one-loop
determinant factorizes to the product of the one-loop determinants of the
vector multiplet and the hypermultiplet:

$\displaystyle
Z^{\text{5d}}_{1\text{-loop}}=Z^{\text{5d.vec}}_{1\text{-loop}}Z^{\text{5d.hyp}}_{1\text{-loop}}\,,$
(3.70)

with

$\displaystyle Z^{\text{5d.vec}}_{1\text{-loop}}$
$\displaystyle=\left[\frac{\mathrm{Det}_{\mathrm{coker}(D_{\text{vec}})}\widehat{\sf
Q}^{2}}{\mathrm{Det}_{\mathrm{ker}(D_{\text{vec}})}\widehat{\sf
Q}^{2}}\right]^{\frac{1}{2}},\quad
Z^{\text{5d.hyp}}_{1\text{-loop}}=\left[\frac{\mathrm{Det}_{\mathrm{coker}(D_{\text{hyp}})}\widehat{\sf
Q}^{2}}{\mathrm{Det}_{\mathrm{ker}(D_{\text{hyp}})}\widehat{\sf
Q}^{2}}\right]^{\frac{1}{2}}\,.$ (3.71)

The important point here is that the ratio of the functional determinants is
evaluated in term of the index theorem of transversally elliptic operators.
Let $\mathrm{ind}(D)$ be the equivariant index of a transversally elliptic
operator $D=D_{\text{vec}}\text{ or }D_{\text{hyp}}$ defined by

$\displaystyle\mathrm{ind}(D)=\mathrm{tr}_{\mathrm{ker}(D)}(e^{\widehat{\sf
Q}^{2}})-\mathrm{tr}_{\mathrm{coker}(D)}(e^{\widehat{\sf Q}^{2}})\,.$ (3.72)

Since $\widehat{\sf Q}^{2}$ generates a linear combination of torus actions,
$e^{\widehat{\sf Q}^{2}}$ is thought as a group element. When the equivariant
index is written as a form $\text{ind}(D)=\sum_{i}n_{i}e^{w_{i}}$, where
$w_{i}$ is a weight of a representation of $\widehat{\sf Q}^{2}$ and $n_{i}$
is the multiplicity of the representation, the ratio of the functional
determinants is obtained by the following rule:

$\displaystyle\text{ind}(D)=\sum_{i}n_{i}e^{w_{i}}\to\frac{\mathrm{Det}_{\mathrm{coker}(D)}\widehat{\sf
Q}^{2}}{\mathrm{Det}_{\mathrm{ker}(D)}\widehat{\sf
Q}^{2}}=\prod_{i}w_{i}^{n_{i}}\,.$ (3.73)

Thus remained task to obtain the one-loop determinant is to evaluate the
weights $w_{i}$ and the multiplicities $n_{i}$ for the representations of
$D_{\text{vec}}$ and $D_{\text{hyp}}$. The expression of $D_{10}$ evaluated in
Appendix C shows that the differential operators $D_{\text{vec}}$ and
$D_{\text{hyp}}$ are same type as the ones appeared in the evaluation of the
one-loop determinant for ’t Hooft loops on $S^{1}\times\mathbb{R}^{3}$ [1].
The difference between the differential operators for ’t Hooft surface
operators here and the ones for BPS ’t Hooft loop operators in [1] only comes
from the spacetime derivative in $\widehat{\sf Q}^{2}$; $\widehat{\sf Q}^{2}$
in (3.38) contains the derivative of the $T^{2}$-coordinate, whereas
$\widehat{\sf Q}^{2}$ for ’t Hooft loops contains the derivative of the
$S^{1}$-coordinate [1]. Therefore, for the each Kaluza-Klein (KK) mode along
the $T^{2}$-direction, the index is same as the equivariant indices
$\text{ind}(D_{\text{Bogo}})$ and $\text{ind}(D_{\text{DH},\mathcal{R}})$
evaluated in [17, 1]:

$\displaystyle\text{ind}(D_{\text{Bogo},\mathbb{C}})$
$\displaystyle=-\frac{e^{\pi{\rm i}\epsilon}+e^{-\pi{\rm
i}\epsilon}}{2}\sum_{\alpha\in\text{rt}(\mathfrak{g})}e^{2\pi{\rm
i}\alpha\cdot{\bm{a}}}\sum_{k=0}^{|\alpha\cdot{\bm{p}}|-1}e^{\pi{\rm
i}(|\alpha\cdot{\bm{p}}|-2k-1)\epsilon}\,,$ (3.74)
$\displaystyle\text{ind}(D_{\text{DH},\mathcal{R}})$
$\displaystyle=-\frac{1}{2}\sum_{w\in\Delta(\mathcal{R})}e^{2\pi{\rm
i}w\cdot{\bm{a}}}\sum_{k=0}^{|w\cdot{\bm{p}}|-1}e^{\pi{\rm
i}(|w\cdot{\bm{p}}|-2k-1)\epsilon}\,,$ (3.75)

Here we choose a saddle point specified by a magnetic charge ${\bm{p}}$. By
summing over the KK modes along $T^{2}$ we obtain the indices of
$D_{\text{vec}}$ and $D_{\text{hyp}}$ on $T^{2}\times\mathbb{R}^{3}$:

$\displaystyle\mathrm{ind}(D_{\text{vec}})$
$\displaystyle=\sum_{m,n\in\mathbb{Z}}e^{2\pi{\rm
i}(m+n\tau)}\text{ind}(D_{\text{Bogo},\mathbb{C}})\,,$ (3.76)
$\displaystyle\mathrm{ind}(D_{\text{hyp}})$
$\displaystyle=-\sum_{m,n\in\mathbb{Z}}e^{2\pi{\rm
i}(m+n\tau)}\sum_{f=1}^{N_{F}}\left(e^{-2\pi{\rm
i}m_{f}}\text{ind}(D_{\text{DH},\mathcal{R}})+e^{2\pi{\rm
i}m_{f}}\text{ind}(D_{\text{DH},\mathcal{R}})|_{{\bm{a}}\to-{\bm{a}}}\right)\,.$
(3.77)

Note that if the KK momentum contributions $\sum_{m,n\in\mathbb{Z}}e^{2\pi{\rm
i}(m+n\tau)}$ for $T^{2}$ in (3.76) and (3.77) are replaced by
$\sum_{m\in\mathbb{Z}}e^{2\pi{\rm i}m}$ for $S^{1}$, we obtain the equivariant
indices for the one-loop determinants of the BPS ’t Hooft loop operators on
$S^{1}\times\mathbb{R}^{3}$. By using the relation (3.73) with (3.76) and
(3.77), up to overall normalization constants, we obtain the non-regularized
expression of the one-loop determinants around a saddle point specified by a
magnetic charge ${\bm{p}}$:

$\displaystyle Z^{\text{5d.vec}}_{1\mathchar
45\relax\text{loop}}({\bm{a}},\bm{p},\epsilon,\tau)=\left[\prod_{\alpha\in\mathrm{rt}(\mathfrak{g})}\prod_{k=0}^{|\alpha\cdot\bm{p}|-1}\prod_{m,n\in\mathbb{Z}}(m+n\tau+\alpha\cdot{\bm{a}}+\left(\frac{|\alpha\cdot\bm{p}|}{2}-k\right)\epsilon\right]^{-\frac{1}{2}},$
(3.78) $\displaystyle Z^{\text{5d.hyp}}_{1\mathchar
45\relax\text{loop}}({\bm{a}},{\bm{m}},\bm{p},\epsilon,\tau)=\left[\prod_{{w}\in\Delta(\mathcal{R})}\prod_{f=1}^{N_{F}}\prod_{k=0}^{|{w}\cdot\bm{p}|-1}\prod_{m,n\in\mathbb{Z}}\left(m+n\tau+{w}\cdot{\bm{a}}-{m}_{f}+\left(\frac{|{w}\cdot\bm{p}|-1}{2}-k\right)\epsilon\right)\right]^{\frac{1}{2}},$
(3.79)

By using the zeta function regularization, we obtain the one-loop determinants
(3.4), (3.5).

##  4 Monopole bubbling effects via branes and elliptic genera

For BPS ’t Hooft loops in 4d $\mathcal{N}=2$ $U(N)$ and $SU(N)$ gauge theories
on $S^{1}\times\mathbb{R}^{3}$, monopole bubbling effects are given by Witten
indices for supersymmetric quantum mechanics (SQM) arising from the low energy
world volume theories on D1-branes [24, 28, 29] in the type IIB string theory.
Monopole bubbling effects were also studied in [26] in terms of the type IIA
string theory. By taking a T-dual of the brane configuration of [26] in the
$x^{4}$-direction, we obtain monopole bubbling effects $Z_{\text{mono}}$ for
the surface operators on $T^{2}\times\mathbb{R}^{3}$ which are given by
elliptic genera for 2d gauged linear sigma models (GLSMs) living on $T^{2}$.

###  4.1 Brane construction for t’ Hooft surface operators and monopole
bubbling

| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
D5 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  | $\times$ |  |  |
NS5 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  |  |
D3 | $\times$ |  |  |  | $\times$ | $\times$ | $\times$ |  |  |
NS5’ | $\times$ |  |  |  | $\times$ |  | $\times$ | $\times$ | $\times$ | $\times$
Table 1: The symbol $\times$ denotes the directions in which D-branes and
NS5-branes extend.

