Expanding the Bethe/Gauge dictionary
HJE
Expanding the Bethe/Gauge dictionary
Mathew Bullimore 0 1 2 5
HeeCheol Kim 0 1 2 3 4
Tomasz Lukowski 0 1 2 5
Andrew Wiles Building 0 1 2
Radcli e Observatory Quarter 0 1 2
0 in Lower Dimensions , Topological Field Theories
1 Cambridge , MA 02138 , U.S.A
2 Woodstock Road , Oxford, OX2 6GG , U.K
3 Je erson Physical Laboratory, Harvard University
4 Department of Physics , Postech
5 Mathematical Institute, University of Oxford
We expand the Bethe/Gauge dictionary between the XXX Heisenberg spin chain and 2d N = (2; 2) supersymmetric gauge theories to include aspects of the algebraic Bethe ansatz. We construct the wave functions of o shell Bethe states as orbifold defects in the Atwisted supersymmetric gauge theory and study their correlation functions. We also present an alternative description of o shell Bethe states as boundary conditions = 4 supersymmetric quantum mechanics. Finally, we interpret spin chain Rmatrices as correlation functions of Janus interfaces for mass parameters in the supersymmetric quantum mechanics.
Supersymmetric Gauge Theory; Supersymmetry and Duality; Field Theories

in an e ective N
1 Introduction
2 Spin chain primer
3 Setup
Heisenberg spin chain
Rmatrices
Bethe states
The model
Sphere partition function
Vortex partition function
Factorization
4 Defect operators in 2d
Abelian theories
Orbifold construction
Bethe wavefunctions
5 Quantum mechanical description
6 The Rmatrix
7 Discussion
A Conventions
N = 4 quantum mechanics
Empty boundary condition
Stable boundary conditions
Thimble boundary conditions
Orthonormality of stable basis
Rmatrix from Janus interface
YangBaxter equation
A.1 2d N = (2; 2) supersymmetry
A.2 N = 4 quantum mechanics
2.1
2.2
2.3
3.1
3.2
3.3
3.4
4.1
4.2
4.3
5.1
5.2
5.3
5.4
6.1
6.2
6.3
1
Introduction
The aim of this paper is to extend the dictionary between quantum integrable systems and
supersymmetric gauge theories introduced and studied in [1{3], the socalled Bethe/Gauge
correspondence. We focus on an elementary example of this phenomenon: the
correspondence between the XXX 1 Heisenberg spin chain and a family of 2d N = (2; 2)
super2
symmetric gauge theories. Some basic aspects of this correspondence are summarized
in gure 1.
{ 1 {
U (k) gauge group
N flavours
periodic boundary condition and ~ is Planck's constant.
On the supersymmetric side of the correspondence, mj and ~ are complex mass
parameters associated to
avour symmetries of the gauge theory in
gure 1, while q is the
exponential of a complexi ed FI parameter. For generic values of the mass parameters,
the theory has a low energy description as a U(1)k gauge theory with an e ective twisted
superpotential Wf( ) depending on complex vectormultiplet scalars a. This function is
determined exactly by a oneloop calculation. The equations for supersymmetric vacua,
exp
a = 1; : : : ; k ;
(1.2)
coincide with the Bethe equations for the spin chain (1.1). The ring generated by
gaugeinvariant functions of the vectormultiplet scalars
1; : : : ; k modulo the relations (1.2) is
the twisted chiral ring of the supersymmetric gauge theory. The twisted superpotential
itself can be identi ed with YangYang function of the spin chain.
A powerful approach to computing a wide range of observables in quantum integrable
systems is the algebraic Bethe ansatz, as explained in [5]. In this paper, we will understand
how elements of this approach arise in the Bethe/Gauge correspondence. For this purpose,
we will perform exact computations in the original supersymmetric gauge theory shown in
gure 1, rather than the e ective abelian description. In particular, we will interpret aspects
of the algebraic Bethe ansatz in terms of correlation functions in the Atype topological
twist of the supersymmetric gauge theory, using techniques from supersymmetric
localization [6, 7]. Investigations of the Bethe/Gauge correspondence in this context have appeared
in [8, 9]. The remainder of the introduction is dedicated to summarizing our results.
An important part of the algebraic Bethe ansatz is the construction of o shell Bethe
states j 1; : : : ; ki, which are elements of the spin chain Hilbert space depending on
auxiliary parameters 1 : : : ; k. The inner product hf j 1; : : : ; ki with another state jf i is
{ 2 {
hf gi
=
functions in the Atwisted supersymmetric gauge theory on CP1.
a symmetric function f ( 1; : : : ; k) of the auxiliary parameters, which can be identi ed
with a gaugeinvariant function of the vectormultiplet scalar in the supersymmetric gauge
theory in
gure 1. The correlation functions of such operators in the Atype topological
twist depend only on the class [f ( 1; : : : ; k)] of the function modulo the twisted chiral
ring relations (1.2). The map
jf i ! [f ( 1; : : : ; k)] ;
then sets up a correspondence between states in the spin chain Hilbert space and invariant
functions of 1; : : : ; a modulo relations, such that the inner product hf jgi on the spin
chain Hilbert space coincides with the twopoint correlation function of f ( 1; : : : ; k) and
g( 1; : : : ; k) in the Atwisted theory on CP1. This is illustrated in gure 2.
