Yutaka<EMAIL_ADDRESS>
Department of Current Legal Study, Faculty of Law, Meiji Gakuin University,
Yokohama, 244-8539, Japan.

#  Euler transformation for multiple $q$-hypergeometric series from wall-
crossing formula of $K$-theoretic vortex partition function

###### Contents

  1. 1 Introduction and summary
  2. 2 Kajihara transformation as wall-crossing formula
    1. 2.1 $K$-theoretic vortex partition function and handsaw quiver variety
    2. 2.2 Derivation of wall-crossing formula of vortex partition function
    3. 2.3 Kajihara transformation as WC formula in 3d $\mathcal{N}=2$ theory
  3. 3 Hallnäs-Langmann-Noumi-Rosengren formula as wall-crossing formula
    1. 3.1 Vortex partition function in 3d $\mathcal{N}=4$ gauge theory
    2. 3.2  Hallnäs-Langmann-Noumi-Rosengren formula as WC formula in 3d $\mathcal{N}=4$ theory
    3. 3.3 $K$-theoretic vortex partition function as equivariant $\chi_{t}$-genus
    4. 3.4 Elliptic analogue of wall-crossing formula
  4. 4 Two-dimensional/cohomological/rational limit
  5. 5 Future directions
  6. A $q$-Pochhammer symbol and useful identities

##  1 Introduction and summary

Dualities relate supersymmetric quantum theories, which typically have the
same fixed point in the renormalization group flow. In the level of
supersymmetric indices and also partition functions, a duality implies the
agreement of supersymmetric indices of the dual pair. Combined with
supersymmetric localization formula, the agreement of indices predicts many
non-trivial identities between special functions. For example, in four
dimensions, the equality between superconformal indices [4, 5] of a dual pair
predicts a non-trivial identity between integrals of elliptic gamma functions.

In three dimensions, supersymmetric localization formulas for supersymmetric
indices/partition functions [6, 7, 8, 9] of $U(N)$ gauge theories on a closed
three manifold $M_{3}$ have a remarkable factorization into a pair of
$K$-theoretic vortex partition functions [10, 11, 12, 13, 14, 15]:

$\displaystyle Z_{M_{3}}=\sum Z_{1\text{-loop}}Z_{\rm vortex}Z_{\rm
vortex},\qquad(M_{3}=S^{3}_{b},S^{1}\times S^{2})\,.$ (1.1)

Here $Z_{\rm vortex}$ called the $K$-theoretic vortex partition function is
the generating function of Witten indices of supersymmetric quantum mechanics
(SQM) associated with handsaw quiver of type $A_{1}$. Since a moduli space of
vortex given by the Higgs branch vacua of SQM is isomorphic to a handsaw
quiver of type $A_{1}$, the $K$-theoretic vortex partition function agrees
with the generating function of an equivariant index such as the
$\chi_{t}$-genus of handsaw quiver varieties of type $A_{1}$.

The problem to find the precise relation of $Z_{M_{3}}$’s between a three-
dimensional (3d) Seiberg-like dual pair reduces to find the relation between
the $K$-theoretic vortex partition functions of the Seiberg-like dual pair
[16]. Later it is found [1] that the relation between the $K$-theoretic vortex
partition functions of a Seiberg-like dual pair is the same as the wall-
crossing (WC) formula of the $K$-theoretic vortex partition function with
respect to the variation of the Fayet-Iliopoulos (FI) parameter of SQM.

In this article we will show that the wall-crossing formulas of vortex
partition functions in 3d $\mathcal{N}=2$ and $\mathcal{N}=4$ $U(N)$ gauge
theories obtained in [16, 1] are the same as Euler transformations of
$q$-hypergeometric series studied by Kajihara [2], and Hallnäs, Langmann,
Noumi and Rosengren [3], respectively. These Euler transformations are thought
of as generalizations of Euler transformation of Gauss hypergeometric series:

$\displaystyle{}_{2}F_{1}(a,b;c;z)=(1-z)^{c-a-b}{}_{2}F_{1}(c-a,c-b;c;z)\,.$
(1.2)

Since the $K$-theoretic vortex partition functions are the indices for handsaw
quiver varieties, wall-crossing formulas give geometric interpretations of
Euler transformations of $q$-hypergeometric series.

This article is organized as follows. In section 2.1, we review the
$K$-theoretic vortex partition functions in 3d $\mathcal{N}=2$ Chern-Simons-
matter theory (gauge theory with Chern-Simons term) and explain the relation
between an ADHM-like description of the moduli space of vortices and a handsaw
quiver variety of $A_{1}$. Then we obtain two distinct expressions of the
supersymmetric localization formula in the positive and negative 1d FI-
parameter region, respectively. In section 2.2, according to [1], we prove the
wall-crossing formula which relates the vortex partition function in the
positive FI-parameter and the one in the negative FI-parameter. In section 2.3
we give the precise parameter identification of the vortex partition function
and Kajihara transformation.

We perform a similar computation in a 3d $\mathcal{N}=4$ gauge theory in
section 3. In section 3.1, we review the $K$-theoretic vortex partition
function for the 3d $\mathcal{N}=4$ theory. In section 3.2, we show that the
wall-crossing formula of the vortex partition function agrees with a
transformation formula [3] of trigonometric type. Next we explain the
geometric interpretation of the vortex partition functions in section 3.3. For
3d $\mathcal{N}=4$ gauge theory, we find that the $K$-theoretic vortex
partition function is the same as the equivariant $\chi_{t}$-genus up to a
trivial factor. For 3d $\mathcal{N}=2$ gauge theory, although the
$K$-theoretic $k$-vortex partition function and an index are related to each
other up to some factor, the generating functions are different. We give a
brief comment on an elliptic analogue of the wall-crossing formula in section
3.4.

In section 4, we perform the dimensional reduction from three dimensions to
two dimension, and derive the wall-crossing formula of vortex partition
functions in 2d $\mathcal{N}=(2,2)$ and $\mathcal{N}=(2,2)^{*}$ gauge theory.
In section 5, we give comments on future directions of this work.

##  2 Kajihara transformation as wall-crossing formula

In this section we will show that the wall-crossing formula of $K$-theoretic
vortex partition derived in [1] is the same as the Kajihara transformation of
multiple $q$-hypergeometric series [2].

###  2.1 $K$-theoretic vortex partition function and handsaw quiver variety

Figure 1: The quiver diagram representing 1d $\mathcal{N}=2$ SQM associated
with the ADHM-like description of the $k$-vortex moduli for the 3d
$\mathcal{N}=2$ gauge theory in Table 1. The circle with ’$k$’ denotes the 1d
$\mathcal{N}=2$ $U(k)$ vector multiplet. An arrow with solid line denotes a 1d
$\mathcal{N}=2$ chiral multiplet. The dashed line denotes the 1d
$\mathcal{N}=2$ fermi multiplet. ’$I$’ and ’$J$’ denote the scalars in each
chiral multiplet. ’$\psi$’ denotes the fermion in the fermi multiplet. The
representation of the multiplets is summarized in Table 2

First, we review the vortex partition function of 3d $\mathcal{N}=2$ Chern-
Simons-matter theory with the unitary the gauge group $U(N_{1})$ coupled to
$N_{2}$ chiral multiplets in the fundamental representation of $U(N_{1})$ and
with $N_{0}$ chiral multiplets in the anti-fundamental representation of
$U(N_{1})$. In this paper, we assume $N_{0}\leq N_{2}$. The precise matter
content of the 3d gauge theory is summarized in Table 1. As we will show the
case for $N_{2}=N_{0}$ and zero Chern-Simons level: $\kappa=0$ is relevant to
the Kajihara transformation.

Let us consider the partition function (supersymmetric index) of the 3d gauge
theory on $S^{1}\times\mathbb{R}^{2}$ around the solutions of the vortex
equation:

$\displaystyle F_{12}=\frac{g^{2}}{2}(\phi\phi^{\dagger}-\xi 1_{N_{1}\times
N_{1}}),$ $\displaystyle(D_{1}+{\rm i}D_{2})\phi=0\,.$ (2.1)

with $k$-vortex number:
$k=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\mathrm{Tr}F_{12}$. Here
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+{\rm
i}[A_{\mu},A_{\nu}]$ is the field strength of the $U(N_{1})$ gauge field
$\\{A_{\mu}\\}_{\mu=0}^{2}$. $D_{\mu}=\partial_{\mu}+{\rm i}A_{\mu}$ is the
covariant derivative. The suffix $\mu=1,2$ denotes the space-time directions
along $\mathbb{R}^{2}$. As explained in Table 1, $\phi$ is the $N_{2}$-tuple
scalars in the fundamental representation of the gauge group $U(N_{1})$. $\xi$
is the FI-parameter for the 3d gauge theory. Since the moduli data of the
vortex equation has an ADHM-like description in terms of the D-brane
construction [17] and the field theoretic construction [18, 19], the partition
function of 3d gauge theory around the $k$-vortex background on
$S^{1}\times\mathbb{R}^{2}$ reduces to the Witten index of 1d $\mathcal{N}=2$
SQM on $S^{1}$ associated with the handsaw quiver of type $A{}_{1}$ depicted
by the Figure 1. The matter content of the SQM is depicted by Table 2.

In the SQM, the moduli space of the Higgs branch vacua is given by a Kähler
quotient which is isomorphic to a handsaw quiver variety of type $A_{1}$:

$\displaystyle\mathcal{M}^{(\zeta)}_{k,N_{1},N_{2}}$
$\displaystyle=\Bigl{\\{}(B,I,J)\Big{|}[B,B^{\dagger}]+II^{\dagger}-J^{\dagger}J=\zeta
1_{k\times k}\Bigr{\\}}\Big{/}U(k)\,,$ (2.2)
$\displaystyle\simeq\\{(B,I,J)\\}/GL(N,\mathbb{C})\,.$ (2.3)

Here

$\displaystyle
B\in\mathrm{End}(V),\,\,I\in\mathrm{Hom}(W^{(1)},\,\,V),\,\,J\in\mathrm{Hom}(V,W^{(2)})$
(2.4)

with $V\simeq\mathbb{C}^{k},W^{(1)}\simeq\mathbb{C}^{N_{1}}$ and
$W^{(2)}\simeq\mathbb{C}^{N_{2}-N_{1}}$ are the vacuum expectation values of
the scalars in 1d $\mathcal{N}=2$ chiral multiplets. In (2.2) and (2.3), $g\in
U(k)$ or $GL(k,\mathbb{C})$ acts on $(B,I,J)$ as
$(B,I,J)\mapsto(gBg^{-1},gI,Jg^{-1})$. The quotients in (2.2) and (2.3) denote
the Kähler quotient and the quotient with a stability condition.

The positive and the negative FI-parameter regions in (2.2) correspond to two
different stability conditions of the handsaw quiver variety, respectively.
According to the D-brane construction [17] and the field theoretic
construction [18, 19] of the vortex moduli, the moduli space of the Higgs
branch vacua $\mathcal{M}^{(\zeta>0)}_{k,N_{1},N_{2}}$ is isomorphic to the
moduli space of $k$-vortex solutions of (2.1).

The Witten index for the 1d $\mathcal{N}=2$ SQM depicted by the handsaw quiver
of type $A_{1}$ is called the $K$-theoretic $k$-vortex partition function for
the 3d $\mathcal{N}=2$ gauge theory. Since the Witten index is related to an
(equivariant) index of the moduli space of Higgs branch vacua of the SQM,
$K$-theoretic vortex partition functions give indices of the handsaw quiver
variety. Note that the fermion $\psi$ in the fermi multiplet corresponds to a
characteristic class integrated over the handsaw quiver variety. This
situation is analogous to the instanton case: the Witten indices for SQM
depicted by the ADHM (Jordan) quiver are called $K$-theoretic instanton
partition [20, 21], which give indices of the instanton moduli spaces. We will
explain the geometric interpretation of the vortex partition functions in
section 3.3.

| $U(N_{0})$ | $U(N_{1})$ | $U(N_{2})$
---|---|---|---
$\phi$ | ${\bm{1}}$ | ${\bm{N}_{1}}$ | $\overline{\bm{N}_{2}}$
$\tilde{\phi}$ | ${\bm{N}_{0}}$ | $\overline{\bm{N}_{1}}$ | ${\bm{1}}$
Table 1: The representation of the scalars in the 3d $\mathcal{N}=2$ chiral multiplets. $G=U(N_{1})$ is the gauge group of the 3d theory. $G_{F}=(U(N_{0})\times U(N_{2}))/U(1)$ is the flavor symmetry group. $U(1)_{R}$ is the R-symmetry group. Here $\phi\in\mathrm{Hom}(\mathbb{C}^{N_{1}},\mathbb{C}^{N_{2}})$ and $\tilde{\phi}\in\mathrm{Hom}(\mathbb{C}^{N_{0}},\mathbb{C}^{N_{1}})$ are the scalars in the chiral multiplets. ${\bm{N}_{s}}$, (resp. $\overline{\bm{N}_{s}}$) denotes the fundamental (resp. anti-fundamental) representation of $U(N_{s})$. ${\bm{1}}$ denotes the trivial representation.  | $U(k)$ | $U(N_{1})$ | $U(N_{2}-N_{1})$ | $U(N_{0})$ | $U(1)_{\varepsilon}$
---|---|---|---|---|---
$B$ | $\bm{adj}$ | ${\bm{1}}$ | ${\bm{1}}$ | ${\bm{1}}$ | $-1$
$I$ | ${\bm{k}}$ | $\overline{\bm{N}_{1}}$ | ${\bm{1}}$ | ${\bm{1}}$ | $-\frac{1}{2}$
$J$ | $\bar{\bm{k}}$ | ${\bm{1}}$ | ${{\bm{N}_{2}}-{\bm{N}_{1}}}$ | ${\bm{1}}$ | $-\frac{1}{2}$
$\psi$ | $\bar{\bm{k}}$ | ${\bm{1}}$ | ${\bm{1}}$ | ${\bm{N}_{0}}$ | $-\frac{1}{2}$
Table 2: The representation of the scalar and fermion in the 1d
$\mathcal{N}=2$ multiplets describing $k$-vortex moduli in the 3d
$\mathcal{N}=2$ $U(N_{1})$ gauge theory depicted by the quiver diagram in
Figure 1 . The fields $B,I$ and $J$ are scalars in the 1d $\mathcal{N}=2$
chiral multiplets. A field $\psi$ is the fermion in the 1d $\mathcal{N}=2$
fermi multiplet. $U(k)$ is the gauge group. $U(N_{1})$, $U(N_{2}-N_{1})$ and
$U(N_{0})$, as well as $U(1)_{\varepsilon}$ are the flavor symmetry group.
’$\bm{adj}$’ denotes the adjoint representation.