Figure 1: (a): A brane configuration for a magnetic charge
${\bm{B}}={\bm{e}}_{1}$. (b): A brane configuration for a magnetic charge
${\bm{B}}=-{\bm{e}}_{N}$. (c): A brane configuration for
${\bm{B}}={\bm{e}}_{1}+{\bm{e}}_{2}$. (d): A brane configuration for a
magnetic charge ${\bm{B}}=2{\bm{e}}_{1}$.

Figure 2: (a): The brane configuration for the surface operator with the
magnetic charge ${\bm{B}}={\bm{e}}_{1}-{\bm{e}}_{N}$. (b) A D3-brane suspended
between the leftmost D5-brane and the rightmost D5-brane is introduced to
describe the ’t Hooft-Polyakov monopole with the charge
${\bm{B}}=-{\bm{e}}_{1}+{\bm{e}}_{N}$. (c): When the positions of the three
D3-branes in the $x^{3}$-direction coincide, the D3-branes reconnect and form
a single D3-brane. Then the charge is reduced to ${\bm{p}}=0$ which is the
brane interpretation of the monopole bubbling effect. (d): The quiver diagram
for the world volume theory on the D3-brane in Figure (c). The circle with $1$
denotes 2d $\mathcal{N}=(4,4)$ $U(1)$ vector multiplet. The solid line
connected between the circle and the box with $N$ denotes the $N$-tuple 2d
$\mathcal{N}=(4,4)$ hypermultiplets with the gauge charge $+1$.

First we explain a 5d $\mathcal{N}=1$ $U(N)$ gauge theory arising from the low
energy world volume theory on D5-NS5-brane system. The brane configuration is
specified by Figure 1. When $N$ D5-branes are suspended between two NS5-branes
in the $x^{6}$-direction, a 5d $\mathcal{N}=1$ $U(N)$ vector multiplet arises
from the low energy world volume theory on the D5-branes. When $N$ D5-branes
and $N^{\prime}$ D5-branes are attached to an NS5-brane from the left side and
the right side in the $x^{6}$-direction, respectively, a 5d $\mathcal{N}=1$
hypermultiplet in the $U(N)\times U(N^{\prime})$ bi-fundamental representation
arises from the open strings ended on the $N$ D5-branes and the $N^{\prime}$
D5-branes separated by the NS5-brane. If these $N$ (resp. $N^{\prime}$)
D5-branes are suspended between two NS5-branes, $U(N)$ (resp. $U(N^{\prime})$)
is a gauge group. On the other hand, if the $N$ (resp. $N^{\prime}$) D5-branes
are semi-infinite in the $x^{6}$-direction, $U(N)$ (resp. $U(N^{\prime})$) is
a flavor symmetry group. If the $x^{6}$-direction is compactified by a circle,
we obtain circular quiver gauge theories with $U(N)$ gauge groups. In this
paper, we compute explicitly monopole bubbling effects for 5d $U(N)$
$\mathcal{N}=1^{*}$ gauge theories. The $\mathcal{N}=1^{*}$ $U(N)$ gauge
theory is obtained by the $N$ D5-branes are ended on a single NS5-brane and
$x^{6}$-direction is compactified by the circle. Monopole bubbling effects for
other gauge theories will be studied in elsewhere.

An ’t Hooft surface operator is described by D3-branes stretched between
D5-branes and NS5’-branes in the $x^{5}$-direction. Here we write an NS5-brane
extending in the $x^{0,4,6,7,8,9}$-directions as an NS5’-brane to distinguish
it from an NS5-brane extending in the $x^{0,1,2,3,4,5}$-directions. For later
convenience, we call $N$ D5-branes as the first, the second, …, the $N$-th
D5-brane from the right to the left in the $x^{5}$-direction. The magnetic
charge ${\bm{B}}$ is read off from brane configurations as

  * •

A D3-brane attached to the $i$-th D5-brane from the right side corresponds to
a magnetic charge ${\bm{B}}={\bm{e}}_{i}$.

  * •

A D3-brane attached to the $i$-th D5-brane from the left side corresponds to a
magnetic charge ${\bm{B}}=-{\bm{e}}_{i}$.

Here ${\bm{e}}_{i}=\mathrm{diag}(0,\cdots,0,\stackrel{{\scriptstyle
i}}{{1}},0\cdots,0)$. Examples of brane configurations for surface operators
and their magnetic charges are listed in Figure 1.

To describe monopole bubbling effects in the brane set up, we introduce a
D3-brane suspended between the $i$-th D5-brane and the $j$-th D5-brane which
is interpreted as a smooth ’t Hooft-Polyakov monopole with
${\bm{B}}={\bm{e}}_{i}-{\bm{e}}_{j}$ [30], see Figure 2. When the
$x^{3}$-coordinate of the D3-branes for the surface operators and the one of a
D3-brane for the ’t Hooft-Polyakov monopole coincide, the magnetic charge is
screened and monopole bubbling occurs, see 2. The low energy world volume
theory on D3-branes suspended between NS5’-branes is 2d supersymmetric GLSM.
This T-dual picture of the proposal in [28, 26] suggests that the
supersymmetric indices, i.e., elliptic genera for the world volume theories of
D3-branes ended on NS5’-branes give the monopole bubbling effects
$Z_{\text{mono}}$.

###  4.2 Elliptic genera for monopole bubbling effects

By using the localization formula for elliptic genera in [31, 32], we can
express the elliptic genus for the monopole bubbling effects in terms of
residue integrals called Jeffrey-Kirwan (JK) residues:

$\displaystyle
Z^{(\eta)}_{\text{mono}}({\bm{a}},{\bm{m}},{\bm{p}},{\bm{B}},\epsilon,\tau)$
$\displaystyle=\frac{1}{|W_{G_{\text{2d}}}|}\sum_{u_{\ast}\in\mathfrak{M}_{\text{sing}}}\mathop{\text{JK-
Res}}_{u={u}_{*}}({Q}_{*},{\eta})$ $\displaystyle\quad\times\prod
Z_{\text{2d.vec}}\prod Z_{\text{2d.hyp}}\,du^{1}\wedge\cdots\wedge
du^{\mathrm{rk}(\mathfrak{g}_{\text{2d}})}\,.$ (4.1)

Here $G_{\text{2d}}$ and $\mathfrak{g}_{\text{2d}}$ are the gauge group and
its Lie algebra of 2d GLSM, respectively. $\text{rk}(\mathfrak{g})$ is the
rank of the Lie algebra $\mathfrak{g}$.

For 5d $\mathcal{N}=1^{*}$ $U(N)$ gauge theory, the GLSMs for the monopole
bubbling preserve the 2d $\mathcal{N}=(4,4)$ supersymmetry ( more precisely
the mass deformation of $\mathcal{N}=(4,4)$ supersymmetry). For
$G_{\text{2d}}=U(n)$ 2d $\mathcal{N}=(4,4)$ GLSM, the one-loop determinant for
the vector multiplet is

$\displaystyle
Z_{\text{2d.vec}}=\left(2\pi\eta(\tau)^{3}\right)^{n}\prod_{1\leq a\neq b\leq
n}\vartheta_{1}(u_{a}-u_{b})\cdot\prod_{a,b=1}^{n}\frac{\vartheta_{1}(u_{a}-u_{b}+\epsilon)}{\vartheta_{1}(u_{a}-u_{b}\pm{m}_{\text{ad}}+\frac{1}{2}\epsilon)}\,.$
(4.2)

A fugacity $m_{\text{ad}}$ corresponds to the flavor fugacity for 5d
$\mathcal{N}=1^{*}$ theory. The one-loop determinant for a 2d
$\mathcal{N}=(4,4)$ hypermultiplet in the bifundamental representation of
$U(n)\times U(n^{\prime})$ is

$\displaystyle
Z_{\text{2d.hyp}}=\prod_{a=1}^{n}\prod_{b=1}^{n^{\prime}}\frac{\vartheta_{1}(\pm(u_{a}-u^{\prime}_{b})-m_{\text{ad}})}{\vartheta_{1}(\pm(u_{a}-u^{\prime}_{b})+\frac{1}{2}\epsilon)}\,.$
(4.3)

When $U(n^{\prime})$ is a flavor symmetry of 2d GLSM, fugacity $u^{\prime}$
corresponds to components of the flat connection ${\bm{a}}$ of the five-
dimensional gauge field.