In order to investigate this relation, it is convenient to introduce an orthonormal
`updown' basis for the spin chain Hilbert space. The basis elements are labelled by subsets
I = fi1; : : : ; ikg
f1; : : : ; N g such that jIi is the state with spin " at positions i1; : : : ; ik
and spin # everywhere else. We can then introduce the wavefunctions of o shell Bethe
states in this basis,
SI ( 1; : : : ; k) / hIj 1; : : : ; ki ;
which provide a set of generators for the twisted chiral ring. Nekrasov has proposed a
physical de nition of the corresponding twisted chiral ring elements as `orbifold defects' [10].
In this paper, we explain how to implement this orbifold construction in the Atwisted
supersymmetric gauge theory to compute correlation functions of the twisted chiral
operators (1.4). We furthermore demonstrate that these operators are orthonormal with respect
to the Amodel twopoint functions, corresponding to the fact that hIjJ i = I;J in the spin
chain Hilbert space.
In the algebraic Bethe ansatz, the eigenstates of the spin chain Hamiltonian are
obtained by evaluating the o shell Bethe state j 1; : : : ; ki on a solution of the Bethe
equations (1.1). The functions SI ( 1; : : : ; k) evaluated on solutions of the Bethe equations are
therefore the wavefunctions of the eigenstates in the `updown' basis jIi. We will show
that this wavefunction can be obtained directly from the supersymmetric gauge theory by
computing the Amodel in a cigar geometry with a vacuum corresponding to I at in
nity. More precisely, we rst introduce an
background and then compute a normalized
correlation function that is nite in the limit
! 0. This is shown in gure 3.
(1.3)
(1.4)
{ 3 {
SI ( )
✏
vJ
f ( )
g( )
Bf
Bg
equations, the steps in the algebraic Bethe ansatz are independent of the parameter q. It is
therefore su cient to understand these aspects in the limit q ! 0, which corresponds to
discarding instanton corrections in the Atwisted supersymmetric gauge theory. In this limit,
correlation functions can be understood in a
nitedimensional N
= 4 supersymmetric
quantum mechanics with boundary conditions preserving the same pair of supercharges as
the Atwist. In particular, each twisted chiral operator generates a boundary condition in
the supersymmetric quantum mechanics, and twopoint functions are computed by interval
partition functions  as shown in gure 4. In particular, we will provide two independent
constructions of the boundary conditions generated by the operators SI ( 1; : : : ; k), either
by coupling Neumann boundary conditions to additional degrees of freedom or as `thimble'
boundary conditions.
The setup described above is compatible with turning on background holonomies for
avour symmetries. In the supersymmetric quantum mechanics description, background
avour holonomies around the circle become `real mass parameters' for avour symmetries.
The ordering of the holonomy eigenvalues or real masses can be identi ed with an ordering
of sites on the spin chain. It is therefore natural to consider `Janus' interfaces which
permute the ordering of the masses. We will show that Amodel correlation functions of
such Janus interfaces in between the elements SI ( 1; : : : ; k) reproduce matrix elements of
the spin chain Rmatrix. The YangBaxter relation is interpreted as the statement that a
given permutation of real mass parameters can be decomposed in a number of ways into
elementary Janus interfaces permuting a pair of real mass parameters.
{ 4 {
HJEP1(207)5
Finally, the Bethe/Gauge correspondence provides a physical realization of a parallel
developments in geometry and representation theory, and many of the objects we
consider here have already appeared in this context. The starting point is the statement that
the twisted chiral ring is the equivariant quantum cohomology ring of the vacuum
manifold of the supersymmetric gauge theory, T G(k; N ). The Amodel correlation functions
considered here can be formulated in the language of quasimaps to the vacuum
manifold. In particular, the functions SI ( 1; : : : ; k) were introduced in work of Maulik and
Okounkov [11] as the `stable basis' in the quantum equivariant cohomology (see [12] for
connections to Bethe wavefunctions). This paper is largely motivated by understanding
these mathematical constructions in the language of supersymmetric gauge theory.
The paper is organized as follows. In section 2 we collect some relevant properties of
the Heisenberg spin chain. In section 3 we describe 2d N = (2; 2) supersymmetric gauge
theory and explain how to calculate their Atwisted sphere and cigar partition functions
using supersymmetric localization. Section 4 focuses on the de nition of the distinguished
twisted chiral operators and their correlation functions. In section 5 we reduce the problem
to a 1d quantum mechanics and explain how the twisted chiral operators can arise from
appropriate boundary conditions in this quantum mechanics. Finally, in section 6 we show
how the spin chain Rmatrices can be obtained as correlation functions of twisted chiral
operators in the supersymmetric gauge theory. Our conventions and more technical details
of the calculation are postponed to the appendix.
2
Spin chain primer
In this section we collect some basic information on the Heisenberg XXX 1 spin chain,
where all spins transform in the fundamental representation of su(2). Many of the
statements we present here and in subsequent sections have a natural generalization to spin
chains with higher representations, as well as to higher rank algebras. Our notation is
designed to match that of supersymmetric gauge theory and therefore di ers from standard
2
integrability conventions.
2.1
Heisenberg spin chain
In order to de ne the Heisenberg spin chain we need to specify a Hamiltonian and a Hilbert
space on which it acts. The Hilbert space of the spin chain is the N fold tensor product of
the fundamental representation of su(2),
We introduce standard basis elements j " i and j # i for each spin chain site C2. There is then
a natural basis for V that is labelled by subsets I = fI1; : : : ; Ikg
f1; : : : ; N g such that
V = C
2

C
2
N {tizmes
: : :
C2 :
}
jIi = j # : : : " : : : " : : : #i :
{I1z}
{Ikz}
{ 5 {
(2.1)
(2.2)
We de ne an inner product by demanding that the basis vectors at each site are orthonormal
h # j # i = h " j " i = 1;
h # j " i = h " j # i = 0 ;
a 2N
and naturally extending this de nition to V. Any operator A : V ! V can be represented
2N matrix of its expectation values between tensor products of j " i and j # i.