The $K$-theoretic $k$-vortex partition function $Z^{(\zeta)}_{k}$ is evaluated
by applying the supersymmetric localization formula of the Witten index [21,
22]. Then we obtain the following residue integral expression

$\displaystyle Z^{(\zeta)}_{k}$
$\displaystyle=\frac{1}{k!(2\sinh\frac{-\varepsilon}{2})^{k}}\oint_{JK(\zeta)}\prod_{\alpha=1}^{k}\frac{du_{\alpha}}{2\pi{\rm
i}}\prod_{1\leq\alpha\neq\beta\leq
k}\frac{2\sinh\frac{u_{\alpha}-u_{\beta}}{2}}{2\sinh\frac{u_{\alpha}-u_{\beta}-\varepsilon}{2}}$
$\displaystyle\qquad\times\prod_{\alpha=1}^{k}\frac{e^{\kappa
u_{\alpha}}\prod_{j=1}^{N_{0}}2\sinh\left({\frac{u_{\alpha}-\tilde{m}_{j}-\varepsilon/2}{2}}\right)}{\prod_{j=1}^{N_{1}}2\sinh\left(\frac{u_{\alpha}-m_{j}-\varepsilon/2}{2}\right)\cdot\prod_{j=N_{1}+1}^{N_{2}}2\sinh\left(\frac{-u_{\alpha}+m_{j}-\varepsilon/2}{2}\right)}\,.$
(2.5)

In (2.5), complex parameters $\\{m_{i}\\}_{i=1}^{N_{1}}$,
$\\{m_{i}\\}_{i=N_{1}+1}^{N_{2}}$ and $\\{\tilde{m}_{i}\\}_{i=1}^{N_{0}}$ are
chemical potentials of 1d flavor symmetry groups $U(N_{1})$ $U(N_{2}-N_{1})$
and $U(N_{0})$ acting on $I,J$ and $\psi$, respectively. A chemical potential
$\varepsilon$ is called an $\Omega$-background parameter.

Later $e^{m_{i}}$ and $e^{\varepsilon}$ are identified with the
$(\mathbb{C}^{*})^{N_{2}+1}$-equivariant parameters associated with
$U(N_{1})\times U(N_{2}-N_{1})\times U(1)_{\varepsilon}$-action on
$\mathcal{M}^{(\zeta)}_{k,N_{1},N_{2}}$ defined by

$\displaystyle(B,I,J)\mapsto(B,Ig^{-1}_{1},J),\quad g\in U(N_{1}),$
$\displaystyle(B,I,J)\mapsto(B,I,g_{2}J),\quad g_{2}\in U(N_{2}-N_{1}),$ (2.6)
$\displaystyle(B,I,J)\mapsto(g_{3}B,g_{3}I,g_{3}J),\quad g_{3}\in
U(1)_{\varepsilon}\,.$

$\kappa\in\mathbb{Z}$ or $\mathbb{Z}+\frac{1}{2}$ is the 1d Chern-Simons level
induced by the 3d supersymmetric Chern-Simons term in the $U(N_{1})$ gauge
theory:

$\displaystyle{S}_{CS}$ $\displaystyle=\frac{\kappa}{4\pi{\rm
i}}\int_{S^{1}\times\mathbb{R}^{2}}\mathrm{Tr}\left(A\wedge dA-\frac{2{\rm
i}}{3}A^{3}\right)+\cdots\,.$ (2.7)

The trace in (2.7) is taken for $\mathfrak{u}(N_{1})$. In this article we
assume $-\frac{N_{2}-N_{0}}{2}\leq\kappa\leq\frac{N_{2}-N_{0}}{2}$ and
$\kappa+\frac{N_{2}-N_{0}}{2}\in\mathbb{Z}$.

Let us evaluate the contour integral in (2.5). The contour is determined by
the Jeffrey-Kirwan (JK) residues which correlated with the sign of the 1d FI-
parameter $\zeta$ in the SQM [22]. The JK residues are evaluated as follows
[1].

When the 1d FI parameter is positive $\zeta>0$, the JK residues in (2.5) are
evaluated at the poles given by

$\displaystyle u_{\alpha}-u_{\beta}-\varepsilon$
$\displaystyle=0,\quad\alpha,\beta\in\\{1,\cdots,k\\},$ $\displaystyle
u_{\alpha}-m_{i}-\frac{\varepsilon}{2}$
$\displaystyle=0,\quad\alpha\in\\{1,\cdots,k\\},\,\,i\in\\{1,\cdots,N_{1}\\}.$
(2.8)

Then the intersection points of the hyperplanes in (2.8) are classified as

$\displaystyle
u_{\alpha=\ell_{i}}=m_{i}+(\ell_{i}-1/2)\varepsilon,\quad\ell_{i}=1,\cdots,k_{i},\,\,\,i\in\\{1,\cdots,N_{1}\\},$
(2.9)

where $k_{i}\geq 0$ and $\sum_{i=1}^{N_{1}}k_{i}=k$.

On the other hand, the JK residues in the negative FI-parameter region
$\zeta<0$ are evaluated at

$\displaystyle u_{\alpha}-u_{\beta}-\varepsilon$
$\displaystyle=0,\quad\alpha,\beta\in\\{1,\cdots,k\\},$ $\displaystyle-
u_{\alpha}+m_{j}-\frac{\varepsilon}{2}$
$\displaystyle=0,\quad\alpha\in\\{1,\cdots,k\\},\,\,j\in\\{N_{1}+1,\cdots
N_{2}\\}.$ (2.10)

The intersection points of the hyperplanes in (2.10) are classified as

$\displaystyle
u_{\alpha=\ell_{i}}=m_{i}-(\ell_{i}-1/2)\varepsilon,\quad\ell_{i}=1,\cdots,k_{i},\,\,\,i\in\\{N_{1}+1,\cdots,N_{2}\\},$
(2.11)

where $k_{i}\geq 0$ and $\sum_{i=N_{1}+1}^{N_{2}}k_{i}=k$. Then the
$K$-theoretic $k$-vortex partition function in the positive and negative FI-
parameter regions are given by

$\displaystyle Z^{(\zeta>0)}_{k}$
$\displaystyle=\sum_{k_{1}+\cdots+k_{N_{1}}=k}\frac{(-1)^{N_{1}k}e^{\kappa\sum_{i=1}^{N_{1}}(k_{i}m_{i}+k^{2}_{i}\varepsilon/2)}\prod_{i=1}^{N_{1}}\prod_{j=1}^{N_{0}}\prod_{\ell_{i}=1}^{k_{i}}2\sinh\left({\frac{m_{i}-\tilde{m}_{j}+(\ell_{i}-1)\varepsilon}{2}}\right)}{\prod_{i,j=1}^{N_{1}}\prod_{\ell_{i}=1}^{k_{i}}2\sinh\left(\frac{m_{j}-m_{i}-(\ell_{i}-k_{j}-1)\varepsilon}{2}\right)\prod_{i=1}^{N_{1}}\prod_{j=N_{1}+1}^{N_{2}}\prod_{\ell_{i}=1}^{k_{i}}2\sinh\left(\frac{-m_{i}+m_{j}-\ell_{i}\varepsilon}{2}\right)}$
(2.12) $\displaystyle Z^{(\zeta<0)}_{k}$
$\displaystyle=\sum_{{k}_{N_{1}+1}+\cdots+{k}_{N_{2}}={k}}\frac{(-1)^{(N_{2}-N_{1}+N_{0})k}e^{-\kappa\sum_{i=N_{1}+1}^{N_{2}}(-k_{i}m_{i}+k^{2}_{i}\varepsilon/2)}\prod_{i=N_{1}+1}^{N_{2}}\prod_{j=1}^{N_{0}}\prod_{\ell_{i}=1}^{k_{i}}2\sinh\left({\frac{-m_{i}+\tilde{m}_{j}+\ell_{i}\varepsilon}{2}}\right)}{\prod_{i,j=N_{1}+1}^{N_{2}}\prod_{\ell_{i}=1}^{k_{i}}2\sinh\left(\frac{-m_{j}+m_{i}-(\ell_{i}-k_{j}-1)\varepsilon}{2}\right)\prod_{i=N_{1}+1}^{N_{2}}\prod_{j=1}^{N_{1}}\prod_{\ell_{i}=1}^{k_{i}}2\sinh\left(\frac{m_{i}-m_{j}-\ell_{i}\varepsilon}{2}\right)}$
(2.13)

Since the sign of the FI-parameter corresponds to the two distinct stability
condition defining the handsaw quiver variety, the supersymmetric localization
formula for the $K$-theoretic $k$-vortex partition functions (2.12) and (2.13)
should be related to indices of handsaw quiver varieties with two different
stability conditions. The vortex partition functions in the positive and
negative FI-parameter regions possibly have different values:
$Z^{(\zeta>0)}_{k}\neq Z^{(\zeta<0)}_{k}$, and such a situation is called a
wall-crossing phenomenon.

###  2.2 Derivation of wall-crossing formula of vortex partition function

According to [1], we derive the relation between $Z^{(\zeta>0)}_{k}$ and
$Z^{(\zeta<0)}_{k}$, i.e., the wall-crossing formula for $K$-theoretic vortex
partition function, which is closely related to the wall-crossing formula for
indices of handsaw quiver varieties of type $A_{1}$.

In order to derive the wall-crossing formula, first we introduce a multi-
contour integral defined by

$\displaystyle\mathcal{Z}_{k}$
$\displaystyle:=\oint_{T^{k}_{x}}\frac{dx_{k}}{2\pi{\rm
i}x_{k}}\cdots\frac{dx_{2}}{2\pi{\rm i}x_{2}}\frac{dx_{1}}{2\pi{\rm
i}x_{1}}\mathcal{I}_{k}(x_{1},x_{2},\cdots x_{k})\,,$ (2.14)

where

$\displaystyle\mathcal{I}_{k}(x_{1},\cdots,x_{k})$
$\displaystyle=\frac{q^{\frac{k^{2}}{2}}}{k!(1-q)^{k}}(\prod_{i=1}^{N_{2}}x^{(2)}_{i})^{\frac{k}{2}}(\prod_{i=1}^{N_{0}}x^{(0)}_{i})^{-\frac{k}{2}}q^{\frac{(2N_{1}-N_{2}-N_{0})k}{4}}\prod_{1\leq\alpha\neq\beta\leq
k}\frac{x_{\alpha}-x_{\beta}}{x_{\alpha}-qx_{\beta}}$
$\displaystyle\times\prod_{\alpha=1}^{k}\frac{x^{\kappa+\frac{N_{2}-N_{0}}{2}}_{\alpha}\prod_{j=1}^{N_{0}}(x_{\alpha}-x^{(0)}_{j}q^{\frac{1}{2}})}{\prod_{i=1}^{N_{1}}(x_{\alpha}-x^{(2)}_{i}q^{\frac{1}{2}})\cdot\prod_{j=N_{1}+1}^{N_{2}}(x^{(2)}_{j}q^{-\frac{1}{2}}-x_{\alpha})}\,.$
(2.15)

Here ${T}^{k}_{x}$ is a $k$-dimensional torus defined by
${T}^{k}_{x}=\\{(x_{1},\cdots,x_{k})\in\mathbb{C}^{k}||x_{1}|=\cdots=|x_{k}|=1\\}$.
We assume that the variables $q,x^{(2)}_{i}\in\mathbb{C}$ are in the following
regions:

$\displaystyle|q|<1,\,\,|x^{(2)}_{i}q^{n+\frac{1}{2}}|<1,\,\,|x^{(2)}_{j}q^{-n-\frac{1}{2}}|>1,$
(2.16)

with $n=0,1,2,\cdots,$ $i=1,2,\cdots,N_{1},$ and $j=N_{1}+1,\cdots,N_{2}$.

The multi-contour integral in (2.14) is evaluated either by the residues
inside the torus ${T}^{k}_{x}$, or by the residues outside of the torus. First
we consider evaluating the residues inside the torus. In this case, the
sequence of poles that contributing to the iterated residues are classified
into the following two types. The first type includes at least one pole at the
origin $x_{i}=0$ such that:

$\displaystyle\\{x_{i_{1}}=qx_{i_{2}},\,x_{i_{2}}=qx_{i_{3}},\,\cdots,x_{i_{n-1}}=qx_{i_{n}},\,x_{i_{n}}=0\\}\,,$
$\displaystyle\\{x_{i_{1}}=qx_{i_{3}},\,x_{i_{2}}=q^{2}x_{i_{3}},\,\cdots,x_{i_{n-1}}=qx_{i_{n}},\,x_{i_{n}}=0\\}\,,$
$\displaystyle\cdots$
$\displaystyle\\{x_{i_{1}}=0,\,x_{i_{2}}=0,\cdots,x_{i_{n-1}}=0,\,x_{i_{n}}=0\\}\,.$
(2.17)

Here $1\leq i_{1}<i_{2},<\cdots<i_{n}\leq k$ and $0\leq n\leq k$. The second
type is given by the sequences of poles which do not include the pole at the
origin:

$\displaystyle\\{x_{i_{n+1}}=qx_{i_{n+2}},\,x_{i_{n+2}}=qx_{i_{n+3}},\,\cdots,x_{i_{k-1}}=qx_{i_{k}},\,x_{i_{k}}=x^{(2)}_{j}q^{\frac{1}{2}}\\},$
$\displaystyle\\{x_{i_{n+1}}=qx_{i_{n+3}},\,x_{i_{n+2}}=q^{2}x_{i_{n+3}},\,\cdots,x_{i_{k-1}}=qx_{i_{k}},\,x_{i_{k}}=x^{(2)}_{j}q^{\frac{1}{2}}\\},$
$\displaystyle\cdots$
$\displaystyle\\{x_{i_{n+1}}=x^{(2)}_{j_{1}}q^{\frac{1}{2}},\,x_{i_{n+2}}=x^{(2)}_{j_{2}}q^{\frac{1}{2}},\cdots,x_{i_{k-1}}=x^{(2)}_{j_{k-n-1}}q^{\frac{1}{2}},\,x_{i_{k}}=x^{(2)}_{j_{k-n}}q^{\frac{1}{2}}\\}.$
(2.18)

Here $j_{1},\cdots j_{k-n}\in\\{1,\cdots,N_{1}\\}$. Except for the case where
$\kappa=\frac{N_{2}-N_{0}}{2}$ is satisfied, the residues at the poles of the
first type become zero. In this case, $\mathcal{Z}_{k}$ is given by the
residues of the second type, which agrees with the $k$-vortex partition
function (2.12). This is because that $\mathcal{I}_{k}(x_{1}\cdots,x_{k})$ is
same as the integrand of (2.5) under the following identification of the
parameters

$\displaystyle x_{\alpha}$
$\displaystyle=e^{u_{\alpha}},\,\,x^{(0)}_{a}=e^{\tilde{m}_{a}},\,\,x^{(2)}_{a}=e^{m_{a}},\,\,q=e^{\varepsilon},$
(2.19)

and also that the second type poles are same as the poles (2.8) appeared in
the JK residues operation under the identification (2.19).