$\mathop{\text{JK-Res}}_{u={u}_{*}}({Q}_{*},{\eta})$ in (4.1) denotes the
Jeffrey–Kirwan residue at a point $u_{*}$ defined as follows. We consider a
situation that $\mathrm{rk}(\mathfrak{g}_{\text{2d}})$ hyperplanes of
codimension one, called singular hyperplanes
${Q}_{i}\cdot({u}-{u}_{*})=\sum_{a=1}^{\mathrm{rk}(\mathfrak{g}_{\text{2d}})}{Q}^{a}_{i}({u}^{a}-{u}^{a}_{*})=0$
$(i=1,\cdots,\mathrm{rk}(\mathfrak{g}_{\text{2d}}))$ intersect at a point
$u_{*}=(u^{1}_{*},\cdots,u^{\mathrm{rk}(\mathfrak{g}_{\text{2d}})}_{*})$ in
the $u$-space. In our case,
$Q_{i}=(Q^{1}_{i},\cdots,Q^{\text{rk}(\mathfrak{g}_{\text{2d}})}_{i})\in\mathbb{R}^{\mathrm{rk}(\mathfrak{g}_{\text{2d}})}$
is a weight vector appearing in the denominators of the one-loop determinants
(4.2) and (4.3). Then the JK residue at the point $u_{*}$ is defined by

$\displaystyle\mathop{\text{JK-
Res}}_{u={u}_{*}}({Q}_{*},{\eta})\frac{du^{1}\wedge\cdots\wedge
du^{\mathrm{rk}(\mathfrak{g}_{\text{2d}})}}{{Q}_{1}\cdot({u}-{u}_{*})\cdots
Q_{\mathrm{rk}(\mathfrak{g}_{\text{2d}})}\cdot({u}-{u}_{*})}$ (4.4)
$\displaystyle\qquad\qquad\qquad\qquad=\left\\{\begin{array}[]{cl}\frac{1}{|\det({Q}_{1},\ldots,{Q}_{\text{rk}(\mathfrak{g}_{\text{2d}})})|}&\text{
if
}{\eta}\in\text{Cone}({Q}_{1},\ldots,{Q}_{\text{rk}(\mathfrak{g}_{\text{2d}})})\,,\\\
0&\text{ otherwise}\,.\end{array}\right.$

Here
$\text{Cone}({Q}_{1},\ldots,{Q}_{\text{rk}(\mathfrak{g}_{\text{2d}})})=\sum_{i=1}^{\text{rk}(\mathfrak{g}_{\text{2d}})}\mathbb{R}_{>0}Q_{i}$
and each $Q_{i}$ is a $\text{rk}(\mathfrak{g}_{\text{2d}})$-dimensional
vector. The sum $\sum_{u_{\ast}}$ runs over all the points $u_{\ast}$, where
$N^{\prime}$ with $N^{\prime}=\text{rk}(\mathfrak{g}_{\text{2d}})$ singular
hyperplanes meet at a point and the condition
${\eta}\in\text{Cone}({Q}_{1},\ldots,{Q}_{\text{rk}(\mathfrak{g}_{\text{2d}})})$
is satisfied. If $N^{\prime}$ singular hyperplanes with
$N^{\prime}>\text{rk}(\mathfrak{g}_{\text{2d}})$ intersect at a point, we
apply the constructive definition of the JK residue in [32]. All the examples
treated in Section 6 satisfy the condition
$N^{\prime}=\text{rk}(\mathfrak{g}_{\text{2d}})$.

##  5 Deformation quantization and elliptic Ruijsenaars operators

In [2] we have shown that the deformation quantization of SUSY localization
formula of the ’t Hooft loops in the anti-symmetric representation in 4d
$\mathcal{N}=2^{*}$ $U(N)$ gauge theory can be identified with trigonometric
Macdonald operators by redefinitions of variables. We also gave the
identification between SUSY localization formulas for monopole operators and
representation of monopole operators as difference operators of rational type
in [9]. These observations motivate us to relate a deformation quantization of
vevs of ’t Hooft surface operators to difference operator of elliptic type.

We regard ${\bm{a}}=(a_{1},\cdots,a_{\mathrm{rk}(\mathfrak{g})})$,
${\bm{b}}=(b_{1},\cdots,b_{\mathrm{rk}(\mathfrak{g})})$ as the coordinate and
momentum variables, respectively, and define the quantization by

$\displaystyle[\hat{a}_{i},\hat{a}_{j}]=0,\quad[\hat{b}_{i},\hat{b}_{j}]=0,\quad[\hat{b}_{i},\hat{a}_{i}]=\epsilon
C_{ij}\,.$ (5.1)

where, and $C_{ij}$ is the $(i,j)$-component of the inverse matrix of the
Killing form $C^{ij}=\mathrm{Tr}(H_{i}H_{j})$, where
$\\{H_{i}\\}_{i=1}^{\mathrm{rk}(\mathfrak{g})}$ is bases of the Cartan
subalgebra of $\mathfrak{g}$.111 For the simple Lie algebras, we normalize
$H_{i}$ so that $C^{ij}$ agrees with the Cartan matrix. For $U(N)$ gauge
theory, $C^{ij}$ is normalized as $C^{ij}=\delta^{ij}$.

We define the quantization of a function $f({\bm{a}},{\bm{b}})$ by the Weyl-
Wigner transform:

$\displaystyle\widehat{f}(\widehat{\bm{a}},\widehat{\bm{b}}):=\left[\exp\left(\frac{\epsilon}{2}\sum_{i,j=1}^{N}C_{ij}\partial_{{b}_{i}}\partial_{{a}_{j}}\right)f({\bm{a}},{\bm{b}})\right]\Bigg{|}_{{\bm{a}}\mapsto\widehat{\bm{a}},{\bm{b}}\mapsto\widehat{\bm{b}}}.$
(5.2)

After the derivative is taken in (5.2), the ordering of $\hat{a}_{i}$ and
$\hat{b}_{i}$ is defined as follows. All the $a_{i}$’s should be on left side
and all $b_{i}$’s shoud be on the right side. Next we define the Moyal product
$f*g$ of $f$ and $g$ by

$\displaystyle
f({\bm{a}},{\bm{b}})*g({\bm{a}},{\bm{b}}):=\exp\left(\frac{\epsilon}{2}\sum_{i,j=1}^{N}C^{ij}(\partial_{\underline{a}_{i}}\partial_{b_{i}}-\partial_{a_{i}}\partial_{\underline{b}_{i}})\right)f({\bm{a}},{\bm{b}})g(\underline{\bm{a}},\underline{\bm{b}})|_{\underline{\bm{a}}={\bm{a}},\underline{\bm{b}}={\bm{b}}}\,.$
(5.3)

Since the Weyl-Wigner transform satisfies the following relation

$\displaystyle\widehat{f*g}=\hat{f}\hat{g},$ (5.4)

the algebra of the deformation quantization of surface operators can be
studied in term of the Moyal products. For example, the commutativity of the
operators $\hat{f}$ and $\hat{g}$ is equivalent to that of $f$ and $g$ with
respect to the Moyal product:

$\displaystyle[\hat{f},\hat{g}]=0\leftrightarrow f*g-g*f=0.$ (5.5)

In particular, we will see that the commutativity of elliptic Ruijsenaars
operators is rephrased as the absence of wall-crossing phenomena
($\eta$-independence) of elliptic genera for the monopole bubbling effects.
Recently the author of [33] observed that the localization formula [1] of ’t
Hooft loop operators with particular charges in 4d $\mathcal{N}=2^{*}$ gauge
theory agrees with a trigonometric limit of type-$A$ elliptic Ruijsenaars
operators [34]. Since the KK modes along the $T^{2}$-direction give an
elliptic deformation of the localization formula of the ’t Hooft loops, we
expect the deformation quantization of ’t Hooft surface operator itself agrees
with elliptic Ruisenaars operators. In fact, we show that the deformation
quantization of ’t Hooft surface operators with specific magnetic charge in 5d
$\mathcal{N}=1^{*}$ theory is identified with elliptic Ruijsenaars operators.
We also study the algebra of ’t Hooft surface operators defined by the Moyal
product .

From the the localization formula (3.1), the vevs of ’t Hooft surface
operators $S_{\bm{B}}$ with ${\bm{B}}=\sum_{k=1}^{\ell}{\bm{e}}_{k}$ and
${\bm{B}}=-\sum_{k=\ell+1}^{N}{\bm{e}}_{k}$ in the $\mathcal{N}=1^{*}$ $U(N)$
gauge theory are given by222A dominant coweight ${\bm{B}}$ of $U(N)$ is
$\text{diag}(B_{1},B_{2},\cdots,B_{N})$ with $B_{1}\geq B_{2}\geq\cdots\geq
B_{N}$ and $B_{i}\in\mathbb{Z}$.

$\displaystyle\langle S_{\sum_{k=1}^{\ell}{\bm{e}}_{k}}\rangle$
$\displaystyle=\sum_{I\subset\\{1,\cdots,N\\}\atop|I|=\ell}\left(\prod_{i\in
I,j\notin
I}\frac{\vartheta_{1}\left(a_{i}-a_{j}-m_{\mathrm{ad}}\right)\vartheta_{1}\left(a_{j}-a_{i}-m_{\mathrm{ad}}\right)}{\vartheta_{1}\left(a_{i}-a_{j}+\frac{\epsilon}{2}\right)\vartheta_{1}\left(a_{j}-a_{i}+\frac{\epsilon}{2}\right)}\right)^{\frac{1}{2}}\prod_{i\in
I}e^{b_{i}},$ (5.6) $\displaystyle\langle
S_{-\sum_{k=\ell+1}^{N}{\bm{e}}_{k}}\rangle$
$\displaystyle=\sum_{I\subset\\{1,\cdots,N\\}\atop|I|=\ell}\left(\prod_{i\in
I,j\notin
I}\frac{\vartheta_{1}\left(a_{i}-a_{j}-m_{\mathrm{ad}}\right)\vartheta_{1}\left(a_{j}-a_{i}-m_{\mathrm{ad}}\right)}{\vartheta_{1}\left(a_{i}-a_{j}+\frac{\epsilon}{2}\right)\vartheta_{1}\left(a_{j}-a_{i}+\frac{\epsilon}{2}\right)}\right)^{\frac{1}{2}}\prod_{i\in
I}e^{-b_{i}}$ (5.7)