The (twisted, homogeneous) Heisenberg spin chain is de ned by the Hamiltonian,
with the twisted boundary condition,
Later on, we will also introduce inhomogeneities for each spin chain site.
The Hamiltonian commutes with the operator counting up spins and therefore the
Hilbert space can be decomposed into a direct sum of spaces with
xed number of
excitations,
sion relation
The spectrum of the Heisenberg spin chain can be then found using the celebrated Bethe
ansatz. In particular, the eigenvalues of the Hamiltonian H are obtained from the
dispera
a
H =
include inhomogeneities mi at each site of the spin chain.2 In that case we denote the spin
chain Hilbert space as
and the Bethe equations turn into
mi + ~~2 = q Y
2
where we sum over the rapidities1 of excitations a, a = 1; : : : ; k, which are solutions to
the Bethe equations
a + ~2 !N
~
2
= q Y
a
For a given number of excitations k, there are Nk solutions distinct solutions of the Bethe
equations. The solutions aI can be labelled by a subset I = fI1; : : : ; Ikg
that, expanding around q ! 0, the solutions are of the form
aI = mIa
~2 + O(q).
f1; : : : ; N g such
1It is common to use the letter u to denote rapidities. In this paper we use the letter instead in order
to make connection with the gauge theory side of our story in the following sections.
2It is common to use the letters vi to denote inhomogeneities. In this paper we use the letters mi instead
in order to make connection with the gauge theory side of our story in the following sections.
{ 6 {
(a) YangBaxter equation.
(b) Unitarity of the Rmatrix.
most powerful method is the algebraic Bethe ansatz, which is based on the construction of
an Rmatrix. For the inhomogeneous spin chain, this is an operator acting on two sites,
Rij (mj
mi) : C2mi
2
Cmj ! Cmj
2
C2mi :
It has rational dependence on mj
mi and satis es the regularity property Rij (0)
Pij
where Pij is the permutation operator, together with the YangBaxter equation (shown
graphically in gure 5a)
R12(m2
m1)R13(m3
m1)R23(m3
m2) = R23(m3
m2)R13(m3
m1)R12(m2
where Rij acts nontrivially only on C2mi and C2mj .
An explicit form of the Rmatrix is
(2.11)
m1) ;
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
where each matrix element is an operator acting on the Hilbert space (2.9). Namely, each
matrix element of M ( ) in the auxiliary space can itself be represented by a 2N
2N matrix.
M ( ) =
A( ) B( )
C( ) D( )
!
;
{ 7 {
1
mji + ~
Rij (mji) =
(mji Iij + ~ Pij ) ;
R12(m)R21( m) = I :
where mji = mj
mi and we xed the normalisation by the unitarity condition (shown
graphically in gure 5b)
2.3
Bethe states
odromy matrix
We now introduce an auxiliary space C
2 with spectral parameter
and de ne the
monM ( ) = R10(
m1)R20(
m2) : : : RN0(
mN ) ;
where each Ri0(
2
2 matrix in the auxiliary space
mi) acts nontrivially only on the auxiliary space and C2mi . This is a
For a given k, we de ne an o shell Bethe state by
j 1; : : : ; ki = B( 1) : : : B( k)j i
;
where j i = j # : : : #i. Additionally, for given subset I = fI1; : : : ; Ikg of f1; : : : ; N g, we
de ne the functions SI ( ) as the overlaps of o shell Bethe state j 1; : : : ; ki with the basis
vectors:
SI ( ) = ( 1)jIjN ( )hIj 1; : : : ; ki :
Here we have introduced a normalization factor
HJEP1(207)5
N ( ) = ( 1) 2
k(k 1) +kN
Qa;i( a
Qa;b( a
mi + ~2 ) ;
b + ~)
which is independent of I. These functions can be computed explicitly with the result,
SI ( ) = Sym
k
Q
a=1
Ia 1
Q ( a
i=1
It can be shown that the states j 1; : : : ; ki become eigenstates of the spin chain
Hamiltonian provided
a are evaluated on a solution
aJ of the Bethe equations. The functions
SI ( J ) are then (up to normalization) the wavefunctions of the Bethe eigenstates in the
position basis jJ i. In the following sections, we will explain how to construct such
wavefunctions in the Bethe/Gauge correspondence.
3
Setup
In this section, we review the computation of correlation functions in Atwisted
supersymmetric gauge theories on CP1 and a cigar. We review two approaches to computing such
correlation functions using supersymmetric localization. The rst leads to a contour
integral in the complex Cartan subalgebra of the gauge group. The second is via equivariant
localization on the moduli space of quasimaps into the vacuum manifold. This will provide
a foundation for the results presented in the following sections.
3.1
The model
We consider 2d N = (2; 2) supersymmetric gauge theories with Rsymmetry U(1)V
that ow to sigma models onto cotangent bundles to complex Grassmannians, T G(k; N ).
Such a theory has gauge symmetry G = U(k) and avour symmetry Gf = PSU(N )
The eld content is depicted in gure 6 and can be summarized as follows:
U(1)A
U(1)~.
A vectormultiplet containing bosonic elds (A ; ; D) transforming in the adjoint
representation of U(k), where A is the gauge eld,
is a complex scalar and D is
an auxiliary scalar.