On the other hand, the condition $\kappa=\frac{N_{2}-N_{0}}{2}$ is satisfied,
residues at poles of the first type (2.17) are non-zero. The summation of the
residues of the first type for $x_{i_{1}},\cdots,x_{i_{n}}$ is evaluated as
follows. Since the second line in (2.15) is regular at the origin of
$(x_{i_{1}},\cdots,x_{i_{n}})$, we may replace in the residue computation as

$\displaystyle\prod_{\alpha\in\mathcal{J}}\prod_{\beta\in\mathcal{J}^{c}}\frac{x_{\alpha}-x_{\beta}}{x_{\alpha}-qx_{\beta}}\frac{x_{\beta}-x_{\alpha}}{x_{\beta}-qx_{\alpha}}\cdot\prod_{\alpha\in\mathcal{J}}\frac{x^{\kappa+\frac{N_{2}-N_{0}}{2}}_{\alpha}\prod_{j=1}^{N_{0}}(x_{\alpha}-x^{(0)}_{j}q^{\frac{1}{2}})}{\prod_{i=1}^{N_{1}}(x_{\alpha}-x^{(2)}_{i}q^{\frac{1}{2}})\cdot\prod_{j=N_{1}+1}^{N_{2}}(x^{(2)}_{j}q^{-\frac{1}{2}}-x_{\alpha})}$
$\displaystyle\underset{\\{x_{i}=0\\}_{i\in\mathcal{J}}}{\longrightarrow}(-1)^{N_{0}-N_{1}}q^{-2n(k-n)+(N_{0}+N_{2}-N_{1})\frac{n}{2}}(\prod_{a=1}^{N_{0}}x^{(0)}_{a})^{n}(\prod_{a=1}^{N_{2}}x^{(2)}_{a})^{-n}$
(2.20)

Here $\mathcal{J}=\\{i_{1},i_{2},\cdots,i_{n}\\}$ and the set
$\mathcal{J}^{c}$ is the complement of $\mathcal{J}$ in $\\{1,\cdots,k\\}$.
Then the summation over the residues of $(x_{i_{1}},\cdots,x_{i_{n}})$ at the
first type poles (2.17) is given by

$\displaystyle\sum\underset{x_{i_{n}}}{\mathrm{Res}}\cdots\underset{x_{i_{1}}}{\mathrm{Res}}\,\frac{\mathcal{I}_{k}(x_{1},\cdots,x_{k})}{\prod_{\alpha=1}^{k}x_{\alpha}}$
$\displaystyle=\frac{n!(k-n)!}{k!}(-1)^{(N_{1}-N_{0})n}(\prod_{a=1}^{N_{0}}x^{(0)}_{a})^{\frac{n}{2}}(\prod_{a=1}^{N_{2}}x^{(2)}_{a})^{-\frac{n}{2}}q^{\frac{n}{4}(N_{0}+N_{2}-2N_{1})}\
\frac{\mathcal{I}_{k-n}(\\{x_{i}\\}_{i\in\mathcal{J}^{c}})}{\prod_{\alpha\in\mathcal{J}^{c}}x_{\alpha}}$
$\displaystyle\times\frac{q^{\frac{n^{2}}{2}}}{n!(1-q)^{n}}\oint_{T^{n}}\prod_{\alpha=1}^{n}\frac{dx_{i_{\alpha}}}{2\pi{\rm
i}x_{i_{\alpha}}}\prod_{1\leq\alpha\neq\beta\leq
n}\frac{x_{i_{\alpha}}-x_{i_{\beta}}}{x_{i_{\alpha}}-qx_{i_{\beta}}}$
$\displaystyle=\frac{n!(k-n)!}{k!}(-1)^{(N_{1}-N_{0})n}(\prod_{a=1}^{N_{0}}x^{(0)}_{a})^{\frac{n}{2}}(\prod_{a=1}^{N_{2}}x^{(2)}_{a})^{-\frac{n}{2}}q^{\frac{n}{4}(N_{0}+N_{2}-2N_{1})}\
\frac{\mathcal{I}_{k-n}(\\{x_{i}\\}_{i\in\mathcal{J}^{c}})}{\prod_{\alpha\in\mathcal{J}^{c}}x_{\alpha}}\frac{(-q^{\frac{1}{2}})^{n}}{(q;q)_{n}}$
(2.21)

Here $(q;q)_{n}$ is a $q$-Pochhammer symbol defined by (A.4). In the last line
in (2.21), we used the following formula:

$\displaystyle\frac{q^{\frac{n^{2}}{2}}}{n!(1-q)^{n}}\sum\underset{x_{i_{n}}}{\mathrm{Res}}\cdots\underset{x_{i_{1}}}{\mathrm{Res}}\,\prod_{\alpha=1}^{n}x^{-1}_{i_{\alpha}}\cdot\prod_{1\leq\alpha\neq\beta\leq
n}\frac{x_{i_{\alpha}}-x_{i_{\beta}}}{x_{i_{\alpha}}-qx_{i_{\beta}}}$
$\displaystyle=\frac{q^{\frac{n^{2}}{2}}}{n!(1-q)^{n}}\oint_{T^{n}_{x}}\prod_{\alpha=1}^{n}\frac{dx_{i_{\alpha}}}{2\pi{\rm
i}x_{i_{\alpha}}}\prod_{1\leq\alpha\neq\beta\leq
n}\frac{x_{i_{\alpha}}-x_{i_{\beta}}}{x_{i_{\alpha}}-qx_{i_{\beta}}}=\frac{(-q^{\frac{1}{2}})^{n}}{(q;q)_{n}}\,.$
(2.22)

The iterated residues of the second type for
$\\{x_{i}\\}_{\in\mathcal{J}^{c}}$ in the (2.21) gives $Z^{(\zeta>0)}_{n-k}$.
By taking the summation over $1\leq i_{1}<\cdots<i_{n}\leq k$ and
$n=0,1,\cdots,k$ , and taking all the possible residues at the first type
poles for $\\{x_{i}\\}_{i\in\mathcal{J}}$ and also at the second type poles
for $\\{x_{i}\\}_{i\in\mathcal{J}^{c}}$ give the relation between
$\mathcal{Z}_{k}$ and $K$-theoretic vortex partition functions:

$\displaystyle\mathcal{Z}_{k}=\left\\{\begin{array}[]{cl}\sum\limits_{n=0}^{k}(-1)^{(N_{1}-N_{0})n}\left(\prod_{a=1}^{N_{0}}x^{(0)}_{a}\right)^{\frac{n}{2}}\left(\prod_{a=1}^{N_{2}}x^{(2)}_{a}\right)^{-\frac{n}{2}}q^{\frac{n}{4}(N_{0}+N_{2}-2N_{1})}\frac{(-q^{\frac{1}{2}})^{n}}{(q;q)_{n}}{Z}^{(\zeta>0)}_{k-n},&(\kappa=\frac{N_{0}-N_{2}}{2})\\\
{Z}^{(\zeta>0)}_{k},&(\kappa\neq\frac{N_{0}-N_{2}}{2})\end{array}\right.$
(2.25)

Next we consider evaluating the residues outside of the torus $T^{k}_{x}$. By
changing the integration variables as $w_{i}=1/x_{i}$ for $i=1,\cdots,k$, the
contour integral is evaluated by the residue inside of the torus $T^{k}_{w}$.

As in the case of the residues inside of $T^{k}_{x}$, the set of poles
contributing to the residue for $w_{i}$ are classified into the following two
types. The first one is given by

$\displaystyle\\{w_{i_{1}}=qw_{i_{2}},\,x_{i_{2}}=qw_{i_{3}},\,\cdots,w_{i_{n-1}}=qw_{i_{n}},\,w_{i_{n}}=0\\}\,,$
$\displaystyle\\{w_{i_{1}}=qw_{i_{3}},\,w_{i_{2}}=q^{2}w_{i_{3}},\,\cdots,w_{i_{n-1}}=qw_{i_{n}},\,w_{i_{n}}=0\\}\,,$
$\displaystyle\cdots$
$\displaystyle\\{w_{i_{1}}=0,\,w_{i_{2}}=0,\cdots,w_{i_{n-1}}=0,\,w_{i_{n}}=0\\}\,.$
(2.26)

The poles of second type are given by

$\displaystyle\\{w_{i_{n+1}}=qw_{i_{n+2}},\,w_{i_{n+2}}=qw_{i_{n+3}},\,\cdots,w_{i_{k-1}}=qw_{i_{k}},\,w_{i_{k}}=(x^{(2)}_{j})^{-1}q^{\frac{1}{2}}\\},$
$\displaystyle\\{w_{i_{n+1}}=qw_{i_{n+3}},\,w_{i_{n+2}}=q^{2}w_{i_{n+3}},\,\cdots,w_{i_{k-1}}=qw_{i_{k}},\,w_{i_{k}}=(x^{(2)}_{j})^{-1}q^{\frac{1}{2}}\\},$
$\displaystyle\cdots$
$\displaystyle\\{w_{i_{n+1}}=(x^{(2)}_{j_{1}})^{-1}q^{\frac{1}{2}},\,w_{i_{n+2}}=(x^{(2)}_{j_{1}})^{-1}q^{\frac{1}{2}},\cdots,w_{i_{k-1}}=(x^{(2)}_{j_{1}})^{-1}q^{\frac{1}{2}},\,w_{i_{k}}=(x^{(2)}_{j})^{-1}q^{\frac{1}{2}}\\}.$
(2.27)

Note that, if $-\frac{N_{2}-N_{0}}{2}\leq\kappa<\frac{N_{2}-N_{0}}{2}$, the
residues at (2.26) are zero. In this case, the residues at (2.27) agrees with
$Z^{(\zeta<0)}_{k}$ with the identification of the parameters (2.19). On the
other hand, if $\kappa=\frac{N_{2}-N_{0}}{2}$ is satisfied, the residues at
the first type poles gives non-zero contribution. In a similar way as (2.21),
this residues can be calculated as follows.

$\displaystyle\sum\underset{w_{i_{n}}}{\mathrm{Res}}\cdots\underset{w_{i_{1}}}{\mathrm{Res}}\frac{\mathcal{I}_{k}(w^{-1}_{1},\cdots,w^{-1}_{k})}{\prod_{\alpha=1}^{k}w_{\alpha}}$
$\displaystyle=\frac{n!(k-n)!}{k!}(-1)^{(N_{2}-N_{1})n}(\prod_{a=1}^{N_{0}}x^{(0)}_{a})^{-\frac{n}{2}}(\prod_{a=1}^{N_{2}}x^{(2)}_{a})^{\frac{n}{2}}q^{\frac{n}{4}(2N_{1}-N_{0}-N_{2})}\frac{\mathcal{I}_{k-n}(\\{w^{-1}_{\alpha}\\}_{\alpha\in\mathcal{J}^{c}})}{\prod_{\alpha\in\mathcal{J}^{c}}w^{-1}_{\alpha}}$
$\displaystyle\times\frac{q^{\frac{n^{2}}{2}}}{n!(1-q)^{n}}\oint_{T^{n}_{w}}\prod_{\alpha=1}^{n}\frac{dw_{i_{\alpha}}}{2\pi{\rm
i}w_{i_{\alpha}}}\prod_{1\leq\alpha\neq\beta\leq
n}\frac{w_{i_{\alpha}}-w_{i_{\beta}}}{w_{i_{\alpha}}-qw_{i_{\beta}}}$
$\displaystyle=\frac{n!(k-n)!}{k!}(-1)^{(N_{2}-N_{1})n}(\prod_{a=1}^{N_{0}}x^{(0)}_{a})^{-\frac{n}{2}}(\prod_{a=1}^{N_{2}}x^{(2)}_{a})^{\frac{n}{2}}q^{\frac{n}{4}(2N_{1}-N_{0}-N_{2})}\
\frac{\mathcal{I}_{k-n}(\\{w^{-1}_{\alpha}\\}_{\alpha\in\mathcal{J}^{c}})}{\prod_{\alpha\in\mathcal{J}^{c}}w^{-1}_{\alpha}}\frac{(-q^{\frac{1}{2}})^{n}}{(q;q)_{n}}\,.$
(2.28)

We evaluate the residues at (2.27) and take the summation over all the poles
in the first and the second type. Then we obtain

$\displaystyle\mathcal{Z}_{k}=\left\\{\begin{array}[]{cl}\sum\limits_{n=0}^{k}(-1)^{(N_{2}-N_{1})n}(\prod_{a=1}^{N_{0}}x^{(0)}_{a})^{-\frac{n}{2}}(\prod_{a=1}^{N_{2}}x^{(2)}_{a})^{\frac{n}{2}}q^{\frac{n}{4}(2N_{1}-N_{0}-N_{2})}\frac{(-q^{\frac{1}{2}})^{n}}{(q;q)_{n}}{Z}^{(\zeta<0)}_{k-n},&(\kappa=\frac{N_{2}-N_{0}}{2})\\\
{Z}^{(\zeta<0)}_{k},&(\kappa\neq\frac{N_{2}-N_{0}}{2})\end{array}\right.$
(2.31)

The equality of (2.25) and (2.31) gives the relation between
$Z^{(\zeta>0)}_{k}$’s and $Z^{(\zeta<0)}_{k}$’s, i.e., the wall-crossing
formula for the $K$-theoretic vortex partition functions.

In order to express the wall-crossing formula in a compact manner, we define a
generating function of $k$-vortex partition functions by

$\displaystyle Z^{(\zeta)}=1+\sum_{k=1}^{\infty}z^{k}Z^{(\zeta)}_{k}\,.$
(2.32)

For simplicity we often refer to the generating function of vortex partition
functions as the vortex partition function. Then the wall-crossing formula is
expressed by

$\displaystyle\sum_{k=0}^{\infty}z^{k}\mathcal{Z}_{k}$
$\displaystyle=\left(\sum_{n=0}^{\infty}\frac{z^{n}_{+}}{(q;q)_{n}}\right)^{\delta_{2\kappa,N_{0}-N_{2}}}Z^{(\zeta>0)}$
$\displaystyle=\left(\sum_{n=0}^{\infty}\frac{z^{n}_{-}}{(q;q)_{n}}\right)^{\delta_{2\kappa,N_{2}-N_{0}}}Z^{(\zeta<0)}\,,$
(2.33)

where $z_{+}$ and $z_{-}$ are defined by

$\displaystyle z_{+}$
$\displaystyle:=(-1)^{N_{1}-N_{0}+1}(\prod_{a=1}^{N_{0}}x^{(0)}_{a})^{\frac{1}{2}}(\prod_{a=1}^{N_{2}}x^{(2)}_{a})^{-\frac{1}{2}}q^{\frac{1}{4}(N_{0}+N_{2}-2N_{1}+2)}z\,,$
(2.34) $\displaystyle z_{-}$
$\displaystyle:=(-1)^{N_{1}-N_{2}+1}(\prod_{a=1}^{N_{0}}x^{(0)}_{a})^{-\frac{1}{2}}(\prod_{a=1}^{N_{2}}x^{(2)}_{a})^{\frac{1}{2}}q^{-\frac{1}{4}(N_{0}+N_{2}-2N_{1}-2)}z\,.$
(2.35)

Using the $q$-binomial theorem (A.7), we obtain the following expression of
the wall-crossing formula of the $K$-theoretic vortex partition function in 3d
$\mathcal{N}=2$ gauge theory with the Chern-Simons term [1]:

$\displaystyle{(z_{+};q)_{\infty}^{{\delta_{2\kappa,N_{0}-N_{2}}}}}Z^{(\zeta>0)}={(z_{-};q)_{\infty}^{{\delta_{2\kappa,N_{2}-N_{0}}}}}\,Z^{(\zeta<0)}.$
(2.36)

###  2.3 Kajihara transformation as WC formula in 3d $\mathcal{N}=2$ theory

Let us show that the wall-crossing formula (2.36) for $N_{2}=N_{0}$ and
$\kappa=0$ is identical to the Kajihara transformation [2]. Using the
identities (A.5) and (A.6), the $K$-theoretic vortex partition functions
(2.12) and (2.13) with (2.19) are expressed as