Here $m_{\mathrm{ad}}$ is the flavor fugacity for the adjoint hypermultiplet
in five dimensions. $I$ is a subset of $\\{1,\cdots,N\\}$ with the cardinality
$|I|=\ell$. Then the Weyl-Wigner transform of the vevs of surface operators
are written as

$\displaystyle\widehat{S}_{\sum_{k=1}^{\ell}{\bm{e}}_{k}}$
$\displaystyle=\sum_{I\subset\\{1,\cdots,N\\}\atop|I|=\ell}\left(\prod_{i\in
I,j\notin
I}\frac{\vartheta_{1}\left(a_{i}-a_{j}-m_{\mathrm{ad}}+\frac{\epsilon}{2}\right)\vartheta_{1}\left(a_{j}-a_{i}-m_{\mathrm{ad}}-\frac{\epsilon}{2}\right)}{\vartheta_{1}\left(a_{i}-a_{j}+\epsilon\right)\vartheta_{1}\left(a_{j}-a_{i}\right)}\right)^{\frac{1}{2}}\prod_{i\in
I}e^{\epsilon\frac{\partial}{\partial a_{i}}}\,,$ (5.8)
$\displaystyle\widehat{S}_{-\sum_{k=\ell+1}^{N}{\bm{e}}_{k}}$
$\displaystyle=\sum_{I\subset\\{1,\cdots,N\\}\atop|I|=\ell}\left(\prod_{i\in
I,j\notin
I}\frac{\vartheta_{1}\left(a_{i}-a_{j}-m_{\mathrm{ad}}-\frac{\epsilon}{2}\right)\vartheta_{1}\left(a_{j}-a_{i}-m_{\mathrm{ad}}+\frac{\epsilon}{2}\right)}{\vartheta_{1}\left({a}_{i}-{a}_{j}\right)\vartheta_{1}\left(a_{j}-a_{i}+\epsilon\right)}\right)^{\frac{1}{2}}\prod_{i\in
I}e^{-\epsilon\frac{\partial}{\partial a_{i}}}\,.$ (5.9)

Here we use a representation $\hat{a}_{i}=a_{i}$ and
$\hat{b}_{i}=\epsilon\frac{\partial}{\partial a_{i}}$, and find that the
deformation quantization of the ’t Hooft surface operators (5.8) and (5.9)
agree with type-$A$ elliptic Ruijsenaars operators [34].

##  6 Products of ’t Hooft surface operators and monopole bubbling

In this section, we evaluate explicitly elliptic genera for monopole bubbling
effects with small values of magnetic charges and compare them with results
from the Moyal product. We consider monopole bubbling effects in the monopole
surface operator $S_{\bm{B}}$ for
${\bm{B}}=(1,0,\cdots,0,-1)=\mathrm{diag}(1,0,\cdots,0,-1)$ and
${\bm{B}}=(1,1,0,\cdots,0,-1,-1)=\mathrm{diag}(1,1,0,\cdots,0,-1,-1)$ in
$\mathcal{N}=1^{*}$ $U(N)$ gauge theory.

###  6.1 Surface operator $S_{(1,0,\cdots,0,-1)}$

From the localization formula (3.1) and the definition of the Moyal product
(5.3), two different orderings of the Moyal product of $\langle
S_{{\bm{e}}_{1}}\rangle$ and $\langle S_{-{\bm{e}}_{N}}\rangle$ are evaluated
as

$\displaystyle\langle S_{{\bm{e}}_{1}}\rangle*\langle
S_{-{\bm{e}}_{N}}\rangle$ $\displaystyle=\sum_{1\leq i\neq j\leq
N}e^{{b_{i}-b_{j}}}Z^{\text{5d}}_{1\text{-loop}}({\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j})$
$\displaystyle+\sum_{i=1}^{N}\prod_{j=1\atop j\neq
i}^{N}\frac{\vartheta_{1}(a_{i}-a_{j}-m_{\text{ad}}+\frac{1}{2}\epsilon)\vartheta_{1}(a_{j}-a_{i}-m_{\text{ad}}-\frac{1}{2}\epsilon)}{\vartheta_{1}(a_{i}-a_{j}+\epsilon)\vartheta_{1}(a_{j}-a_{i})}\,,$
(6.1) $\displaystyle\langle S_{-{\bm{e}}_{N}}\rangle*\langle
S_{{\bm{e}}_{1}}\rangle$ $\displaystyle=\sum_{1\leq i\neq j\leq
N}e^{{b_{i}-b_{j}}}Z^{\text{5d}}_{1\text{-loop}}({\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j})$
$\displaystyle+\sum_{i=1}^{N}\prod_{j=1\atop j\neq
i}^{N}\frac{\vartheta_{1}(a_{i}-a_{j}-m_{\text{ad}}-\frac{1}{2}\epsilon)\vartheta_{1}(a_{j}-a_{i}-m_{\text{ad}}+\frac{1}{2}\epsilon)}{\vartheta_{1}(a_{i}-a_{j})\vartheta_{1}(a_{j}-a_{i}+\epsilon)}\,.$
(6.2)

Here the explicit form of the one-loop determinant
$Z^{\text{5d}}_{1\text{-loop}}({\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j})$ is
summarized in Appendix B. Although the second line in (6.1) looks different
from the one in (6.2), we will show these terms are actually same by using the
contour integral expression of an elliptic genus. This leads to the Moyal
products of $\langle S_{-{\bm{e}}_{N}}\rangle$ and $\langle
S_{{\bm{e}}_{1}}\rangle$ commute each other. $\langle
S_{{\bm{e}}_{1}}\rangle*\langle S_{-{\bm{e}}_{N}}\rangle=\langle
S_{{\bm{e}}_{1}}\rangle*\langle S_{-{\bm{e}}_{N}}\rangle$.

We evaluate the monopole bubbling effect in the expectation value of
$S_{\bm{B}}$ with ${\bm{B}}={\bm{e}}_{1}-{\bm{e}}_{N}=(1,0\cdots,0,-1)$ and
study the algebra of ’t Hooft surface operators. From the localization formula
(3.1), $\langle S_{{\bm{e}}_{1}-{\bm{e}}_{N}}\rangle$ is given by

$\displaystyle\langle S_{{\bm{e}}_{1}-{\bm{e}}_{N}}\rangle$
$\displaystyle=\sum_{1\leq i\neq j\leq
N}e^{{b_{i}-b_{j}}}Z^{\text{5d}}_{1\text{-loop}}({\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j})+Z_{\text{mono}}({\bm{p}}={\bm{0}},{\bm{B}}={\bm{e}}_{1}-{\bm{e}}_{N}).$
(6.3)

The monopole bubbling effect in (6.3) is evaluated by the brane construction
and localization formula for the elliptic genus. As explained in Section 4.1,
the brane configuration for the monopole bubbling effect
${\bm{B}}={\bm{e}}_{1}-{\bm{e}}_{N}$ and ${\bm{p}}=0$ is depicted in Figure 2.
The quiver diagram of the bubbling GLSM is specified by Figure 2. From the
localization formula, the JK residue for the elliptic genus of the GLSM in
Figure 2 is same as the following contour integral:

$\displaystyle
Z^{(\eta)}_{\text{mono}}({\bm{p}}={\bm{0}},{\bm{B}}={\bm{e}}_{1}-{\bm{e}}_{N})=\text{sign}(\eta)\frac{2\eta(\tau)^{3}\vartheta_{1}(\epsilon)}{\vartheta_{1}(\pm
m_{\text{ad}}+\frac{1}{2}\epsilon)}\oint{du}\prod_{i=1}^{N}\frac{\vartheta_{1}(\pm(u-a_{i})-m_{\text{ad}})}{\vartheta_{1}(\pm(u-a_{i})+\frac{1}{2}\epsilon)}.$
(6.4)

Here $\text{sign}(\eta)$ is $+1$ for $\eta>0$ and $-1$ for $\eta<0$,
respectively. When $\eta>0$, the JK residue operation is the residues at poles
$u=a_{i}-\frac{1}{2}\epsilon$ for $i=1,\cdots,N$. On the other hand, when
$\eta<0$, the JK residue operation is the residues at poles
$u=a_{i}+\frac{1}{2}\epsilon$ for $i=1,\cdots,N$. The elliptic genus for the
the monopole bubbling effect is given by

$\displaystyle
Z^{(\eta>0)}_{\text{mono}}({\bm{p}}={\bm{0}},{\bm{B}}={\bm{e}}_{1}-{\bm{e}}_{N})=\sum_{i=1}^{N}\prod_{j=1\atop
j\neq
i}^{N}\frac{\vartheta_{1}(a_{i}-a_{j}-m_{\text{ad}}-\frac{1}{2}\epsilon)\vartheta_{1}(a_{j}-a_{i}-m_{\text{ad}}+\frac{1}{2}\epsilon)}{\vartheta_{1}(a_{i}-a_{j})\vartheta_{1}(a_{j}-a_{i}+\epsilon)},$
(6.5) $\displaystyle
Z^{(\eta<0)}_{\text{mono}}({\bm{p}}={\bm{0}},{\bm{B}}={\bm{e}}_{1}-{\bm{e}}_{N})=\sum_{i=1}^{N}\prod_{j=1\atop
j\neq
i}^{N}\frac{\vartheta_{1}(a_{j}-a_{i}-m_{\text{ad}}+\frac{1}{2}\epsilon)\vartheta_{1}(a_{i}-a_{j}-m_{\text{ad}}-\frac{1}{2}\epsilon)}{\vartheta_{1}(a_{i}-a_{j}+\epsilon)\vartheta_{1}(a_{j}-a_{i})}.$
(6.6)

Note that $Z^{(\eta>0)}_{\text{mono}}=Z^{(\eta<0)}_{\text{mono}}$. This
follows from the fact that the sum of the residues of a rational form on the
torus is zero.