{ 8 {
(2.17)
(2.18)
(2.19)
(2.20)
Chiral multiplets ( ; X; Y ) transforming as shown in the table below.
X
Y
U(k)
adj
U(1)~
1
and theta angle .
Our conventions are summarized in appendix A.
space of solutions the vacuum equations
For positive FI parameter, r > 0, the theory ows to a sigma model onto the moduli
modulo constant gauge transformations. We de ne
R
to be the moment map for the gauge symmetry. It can be shown that solutions require
= 0 and
= 0, and that the remaining equations reproduce the hyperKahler quotient
construction of T G(k; N ) where r is the Kahler parameter of the base Grassmannian
G(k; N ). We refer to this as the vacuum manifold V.
It is useful to provide an algebraic description of the vacuum manifold. For r > 0,
we can replace the Dterm equation (3.1) by the stability condition that the matrix X has
maximal rank and divide by complex gauge transformations,
V = fX; Y jX Y = 0; rk(X) = kg=GL(k; C) = T G(k; N ) :
(3.5)
From this perspective, X de nes a kplane in CN corresponding to a point in the base
Grassmannian G(k; N ). For example, in the case k = 1, we have T CPN 1 with
homogeneous coordinates [X1; : : : ; XN ] on the base. For negative FI parameter r < 0, the roles of
X and Y would be interchanged.
We can also introduce complex mass parameters for the
avour symmetry Gf by
coupling to a background vectormultiplet and introducing nonzero vacuum expectation
{ 9 {
(3.1)
(3.2)
(3.3)
(3.4)
.
{+}
{ }
values (m1; : : : ; mN ; ~) to the complex scalar in the vectormultiplet in a Cartan subalgebra
equations (3.3) are replaced by
j mj = 0. In the presence of complex masses,
a
mj +
( 1; : : : ; k) denote the eigenvalues of the vectormultiplet scalar
. For generic values
of the complex masses (m1; : : : ; mN ; ~), the vacuum manifold V is lifted, leaving behind
Xaj = p
r j;Ia ;
Y j a = 0 ;
ab = 0 ;
(3.7)
labelled by subsets I = fI1; : : : ; Ikg
f1; : : : ; N g of size jIj = k. The massive vacua
can be identi ed with the
xed points of the in nitesimal Tf action on the vacuum
manifold V = T G(k; N ) generated by (m1; : : : ; mN ; ~) and correspond geometrically to the
coordinate hyperplanes in the base G(k; N ).
3.2
Sphere partition function
We will now consider correlation functions in the deformed Amodel on C = CP1,
introduced in [6, 7]. We introduce homogeneous coordinates [z : w] on CP1 and de ne a
U(1)J isometry that transforms the homogeneous coordinates by (z; w) ! (e =2z; e =2w)
with
xed points f+g = f
z = 0g and f g = fw = 0g, as shown in
gure 7. The
background preserves a pair of supercharges Q , Q+ that commute with the combination
U(1) := U(1)J + U(1)V .
3.2.1
Contour integral
Partition functions in the deformed Amodel can be computed exactly using
supersymmetric localization for the supercharge Q = Q
+ Q+ [6, 7]. This reduces the path integral
to a contour integral over the complex Cartan subalgebra of G parametrized by the
eigenvalues ( 1; : : : ; N ) of the vectormultiplet scalar .
In order to express the contributions to the integrand of the contour integral from the
various multiplets, it is convenient to introduce the following function
Z(n)( ) =
1(
is Barnes' gamma function. Due to the functional equation 1(x + ) = x 1(x), this ratio
of Barnes' gamma functions is in fact a rational function of .
The contribution to the integrand from a chiral multiplet of charge r under the U(1)V
vector Rsymmetry and charge qf under a U(1)f
avour symmetry is Z(qf n r)(qf ), where
is the vectormultiplet scalar and n 2 Z is the quantized
ux through CP1.
Coming back to the model introduced in section 3.1, partition functions are expressed
as a contour integral over the complex vectormultiplet scalar ~ = ( 1; : : : ; k) together
with a summation over the ux ~n = (n1; : : : ; nk) 2 Zk. The contributions to the integrand
from the chiral multiplets are
k
Y
a;b=1
N
N
where we introduce a shorthand notation
ab =
a
b and nab = na
nb. There is an
additional contribution from the vectormultiplet
V
Z(~n)(~ ) = Y Z
( nab 2)( ab)
a6=b
a<b
= Y( 1)nab+1 ( a
b)
2
(na
nb)2 :
The partition function is then given by
where
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
the contributions from the chiral multiplets [6] and (k + N + 1) P
combines the contributions from the vector and chiral multiplets. We include an additional
sign ( 1)P with P = k2 + (k + N + 1) P
a na, where the factor k2 xes a sign ambiguity in
a na is an additional sign
that can be absorbed into the de nition of q. The contour
is given by the Je reyKirwan
prescription, which reduces for r > 0 to the contour surrounding poles at
a = mi
~
2
na
2
`
;
scalar. As explained in [6, 7], there are then additional contributions
f+g
f g
:
:
f ~
f ~ +
~n
2
~n
2
;
;
notation n := P
a na.