$\displaystyle Z^{(\zeta>0)}$
$\displaystyle=\sum_{k=0}^{\infty}z_{+}^{k}\sum_{k_{1}+\cdots+k_{N_{1}}=k}\,\,\prod_{1\leq
i<j\leq
N_{1}}\frac{x^{(2)}_{i}q^{k_{i}}-x^{(2)}_{j}q^{k_{j}}}{x^{(2)}_{i}-x^{(2)}_{j}}$
$\displaystyle\qquad\times\prod_{i,j=1}^{N_{1}}\frac{(x^{(2)}_{i}/x^{(0)}_{j};q)_{k_{i}}}{(qx^{(2)}_{i}/x^{(2)}_{j};q)_{k_{i}}}\cdot\prod_{i=1}^{N_{1}}\prod_{j=N_{1}+1}^{N_{2}}\frac{(x^{(2)}_{i}/x^{(0)}_{j};q)_{k_{i}}}{(qx^{(2)}_{i}/x^{(2)}_{j};q)_{k_{i}}}\,,$
(2.37) $\displaystyle Z^{(\zeta<0)}$
$\displaystyle=\sum_{k=0}^{\infty}z_{-}^{{k}}\sum_{{k}_{N_{1}+1}+\cdots+{k}_{N_{2}}={k}}\,\,\prod_{N_{1}+1\leq
i<j\leq
N_{2}}\frac{(x^{(2)}_{i})^{-1}q^{{k}_{i}}-(x^{(2)}_{j})^{-1}q^{{k}_{j}}}{(x^{(2)}_{i})^{-1}-(x^{(2)}_{j})^{-1}}$
$\displaystyle\qquad\times\prod_{i,j=N_{1}+1}^{N_{2}}\frac{(qx^{(0)}_{i}/x^{(2)}_{j};q)_{{k}_{i}}}{(qx^{(2)}_{i}/x^{(2)}_{j};q)_{{k}_{i}}}\cdot\prod_{i=N_{1}+1}^{N_{2}}\prod_{j=1}^{N_{1}}\frac{(qx^{(0)}_{j}/x^{(2)}_{i};q)_{{k}_{i}}}{(qx^{(2)}_{j}/x^{(2)}_{i};q)_{{k}_{i}}}\,.$
(2.38)

Next we define ${\sf x}_{i},{\sf y}_{i},{\sf a}_{i},{\sf b}_{i},{\sf c}$, and
${\sf z}$ by

$\displaystyle x^{(2)}_{i}$
$\displaystyle=\left\\{\begin{array}[]{cl}\frac{{\sf x}_{i}}{{\sf
x}_{N_{1}}},&i\in\\{1,\cdots,N_{1}\\}\,,\\\ \frac{q\,{\sf
y}_{N_{2}-N_{1}}}{c\,{\sf
y}_{i-N_{1}}},&i\in\\{N_{1}+1,\cdots,N_{2}\\}\end{array}\right.$ (2.41)
$\displaystyle x^{(0)}_{i}$
$\displaystyle=\left\\{\begin{array}[]{cl}\frac{{\sf x}_{i}}{{\sf
x}_{N_{1}}{\sf a}_{i}},&i\in\\{1,\cdots,N_{1}\\}\,,\\\ \frac{{\sf
y}_{N_{2}-N_{1}}}{{\sf b}_{i-N_{1}}\,{\sf
y}_{i-N_{1}}}\,,&i\in\\{N_{1}+1,\cdots,N_{2}\\}\,,\end{array}\right.$ (2.44)
$\displaystyle(-1)^{N_{1}-N_{2}+1}$
$\displaystyle(\prod_{a=1}^{N_{2}}x^{(0)}_{a}/x^{(2)}_{a})^{\frac{1}{2}}q^{\frac{1}{2}(N_{2}-N_{1}+1)}z=z_{+}={\sf
z}\,.$ (2.45)

Then the wall-crossing formula (2.36) is written as

$\displaystyle\sum_{k=0}^{\infty}{\sf
z}^{k}\sum_{k_{1}+\cdots+k_{N_{1}}=k}\prod_{1\leq i<j\leq N_{1}}\frac{{\sf
x}_{i}q^{k_{i}}-{\sf x}_{j}q^{k_{j}}}{{\sf x}_{i}-{\sf
x}_{j}}\cdot\prod_{i,j=1}^{N_{1}}\frac{({\sf a}_{j}{\sf x}_{i}/{\sf
x}_{j};q)_{k_{i}}}{(q\,{\sf x}_{i}/{\sf
x}_{j};q)_{k_{i}}}\cdot\prod_{i=1}^{N_{1}}\prod_{j=1}^{N_{2}-N_{1}}\frac{({\sf
b}_{j}{\sf x}_{i}{\sf y}_{j}/{\sf x}_{N_{1}}{\sf
y}_{N_{2}-N_{1}};q)_{k_{i}}}{(c{\sf x}_{i}{\sf y}_{j}/{\sf x}_{N_{1}}{\sf
y}_{N_{2}-N_{1}};q)_{k_{i}}}$ $\displaystyle=\frac{({\sf
z}\,c^{N_{2}-N_{1}}\prod_{i=1}^{N_{1}}{\sf a}_{i}\prod_{j=1}^{N_{2}-N_{1}}{\sf
b}_{j};q)_{\infty}}{({\sf z};q)_{\infty}}\sum_{k=0}^{\infty}\Bigl{(}{\sf
z}\,c^{N_{2}-N_{1}}\prod_{i=1}^{N_{1}}{\sf a}_{i}\prod_{j=1}^{N_{2}-N_{1}}{\sf
b}_{j}\Bigr{)}^{{k}}\sum_{{k}_{1}+\cdots+{k}_{N_{2}-N_{1}}={k}}$
$\displaystyle\times\prod_{1\leq i<j\leq N_{2}-N_{1}}\frac{{\sf
y}_{i}q^{{k}_{i}}-{\sf y}_{j}q^{k_{j}}}{{\sf y}_{i}-{\sf
y}_{j}}\cdot\prod_{i,j=1}^{N_{2}-N_{1}}\frac{((c/{\sf b}_{j}){\sf y}_{i}/{\sf
y}_{j};q)_{{k}_{i}}}{(q\,{\sf y}_{i}/{\sf
y}_{j};q)_{{k}_{i}}}\cdot\prod_{i=1}^{N_{2}-N_{1}}\prod_{j=1}^{N_{1}}\frac{((c/{\sf
a}_{j}){\sf y}_{i}{\sf x}_{j}/{\sf y}_{N_{2}-N_{1}}{\sf
x}_{N_{1}};q)_{{k}_{i}}}{(c{\sf y}_{i}{\sf x}_{j}/{\sf y}_{N_{2}-N_{1}}{\sf
x}_{N_{1}};q)_{{k}_{i}}}$

This agrees with Kajihara transformation [2] of Kajihara-Noumi multiple
hypergeometric series [23]. The wall-crossing formula for the general
$N_{2},N_{0}$ and $\kappa$ is thought as a generalization of Kajihara
transformation.

##  3 Hallnäs-Langmann-Noumi-Rosengren formula as wall-crossing formula

Figure 2: The quiver diagram representing the 1d $\mathcal{N}=4$ SQM associated with the $k$-vortex moduli for the 3d $\mathcal{N}=4$ gauge theory. The circle with ’$k$’ denotes the 1d $\mathcal{N}=4$ $U(k)$ vector multiplet. An arrow with solid line denotes a 1d $\mathcal{N}=4$ chiral multiplet. ’$B$’, ’$I$’ and ’$J$’ denote the scalars in each 1d $\mathcal{N}=4$ chiral multiplet. The representation of the 1d multiplets are the same as the 1d $\mathcal{N}=2$ case and summarized in Table 3 | $U(k)$ | $U(N_{1})$ | $U(N_{2}-N_{1})$ | $U(N_{0})$ | $U(1)_{\varepsilon}$
---|---|---|---|---|---
$B$ | $\bm{adj}$ | ${\bm{1}}$ | ${\bm{1}}$ | ${\bm{1}}$ | $-1$
$I$ | ${\bm{k}}$ | $\overline{\bm{N}_{1}}$ | ${\bm{1}}$ | ${\bm{1}}$ | $-\frac{1}{2}$
$J$ | $\bar{\bm{k}}$ | ${\bm{1}}$ | ${{\bm{N}_{2}}-{\bm{N}_{1}}}$ | ${\bm{1}}$ | $-\frac{1}{2}$
Table 3: The representation of the scalars in the 1d $\mathcal{N}=4$ chiral
multiplets describing $k$-vortex moduli in the 3d $\mathcal{N}=4$ $U(N_{1})$
gauge theory depicted by the quiver diagram in Figure 2 . $B,I$ and $J$ are
scalar in the chiral multiplets. $U(k)$ is the gauge group of the SQM.
$U(N_{1})$, $U(N_{2}-N_{1})$ and $U(N_{0})$ $U(1)_{\varepsilon}$ are flavor
symmetry groups.

In the previous section we have shown that the wall-crossing formula of vortex
partition function in 3d $\mathcal{N}=2$ supersymmetric gauge theory agrees
with the Kajihara transformation. In this section we will show that the wall-
crossing formula of vortex partition function in 3d $\mathcal{N}=4$ $U(N_{1})$
supersymmetric gauge theory agrees with the trigonometric limit of
transformation formula by Hallnäs-Langmann-Noumi-Rosengren formula [3]. The
hypermultiplets in the $U(N_{1})$ gauge theory are the same as the chiral
multiplets in Table 1 with $N_{0}=N_{2}$. The 3d $\mathcal{N}=4$ $U(N_{1})$
vector multiplet is given by 3d $\mathcal{N}=2$ with an additional chiral
multiplet in the adjoint representation of $U(N_{1})$.

###  3.1 Vortex partition function in 3d $\mathcal{N}=4$ gauge theory

First we briefly review the $K$-theoretic vortex partition function in 3d
$\mathcal{N}=4$ gauge theory. As explained in the section 2.1, the partition
function of the $\mathcal{N}=2$ gauge theory on $S^{1}\times\mathbb{R}^{2}$
reduce to the $K$-theoretic vortex partition function given by the Witten
index of 1d $\mathcal{N}=2$ SQM. Since the vortex equation is a half-BPS
equation, the partition function of the 3d $\mathcal{N}=4$ $U(N_{1})$ gauge
theory on $S^{1}\times\mathbb{R}^{2}$ reduce to the $K$-theoretic vortex
partition function given by the Witten index of 1d $\mathcal{N}=4$ SQM
depicted by the handsaw quiver in Figure 2.

From the supersymmetric localization formula, the $K$-theoretic $k$-vortex
partition function of 3d $\mathcal{N}=4$ gauge theory is given by

$\displaystyle Z^{(\zeta)}_{k,(N_{1},N_{2})}$
$\displaystyle=\frac{1}{k!(2\sinh\frac{-m}{2})^{k}}\oint_{JK(\zeta)}\prod_{\alpha=1}^{k}\frac{du_{\alpha}}{2\pi{\rm
i}}\left(\prod_{1\leq\alpha\neq\beta\leq
k}\frac{2\sinh\frac{u_{\alpha}-u_{\beta}}{2}}{2\sinh\frac{u_{\alpha}-u_{\beta}-m}{2}}\right)\left(\prod_{\alpha,\beta=1}^{k}\frac{2\sinh\frac{u_{\alpha}-u_{\beta}-m-\varepsilon}{2}}{2\sinh\frac{u_{\alpha}-u_{\beta}-\varepsilon}{2}}\right)$
$\displaystyle\qquad\times\prod_{\alpha=1}^{k}\prod_{j=1}^{N_{1}}\frac{2\sinh\left(\frac{u_{\alpha}-m_{j}-m-\varepsilon/2}{2}\right)}{2\sinh\left(\frac{u_{\alpha}-m_{j}-\varepsilon/2}{2}\right)}\prod_{j=N_{1}+1}^{N_{2}}\frac{2\sinh\left(\frac{-u_{\alpha}+m_{j}-m-\varepsilon/2}{2}\right)}{2\sinh\left(\frac{-u_{\alpha}+m_{j}-\varepsilon/2}{2}\right)}\,.$
(3.1)

Here again parameters $\\{m_{i}\\}_{i=1}^{N_{1}}$ and
$\\{m_{i}\\}_{i=N_{1}+1}^{N_{2}}$ and $\varepsilon$ are the chemical
potentials of 1d flavor symmetry groups $U(N_{1})$, $U(N_{2}-N_{1})$ and
$U(1)_{\varepsilon}$, respectively. A parameter $m$ is a chemical potential
for a linear combination of the generators of the $U(1)$-part in the $SU(2)$
R-symmetry and the $U(1)$-rotation of the time circle in 1d $\mathcal{N}=4$
SQM.

Since the residues coming from the pole $u_{\alpha}-u_{\beta}-m=0$ is zero, in
the positive FI-parameter region, the poles contributing to the JK residues of
(3.1) are the same as (2.8). In a similar way, the JK residue of (3.1) in the
negative FI-parameter region is evaluated at (2.10). Then the $k$-vortex
partition function for 3d $\mathcal{N}=4$ gauge theory is given by

$\displaystyle Z^{(\zeta>0)}_{k,(N_{1},N_{2})}$
$\displaystyle=\sum_{k_{1}+\cdots+k_{N_{1}}=k}\prod_{i,j=1}^{N_{1}}\prod_{\ell_{i}=1}^{k_{i}}\frac{2\sinh\left(\frac{m_{i}-m_{j}-m+(\ell_{i}-k_{j}-1)\varepsilon}{2}\right)}{2\sinh\left(\frac{m_{i}-m_{j}+(\ell_{i}-k_{j}-1)\varepsilon}{2}\right)}\cdot\prod_{i=1}^{N_{1}}\prod_{j=N_{1}+1}^{N_{2}}\prod_{\ell_{i}=1}^{k_{i}}\frac{2\sinh\left(\frac{-m_{i}+m_{j}-m-\ell_{i}\varepsilon}{2}\right)}{2\sinh\left(\frac{-m_{i}+m_{j}-\ell_{i}\varepsilon}{2}\right)}$
(3.2) $\displaystyle Z^{(\zeta<0)}_{k,(N_{1},N_{2})}$
$\displaystyle=\sum_{k_{N_{1}+1}+\cdots+k_{N_{2}}=k}\prod_{i,j=N_{1}+1}^{N_{2}}\prod_{\ell_{i}=1}^{k_{i}}\frac{2\sinh\left(\frac{-m_{i}+m_{j}-m+(\ell_{i}-k_{j}-1)\varepsilon}{2}\right)}{2\sinh\left(\frac{-m_{i}+m_{j}+(\ell_{j}-k_{i}-1)\varepsilon}{2}\right)}\cdot\prod_{i=N_{1}+1}^{N_{2}}\prod_{j=1}^{N_{1}}\prod_{\ell_{i}=1}^{k_{i}}\frac{2\sinh\left(\frac{m_{i}-m_{j}-m-\ell_{i}\varepsilon}{2}\right)}{2\sinh\left(\frac{m_{i}-m_{j}-\ell_{i}\varepsilon}{2}\right)}$
(3.3)

###  3.2  Hallnäs-Langmann-Noumi-Rosengren formula as WC formula in 3d
$\mathcal{N}=4$ theory

In section 2.2, we derived the wall-crossing formula in terms of contour
integral (2.15). In this subsection we review the large mass limit applied in
[16] to derive the wall-crossing formula of the 3d $\mathcal{N}=4$ gauge
theory. First we define the generating function of the $K$-theoretic
$k$-vortex partition function for 3d $\mathcal{N}=4$ gauge theory by

$\displaystyle
Z^{(\zeta)}_{(N_{1},N_{2})}(x,\tilde{x},t,q,z)=1+\sum_{k=1}^{\infty}z^{k}Z^{(\zeta)}_{k,(N_{1},N_{2})}(x,\tilde{x},t,q)\,,$
(3.4)

where $x_{i},\tilde{x}_{i},t$ are defined by

$\displaystyle
x_{i}=e^{m_{i}},\,\,\tilde{x}_{i}=e^{m_{N_{1}+i}},\,\,t=e^{m}\,,$ (3.5)

for $i\in\\{1,\cdots,N_{1}\\}$. $q$ is again the exponentiated
$\Omega$-background parameter $q=e^{\varepsilon}$.