By comparing the expressions of $\langle S_{{\bm{e}}_{1}}\rangle*\langle
S_{-{\bm{e}}_{N}}\rangle$, $\langle S_{-{\bm{e}}_{N}}\rangle*\langle
S_{{\bm{e}}_{1}}\rangle$ and $\langle S_{{\bm{e}}_{1}-{\bm{e}}_{N}}\rangle$,
we find that the algebraic relation of the ’t Hooft surface operators:

$\displaystyle\langle{S}_{{\bm{e}}_{1}-{\bm{e}}_{N}}\rangle=\langle{S}_{{\bm{e}}_{1}}\rangle*\langle{S}_{-{\bm{e}}_{N}}\rangle=\langle{S}_{-{\bm{e}}_{N}}\rangle*\langle{S}_{{\bm{e}}_{1}}\rangle$
(6.7)

Therefore we find that a surface operator with a higher charge is generated by
surface operators with the minimal charges. Applying the Weyl-Wigner transform
to
$\langle{S}_{{\bm{e}}_{1}}\rangle*\langle{S}_{-{\bm{e}}_{N}}\rangle=\langle{S}_{-{\bm{e}}_{N}}\rangle*\langle{S}_{{\bm{e}}_{1}}\rangle$,
we obtain the commutativity of two elliptic Ruijsenaars operators
$[\widehat{S}_{{\bm{e}}_{1}},\widehat{S}_{-{\bm{e}}_{N}}]=0$.

###  6.2 Surface operator $S_{(1,1,0,\cdots,0,-1,-1)}$

Figure 3: (a): The brane configuration for the surface operator with the
magnetic charge
${\bm{B}}={\bm{e}}_{1}+{\bm{e}}_{2}-{\bm{e}}_{N-1}-{\bm{e}}_{N}$. (b) A
D3-brane suspended between the second D5-brane and the $(N-1)$-th D5-brane is
introduced to describe the ’t Hooft-Polyakov monopole with the charge
${\bm{B}}=-{\bm{e}}_{2}+{\bm{e}}_{N-1}$. (c): After Hanany-Witten effect, we
obtain the bran configuration for describing the monopole bubbling effect
${\bm{p}}={\bm{e}}_{1}-{\bm{e}}_{N}$.

Next we study the monopole bubbling effects in $\langle
S_{(1,1,0,\cdots,0,-1,-1)}\rangle=\langle
S_{{\bm{e}}_{1}+{\bm{e}}_{2}-{\bm{e}}_{N-1}-{\bm{e}}_{N}}\rangle$, and study
the algebra of ’t Hooft surface operators. The Moyal products of $\langle
S_{{\bm{e}}_{1}+{\bm{e}}_{2}}\rangle$ and $\langle
S_{-{\bm{e}}_{N-1}-{\bm{e}}_{N}}\rangle$ are evaluated as

$\displaystyle\langle S_{{\bm{e}}_{1}+{\bm{e}}_{2}}\rangle*\langle
S_{-{\bm{e}}_{N-1}-{\bm{e}}_{N}}\rangle=\sum_{1\leq i<j\leq N,1\leq k<l\leq
N\atop\\{k,l\\}\cap\\{i,j\\}=\emptyset}e^{b_{i}+b_{j}-b_{k}-b_{l}}Z^{\text{5d}}_{1\text{-loop}}({\bm{p}}={\bm{e}}_{i}+{\bm{e}}_{j}-{\bm{e}}_{k}-{\bm{e}}_{l})$
$\displaystyle+\sum_{1\leq i\neq j\leq
N}e^{{b_{i}-b_{j}}}Z^{\text{5d}}_{1\text{-loop}}({\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j})\sum_{l=1\atop
l\neq i,j}^{N}\prod_{k=1\atop k\neq
l}^{N}\frac{\vartheta_{1}(a_{l}-a_{k}-m_{\text{ad}}+\frac{1}{2}\epsilon)\vartheta_{1}(a_{k}-a_{l}-m_{\text{ad}}-\frac{1}{2}\epsilon)}{\vartheta_{1}(a_{l}-a_{k}+\epsilon)\vartheta_{1}(a_{k}-a_{l})}$
$\displaystyle\quad+\sum_{i,j=1}^{N}\prod_{l=i,j}\prod_{k=1\atop k\neq
i,j}^{N}\frac{\vartheta_{1}(a_{l}-a_{k}-m_{\text{ad}}+\frac{1}{2}\epsilon)\vartheta_{1}(a_{k}-a_{l}-m_{\text{ad}}-\frac{1}{2}\epsilon)}{\vartheta_{1}(a_{l}-a_{k}+\epsilon)\vartheta_{1}(a_{k}-a_{l})},$
(6.8) $\displaystyle\langle S_{-{\bm{e}}_{N-1}-{\bm{e}}_{N}}\rangle*\langle
S_{{\bm{e}}_{1}+{\bm{e}}_{2}}\rangle=\sum_{1\leq i<j\leq N,1\leq k<l\leq
N\atop\\{k,l\\}\cap\\{i,j\\}=\emptyset}e^{b_{i}+b_{j}-b_{k}-b_{l}}Z^{\text{5d}}_{1\text{-loop}}({\bm{p}}={\bm{e}}_{i}+{\bm{e}}_{j}-{\bm{e}}_{k}-{\bm{e}}_{l})$
$\displaystyle+\sum_{1\leq i\neq j\leq
N}e^{{b_{i}-b_{j}}}Z^{\text{5d}}_{1\text{-loop}}({\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j})\sum_{l=1\atop
l\neq i,j}^{N}\prod_{k=1\atop k\neq
l}^{N}\frac{\vartheta_{1}(a_{l}-a_{k}-m_{\text{ad}}-\frac{1}{2}\epsilon)\vartheta_{1}(a_{k}-a_{l}-m_{\text{ad}}+\frac{1}{2}\epsilon)}{\vartheta_{1}(a_{l}-a_{k})\vartheta_{1}(a_{k}-a_{l}+\epsilon)}$
$\displaystyle\quad+\sum_{i,j=1}^{N}\prod_{l=i,j}\prod_{k=1\atop k\neq
i,j}^{N}\frac{\vartheta_{1}(a_{l}-a_{k}-m_{\text{ad}}-\frac{1}{2}\epsilon)\vartheta_{1}(a_{k}-a_{l}-m_{\text{ad}}+\frac{1}{2}\epsilon)}{\vartheta_{1}(a_{l}-a_{k})\vartheta_{1}(a_{k}-a_{l}+\epsilon)}.$
(6.9)

We evaluate $\langle
S_{{\bm{e}}_{1}+{\bm{e}}_{2}-{\bm{e}}_{N-1}-{\bm{e}}_{N}}\rangle$ and compare
it with the Moyal products (6.8) and (6.9). The localization formula for
$\langle S_{{\bm{e}}_{1}+{\bm{e}}_{2}-{\bm{e}}_{N-1}-{\bm{e}}_{N}}\rangle$ has
the following expression.

$\displaystyle\langle
S_{{\bm{e}}_{1}+{\bm{e}}_{2}-{\bm{e}}_{N-1}-{\bm{e}}_{N}}\rangle$
$\displaystyle=\sum_{1\leq i<j\leq N,1\leq k<l\leq
N\atop\\{k,l\\}\cap\\{i,j\\}=\emptyset}e^{b_{i}+b_{j}-b_{k}-b_{l}}Z^{\text{5d}}_{1\text{-loop}}({\bm{p}}={\bm{e}}_{i}+{\bm{e}}_{j}-{\bm{e}}_{k}-{\bm{e}}_{l})$
$\displaystyle+\sum_{1\leq i\neq j\leq
N}e^{{b_{i}-b_{j}}}Z^{\text{5d}}_{1\text{-loop}}({\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j})Z_{\text{mono}}({\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j})$
$\displaystyle\quad+Z_{\text{mono}}({\bm{p}}={\bm{0}}).$ (6.10)

Here we suppressed
${\bm{B}}={\bm{e}}_{1}+{\bm{e}}_{2}-{\bm{e}}_{N-1}-{\bm{e}}_{N}$ in
$Z_{\text{mono}}$’s to shorten the expression. The one-loop determinants
$Z^{\text{5d}}_{1\text{-loop}}$ are given by (B) and (B.2). In this case,
there are two monopole bubbling sectors specified by
${\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j}$ and ${\bm{p}}={\bm{0}}$.