multiplets Xai in the fundamental representation of the gauge group. The summation over
uxes can therefore be restricted to the region ~n 2 Z 0
k . We will often use the shorthand
The partition function is enriched by inserting twisted chiral operators annihilated by
. We will consider gaugeinvariant functions f (~ ) of the vectormultiplet
to the integrand in equation (3.12). We denote a correlation function with f (~ ) inserted
at f+g and g(~ ) inserted at f g by
hf (~ ); g(~ )iS2 =
X
~n2Zk 0
qPi ni Z d
k
k! ZN(~n;)k(~ )f ~
~n
2
~n
2
g ~ +
:
(3.16)
Importantly, the contributions from na > 0 vanish unless certain conditions are
satised. For example, in the abelian case instanton corrections to hf ( ); g( )iS2 vanish unless
the combined degree of the polynomials is greater than or equal to 2N
1. This follows
from the fact that for n > 0 the only potential pole outside of the contour is at
which only exists if deg(f ) + deg(g)
2N
1. This phenomenon can be understood as
the condition to cancel the U(1)A axial anomaly. In all cases, the partition function h1iS2
receives contributions only from ~n = 0 and is therefore independent of q.
! 1,
Moreover, correlation functions involving particular combinations of twisted chiral
operators vanish, re ecting the structure of the twisted chiral ring. For example, in the
abelian case
(3.14)
(3.15)
(3.17)
(3.18)
hf ( ) Y
N
j=1
hf ( ); g( ) Y
N
j=1
mj +
N
Y
j=1
~
2
mj +
q f (
~
2
mj +
~
2
for any f ( ) and g( ). In the limit
! 0, we recover the twisted chiral ring relations,
N
) Y
j=1
q g( + ) Y
N
j=1
N
q Y
j=1
; g( )iS2 = 0 ;
~
2 iS2 = 0 ;
~
2
mj
~
2
mj
= 0 ;
which coincide with the equivariant quantum cohomology ring of the vacuum manifold
V = T CPN 1
.
that hold inside correlation functions are
! 0, the general twisted chiral ring relations of a nonabelian theory
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
(3.25)
together with
R
r1 =
modulo gauge transformations.
The solutions of such `generalized vortex equations' are known as freckled
instantons [17, 18]. The moduli space of solutions has an algebraic description by dropping the
Dterm equation (3.20) in favour of a stability condition and dividing by complex gauge
transformations. This leads to a description in terms of stable `quasimaps' from C = CP1
into the vacuum moduli space V = T G(k; N ). The moduli space of solutions decomposes
into a union of components
labelled by the vortex number or ux n 2 Z through CP1, which coincides with the degree
of the quasimap.
YN ( a
j=1 ( a
mj + ~2 )
mj
~ )
2
= q Y
a
This coincides with the quantum equivariant cohomology ring of V = T G(k; N ) and the
Bethe equations (2.10) for an inhomogeneous XXX 1 spin chain of length N with
quasiperiodic boundary conditions.
We now consider alternative approach to computing correlation functions in Atwisted
gauged linear sigma models introduced in [
13, 14
] in terms of vortex counting.
This
approach was derived rigorously from supersymmetric localization in [6].
The mathe
matical formulation of this construction involves equivariant integrals over holomorphic
`quasimaps' to the vacuum manifold V, which may be computed by equivariant
localization [15, 16]. This provides the link with recent mathematical work on the Bethe/Gauge
correspondence [12].
In this approach, we rst set the complex mass parameters (m1; : : : ; mN ; ~) and the
to zero, and consider con gurations preserving both Q
and Q+. Such
con gurations are given by
b + ~
b
~
;
2
M =
Mn ;
[
n2Z
Now turning on the mass parameters (m1; : : : ; mN ; ~) and the
deformation
parameter
deforms the equations (3.23){(3.24) that determine
by replacing
!
+ m + ~ +
LV ;
where (m1; : : : ; mN ; ~) are understood to mean the in nitesimal Tf avour transformation
generated by these parameters and LV is the Lie derivative along the vector eld V
generating U(1) rotations. This restricts the system to the xed points of the corresponding
Tf action on the moduli space M.
This can be understood as working equivariantly with respect to the action of Tf
U(1)
on the moduli space M with equivariant di erential Q = Q
+ Q+. In particular,
localization of the path integral to Gaussian
uctuations around Q , Q+invariant con gurations
is equivalent to computing the following sum of equivariant integrals
where [Mn]vir is the virtual fundamental class. The correlation functions hf (~ ); g(~ )iS2
correspond to computing the equivariant integrals of certain virtual equivariant cohomology
classes [f ] and [g] on Mn.
We will rst explain how to compute the partition function in this manner in the
abelian case, before considering the general case.
Abelian case.
We rst set the mass parameters (m1; : : : ; mN ; ~) and
deformation
to vanish. Assuming r > 0, we then have
=
= 0 and the remaining equations become
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
Solutions are labelled by the ux
N
X(jXj2
j=1
jYj j2)
r =
2i
through C = CP1 and we denote the corresponding moduli space by Mn.
It is convenient to introduce the following algebraic description of the moduli space
Mn. We rst remove the Dterm equation (3.28) and replace it for r > 0 by the stability
condition that Xj 6= 0 for all j = 1; : : : ; N except at a
nite number of points on CP1.
In addition, we divide by complex gauge transformations that leave the remaining
equations (3.29) invariant. A point in Mn is now speci ed by N holomorphic sections (Xj ; Yj )
of O(n)
O( n), such that P
straightforward to compute the moduli spaces explicitly:
j Xj Yj = 0 and the sections Xj are not all zero. It is now
If n < 0, the moduli space is empty Mn = ;.
If n = 0,
we recover the algebraic description of the vacuum
manifold
M0 = V = T CPN 1
.