In order to prove the wall-crossing formula, we consider the case for
$N_{2}=2N_{1}$. From the expressions of (3.2) and (3.3), the following
relation holds under the exchange of the variables $m_{i}\leftrightarrow-
m_{i+N_{1}}$ for $i=1,\cdots,N_{1}$ between the $K$-theoretic vortex partition
functions:

$\displaystyle
Z^{(\zeta>0)}_{(N_{1},2N_{1})}(x,\tilde{x},t,q,z)=Z^{(\zeta>0)}_{(N_{1},2N_{1})}(\tilde{x}^{-1},x^{-1},t,q,z)$
(3.6)

We take a large negative mass limit: ${\rm Re}(m_{2N_{1}})\to-\infty$ in
(3.6). The limit ${\tilde{x}_{N_{1}}\to 0}$ in the left hand side of (3.6) is
given by

$\displaystyle\lim_{\tilde{x}_{N_{1}}\to
0}Z^{(\zeta>0)}_{(N_{1},2N_{1})}(x,\tilde{x},t,q,z)=Z^{(\zeta>0)}_{(N_{1},2N_{1}-1)}(x,\tilde{x},t,q,zt^{\frac{1}{2}})$
(3.7)

On the other hand, the limit $\tilde{x}_{N_{1}}\to 0$ in the right hand side
of (3.6) is given by

$\displaystyle\lim_{\tilde{x}_{N_{1}}\to\infty}Z^{(\zeta>0)}_{(N_{1},2N_{1})}(\tilde{x}^{-1},{x}^{-1},t,q,z)$
$\displaystyle=Z^{(\zeta)}_{1,1}(t,q,zt^{\frac{1}{2}})Z^{(\zeta>0)}_{(N_{1}-1,2N_{1}-1)}(\tilde{x}^{-1},{x}^{-1},t,q,zt^{\frac{1}{2}})$
$\displaystyle=\frac{(zqt;q)_{\infty}}{(z;q)_{\infty}}Z^{(\zeta>0)}_{(N_{1}-1,2N_{1}-1)}(\tilde{x}^{-1},{x}^{-1},t,q,zt^{\frac{1}{2}})$
(3.8)

Here we used the following identity

$\displaystyle Z^{(\zeta)}_{(1,1)}(t,q,z)$
$\displaystyle=\sum_{k=0}^{\infty}z^{k}\prod_{\ell=1}^{k}\frac{2\sinh\left(\frac{-m+(\ell-k-1)\varepsilon}{2}\right)}{2\sinh\left(\frac{(\ell-k-1)\varepsilon}{2}\right)}=\sum_{k=0}^{\infty}(t^{-\frac{1}{2}}z)^{k}\frac{(tq;q)_{k}}{(q;q)_{k}}$
$\displaystyle=\frac{(zqt^{\frac{1}{2}};q)_{\infty}}{(zt^{-\frac{1}{2}};q)_{\infty}}\,.$
(3.9)

Note that the vortex partition function $Z^{(\zeta)}_{(1,1)}(t,q,z)$ is
independent of the sign of 1d FI-parameter $\zeta$. In a similar way, by
repeating this procedure $\lim_{\tilde{x}_{s}\to 0}$ for
$s=2N_{1},2N_{1}-1,\cdots 2N_{1}-N_{2}$ in (3.6), we obtain the following
relation

$\displaystyle
Z^{(\zeta>0)}_{N_{1},N_{2}}(x,\tilde{x},t,q,zt^{\frac{2N_{1}-N_{2}}{2}})$
$\displaystyle=\prod_{s=1}^{2N_{1}-N2}\frac{(zq\,t^{s};q)_{\infty}}{(z\,t^{s-1};q)_{\infty}}\,\cdot
Z^{(\zeta>0)}_{N_{2}-N_{1},N_{2}}(\tilde{x}^{-1},x^{-1},t,q,zt^{\frac{2N_{1}-N_{2}}{2}})\,.$
(3.10)

Therefore we obtain the wall-crossing formula for 3d $\mathcal{N}=4$
$U(N_{1})$ gauge theory with the $N_{2}$ hypermultiplets [16]

$\displaystyle Z^{(\zeta>0)}_{N_{1},N_{2}}(x,\tilde{x},t,q,z)$
$\displaystyle=\prod_{s=1}^{2N_{1}-N2}\frac{(zq\,t^{s+\frac{N_{2}-2N_{1}}{2}};q)_{\infty}}{(z\,t^{s-1+\frac{N_{2}-2N_{1}}{2}};q)_{\infty}}\,\cdot
Z^{(\zeta<0)}_{N_{1},N_{2}}(x,\tilde{x},t,q,z),$ (3.11)

where we refine $z\mapsto zt^{N_{2}-2N_{1}}$ and used the relation:

$\displaystyle
Z^{(\zeta>0)}_{(N_{2}-N_{1},N_{2})}(\tilde{x}^{-1},{x}^{-1},t,q)=Z^{(\zeta<0)}_{(N_{1},N_{2})}(x,\tilde{x},t,q)\,.$
(3.12)

In (3.11), we set

$\displaystyle x_{i}$ $\displaystyle=e^{m_{i}}={\sf
x}_{i},\,\,i\in\\{1,\cdots,N_{1}\\}\,,$ (3.13)
$\displaystyle\tilde{x}^{-1}_{i-N_{1}}tq$
$\displaystyle=e^{-m_{i}+m-\varepsilon}={\sf
y}_{i-N_{1}},\,\,i\in\\{N_{1}+1,\cdots,N_{2}\\}\,,$ (3.14) $\displaystyle t$
$\displaystyle={\sf t}/{q}\,,$ (3.15) $\displaystyle
zt^{\frac{2N_{1}-N_{2}}{2}}$ $\displaystyle=ze^{\frac{2N_{1}-N_{2}}{2}m}={\sf
z}\,.$ (3.16)

Then the wall-crossing formula (3.11) is expressed by

$\displaystyle\sum_{k=0}^{\infty}{\sf
z}^{k}\sum_{k_{1}+\cdots+k_{N_{1}}=k}\prod_{i,j=1}^{N_{1}}\frac{(q^{1-k_{j}}{\sf
x}_{i}/({\sf t}{\sf x}_{j});q)_{k_{i}}}{(q^{-k_{j}}{\sf x}_{i}/{\sf
x}_{j};q)_{k_{i}}}\prod_{i=1}^{N_{1}}\prod_{j=1}^{N_{2}-N_{1}}\frac{({\sf
x}_{i}{\sf y}_{j};q)_{k_{i}}}{(q{\sf x}_{i}{\sf y}_{j}/{\sf t};q)_{k_{i}}}$
$\displaystyle=\prod_{s=1}^{2N_{1}-N_{2}}\frac{({\sf z}q^{s}/{\sf
t}^{s-1};q)_{\infty}}{({\sf z}q^{s}/{\sf
t}^{s};q)_{\infty}}\sum_{k=0}^{\infty}\left(\frac{q^{2N_{1}-N_{2}}}{{\sf
t}^{2N_{1}-N_{2}}}{\sf z}\right)^{k}$
$\displaystyle\qquad\times\sum_{k_{1}+\cdots+k_{N_{2}-N_{1}}=k}\prod_{i,j=1}^{N_{2}-N_{1}}\frac{(q^{1-k_{j}}{\sf
y}_{i}/({\sf t}{\sf y}_{j});q)_{k_{i}}}{(q^{-k_{j}}{\sf y}_{i}/{\sf
y}_{j};q)_{k_{i}}}\prod_{i=1}^{N_{2}-N_{1}}\prod_{j=1}^{N_{1}}\frac{({\sf
y}_{i}{\sf x}_{j};q)_{k_{i}}}{(q{\sf y}_{i}{\sf x}_{j}/{\sf t};q)_{k_{i}}}\,.$
(3.17)

This agrees with the trigonometric limit of the formula in [3]. For example
see eqs.(14) in [24].

###  3.3 $K$-theoretic vortex partition function as equivariant
$\chi_{t}$-genus

Since the fermi multiplet is absent in the 1d $\mathcal{N}=4$ SQM, and a 1d
$\mathcal{N}=4$ chiral multiplet contains an additional fermions compared with
1d a $\mathcal{N}=2$ chiral multiplet, the numerator of the integrand in (3.1)
has changed from the $\mathcal{N}=2$ case (2.5). The fermions determine a
characteristic class to define an index of the handsaw quiver variety. In
particular, it is known that the Witten index of the 1d $\mathcal{N}=4$ SQM
gives the equivariant $\chi_{t}$-genus of the Higgs branch vacua of the SQM.

Let us explicitly look at the agreement between the $K$-theoretic vortex
partition function and $\chi_{t}$-genus of the handsaw quiver variety. We
evaluate the equivariant character of the handsaw quiver variety of $A_{1}$
[25, 26, 14]. Elements of the complexification of the maximal torit of $U(k)$
and $U(N_{1})\times U(N_{1}-N_{2})\times U(1)_{\varepsilon}$ acts on (2.3) as

$\displaystyle g:(B,I,J)\mapsto(gBg^{-1},gI,Jg^{-1}),$ $\displaystyle
g_{1}:(B,I,J)\mapsto(B,Ig_{1},J),$ $\displaystyle
g_{2}:(B,I,J)\mapsto(B,I,g_{2}J),$ (3.18) $\displaystyle
g_{3}:(B,I,J)\mapsto(g_{3}B,I,J),$

where

$\displaystyle g$
$\displaystyle=\mathrm{diag}(e^{u_{1}},\cdots,e^{u_{k}})\in(\mathbb{C}^{*})^{k}\,,$
(3.19) $\displaystyle g_{1}$
$\displaystyle=\mathrm{diag}(e^{{m_{1}}+\varepsilon/2},\cdots,e^{m_{N_{1}}+\varepsilon/2})\in(\mathbb{C}^{*})^{N_{1}}\,,$
(3.20) $\displaystyle g_{2}$
$\displaystyle=\mathrm{diag}(e^{m_{N_{1}}-\varepsilon/2},\cdots,e^{m_{N_{2}}-\varepsilon/2})\in(\mathbb{C}^{*})^{N_{2}-N_{1}}\,,$
(3.21) $\displaystyle g_{3}$
$\displaystyle=e^{-\varepsilon}\in\mathbb{C}^{*}\,.$ (3.22)

Then the fixed points condition of the infinitesimal transformation generating
$g_{i},i=1,2,3$ up to $(\mathbb{C}^{*})^{k}$-transformation is given by

$\displaystyle\left(u_{\alpha}-u_{\beta}-\varepsilon\right)B_{\alpha,\beta}=0,\,\,\left(u_{\alpha}-m_{i}-\frac{\varepsilon}{2}\right)I_{\alpha,i}=0,\,\,\left(-u_{\alpha}+m_{j}-\frac{\varepsilon}{2}\right)J_{j,\alpha}=0\,,$
(3.23)

with $\alpha,\beta\in\\{1,\cdots,k\\},i\in\\{1,\cdots,N_{1}\\}$ and
$j\in\\{N_{1}+1,\cdots,N_{2}\\}$. Thus the fixed point condition is same as
either (2.8) or (2.10). First, we consider the case for (2.8) in which
$I_{\alpha,i}$’s have non-zero value which means $\zeta>0$. In this case, the
fixed point condition (3.23) is solved as
$u_{\alpha}=m_{i}+(\ell_{i}-1/2)\varepsilon$ with
$\ell\in\\{1,\cdots,k_{i}\\}$, $i\in\\{1,\cdots,N_{1}\\}$. Each fiber of the
tangent space of the handsaw quiver variety
$T\mathcal{M}^{(\zeta>0)}_{k,N_{1},N_{2}}$ is given by

$\displaystyle\mathrm{End}(V)\otimes
Q\oplus\mathrm{Hom}(W^{(1)},V)\oplus\mathrm{Hom}(V,W^{(2)})/\mathrm{End}(V)\,,$
(3.24)

where $Q$ is a one-dimensional vector space on which $e^{-\varepsilon}$ acts.
Then the equivariant character $Ch(T\mathcal{M}^{(\zeta>0)}_{k,N_{1},N_{2}})$
is computed as

$\displaystyle{Ch}(T\mathcal{M}^{(\zeta>0)}_{k,N_{1},N_{2}})$
$\displaystyle=V\times V^{*}\times(Q-1)+V\times(W^{(1)})^{*}+W^{(2)}\times
V^{*}$
$\displaystyle=\sum_{i,j=1}^{N_{1}}\sum_{\ell_{i}=1}^{k_{i}}e^{m_{i}-m_{j}+(\ell_{i}-k_{j}-1)\varepsilon}+\sum_{i=1}^{N_{1}}\sum_{j=N_{1}+1}^{N_{2}}\sum_{\ell_{i}=1}^{k_{i}}e^{-m_{i}+m_{j}-\ell_{i}\varepsilon}$
(3.25)

Here allowing for a slight abuse of notation, we denote each vector space and
its character by the same symbol. The characters are given by

$\displaystyle
V=\sum_{i=1}^{N_{1}}\sum_{\ell_{i}=1}^{k_{i}}e^{m_{i}+(\ell_{i}-1/2)\varepsilon},\,\,W^{(1)}=\sum_{i=1}^{N_{1}}e^{{m_{i}}+\varepsilon/2},\,\,W^{(2)}=\sum_{j=N_{1}+1}^{N_{2}}e^{m_{j}-\varepsilon/2},\,\,Q=e^{-\varepsilon}.$
(3.26)

Note that the fixed points are classified by the partitions of the integer $k$
into $N_{1}$ non-negative integers $k_{i}$, for $i=1,\cdots,N_{1}$.