First let us evaluate the monopole bubbling effect specified by
${\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j}$. The brane configuration for the ’t Hooft
surface operator with the magnetic charge
${\bm{B}}={\bm{e}}_{1}+{\bm{e}}_{2}-{\bm{e}}_{N-1}-{\bm{e}}_{N}$ is depicted
by Figure 3. We introduce a D3-brane suspended between two D5-branes as Figure
3, which correspond to an ’t Hooft-Polyakov monopole. After Hanany-Witten
transition, the matter contents of the low energy world volume theory on the
D3-brane is that the gauge group is $U(1)$ and the number of
$\mathcal{N}=(4,4)$ hypermultiplets is $N-2$. For example the brane
configuration for $i=1,j=N$ depicted by Figure 3. Then the monopole bubbling
effect for ${\bm{B}}={\bm{e}}_{1}+{\bm{e}}_{2}-{\bm{e}}_{N-1}-{\bm{e}}_{N}$
and ${\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j}$ is given by

$\displaystyle
Z^{(\eta)}_{\text{mono}}({\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j})=\text{sign}(\eta)\frac{2\eta(\tau)^{3}\vartheta_{1}(\epsilon)}{\vartheta_{1}(\pm
m_{\text{ad}}+\frac{1}{2}\epsilon)}\oint{du}\prod_{k=1\atop k\neq
i,j}^{N}\frac{\vartheta_{1}(\pm(u-a_{k})-m_{\text{ad}})}{\vartheta_{1}(\pm(u-a_{k})+\frac{1}{2}\epsilon)}.$
(6.11)

(6.11) is evaluate in similar way as (6.4).

$\displaystyle
Z^{(\eta>0)}_{\text{mono}}({\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j})=\sum_{l=1\atop
l\neq i,j}^{N}\prod_{k=1\atop k\neq
l}^{N}\frac{\vartheta_{1}(a_{l}-a_{k}-m_{\text{ad}}-\frac{1}{2}\epsilon)\vartheta_{1}(a_{k}-a_{l}-m_{\text{ad}}+\frac{1}{2}\epsilon)}{\vartheta_{1}(a_{l}-a_{k})\vartheta_{1}(a_{k}-a_{l}+\epsilon)},$
(6.12) $\displaystyle
Z^{(\eta<0)}_{\text{mono}}({\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j})=\sum_{l=1\atop
l\neq i,j}^{N}\prod_{k=1\atop k\neq
l}^{N}\frac{\vartheta_{1}(a_{l}-a_{k}-m_{\text{ad}}+\frac{1}{2}\epsilon)\vartheta_{1}(a_{k}-a_{l}-m_{\text{ad}}-\frac{1}{2}\epsilon)}{\vartheta_{1}(a_{l}-a_{k}+\epsilon)\vartheta_{1}(a_{k}-a_{l})}.$
(6.13)

Again we have
$Z^{(\eta>0)}_{\text{mono}}({\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j})=Z^{(\eta<0)}_{\text{mono}}({\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j})$.

Figure 4: (a): We added two D3-branes suspended between D5-branes to Figure 3.
(b) When the positions of the segments of D3-branes coincide, the screening of
monopole charges charge occurs. The world volume theory on two D3-branes give
the monopole bubbling effect for ${\bm{p}}={\bm{0}}$. (c): The quiver diagram
denotes the matter content of the low energy world volume theory on the
D3-branes of Figure 4. The circle with $2$ denotes $\mathcal{N}=(4,4)$ $U(2)$
vector multiplet, and the solid line denotes the $N$ hypermultiplets in the
fundamental representation of $U(2)$.

Next we evaluate the monopole bubbling effect specified by ${\bm{p}}={\bm{0}}$
from the brane construction. To achieve the monopole bubbling effect, we
introduce two D3-branes suspended between D5-branes depicted as Figure 4. When
the positions of the D3-branes coincides, we obtain the brane configuration in
Figure 4. The low energy world volume theory on two D3-branes in 4 is the 2d
$\mathcal{N}=(4,4)$ $U(2)$ GLSM with $N$ hypermultiplets. The localization
formula for the elliptic genus is given by

$\displaystyle Z^{(\eta)}_{\text{mono}}({\bm{p}}={\bm{0}})$
$\displaystyle=\frac{\left(2\pi\eta(\tau)^{3}\right)^{2}}{2}\sum_{u_{\ast}\in\mathfrak{M}_{\text{sing}}}\mathop{\text{JK-
Res}}_{u={u}_{*}}({Q}_{*},{\eta})\prod_{1\leq a\neq b\leq
2}\vartheta_{1}(u_{a}-u_{b})$
$\displaystyle\times\prod_{a,b=1}^{2}\frac{\vartheta_{1}(u_{a}-u_{b}+\epsilon)}{\vartheta_{1}(u_{a}-u_{b}\pm{m}_{\text{ad}}+\frac{1}{2}\epsilon)}\cdot\prod_{b=1}^{2}\prod_{i=1}^{N}\frac{\vartheta_{1}(\pm(u_{b}-a_{i})-m_{\text{ad}})}{\vartheta_{1}(\pm(u_{b}-a_{i})+\frac{1}{2}\epsilon)}du_{1}\wedge
du_{2}\,.$ (6.14)

When a vector $\eta$ is proportional to $(1,1)$, From the definition of the JK
residues, $\mathfrak{M}_{\text{sing}}$ in the $(u_{1},u_{2})$ space, where JK
residues are evaluated are specified by the intersection point of the
following singular hyperplanes:

$\displaystyle\left\\{u_{1}-a_{i}+\frac{1}{2}\epsilon=0\right\\}\cap\left\\{u_{2}-a_{j}+\frac{1}{2}\epsilon=0\right\\}\text{
for }i,j=1,\cdots,N,$ (6.15)
$\displaystyle\left\\{u_{2}-u_{1}\pm{m}_{\text{ad}}+\frac{1}{2}\epsilon\right\\}\cap\left\\{u_{1}-a_{i}+\frac{1}{2}\epsilon=0\right\\}\text{
for }i=1,\cdots,N,$ (6.16)
$\displaystyle\left\\{u_{1}-u_{2}\pm{m}_{\text{ad}}+\frac{1}{2}\epsilon\right\\}\cap\left\\{u_{2}-a_{i}+\frac{1}{2}\epsilon=0\right\\}\text{
for }i=1,\cdots,N\,.$ (6.17)

The JK residues at the intersection points of (6.16) and (6.17) are zero and
the monopole bubbling effect is given by the JK residue at the intersection
points of (6.15) as

$\displaystyle
Z^{(\eta=(1,1))}_{\text{mono}}({\bm{p}}={\bm{0}},{\bm{B}}={\bm{e}}_{1}-{\bm{e}}_{N})$
$\displaystyle\qquad=\sum_{i,j=1}^{N}\prod_{l=i,j}\prod_{k=1\atop k\neq
i,j}^{N}\frac{\vartheta_{1}(a_{l}-a_{k}-m_{\text{ad}}-\frac{1}{2}\epsilon)\vartheta_{1}(a_{k}-a_{l}-m_{\text{ad}}+\frac{1}{2}\epsilon)}{\vartheta_{1}(a_{l}-a_{k})\vartheta_{1}(a_{k}-a_{l}+\epsilon)}.$
(6.18)

In the similar way, when $\eta$ is proportional to (-1,-1), JK residues are
evaluated at

$\displaystyle\left\\{-(u_{1}-a_{i})+\frac{1}{2}\epsilon=0\right\\}\cap\left\\{-(u_{2}-a_{j})+\frac{1}{2}\epsilon=0\right\\}\text{
for }i,j=1,\cdots,N,$ (6.19)
$\displaystyle\left\\{u_{1}-u_{2}\pm{m}_{\text{ad}}+\frac{1}{2}\epsilon\right\\}\cap\left\\{-(u_{1}-a_{i})+\frac{1}{2}\epsilon=0\right\\}\text{
for }i=1,\cdots,N,$ (6.20)
$\displaystyle\left\\{u_{2}-u_{1}\pm{m}_{\text{ad}}+\frac{1}{2}\epsilon\right\\}\cap\left\\{-(u_{2}-a_{i})+\frac{1}{2}\epsilon=0\right\\}\text{
for }i=1,\cdots,N\,.$ (6.21)

the monopole bubbling effect are given by

$\displaystyle
Z^{(\eta=(-1,-1))}_{\text{mono}}({\bm{p}}={\bm{0}},{\bm{B}}={\bm{e}}_{1}-{\bm{e}}_{N})$
$\displaystyle\qquad=\sum_{i,j=1}^{N}\prod_{l=i,j}\prod_{k=1\atop k\neq
i,j}^{N}\frac{\vartheta_{1}(a_{l}-a_{k}-m_{\text{ad}}+\frac{1}{2}\epsilon)\vartheta_{1}(a_{k}-a_{l}-m_{\text{ad}}-\frac{1}{2}\epsilon)}{\vartheta_{1}(a_{l}-a_{k}+\epsilon)\vartheta_{1}(a_{k}-a_{l})}.$
(6.22)

For the same reason as before,
$Z^{(\eta=(1,1))}_{\text{mono}}=Z^{(\eta=(-1,-1))}_{\text{mono}}$. By
comparing the Moyal products with the SUSY localization formula for (6.10), we
find the algebraic relation between vevs of ’t Hooft surface operators:

$\displaystyle\langle
S_{{\bm{e}}_{1}+{\bm{e}}_{2}-{\bm{e}}_{N-1}-{\bm{e}}_{N}}\rangle=\langle
S_{{\bm{e}}_{1}+{\bm{e}}_{2}}\rangle*\langle
S_{-{\bm{e}}_{N-1}-{\bm{e}}_{N}}\rangle=\langle
S_{-{\bm{e}}_{N-1}-{\bm{e}}_{N}}\rangle*\langle
S_{{\bm{e}}_{1}+{\bm{e}}_{2}}\rangle.$ (6.23)

By applying the Weyl-Wigner transform to (6.23), we obtain the commutation
relation of elliptic Ruijsenaars operators:
$[\widehat{S}_{{\bm{e}}_{1}+{\bm{e}}_{2}},\widehat{S}_{-{\bm{e}}_{N-1}-{\bm{e}}_{N}}]=0$.