If n > 0, we have Yj = 0 and the moduli space is parametrized by N holomorphic
sections Xj of O(n). Using a complex gauge transformation to set Az = 0, the
holomorphic sections are homogeneous polynomials
Xj (z; w) =
X xj;rzn rwr :
n
r=0
(3.31)
HJEP1(207)5
The moduli space is therefore parametrized by the N (n + 1) coordinates xj;r that are
not all zero, modulo residual constant C gauge transformations preserving Az = 0.
We therefore nd that Mn = CPN(n+1) 1.
We now consider the uctuations around a point on the moduli space Mn for n
uctuations decompose into chiral and Fermi multiplets with
respect to the supersymmetry algebra generated by Q
and Q+. A 2d N = (2; 2) chiral
multiplet of U(1)V charge r transforming as a section of a line bundle L contributes:
1. Chiral multiplets: H0(C; KCr=2
2. Fermi multiplets: H1(C; KCr=2
L).
L).
Here, KC is the canonical bundle of the Riemann surface C. For us, KC = O( 2). This can
be summarized by the statement that the
contribute H (C; KCr=2
uctuations of a 2d N = (2; 2) chiral multiplet
L) to the `virtual tangent bundle' of the moduli space.
Turning on (m1; : : : ; mN ; ~) and
corresponds to working equivariantly with respect to
the action of Tf
U(1) on Mn. Let us consider the uctuations from each chiral multiplet
in turn for n > 0, leaving the special case n = 0 until the end.
The
uctuations from each Xj transform in H (C; O(n)).
There are therefore
N (n+1) chiral multiplets corresponding to
uctuations of the coordinates xj;r in
equation (3.31) and no Fermi multiplets. Under a G
Gf
U(1) transformation
generated by parameters ( ; m1; : : : ; mN ; ~; ), they transform with weight
s
;
j = 1; : : : ; N ;
s = 0; : : : ; n :
(3.32)
The uctuations from each Yj transform in H (C; O( n)). There are therefore no
chiral multiplets and N (n
1) Fermi multiplets corresponding to fermion zero modes
in the vortex background. They transform with weight
mj +
+
~
2
n
2
~
2
n
2
2
s
;
j = 1; : : : ; N ;
s = 0; : : : ; n
2 :
(3.33)
The uctuations from
transform in H (C; O( 2)). There is therefore a single Fermi
multiplet transforming with weight
~.
T vir
Mn =
X
N " n
X e
1 comes from the vectormultiplet.
The moduli space Mn = CPN(n+1) 1 has isolated xed points under a generic Tf
U(1)
transformation generated by (m1; : : : ; mN ; ~; ), which correspond to the N (n + 1)
coordinate lines. We can label the xed points by the pair (i; r) with i = 1 : : : ; N and r = 0; : : : ; n.
The xed points correspond to sections
In addition there is a contribution
H (O) from the vectormultiplet. Combining these
contributions, the equivariant index of the virtual tangent bundle is
The contribution to the partition function from
uctuations around each xed point
of Mn is encompassed in the virtual localization formula
Z
[Mn]vir
whose transformation under (m1; : : : ; mN ; ~; ) is compensated by a gauge transformation
r) . The equivariant index at the xed point (i; r),
by
= ji;r := mi
T(vii;rr)Mn =
~
2
N
X
j=1 s=0
(
n
2
n
by the replacement
!
ji;r.
X emi mj+(r s)
X e mi+mj+~ (r s)
e ~
1 ;
(3.36)
!
Z
[M0]vir
d ( ~) Y
N
j=1 (
1
mj + ~2 )(
+ mj + ~2 ) ;
(3.35)
(3.37)
(3.38)
ji;r =
(3.39)
where we have introduced the replacement rule e : Pi niewi !
Qi wini to compute the
equivariant Euler character. This result is most neatly expressed as the following contour
integral
Z
d ( ~) Y s=1
N
j=1
s
s
;
where the contour surrounds the poles corresponding to the
xed points
=
mi
~
2
2
r) . This exactly reproduces the coe cient of qn for n > 0 in the contour
integral formula (3.12). Note that the Je reyKirwan residue corresponds to computing
residues at poles of the integrand corresponding to xed points of Mn.
Let us now consider the special case n = 0. The moduli space now corresponds to
constant maps to the vacuum manifold V = T G(k; N ) with bosonic
uctuations from
both X and Y . As above, the equivariant localization expression is neatly expressed as a
contour integral
where the contour surrounds the poles at
= mj
~2 from the contribution of Xj . This is
a regular equivariant integral of 1 over the vacuum manifold V = T G(k; N ).
The extension to include twisted chiral operators inserted at f g will be discussed in
detail in section 4.
NonAbelian case.
With gauge group U(k), we again pass to an algebraic description
of the moduli space M of solutions to equations (3.20){(3.22) by removing the Dterm
equation in favour of a stability condition and dividing by complex gauge transformations.
We therefore consider only
with the stability condition that the k
N matrix X has maximal rank away from isolated
points on C = CP1 and modulo complex GL(k; C) gauge transformations. A point in Mn
is then speci ed by:
A holomorphic GL(k; C) bundle V .
W ' C
of the PSU(N ) avour symmetry.
Constraints X Y = 0.
Holomorphic sections X and Y of associated vector bundles V
W and V
W where
CN is a trivial vector bundle associated to the fundamental representation
Stability condition that rk(X) = k except at isolated points.