Now we relate the $K$-theoretic vortex partition function obtained by the
supersymmetric localization of the Witten index to the equivariant
$\chi_{t}$-genus:

$\displaystyle\chi_{t}(M)$
$\displaystyle=\int_{M}\mathrm{ch}(\wedge_{-t}TM^{*}){\rm td}(M)$
$\displaystyle=\int_{M}\prod_{i}(1-te^{-{\tt x}_{i}})\frac{{\tt
x}_{i}}{1-e^{-{\tt x}_{i}}}\,.$ (3.27)

Here ${\tt x}_{i}$’s are the Chern roots of the tangent bundle $TM$. ${\rm
ch}(E)$ and ${\rm td}(M)$ are the Chern character of a vector bundle $E$ and
the Todd class of $M$, respectively. $\wedge_{t}E$ is the series of exterior
products of a vector bundle $E$ (in $K$-theory) given by

$\displaystyle\wedge_{t}E=1+tE+t^{2}\wedge^{2}E+\cdots\,.$ (3.28)

From the fixed point formula, for example see [27], the $\chi_{t}$-genus is
written as

$\displaystyle\chi_{t}(M)$ $\displaystyle=\sum_{p:{\rm
fixed}}\prod_{i}\frac{(1-te^{-w_{i,p}})\frac{w_{i,p}}{1-e^{-w_{i,p}}}}{\prod_{i}w_{i,p}}$
$\displaystyle=\sum_{p:{\rm
fixed}}\prod_{i}\frac{1-te^{-w_{i,p}}}{1-e^{-w_{i,p}}}\,.$ (3.29)

Here $Ch(TM)=\sum_{i}e^{w_{i,p}}$ is the equivariant character at a fixed
point $p$. The summation $\sum_{p:{\rm fixed}}$ is taken over the fixed points
$p$’s by the torus action. From the equivariant character (3.25), the
$\chi_{y}$-genus of the handsaw quiver variety is written as

$\displaystyle\chi_{t}(\mathcal{M}^{(\zeta>0)}_{k,N_{1},N_{2}})$
$\displaystyle=\sum_{k_{1}+\cdots+k_{N_{1}}=k}\prod_{i,j=1}^{N_{1}}\prod_{\ell_{i}=1}^{k_{i}}\frac{1-te^{-m_{i}+m_{j}-(\ell_{i}-k_{j}-1)\varepsilon}}{1-e^{-m_{i}+m_{j}-(\ell_{i}-k_{j}-1)\varepsilon}}\cdot\prod_{i=1}^{N_{1}}\prod_{j=N_{1}+1}^{N_{2}}\prod_{\ell_{i}=1}^{k_{i}}\frac{1-te^{m_{i}-m_{j}+\ell_{i}\varepsilon}}{1-e^{m_{i}-m_{j}+\ell_{i}\varepsilon}}$
$\displaystyle=t^{-\frac{N_{2}k}{2}}Z^{(\zeta>0)}_{k,(N_{1},N_{2})},\quad(t=e^{m}).$
(3.30)

Therefore the equivariant $\chi_{t}$-genus of
$\mathcal{M}^{(\zeta>0)}_{k,N_{1},N_{2}}$ agrees with $K$-theoretic $k$-vortex
partition function with the positive FI-parameter up to a factor
$t^{-\frac{N_{2}k}{2}}$.

Next we consider the case where the fixed point condition is given by (2.10),
which is solved as (2.11). In this case, $J_{j,\alpha}$’s have non-zero
components which means that the FI-parameter is negative: $\zeta<0$. Then the
equivariant character of the tangent space is computed as

$\displaystyle{Ch}(T\mathcal{M}^{(\zeta<0)}_{k,N_{1},N_{2}})$
$\displaystyle=V\times V^{*}\times(Q-1)+V\times(W^{(1)})^{*}+W^{(2)}\times
V^{*}$
$\displaystyle=\sum_{i,j=N_{1}+1}^{N_{2}}\sum_{\ell_{i}=1}^{k_{i}}e^{-m_{i}+m_{j}+(\ell_{i}-k_{j}-1)\varepsilon}+\sum_{i=N_{1}+1}^{N_{2}}\sum_{j=1}^{N_{1}}\sum_{\ell_{i}=1}^{k_{i}}e^{m_{i}-m_{j}-\ell_{i}\varepsilon}\,,$
(3.31)

where the characters of space $V$ and $W^{(i)}$ are given by

$\displaystyle
V=\sum_{i=N_{1}+1}^{N_{2}}\sum_{\ell_{i}=1}^{k_{i}}e^{m_{i}-(\ell_{i}-1/2)\varepsilon},\,\,W^{(1)}=\sum_{i=1}^{N_{1}}e^{{m_{i}}+\varepsilon/2},\,\,W^{(2)}=\sum_{j=N_{1}+1}^{N_{2}}e^{m_{j}-\varepsilon/2},\,\,Q=e^{-\varepsilon}\,.$
(3.32)

From the fixed point formula (3.29), again we have an agreement:

$\displaystyle\chi_{t}(\mathcal{M}^{(\zeta<0)}_{k,N_{1},N_{2}})$
$\displaystyle=\sum_{k_{1}+\cdots+k_{N_{1}}=k}\prod_{i,j=N_{1}+1}^{N_{2}}\prod_{\ell_{i}=1}^{k_{i}}\frac{1-te^{m_{i}-m_{j}-(\ell_{i}-k_{j}-1)\varepsilon}}{1-e^{m_{i}-m_{j}-(\ell_{i}-k_{j}-1)\varepsilon}}\cdot\prod_{i=N_{1}+1}^{N_{2}}\prod_{j=1}^{N_{1}}\prod_{\ell_{i}=1}^{k_{i}}\frac{1-te^{-m_{i}+m_{j}+\ell_{i}\varepsilon}}{1-e^{-m_{i}+m_{j}+\ell_{i}\varepsilon}}$
$\displaystyle=t^{-\frac{N_{2}k}{2}}Z^{(\zeta<0)}_{k,(N_{1},N_{2})},\quad(t=e^{m}).$
(3.33)

The difference between between $\chi_{t}$-genera and vortex partition
functions is absorbed into the redefinition $z\to zt^{\frac{N_{2}}{2}}$ in
(3.4), or eliminated by turning on the background flavor 1d Chern-Simons term:
$t^{\frac{N_{2}k}{2}}$. Therefore we have shown that the wall-crossing formula
of $K$-theoretic vortex partition function for 3d $\mathcal{N}=4$ gauge theory
is same as that of $\chi_{t}$-genus of the handsaw quiver variety of type
$A_{1}$.

Next we consider the vortex partition functions for 3d $\mathcal{N}=2$ theory.
Since the computation is parallel to the above 3d $\mathcal{N}=4$ case, we
briefly comment on the geometric interpretation of $K$-theoretic vortex
partition function for 3d $\mathcal{N}=2$ gauge theory. We consider the vector
bundle ${\cal E}$, whose the fiber at each point is given by $V$ quotient by
$GL(k,\mathbb{C})$ and take the Chern character
$\mathrm{ch}(\otimes_{i=1}^{N_{0}}\wedge_{(x^{(0)}_{i}q)^{-1}}{\cal E})$. Then
the index is written as

$\displaystyle\int_{\mathcal{M}^{(\zeta)}_{k,N_{1},N_{2}}}e^{\kappa^{\prime}c_{1}}\mathrm{ch}(\otimes_{i=1}^{N_{0}}\wedge_{-(x^{(0)}_{i}q^{1/2})^{-1}}{\cal
E}){\rm td}(\mathcal{M}^{(\zeta)}_{k,N_{1},N_{2}})$
$\displaystyle\qquad\qquad=q^{\frac{k^{2}}{2}}\Bigl{[}q^{\frac{N_{0}+N_{2}}{2}}\prod_{i=1}^{N_{0}}x^{(0)}_{i}\cdot\prod_{i=1}^{N_{1}}x^{(2)}_{i}\cdot\prod_{i=N_{1}+1}^{N_{2}}(x^{(2)}_{i})^{-1})\Bigr{]}^{\frac{k}{2}}\,Z^{(\zeta)}_{k}\,$
(3.34)

Here we $\kappa^{\prime}=\kappa+\frac{N_{2}-N_{0}-2N_{1}}{2}$ and $c_{1}$ is
the first Chern class. Again the pre-factor given by the $k/2$-th power of
$x^{(0)}_{i},x_{i}^{(2)}$ and $q$ is absorbed into the redefinition of $z$ in
the generating function. On the other hand, $q^{\frac{k^{2}}{2}}$ can not be
absorbed into any redefinition of parameters. Thus the wall-crossing formula
of the index (3.34) is derived by the wall-crossing formula of $k$-vortex
partition function, but not exactly same.

###  3.4 Elliptic analogue of wall-crossing formula

In this section we briefly comment on an elliptic analogue of the wall-
crossing formula. First we consider the 4d $\mathcal{N}=2$ gauge theory on
$T^{2}\times\mathbb{R}^{2}$, whose dimensional reduction gives 3d
$\mathcal{N}=4$ gauge theory considered in section 3. The $k$-vortex partition
function for the 4d $\mathcal{N}=2$ gauge theory is given by the elliptic
genus of 2d $\mathcal{N}=(2,2)$ gauge theory on $T^{2}$, where the matter
content is the 2d lift of the 1d $\mathcal{N}=4$ SQM in Table 2. The elliptic
genus is again evaluated in terms of supersymmetric localization formula in
[28, 29], and written as

$\displaystyle Z^{(\zeta>0)}_{4d,k,(N_{1},N_{2})}$
$\displaystyle=\sum_{k_{1}+\cdots+k_{N_{1}}=k}\prod_{i,j=1}^{N_{1}}\prod_{\ell_{i}=1}^{k_{i}}\frac{\theta_{1}(m_{ij}-m+(\ell_{i}-k_{j}-1)\varepsilon)}{\theta_{1}(m_{ij}+(\ell_{i}-k_{j}-1)\varepsilon)}\cdot\prod_{i=1}^{N_{1}}\prod_{j=N_{1}+1}^{N_{2}}\prod_{\ell_{i}=1}^{k_{i}}\frac{\theta_{1}(m_{ji}-m-\ell_{i}\varepsilon)}{\theta_{1}(m_{j,i}-\ell_{i}\varepsilon)}$
(3.35) $\displaystyle Z^{(\zeta<0)}_{4d,k,(N_{1},N_{2})}$
$\displaystyle=\sum_{k_{N_{1}+1}+\cdots+k_{N_{2}}=k}\prod_{ij=N_{1}+1}^{N_{2}}\prod_{\ell_{i}=1}^{k_{i}}\frac{\theta_{1}(m_{ji}-m+(\ell_{i}-k_{j}-1)\varepsilon)}{\theta_{1}(m_{ji}+(\ell_{j}-k_{i}-1)\varepsilon)}\cdot\prod_{i=N_{1}+1}^{N_{2}}\prod_{j=1}^{N_{1}}\prod_{\ell_{i}=1}^{k_{i}}\frac{\theta_{1}(m_{ij}-m-\ell_{i}\varepsilon)}{\theta_{1}(m_{ij}-\ell_{i}\varepsilon)}$
(3.36)

Here $m_{ij}=m_{i}-m_{j}$ and $\zeta$ is a reference vector which plays the
same pole of the FI-parameter in the JK residue computation. $\theta_{1}(z)$
is a Jacobi theta function defined by

$\displaystyle\theta_{1}(z)=\theta_{1}(\tau|z)=-{\rm
i}y^{\frac{1}{2}}\prod_{k=1}^{\infty}(1-{\sf q}^{k})(1-y{\sf
q}^{k})(1-y^{-1}{\sf q}^{k-1})\,,$ (3.37)

where ${\sf q}=e^{2\pi{\rm i}\tau}$ and $y=e^{2\pi{\rm i}z}$. $\tau$ is the
moduli of two-dimensional torus $T^{2}$. As explained in [28, 29], the
elliptic genus does not have the non-trivial wall-crossing phenomena. Thus the
elliptic analogue of $K$-theoretic vortex partition functions evaluated in two
regions coincides each other up to a signature:

$\displaystyle Z^{(\zeta>0)}_{4d,k,(N_{1},N_{2})}$
$\displaystyle=Z^{(\zeta<0)}_{4d,k,(N_{1},N_{2})}\,.$ (3.38)

Next we give a remark on the elliptic lift of 3d $\mathcal{N}=2$ theory
without Chern-Simons term. In this case, the elliptic lift of 1d
$\mathcal{N}=2$ SQM for the $k$-vortex moduli is given by a 2d
$\mathcal{N}=(0,2)$ gauge theory on $T^{2}$, which has the gauge anomaly and
inconsistent as quantum field theory. It is interesting to find the elliptic
lift of Kajihara transformation realized as an elliptic genus.

##  4 Two-dimensional/cohomological/rational limit

In this section we take the zero-radius limit of the time circle $S^{1}$ and
show that the wall-crossing formulas reduce to the the wall-crossing formulas
for the vortex partition functions in 2d $\mathcal{N}=(2,2)$ and
$\mathcal{N}=(2,2)^{*}$ gauge theories on $\mathbb{R}^{2}$. Since an
exponentiated chemical potential $e^{m_{i}}$ is a BPS background Wilson loop
for a $U(1)$-flavor symmetry, in the 2d limit, we may rescale that
$m_{i}\to-\beta m_{i}$ and take a limit $\beta\to 0$ with keeping the new
$m_{i}$ finite. Here $\beta$ is the circumference of $S^{1}$ and the new
$m_{i}$ is a twisted mass in the 2d theory. Let us rescale all the chemical
potentials and take the zero radius limit.

First we consider the wall-crossing formula of the 2d $\mathcal{N}=(2,2)$
gauge theory obtained by the 2d limit of the 3d $\mathcal{N}=2$ gauge theory
with $N_{2}=N_{0}$ and $\kappa=0$. In the above limit $\beta\to 0$, the
$K$-theoretic vortex partition functions reduce to

$\displaystyle Z^{(\zeta>0)}_{2d,k}:=\lim_{\beta\to 0}Z^{(\zeta>0)}_{k}$
$\displaystyle=\sum_{k_{1}+\cdots+k_{N_{1}}=k}\frac{(-1)^{N_{1}k}\prod_{i=1}^{N_{1}}\prod_{j=1}^{N_{2}}\prod_{\ell_{i}=1}^{k_{i}}\left(m_{i}-\tilde{m}_{j}+(\ell_{i}-1)\varepsilon\right)}{\prod_{i,j=1}^{N_{1}}\prod_{\ell_{i}=1}^{k_{i}}\left(m_{j}-m_{i}-(\ell_{i}-k_{j}-1)\varepsilon\right)\prod_{i=1}^{N_{1}}\prod_{j=N_{1}+1}^{N_{2}}\prod_{\ell_{i}=1}^{k_{i}}\left(-m_{i}+m_{j}-\ell_{i}\varepsilon\right)}\,,$
(4.1) $\displaystyle Z^{(\zeta<0)}_{2d,k}:=\lim_{\beta\to 0}Z^{(\zeta<0)}_{k}$
$\displaystyle=\sum_{{k}_{N_{1}+1}+\cdots+{k}_{N_{2}}={k}}\frac{(-1)^{N_{1}k}\prod_{i=N_{1}+1}^{N_{2}}\prod_{j=1}^{N_{2}}\prod_{\ell_{i}=1}^{k_{i}}\left(-m_{i}+\tilde{m}_{j}+\ell_{i}\varepsilon\right)}{\prod_{i,j=N_{1}+1}^{N_{2}}\prod_{\ell_{i}=1}^{k_{i}}\left(-m_{j}+m_{i}-(\ell_{i}-k_{j}-1)\varepsilon\right)\prod_{i=N_{1}+1}^{N_{2}}\prod_{j=1}^{N_{1}}\prod_{\ell_{i}=1}^{k_{i}}\left(m_{i}-m_{j}-\ell_{i}\varepsilon\right)}\,.$
(4.2)

This expression agrees with the vortex partition functions in 2d
$\mathcal{N}=(2,2)$ $U(N_{1})$ gauge theory with $N_{2}$ fundamental and
$N_{2}$ anti-fundamental chiral multiplets in the positive and negative FI-
parameter regions, respectively.