##  7 Discussion

We discuss the results in our paper and the future directions. We introduced
magnetically charged surface operators on $T^{2}\times\mathbb{R}^{3}$ and
evaluated the expectation values in terms of supersymmetric localization. The
SUSY localization formula obtained in this paper gives an elliptic deformation
of localization formula for BPS ’t Hooft loops on $S^{1}\times\mathbb{R}^{3}$
[1] and the one for BPS bare monopole operators in [2, 3]. We concentrated on
5d $\mathcal{N}=1^{*}$ $U(N)$ gauge theory and concretely studied the algebra
of surface operators and monopole bubbling effects with small magnetic
charges. For general 5d $\mathcal{N}=1$ gauge theories, monopole bubbling
effects are described by elliptic genera of 2d $\mathcal{N}=(0,4)$ GLSMs,
which are obtained by the T-dual picture of the brane configuration studied in
[26]. The gauge anomaly cancellation condition for 2d $\mathcal{N}=(0,4)$
GLSMs give constraints on the matter contents of the five dimensional theory
and also magnetic charge ${\bm{B}}$. So far we do not have a clear
understanding of the role of gauge anomaly cancellation condition from the
five-dimensional view point.

In Section 5, we found that the deformation quantization of ’t Hooft surface
operators in 5d $\mathcal{N}=1^{*}$ gauge theory agrees with the type-A
elliptic Ruijsenaars operators. Although the integrable structure appears in
the ’t Hooft surface operators is not manifest in the supersymmetric gauge
theory, by using dualities in string theory, the brane configuration for ’t
Hooft operators without the monopole bubbling effect is interpreted as defects
in four-dimensional Chern-Simons theory, where the quantum integrable
structure naturally appear [33].

In Section 6, we studied monopole bubbling effects and algebraic relation of
’t Hooft surface operators. The physical interpretation of the Moyal product
and Weyl-Wigner transform are as follows. For ’t Hooft loop operator on
$S^{1}\times\mathbb{R}^{2}_{\epsilon}\times\mathbb{R}$, the Moyal product of
vevs of $n$ ’t Hooft loops is identified with $n$-point correlation functions
of ’t Hooft loop operators [1] [25]. Then we expect a similar result holds,
i.e.,

$\displaystyle\langle S_{{\bm{B}}_{1}}\cdot S_{{\bm{B}}_{2}}\cdots
S_{{\bm{B}}_{n-1}}\cdot S_{{\bm{B}}_{n}}\rangle=\langle
S_{{\bm{B}}_{1}}\rangle*\langle S_{{\bm{B}}_{2}}\rangle*\cdots*\langle
S_{{\bm{B}}_{n-1}}\rangle*\langle S_{{\bm{B}}_{n}}\rangle.$ (7.1)

Here the center of Dirac monopole for the surface operator $S_{{\bm{B}}_{i}}$
for $i=1,\cdots,n$ locates at $(x^{1},x^{2},x^{3})=(0,0,x^{3}_{i})$ with
$x^{3}_{1}>x^{3}_{2}>\cdots>x^{3}_{n}$. If an algebraic relation we find is
written as $\langle S_{{\bm{B}_{1}}+{\bm{B}_{2}}}\rangle=\langle
S_{{\bm{B}_{1}}}\rangle*\langle S_{{\bm{B}_{2}}}\rangle$, (7.1) implies that
the operator product expansion of ’t Hooft surface operators
$S_{{\bm{B}}_{2}}S_{{\bm{B}}_{2}}=S_{{\bm{B}_{1}}+{\bm{B}_{2}}}$ holds in the
correlation function. It is desirable to evaluate correlation functions for
surface operators in terms of supersymmetric localization and to establish the
relation between the correlation functions and the Moyal products of vevs of a
single ’t Hooft surface operator.

Another future direction is to study an elliptic deformation of the Coulomb
branch chiral rings for three dimensional gauge theories. In 3d
$\mathcal{N}=4$ gauge theories, the algebra of the BPS bare monopole
operators, the coulomb branch scalars and the BPS dressed monopole operators
is defined in terms of the Moyal product and the Weyl-Wigner transform [2].
Then it found that the algebra agrees with the Coulomb branch chiral ring and
its deformation quantization (a.k.a quantized Coulomb branch) in [7]. Here the
three dimensional version of algebraic relations studied in Section 6 are
identified with ring relations of bare monopole operators in the quantized
Coulomb branch. Thus the algebraic relation we find is expected to be related
to ring relations in an elliptic deformation of the quantized Coulomb
branches. It is interesting to study the five-dimensional uplifts of the
Coulomb branch scalars and the dressed monopole operators and define an
elliptic deformation of the quantized Coulomb branches.

## Acknowledgements

The author would like to thank Hirotaka Hayashi, Kazunobu Maruyoshi, Hiraku
Nakajima, and Takuya Okuda for helpful discussions.

##  Appendix A Gamma matrices

We summarize the definition of gamma matrices and their useful properties. The
$16\times 16$ gamma matrices $\Gamma^{M}$ are defined as follows.

$\displaystyle{\Gamma}^{M}$
$\displaystyle=\left(\begin{array}[]{cc}0&E^{T}_{M+1}\\\
E_{M+1}&0\end{array}\right),\quad M=1,2,3,5,6,7,$ (A.3)
$\displaystyle{\Gamma}^{4}$
$\displaystyle=\left(\begin{array}[]{cc}0&E^{T}_{1}\\\
E_{1}&0\end{array}\right),\quad{\Gamma}^{8}=\left(\begin{array}[]{cc}0&E^{T}_{5}\\\
E_{5}&0\end{array}\right),$ (A.8) $\displaystyle{\Gamma}^{9}$
$\displaystyle=\left(\begin{array}[]{cc}1_{8\times 8}&0\\\ 0&1_{8\times
8}\end{array}\right),\quad{\Gamma}^{0}=\left(\begin{array}[]{cc}{\rm
i}1_{8\times 8}&0\\\ 0&{\rm i}1_{8\times 8}\end{array}\right).$ (A.13)

Here $E_{i}$ for $i=1,\cdots,8$ are defined by

$\displaystyle E_{\mu}$ $\displaystyle=\left(\begin{array}[]{cc}J_{\mu}&0\\\
0&\bar{J}_{\mu}\end{array}\right),\,\,\mu=1,2,3,4,\quad
E_{A}=\left(\begin{array}[]{cc}0&-J^{T}_{A}\\\
{J}_{A}&0\end{array}\right),A=5,6,7,8.$ (A.18)

with

$\displaystyle(J_{1},J_{2},J_{3},J_{4})$
$\displaystyle=\left(\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{array}\right),\left(\begin{array}[]{cccc}0&-1&0&0\\\
1&0&0&0\\\ 0&0&0&-1\\\
0&0&1&0\end{array}\right),\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&0&0&1\\\
1&0&0&0\\\ 0&-1&0&0\end{array}\right),\left(\begin{array}[]{cccc}0&0&0&-1\\\
0&0&-1&0\\\ 0&1&0&0\\\ 1&0&0&0\end{array}\right)\right),$ (A.35)
$\displaystyle(\bar{J}_{1},\bar{J}_{2},\bar{J}_{3},\bar{J}_{4})$
$\displaystyle=\left(\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{array}\right),\left(\begin{array}[]{cccc}0&-1&0&0\\\
1&0&0&0\\\ 0&0&0&1\\\
0&0&-1&0\end{array}\right),\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&0&0&-1\\\
1&0&0&0\\\ 0&1&0&0\end{array}\right),\left(\begin{array}[]{cccc}0&0&0&-1\\\
0&0&1&0\\\ 0&-1&0&0\\\ 1&0&0&0\end{array}\right)\right),$ (A.52)
$\displaystyle(J_{5},J_{6},J_{7},J_{8})$
$\displaystyle=\left(\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&0\\\
0&0&-1&0\\\ 0&0&0&-1\end{array}\right),\left(\begin{array}[]{cccc}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&1\\\
0&0&-1&0\end{array}\right),\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&-1\\\
1&0&0&0\\\ 0&1&0&0\end{array}\right),\left(\begin{array}[]{cccc}0&0&0&1\\\
0&0&1&0\\\ 0&-1&0&0\\\ 1&0&0&0\end{array}\right)\right)\,.$ (A.69)

Another gamma matrix $\tilde{\Gamma}^{M}$ is defined by

$\displaystyle\tilde{\Gamma}^{M}=\left\\{\
\begin{array}[]{ll}\displaystyle-\Gamma^{0}&\text{ for }M=0,\\\
\displaystyle\Gamma^{M}&\text{ for }M=1,2,\cdots,9.\end{array}\right.$ (A.72)