According to a theorem of Grothendieck, on C = CP1 we can decompose
V = O(n1)
O(nk)
X na = n ;
a
such that Xa become sections of O(na)
W and Ya become sections of O( na)
W . This
leads to a strati cation of the moduli space for ux n 2 Z into components labelled by
integers (n1; : : : ; nk) 2 Z
k with Pa na = n. The moduli space is empty if na < 0 for any
a = 1; : : : ; k. We therefore restrict attention to the region na 2 Z 0
.
Fluctuations around a point on the moduli space Mn decompose into chiral and Fermi
multiplets with respect to the superalgebra generated by Q
and Q+. Following the
discussion above, the contributions can be summarized by the equivariant index
T vir = H
V
W
1
W 2
~
1
W 2
~
+ H
V
W
+ H (KC
V
V
W
1
~ )
H (V
where we have introduced yet another trivial line bundle W~ ' C
C associated to the
fundamental representation of the avour symmetry U(1)~. The rst three contributions
arise from the uctuations of the chiral multiplets X, Y and
respectively. The nal
contribution
H (V
V ) is the contribution from the vectormultiplet.
(3.41)
V ) ;
(3.42)
The equivariant index is straightforward to write down explicitly for any ~n 2 Zk 0. Let
us write the formula for the case when na
1 for all a = 1; : : : ; k, and na 6= nb for a 6= b:
T vir
Mn =
X e b mj+ ~2 +( n2b s)
X e b+mj+ ~2 ( n2b s 1)
#
e bc ~+( n2bc s 1)
X e cb ~+( n2bc s)
#
(3.43)
X e bc ( n2bc s)
e cb ( n2bc s 1)
#
:
~
2
nb 2
s=0
nbc
s=0
The moduli spaces themselves for k > 1 are singular and do not admit an explicit
description as in the abelian case. It is nevertheless possible to compute the equivariant xed
points in terms of the algebraic data and apply the virtual localization formula to compute
the partition function. The xed points are labelled by a decomposition ~n = fn1; : : : ; nkg,
a choice of vacuum I = fI1; : : : ; Ikg
f1; : : : ; N g and a vector ~s = fs1; : : : ; skg where
sa 2 f0; 1; : : : ; nag. The vectormultiplet scalar takes the following value at this point
and the virtual localization formula is
a = ajI;~s
mIa
Z
[Mn]vir
1 =
X
X
j~nj=n (I;~s) e(T(v~nir;I;~s)Mn)
na
2
1
sa
:
;
(3.44)
(3.45)
This reproduces coe cient of qn for n > 0 in the contour integral formula (3.12) where the
data f~n; I; ~sg enumerate poles of the integrand chosen by the Je reyKirwan description.
The case n = 0 should again be treated separately and reproduces a regular equivariant
integral over the vacuum manifold V = T G(k; N ).
3.3
Vortex partition function
We will also consider the vortex partition function or `cigar' partition function with a
xed vacuum vI at in nity. We can equivalently view this as a sphere with the boundary
condition that the system sits in the vacuum vI at f g, as shown in gure 8. In section 4.3,
this partition function will be used to construct the wavefunctions of spin chain Bethe
eigenstates. As above, we present the partition function both as a contour integral over the
complex Cartan subalgebra of the gauge group and its interpretation in terms of counting
quasimaps that are `based' at f g
Let us rst consider the abelian case. The partition function with the vacuum vi at f g
can be expressed as a contour integral in the vectormultiplet scalar ,
h1ivi =
1
1(~)
Z
i
d q~
N
Y
j=1
1
~
2
mj +
1
;
(3.46)
~
2
U (1)J
vI at in nity. This can also be viewed as a sphere with a xed vacuum at f g
where q~ = ( 1)N q. The integrand has poles at
= mj
~
2
` for all j = 1; : : : ; N and
` 2 Z 0. The contour i selects only those poles with j = i arising from the 1loop
determinant for the chiral multiplet Xi that has a nonzero expectation value in the vacuum vi.
The classical and 1loop contributions can be factored out by normalizing by the value
of the partition function at q ! 0
HJEP1(207)5
!0 h1ivI
lim hf (~ )ivI = f (~ I (q)) ;
The result,
h1ivi
h1ivi jq!0
= Vi(q) :
Vi(q) =
1
X q
n=0
N
n
n Y Y mi
j=1 `=1
mj
mi
~
mj
(`
`
1)
;
is the vortex partition function with vacuum vi at in nity.
This can be generalized to the nonabelian case with vacuum vI ,
h1ivI =
Z
d
k
I k!
q~ P
a a=
Q
a6=b
k
Q
a;b=1
1( ab + )
1(
ab +~) a=1 j=1
k n
Y Y
~
2
1
a
mj +
1
a +mj +
;
where q~ = ( 1)N q.
The same integrand appears in the computation of the
hemisphere partition function with the boundary condition supported on the whole of
V = T G(k; N ) [19, 20]. However, the contour I surrounds only the poles arising from
the 1loop determinant of the chiral multiplets Xi for all i 2 I. As above, we can extract
the corresponding vortex partition function VI (q), which we will not write down explicitly.