The 2d limit of the wall-crossing factor in (2.36) is given by

$\displaystyle\lim_{\beta\to
0}\frac{(z_{-};q)_{\infty}}{(z_{+};q)_{\infty}}=(1-z)^{N_{2}-N_{1}+\sum_{a=1}^{N_{2}}\frac{\tilde{m}_{a}-m_{a}}{\varepsilon}}\,.$
(4.3)

Here we used an identity (A.8) to show (4.3). Thus the 2d limit of the wall-
crossing formula of the $K$-theoretic vortex partition function leads to the
relation:

$\displaystyle
Z^{(\zeta>0)}_{2d}=(1-z)^{N_{2}-N_{1}+\sum_{a=1}^{N_{2}}\frac{\tilde{m}_{a}-m_{a}}{\varepsilon}}\,Z^{(\zeta<0)}_{2d}\,,$
(4.4)

where

$\displaystyle
Z^{(\zeta)}_{2d}=1+\sum_{k=1}^{\infty}z^{k}Z^{(\zeta)}_{2d,k}\,.$ (4.5)

This agrees with the 2d wall-crossing formula originally obtained as the
relation between vortex partition functions in the Seiberg duality between 2d
$\mathcal{N}=(2,2)$ $U(N_{1})$ and $U(N_{2}-N_{1})$ gauge theories [30], see
also [31, 32]. Specifically, the 2d wall-crossing formula (4.4) for $N_{1}=1$
and $N_{2}=2$ is same as the Euler transformation of Gauss hypergeometric
series (1.2) by setting

$\displaystyle\frac{m_{1}-\tilde{m}_{1}}{\varepsilon}=a,\,\,\,\frac{m_{1}-\tilde{m}_{2}}{\varepsilon}=b,\,\,\,\frac{m_{1}-m_{2}}{\varepsilon}+1=c.$
(4.6)

Next we consider the 2d limit of the wall-crossing formula for 3d
$\mathcal{N}=4$ gauge theory.

$\displaystyle Z^{(\zeta>0)}_{2d,k,(N_{1},N_{2})}=\lim_{\beta\to
0}Z^{(\zeta>0)}_{k,(N_{1},N_{2})}$
$\displaystyle=\sum_{k_{1}+\cdots+k_{N_{1}}=k}\prod_{i,j=1}^{N_{1}}\prod_{\ell_{i}=1}^{k_{i}}\frac{m_{i}-m_{j}-m+(\ell_{i}-k_{j}-1)\varepsilon}{m_{i}-m_{j}+(\ell_{i}-k_{j}-1)\varepsilon}\cdot\prod_{i=1}^{N_{1}}\prod_{j=N_{1}+1}^{N_{2}}\prod_{\ell_{i}=1}^{k_{i}}\frac{-m_{i}+m_{j}-m-\ell_{i}\varepsilon}{-m_{i}+m_{j}-\ell_{i}\varepsilon}$
(4.7) $\displaystyle Z^{(\zeta>0)}_{2d,k,(N_{1},N_{2})}=\lim_{\beta\to
0}Z^{(\zeta<0)}_{k,(N_{1},N_{2})}$
$\displaystyle=\sum_{k_{N_{1}+1}+\cdots+k_{N_{2}}=k}\prod_{i,j=N_{1}+1}^{N_{2}}\prod_{\ell_{i}=1}^{k_{i}}\frac{-m_{i}+m_{j}-m+(\ell_{i}-k_{j}-1)\varepsilon}{-m_{i}+m_{j}+(\ell_{j}-k_{i}-1)\varepsilon}\cdot\prod_{i=N_{1}+1}^{N_{2}}\prod_{j=1}^{N_{1}}\prod_{\ell_{i}=1}^{k_{i}}\frac{m_{i}-m_{j}-m-\ell_{i}\varepsilon}{m_{i}-m_{j}-\ell_{i}\varepsilon}$
(4.8)

The 2d limit of the wall-crossing factor (3.11) is given by

$\displaystyle\lim_{\beta\to
0}\prod_{s=1}^{2N_{1}-N2}\frac{(zq\,t^{s+\frac{N_{2}-2N_{1}}{2}};q)_{\infty}}{(z\,t^{s-1+\frac{N_{2}-2N_{1}}{2}};q)_{\infty}}=(1-z)^{(N_{2}-2N_{1})(\frac{m}{\varepsilon}-1)}$
(4.9)

Thus the 2d limit of the wall-crossing formula is given by

$\displaystyle
Z^{(\zeta>0)}_{2d,(N_{1},N_{2})}=(1-z)^{(N_{2}-2N_{1})(\frac{m}{\varepsilon}-1)}\,Z^{(\zeta<0)}_{2d,(N_{1},N_{2})}\,,$
(4.10)

where

$\displaystyle
Z^{(\zeta)}_{2d,(N_{1},N_{2})}=1+\sum_{k=1}^{\infty}z^{k}Z^{(\zeta)}_{2d,k,(N_{1},N_{2})}\,.$
(4.11)

This agrees with the 2d wall-crossing formula originally derived in [30], see
also [33] and is recognized as the relation between the vortex partition
functions in the Seiberg duality between 2d $\mathcal{N}=(2,2)^{*}$ $U(N_{1})$
and $U(N_{2}-N_{1})$ gauge theories.

Note that a 2d vortex partition function agrees with the fixed point formula
for the integral of a cohomology class over the handsaw quiver variety. This
is analogous to a 4d instanton partition function, which agrees with an
integral of cohomology class over the instanton moduli space. Later it was
studied in [24] that the 2d wall-crossing formula is re-interpreted in the
context of algebraic geometry and is shown to agree with the rational limits
of the transformation formulas by Kajihara and Hallnäs-Langmann-Noumi-
Rosengren.

##  5 Future directions

In this paper we point out that the wall-crossing formulas of the
$K$-theoretic vortex partition functions [1] for 3d $U(N)$ supersymmetric
gauge theories are identical to the Euler transformations of
$q$-hypergeometric series [2, 3]. We comment on future directions of this
work.

Handsaw quiver of type $A_{n}$
Generalizations to the handsaw quiver varieties of type $A_{n}$ correspond to
the $K$-theoretic vortex partition functions in 3d $\mathcal{N}=2$ or
$\mathcal{N}=4$ $\prod_{s=1}^{n}U(N_{s})$ with $N_{s}<N_{s+1}$ linear quiver
supersymmetric gauge theories [34, 1]. The wall-crossing phenomenon for
$K$-theoretic vortex partition functions associated with the handsaw quiver
varieties of type $A_{n}$ was partly studied in [1]. It is interesting to give
a characterization of the wall-crossing phenomenon studied in [1] in terms of
transformation formulas of hypergeometric series.

Level correspondence of $K$-theoretic $I$-functions
It has been shown [35] that a supersymmetric index on $S^{1}\times_{q}S^{2}$
factorizes to the pair of $K$-theoretic $I$-function with level structure in
[36] of the Higgs branch vacua of 3d theory: a Grassmann manifold. From the
factorization into the vortex partition functions (1.1), the $K$-theoretic
vortex partition function is closely related to $K$-theoretic $I$-function. As
mentioned in this article, since the wall-crossing formula of vortex partition
function is equivalent to the relation between the vortex partition functions
in a Seiberg dual pair, it is expected that the wall-crossing formula gives
the relation of $K$-theoretic $I$-functions between the Higgs branch vacua of
3d Seiberg dual pair. Actually, when the moduli spaces of the vacua of a 3d
Seiberg dual pair are two Grassmannians $\mathrm{Gr}(N_{1},N_{2})$ and
$\mathrm{Gr}(N_{2}-N_{1},N_{2})$, the wall-crossing formula of $k$-vortex
partition function is equivalent to the level correspondence [37], which is
the relation between the $K$-theoretic $I$-functions with level structure for
two Grassmannians. We will study the level correspondence of the $K$-theoretic
$I$-functions for manifolds realized as the moduli space of the Higgs branch
vacua by using the wall-crossing formula [38].

## Acknowledgements

This work is supported by Grant-in-Aid for Scientific Research 21K03382, JSPS.

##  Appendix A $q$-Pochhammer symbol and useful identities

The $q$-Pochhammer symbol is define by

$\displaystyle(x;q)_{n}:=\left\\{\begin{array}[]{cl}\prod_{i=0}^{n-1}(1-xq^{i})&(n\geq
1),\\\ 1&(n=0),\\\
\frac{1}{\prod_{i=1}^{-n}(1-xq^{-i})}&(n\leq-1)\end{array}\right.$ (A.4)

Following identities are used in the main text of this article.

$\displaystyle\prod_{a=1}^{N}\frac{1}{(q;q)_{k_{a}}}\cdot\prod_{1\leq a\neq
b\leq
N}\frac{(x_{a}x_{b}^{-1}q;q)_{k_{a}-k_{b}}}{(x_{a}x_{b}^{-1}q;q)_{k_{a}}}=\prod_{a,b=1}^{N}\frac{1}{(x_{a}x_{b}^{-1}q^{1+k_{a}-k_{b}};q)_{k_{b}}}$
(A.5) $\displaystyle\prod_{1\leq a\neq b\leq
N}(x_{b}x^{-1}_{a}q;q)_{{k_{b}-k_{a}}}=(-1)^{(N-1)\sum_{a=1}^{N}k_{a}}\prod_{a=1}^{N}x_{a}^{Nk_{a}-\sum_{b=1}^{N}k_{b}}$
$\displaystyle\qquad\qquad\qquad\qquad\times
q^{\frac{N}{2}(\sum_{a=1}^{N}k^{2}_{a})-\frac{N-1}{2}\sum_{a=1}^{N}k_{a}-\frac{1}{2}(\sum_{a=1}^{N}k_{a})^{2}}\prod_{1\leq
a<b\leq N}\frac{x_{a}q^{k_{a}}-x_{b}q^{k_{b}}}{x_{a}-x_{b}}\,,$ (A.6)
$\displaystyle\sum_{k=0}^{\infty}z^{k}\frac{(a;q)_{k}}{(q;q)_{k}}=\frac{(az;q)_{\infty}}{(z;q)_{\infty}}\,,$
(A.7)
$\displaystyle\frac{1}{(z;q)_{\infty}}=\prod_{k=1}^{\infty}\exp\left(\frac{z^{k}}{k}\frac{1}{1-q^{k}}\right)\,.$
(A.8)

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# Magnetic reconnection in the wakes of cosmic strings

Dilip Kumar  School of Physics, University of Hyderabad, Gachibowli,
Hyderabad, India 500046  Soma Sanyal  School of Physics, University of
Hyderabad, Gachibowli, Hyderabad, India 500046

###### Abstract

The motion of cosmic strings in the universe leads to the generation of wakes
behind them. We study magnetized wakes of cosmic strings moving in the post
recombination plasma. We show that magnetic reconnection can occur in the post
shock region. Since the width of the cosmic string wake is very small, the
reconnection occurs over a very short lengthscale. The reconnection leads to a
large amount of kinetic energy being released in the post shock region of the
cosmic string wake. This enhances the kinetic energy released during the
reconnection. We make a rudimentary estimate of the kinetic energy released by
the magnetic reconnection in cosmic strings wakes and show that it can account
for low energy Gamma Ray Bursts (GRB) in the post recombination era.

cosmic strings, shocks, magnetic reconnection.

##  I Introduction

The generation and evolution of magnetic fields have been studied quite
extensively in the literature. There are many well established sources for the
generation of these magnetic fields. One of the methods of generation of these
fields involve the motion of cosmic strings. Cosmic strings are topological
defects which are formed due to symmetry breaking phase transitions in the
early universe vilenkin . The motion of these strings through the cosmic
plasma leads to a wake like structure behind them. Primordial magnetic fields
in the early universe can be generated by these cosmic string wakes due to the
Harrison mechanism as well as the Biermann mechanism. In most cases the seed
magnetic field generated in the early universe then grows due to the dynamo
mechanism.

Various signatures of topological defects, especially cosmic strings have been
predicted in the literature CMB2 ; stringsigna1 . Recently, the interaction of
the shocks in the wakes of the cosmic string are being studied in detail to
look for signatures of cosmic string wakes in the 21 cm redshift surveys
stringsigna2 . There are many different kinds of strings based on the nature
of the phase transition that lead to their formation. Some of the cosmological
signals are specific to the kind of cosmic string being considered. For
example, it has been shown that fast radio bursts (FRB) can be generated by
cusps of superconducting strings (SCS)vachaspati ; zadorozhna ; yu . Apart
from this, cusps of superconducting strings can also serve as engines for
generating Gamma Ray Bursts (GRB’s) berezinsky . These are all shown to be
signatures of a superconducting string network in the early universe.

In this work, we show that Gamma Ray Bursts can result from magnetic
reconnection in the wake of an Abelian Higgs strings. The GRB’s occur when the
magnetic field lines come close together in the cosmic string wake and release
a large amount of energy through magnetic reconnection. In this case, the
cosmic string need not be a superconducting cosmic string. Magnetic fields can
be generated in the wake of Abelian Higgs string during the radiation epoch by
the Biermann battery mechanism sovan . The Biermann mechanism generates the
field when there are density inhomogeneities in the wake due to the motion of
neutrinos around the cosmic string. The neutrinos and the electrons interact
through the weak ponderomotive force and an inhomogeneity is generated in the
electron distribution in the wake of the cosmic string. It is well known that
there is a temperature difference across the wake due to the shock structure
layek . The electron inhomogeneity generated by the neutrinos however is not
aligned to the temperature gradient. This misalignment of the electron density
fluctuation and the temperature gradient results in the generation of a
magnetic field. If these fields are enhanced due to the turbulence in the
plasma, they will survive till the matter epoch. Our previous (numerical) work
soumen  has indicated that there is a possibility of magnetic reconnection in
cosmic string wakes, in this work we show how magnetic reconnection occurs in
the wake of cosmic strings and discuss the possibility that they may lead to
Gamma Ray Bursts in the post recombination era.

Magnetic reconnection occurs in highly conducting plasmas. In this process,
the field lines of oppositely directed magnetic fields come close together and
rearrange their topology by breaking and reconnecting. In this process, energy
in the magnetic field is released to the surrounding plasma. Depending upon
the magnitude of the magnetic field and the length of the reconnection region,
the energy released can be very large. The process of magnetic reconnection is
a complex process which involves the motion of charged particles and the
fields generated by these charged particles over a short lengthscale and
timescale. The first model to describe the process of reconnection in detail
was the Sweet-Parker model sweetparker . The model gave an estimate of the
energy released during the magnetic reconnection process. Later several other
models were developed to understand the process in more detail SPreconnection
. In this work, we have used the Sweet Parker model to make an initial
estimate of the energy that can be released by magnetic reconnection in cosmic
string wakes. Though the energy released depends on the magnetic field, we
find that a conservative estimate shows that the energy released may lead to a
Gamma Ray Bursts(GRB).