Note that ${\Gamma}^{M}$ and $\tilde{\Gamma}^{M}$ satisfy the relation

$\displaystyle\tilde{\Gamma}^{M}\Gamma^{N}+\tilde{\Gamma}^{N}\Gamma^{M}=2\delta^{MN},\quad\Gamma^{M}\tilde{\Gamma}^{N}+\Gamma^{N}\tilde{\Gamma}^{M}=2\delta^{MN},$
(A.73)
$\displaystyle(\Gamma^{M})^{T}=\Gamma^{M},\quad(\tilde{\Gamma}^{M})^{T}=\tilde{\Gamma}^{M}.$
(A.74)

$\Gamma^{MN}$, $\tilde{\Gamma}^{MN}$ and $\Gamma^{KLMN}$ are defined by

$\displaystyle\Gamma^{MN}$
$\displaystyle=\tilde{\Gamma}^{[M}\Gamma^{N]},\quad\tilde{\Gamma}^{MN}=\Gamma^{[M}\tilde{\Gamma}^{N]}\,,$
(A.75) $\displaystyle\Gamma^{KLMN}$
$\displaystyle=\tilde{\Gamma}^{[K}\Gamma^{L}\Gamma^{M}\Gamma^{N]}=\frac{1}{4!}\sum_{\sigma\in\mathfrak{S}_{4}}\mathrm{sgn}(\sigma)\tilde{\Gamma}^{\sigma(K)}\Gamma^{\sigma(L)}\Gamma^{\sigma(M)}\Gamma^{\sigma(N)}$
(A.76)

where $\mathfrak{S}_{4}$ is the permutation group of four elements and
$\text{sgn}(\sigma)$ is is the signature of $\sigma\in\mathfrak{S}_{4}$. Here
$[KL]$ and $[KLMN]$ denote the anti-symmetrization of products of gamma
matrices under the exchanges of any two indices.

##  Appendix B One-loop determinants in 5d $\mathcal{N}=1^{*}$ gauge theory

We summarize the explicit forms of the one-loop determinants of the surface
operators studied in Section 6. From the localization computation of the one-
loop determinant (3.3), the one-loop determinants
$Z^{\text{5d}}_{1\text{-loop}}({\bm{p}})$ with
${\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j},$ and
${\bm{e}}_{i}+{\bm{e}}_{j}-{\bm{e}}_{k}-{\bm{e}}_{l}$ in 5d
$\mathcal{N}=1^{*}$ $U(N)$ gauge theory are given by

$\displaystyle
Z^{\text{5d}}_{1\text{-loop}}({\bm{p}}={\bm{e}}_{i}-{\bm{e}}_{j})$
$\displaystyle\qquad=\left(\frac{\vartheta_{1}(\pm(a_{i}-a_{j})-m_{\text{ad}}\pm\frac{1}{2}\epsilon)}{\vartheta_{1}(\pm(a_{i}-a_{j}))\vartheta_{1}(\pm(a_{i}-a_{j})+\epsilon)}\prod_{k=i,j}\prod_{l\neq
i,j}^{N}\frac{\vartheta_{1}(\pm(a_{k}-a_{l})-m_{\text{ad}})}{\vartheta_{1}(\pm(a_{k}-a_{l})+\frac{1}{2}\epsilon)}\right)^{\frac{1}{2}}\,,$
(B.1) $\displaystyle
Z^{\text{5d}}_{1\text{-loop}}({\bm{p}}={\bm{e}}_{i}+{\bm{e}}_{j}-{\bm{e}}_{k}-{\bm{e}}_{l})$
$\displaystyle=\left(\prod_{h=i,j}\prod_{q=k,l}\frac{\vartheta_{1}(\pm(a_{h}-a_{q})-m_{\text{ad}}\pm\frac{\epsilon}{2})}{\vartheta_{1}(\pm(a_{h}-a_{q})+\epsilon)\vartheta_{1}(\pm(a_{h}-a_{q}))}\cdot\prod_{h=i,j,k,l}\prod_{q\neq
i,j,k,l}\frac{\vartheta_{1}(\pm(a_{h}-a_{q})-m_{\text{ad}})}{\vartheta_{1}(\pm(a_{h}-a_{q})+\frac{\epsilon}{2})}\right)^{\frac{1}{2}}\,.$
(B.2)

##  Appendix C The differential operator $D_{10}$

In order to compute the differential operator $D_{10}$, we explicitly write
(3.44).

$\displaystyle V$ $\displaystyle=(\Psi,\overline{\widehat{\sf
Q}\cdot\Psi})+V_{\text{g.f}}$
$\displaystyle=\int_{T^{2}\times\mathbb{R}^{3}}d^{5}x\mathrm{Tr}\left[\sum_{M=1}^{9}(\widehat{\sf
Q}\cdot X_{0,M}-\bar{D}_{M}X_{1,8})\tilde{\Gamma}^{M}{\varepsilon}+{\rm
i}\sum_{j=1}^{7}X_{1,j}\nu^{j}\right]$
$\displaystyle\times\overline{\left[\bar{D}_{M}X_{0,N}(\bar{\nu}_{j}{\Gamma}^{MN}\varepsilon)\nu_{j}+{\rm
i}K_{j}\nu_{j}+(2\bar{D}_{M}X_{0,4}-\bar{D}_{M}\widehat{\sf Q}\cdot
X_{1,8}-\bar{D}_{4}X_{0,M}-{\rm
i}\bar{D}_{0}X_{0,M})\tilde{\Gamma}^{M}{\varepsilon}\right]}$
$\displaystyle+X_{1,9}(\sum_{i=1,2,3,9}\bar{D}_{i}X_{0,i}+\frac{\xi}{2}\widehat{\sf
Q}\cdot X_{1,9})$
$\displaystyle=\int_{T^{2}\times\mathbb{R}^{3}}d^{5}x\mathrm{Tr}\left(\left[\sum_{M=1}^{9}(\widehat{\sf
Q}\cdot X_{0,M}-\bar{D}_{M}X_{1,8})\tilde{\Gamma}^{M}{\varepsilon}+{\rm
i}\sum_{j=1}^{7}X_{1,j}\nu^{j}\right]\right.$
$\displaystyle\qquad\qquad\times\left[2\bar{D}_{M}X_{0,N}(\nu_{i}\widetilde{\Gamma}^{MN}{\varepsilon})\nu^{j}-{\rm
i}\widehat{\sf Q}\cdot
X_{1,j}\nu^{j}-2\bar{D}_{{z}}X_{0,M}\tilde{\Gamma}^{M}{\varepsilon}\right]$
$\displaystyle\left.\qquad\qquad+X_{1,9}({\rm
i}\sum_{i=1,2,3,9}\bar{D}_{i}X_{0,i}+\frac{\xi}{2}\widehat{\sf Q}\cdot
X_{1,9})\right)$
$\displaystyle=\int_{T^{2}\times\mathbb{R}^{3}}\mathrm{Tr}\left[\sum_{M=1}^{9}(\widehat{\sf
Q}\cdot
X_{0,M}-\bar{D}_{M}X_{1,8})(2\bar{D}_{M}X_{0,4}-\bar{D}_{M}\widehat{\sf
Q}\cdot X_{1,8}-\bar{D}_{4}X_{0,M}-{\rm i}\bar{D}_{0}X_{0,M})\right.$
$\displaystyle\left.+\sum_{j=1}^{7}X_{1,j}(\widehat{\sf Q}\cdot X_{1,j}+2{\rm
i}({\nu}_{j}\widetilde{\Gamma}^{MN}{\varepsilon})\bar{D}_{M}X_{0,N})+X_{1,9}\left(\sum_{i=1,2,3,9}\bar{D}_{i}X_{0,i}+\frac{\xi}{2}\widehat{\sf
Q}\cdot X_{1,9}\right)\right]\,.$ (C.1)

From (C.1), the action of $D_{10}$ on the fluctuation fields ${\bm{X}}_{0}$
are read off as

$\displaystyle(D_{10}\cdot{\bm{X}}_{0})_{i}$ $\displaystyle=-2{\rm
i}\sum_{j,k=1}^{3}({\nu}_{i}\widetilde{\Gamma}^{jk}{\varepsilon})\bar{D}_{j}X_{0,k}+2{\rm
i}\overline{D}_{i}X_{0,9}-2{\rm i}\bar{D}_{9}X_{0,i}\quad i\in\\{1,2,3\\}\,.$
(C.2) $\displaystyle(D_{10}\cdot{\bm{X}}_{0})_{i}$ $\displaystyle=2{\rm
i}\sum_{j=1}^{3}\sum_{k=5}^{8}(\nu_{i}\tilde{\Gamma}^{jk}\varepsilon)\bar{D}_{k}X_{0,k}+2{\rm
i}\sum_{k=5}^{9}(\nu_{i}\tilde{\Gamma}^{9k}\varepsilon)\bar{D}_{9}X_{0,k}\quad
i\in\\{4,5,6,7\\}\,,$ (C.3) $\displaystyle(D_{10}\cdot{\bm{X}}_{0})_{8}$
$\displaystyle=\bar{D}_{M}(2\bar{D}_{M}X_{0,4}-\bar{D}_{4}X_{0,M}-{\rm
i}\bar{D}_{0}X_{0,M})\,,$ (C.4) $\displaystyle(D_{10}\cdot{\bm{X}}_{0})_{9}$
$\displaystyle={\rm i}\sum_{i=1,2,3,9}\bar{D}_{i}X_{0,i}\,.$ (C.5)

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