GRB’s are large explosions of energy over a finite period of time. These may
be short duration bursts of very high energy or longer duration bursts of
lower energies. The energy emission mechanism for a GRB is still not well
understood. Most of the GRB’s are attributed to the merger of two neutron
stars or to the collapse of a star. Some of the GRB’s are also attributed to
the process of magnetic reconnection in the plasma. GRB’s from magnetic
reconnection have previously been studied in the context of a turbulent
magnetic plasma with a high magnetization parameter granot . Detailed
numerical simulation are currently being done to investigate the possibility
of generating GRB’s due to magnetic reconnection around jets and collapsing
neutron stars. In this work, we show that GRB’s can also be emitted from
magnetic reconnection in cosmic string wakes.

In section II we revise the formation of shocks in cosmic string wakes and the
generation of magnetic fields in the shock. In section III, we discuss the
possibility of magnetic reconnection occurring in these wakes. In section IV,
we present the cosmological consequences of the magnetic reconnection
occurring in these wakes and predict the possibility of observational signals
arising from these reconnections. Finally we present our conclusions in
section V.

##  II Shocks and magnetic fields in cosmic string wakes

As mentioned in the introduction, cosmic strings generated from symmetry
breaking phase transitions in the early universe, produce a wake like
structure behind them. The wake like structure is due to the conical spacetime
of the cosmic string. Shocks are generated in the wakes of these strings as
they move through the plasma. A detailed analysis of the shock structure can
be obtained from the following references shocks . In a recent work, it was
shown that a magnetic field can be generated in the shock of a cosmic string
due to the Biermann mechanism sovan . We briefly review the generation of the
magnetic field in the shock in this section and show in the next section why
such magnetic fields will lead to magnetic reconnections in the wake of a
cosmic string.

Massive particles moving around cosmic strings often form closed orbits around
the string abhisek . If these particles happen to be neutrinos then they
generate a neutral current close to the string. The neutrinos interact with
the electrons in the plasma by the weak force and cause electron currents in
the plasma. These currents lead to oscillatory density fluctuations of the
electron density in the wake of the cosmic string. It was shown in ref sovan
that these density fluctuations can lead to the generation of a magnetic field
in the cosmic string wake through the Biermann mechanism.

The evolution of the magnetic field by the Biermann battery mechanism is given
by,

$\frac{\partial\vec{B}}{\partial
t}=\nabla\times(\vec{v_{e}}\times\vec{B})+\frac{\eta_{res}}{4\pi}\nabla^{2}\vec{B}-\frac{1}{eN_{e}}\nabla\times(\vec{j}\times\vec{B})\\\
-\frac{1}{N_{e}e}\nabla N_{e}\times\nabla T$ (1)

Here, $\vec{B}$ is the magnetic field generated by the electron current,
$\vec{v_{e}}$ is the velocity of the electrons in the plasma,$\eta_{res}$ is
the resistivity of the plasma, $e$ is the electric charge, $N_{e}$ is the
electron number density and $T$ is the temperature of the plasma. In the
absence of a magnetic field in the plasma, it is the last term which generates
the magnetic field.

We will now establish that due to the temperature difference across the shock
wave, the Biermann term (last term) will generate opposite magnetic fields on
the two sides of the shock. The density gradient in the cosmic string wake is
non uniform and the gradient of the electron number density is given by,

$\nabla N_{e}=\hat{i}\frac{\partial N_{e}}{\partial x}+\hat{j}\frac{\partial
N_{e}}{\partial y}+\hat{k}\frac{\partial N_{e}}{\partial z}$ (2)

In the case of the cosmic string wakes, the temperature gradient is
perpendicular to the flow direction. Though we can have small fluctuations in
other direction the gradient of the temperature is dominated by
$\hat{j}\frac{\partial T}{\partial y}$. If we consider the origin of the
coordinate system to be at the position of the cosmic string then the two
sides of the planar shock generated by the moving cosmic string will have
opposite temperature gradients. Figure 1 illustrates the concept. The upper
half of the shock is referred to as ”A” while the lower part is referred to as
”R”. Therefore, if the temperature gradient at A is $\hat{j}\frac{\partial
T}{\partial y}$ then at R it will be $-\hat{j}\frac{\partial T}{\partial y}$.
The number density gradient remains the same at both A and R sides. This means
that the magnetic field generated by the Biermann mechanism at A and R are
equal and opposite.

Figure 1: Schematic diagram of the magnetic field lines generated by the
cosmic string as it moves through the plasma. The brown (solid) line gives the
outline of the cosmic string wake and the blue (grey) lines are the directions
of the fields that are generated due to the Biermann mechanism.

$\vec{B}_{A}~{}=~{}-\hat{i}\frac{\partial N_{e}}{\partial z}\frac{\partial
T}{\partial y}~{}+~{}\hat{k}\frac{\partial N_{e}}{\partial x}\frac{\partial
T}{\partial y}~{}=~{}-\vec{B}_{R}$ (3)

Since the plasma is highly conducting, the magnetic Reynolds number is very
large. This makes the magnetic diffusivity very large. The two magnetic field
lines then diffuse and squeeze the fluid like two infinitely conducting
sheets. We thus have a situation that was first described by Sweet and Parker
SPreconnection  in their model for magnetic reconnections in conducting
plasmas. This means that there is a strong possibility that the magnetic
reconnections will occur in the shocks generated in the wakes of cosmic
strings.

In recent simulations of cosmic string wakes, lines of magnetic reconnection
have been observed as the string moves through the plasma. These magnetic
reconnection lines have been observed for a magnetic field in the wake of a
cosmic string with some spatial variations. In the next section, we calculate
the reconnection rates and length of the charge sheets that can be generated
in the wakes of cosmic strings based on the Biermann mechanism outlined
previously in ref sovan .

##  III Magnetic reconnection in the wakes

According to the Sweet-Parker model of magnetic reconnection sweetparker , a
current layer is formed between two oppositely directed magnetic fields
generated in a conducting plasma. The electrical current density in these
regions becomes very high. This forces the magnetic lines of force to break
and reconnect, changing the topology of the magnetic field. The curvature
force associated with the changed configuration results in the conversion of
magnetic energy to kinetic energy.

Assuming steady state conditions, the length of the diffusion region between
the two oppositely directed diffusing fields is given by, $l=\frac{uL}{v}$
here $u$ is the velocity with which the field lines merge and $v$ is the
velocity at which the fluid is expelled from the diffusion region. $L$ is the
scale of the magnetic field that we are considering. In the cosmic string
wake, $L$ will be determined by the dimensions of the wake. We are looking at
the motion in two dimensions. Let us assume that the string is moving along
the $x$ direction. The lines of forces will be generating a strain in the wake
of the string. We assume that the magnetic field changes very slowly in the
$y$ direction, so that we can assume the $y$ component of the magnetic field
to be approximately constant. We are then left with the $x$-component of the
magnetic field. The field lines push in the $y$ direction and the pressure
difference due to the streching of the fields leads to the velocity of the
diffusing fields. The velocity of the diffusing field lines will be given by,
$u=\frac{c^{2}}{4\pi\sigma B_{x}}(\frac{\partial B_{x}}{\partial y})$, here
$\sigma$ is the conductivity of the fluid. The quantity $(\frac{\partial
B_{x}}{\partial y})$ is evaluated at $y=0$, this is the Sweet Parker model.
The fluid is expelled with the characteristic velocity $v$ given by,
$v\sim\frac{B}{(4\pi\rho)^{1/2}}$, this is known as the hydromagnetic velocity
and it will be henceforth denoted by $C_{H}$, $\rho$ is the mass density of
the plasma. As can be seen from the expression, this is equivalent to the
Alfven velocity in the magnetized wake. In fig 1, we have briefly sketched the
magnetic field generated by the wakes and shown how it can be mapped onto the
Sweet Parker model.

The length of the diffusion region depends on the dimensions of the cosmic
string wake. For a long single string generated at $t_{f}\leq t_{eq}$, the
dimensions of the wakes in terms of the redshift are given by $l_{f}$ and
$w_{f}$. Here $l_{f}=t_{f}\frac{z_{f}}{z}$ and
$w_{f}=v_{s}\gamma_{s}t_{f}\frac{z_{f}}{z}$. Here, the suffix $f$ stands for
the time at which the string is generated and the suffix $eq$ stands for the
time after which the different particles went out of equilibrium. The
thickness of the wake is given by $4\pi
G\mu\gamma_{s}v_{s}t_{i}\frac{z_{f}}{z}$, where $t_{i}$ is the time at which
the reconnection occurs. According to the Sweet Parker model of magnetic
reconnection, the magnetic energy is converted into the kinetic energy of the
plasma. The sudden increase in the kinetic energy leads to the acceleration of
the particles in the plasma. Thus an estimate of the velocity of the merging
fields gives us an estimate of the increase in the kinetic energy of the
plasma. For this we calculate $(\frac{\partial B_{x}}{\partial y})$. The
details of the derivation can be obtained from sweetparker . The final result
is, $K.E=\frac{1}{2}\rho c^{2}\left(\frac{C_{H}}{\sigma L}\right)$ where $c$
is the velocity of light. So, a large amount of kinetic energy is released for
a dense plasma with a low electrical conductivity and a high magnetic field.
For a cosmic string wake, the density is usually four times the density of the
background plasma. Though the magnetic field in the wake may not be very high
it may be high enough to generate a considerable amount of energy which could
have prospects of detection by various methods.

As the cosmic string moves through the plasma, the emission of kinetic energy
can result in a burst of electromagnetic radiation. Since the wake behind the
cosmic string is quite narrow, this can be detected as a burst of gamma rays.
In the next section, we calculate the kinetic energy that can be released due
to the reconnection in the wakes of the cosmic strings.

##  IV Gamma Ray Bursts from magnetic reconnection

The release of a large amount of energy in the plasma accelerates the
particles in the plasma which may lead to the generation of a significant
amount of radiation. So, it is quite possible that magnetic reconnection in
the shocks of cosmic strings can lead to a burst of energy that could be a
Gamma Ray Progenitor. There exist some GRB models which show magnetic
reconnection to be the progenitor of a GRB but this is the first time that
magnetic reconnections in the cosmic string wakes has been considered as the
progenitor of a GRB. A detailed numerical calculation is beyond the scope of
this work so we do an order of magnitude estimate of the energy that is
released due to magnetic reconnection in the wake.

The kinetic energy that is released due to reconnection depends inversely on
the lengthscale of the reconnecting magnetic field. Since the opening angle of
the cosmic string wake is very small, this lengthscale is also very small. It
will depend on the thickness of the wake and would be of the order of $4\pi
G\mu\gamma_{s}v_{s}t_{i}\frac{z_{f}}{z}$. One of the reasons why magnetic
reconnection are traditionally not considered for GRB’s is because of the
large lengthscale over which they occur. In typical astrophysical plasma, the
lengthscale over which reconnection occurs is considerably large, however in
the case of cosmic string wakes, only the length of the wake is large, the
thickness of the wake is much smaller as it is determined by the deficit angle
of the cosmic string. This depends on the symmetry breaking scale. Since the
GUT symmetry breaking phase transition happened at $10^{16}$GeV, the deficit
angle of the GUT cosmic string is typically of the order of
$10^{-12}-10^{-6}$. Abelian Higgs strings have $G\mu$ of the order of
$10^{-11}-10^{-8}$. Current observational limits from the Planck collaboration
have put the upper limit of Abelian Higgs strings at $3\times 10^{-7}$ planck
. The order of the string velocity is $<\gamma_{s}v_{s}>\sim 0.4$. The time of
the wake formation can be written as,
$t_{f}=\left(\frac{3}{2}H(z)\sqrt{\frac{z_{f}+1}{z+1}}\right)$. Assuming that
the wake formation happened around the recombination epoch $(z\sim 3000)$ and
the magnetic reconnection is happening at lower values of the redshift $(z\sim
30)$, the lengthscale turns out to be $10^{-8}$ in dimensionless units. The
conductivity of the plasma in the matter dominated epoch is given by
$\sigma\sim 10^{10}s^{-1}$ caprini . The kinetic energy is thus dependent
primarily on the magnetic field in the wake and the plasma density. The plasma
density at recombination is given by $3.8\times 10^{-3}eV^{4}$. Since the
value of $C_{H}$ depends upon the magnetic field and is generally greater than
one for strongly magnetized plasma, as a first approximation we consider it to
be of the order of one. An approximate estimate will then put the kinetic
energy generated due to the magnetic reconnection to be of the order of
$10^{14}erg/cm^{2}$.

Since there have been GRB’s detected in the lower energy range $(10-10000)$
keV which have fluences of the order of $10^{-5}erg/cm^{2}$, it is quite
possible that the magnetic reconnection in magnetized wakes of cosmic strings
can be detected as a GRB. Our rudimentary method unfortunately does not allow
us to determine the timescale of the GRB which would have been helpful in
identifying an observational signal in a better way. However, a large number
of GRB’s have been reported with flux densities ranging from
$10^{-5}erg/cm^{2}$ to $2X10^{-4}erg/cm^{2}$, with burst durations ranging
from less than $0.1$ sec to $30$ secs klebesadle . As the time duration is
also similar to the timescales observed in magnetic reconnections, we plan to
look into these GRB’s in greater detail to see if they can be signatures of
magnetized cosmic string wakes.

##  V Conclusions

In this paper, we have shown that it is possible to generate GRB’s from
magnetic reconnection in the wakes of Abelian Higgs cosmic strings. Magnetic
reconnections are currently being studied in both laboratory and astrophysical
plasmas. It is manifested in solar flares, coronal mass ejection and other
solar phenomenon. The search for signatures of cosmic strings and their wakes
are also continuing through both direct detection methods as well as
statistical methods. Cosmic string networks are known to generate both
gravitational and non gravitational signatures. The signatures related to
magnetic fields and electromagnetic phenomenon have however been limited to
superconducting cosmic strings. Our work shows that in any cosmic string wake
if the magnetic field evolves by the Biermann mechanism, it is possible to
have magnetic reconnections. All it requires is a density inhomogeneity which
is not aligned to the temperature gradient of the cosmic string wake. Though
in this work we have used a specific method to show the phenomenon
analytically, it may be extended to many other cases where magnetic fields
occur in cosmic string wakes. We have only done a preliminary calculation for
the energy of the GRB. An important parameter which distinguishes a GRB is the
duration of the burst. This helps in the classification of the observed GRBs.
In this current work we have not estimated the duration of the emitted GRB in
our model.

Since this is a preliminary work, we have not been able to obtain an exact
match to the observed GRB’s. We have used the Sweet-Parker model, and found
that the energy generated by the reconnection is quite significant. There are
various improvements on the Sweet-Parker model. We are currently working on a
more detailed model that will tell us more about the reconnection phenomenon
in the magnetized wakes of cosmic strings.

Acknowledgments

The authors D.K and S.S would like to thank Biswanath Layek for the
interesting discussions which helped to improve the paper considerably.This
work is supported by the DST- SERB Power Grant no. SPG/2021/002228 of the
Govt. of India.

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