We will denote the correlation function of a twisted chiral operator f (~ ) at f+g in the
background with a supersymmetric vacuum vI at in nity f g by hf (~ )ivI . In the limit
that we remove the
deformation,
asymptotic behavior
! 0, such correlation functions have the common
hf (~ )ivI
! e 1 Wf(~ I (q)) + : : : ;
where Wf(~ ) is the e ective twisted superpotential and
particular solution of the Bethe equations (2.10) associated to the
aI(q) = mIa
~2 + O(q) is the
xed vacuum vI at
in nity. Therefore, normalizing by the vortex partition function, we nd that
(3.47)
(3.48)
~
2
(3.49)
(3.50)
(3.51)
We now perform an orbifold construction at either of the xed points f+g or f g by
choosing a subgroup ZN
U(1)J and turning on a discrete holonomy g = !
i 1 for the
U(1)f avour symmetry for some choice of i = 0; : : : ; N
1 where !N = 1. The introduction
of this orbifold is implemented at the level of the equivariant index by replacing
j
! j + (i
;
! N
;
and then averaging over the transformation
xed. Applying this operation to the equivariant indices (4.22) in
1) < N we nd that
eqf j+ if 0 < qf (i
1) < N ;
e(z) = <>`~ 1 `
> P e~ z
>:`~ 0 `
+ `~ if
8> P e~ z
+ `~ if 0 < qf (i
1) < N ;
1)
0 ;
Therefore, the partition function remains unchanged for
g. Otherwise, the partition function is multiplied by an additional
factor qf j . In this case, the orbifold construction is then equivalent to inserting the
twisted chiral operator qf .
Let us now explain this procedure by implementing the orbifold construction directly
on the mode expansion. We focus on the xed point f+g and set the coordinate w = 1 with
the understanding that
transforms with weight j+ =
formations. This is natural because j+ is the vacuum expectation value of the complex
scalar at f+g. With this understanding, the chiral eld is expanded
n under U(1)f
2
avour
trans(z) =
X
` 0
+ `
` z :
e(z) =
X
` 0
`+zqf (i 1)+` :
In the presence of the orbifold, the chiral eld should transform under ZN
transformations as (z) ! gqf (!z), where g = !i 1. It is therefore convenient to de ne a deformed
eld ~(z) := zqf (i 1) (z), which absorbs the e ect of the discrete holonomy and transforms
in the standard way ~(z) ! ~(!z). This has an expansion
z ! z1=N , it is straightforward to see that
The modes that are invariant under the ZN action are `+ such that qf (i
some `~ 2 Z. Projecting onto ZN invariant modes and rede ning the complex coordinate
1) + ` = `~N for
(4.24)
(4.25)
(4.26)
(4.27)
(4.28)
where we have de ned e+~ :=
`
expansion is trivial if
orbifold has removed the mode
+
`~N qf (i 1)
partition function by the corresponding equivariant weight qf j+.
0. However, in the region 0 < qf (i
1) < N the
0+ and we should therefore multiply the integrand of the
. Therefore the e ect of the orbifold on the mode
We now consider the supersymmetric gauge theory introduced in section 3.1 and introduce
a ZN orbifold with discrete holonomy that breaks the U(k) gauge and PSU(N ) gauge and
avour symmetry to a maximal torus [10]. The construction depends on the following data:
HJEP1(207)5
A permutation
symmetry,
of f1; : : : ; N g, which speci es the discrete holonomy for the avour
(gF )ij = ! (i) 1 ij :
(gG)ab = !I (a) 1 ab :
An ordered subset I = fI1; : : : ; Ikg
f1; : : : ; N g with Ia < Ib for a < b, which
speci es the discrete holonomy for the gauge symmetry,
and de ne SI (~ ) := Sf1;:::;Ng(~ ).
I
We rst perform the orbifold construction at the point f+g. We will denote the twisted
chiral operator introduced by this orbifold construction at f+g by SI( )(~ ). In the following,
in order to simplify our notation, we write formulae for the unit permutation
= f1; : : : ; N g
The starting point for the computation is the equivariant index for contributions
X
X
e aj+ mi+ ~2 + e aj++mi+ ~2
+
e abj+ ~+
SI (~ ) = Sym
k
Q
a=1
Ia 1
Q ( a
i=1
(
N
b)( a
b
~)
Q ( a
a<b
(4.29)
(4.30)
(4.31)
(4.32)
(4.34)
3
X e abj+ 5 :
a6=b
!
;
!
The orbifold construction is implemented by shifting the complex avour and gauge
parameters according to the discrete holonomy,
aj
!
!
mi ! mi + (i
1)
N
aj + (Ia
1)
N
! N
;
and averaging over the transformations
+ 2 is for s = 0; : : : ; N
1. This operation leads to a modi cation of the equivariant index by
I+ ! I+
k
X
Ia 1
X e aj+ mi+ ~2 +
and therefore to an insertion of
k
X
a;b=1
X
a<b
e aj++mi+ ~2
e abj+ +e abj+ ~
; (4.33)
?
?
?
?
in the integrand of the partition function. In writing this expression, we have symmetrized
over
1; : : : ; k as the operator is inserted inside a contour integral that is symmetric in
these parameters. Note that in the abelian case k = 1 with I = fig, the above expression
reduces to the abelian formula (4.14) considered above.
This formula can be understood graphically as explained in
gure 12. In order to
read o the numerator of (4.34) one draws a k
N table and indicates the positions of
fI1; : : : ; Ikg by ?. Then one lls in the cells to the right (light grey) with equivariant
weights
weights a
a + mi + ~2 corresponding to elds Y ia, and the cells to the left (dark grey) with
mi + ~2 corresponding to elds Xai. Finally, one multiplies all the weights in
the table. The denominator of (4.34) is universal for all I for a given k.
permutation
inserts the operator SI( )(~ ) given by the expression
Performing the orbifold construction at f+g with a holonomy labelled by a general
SI( )(~ ) = S 1(I)(~ )jmj7!m (j) :
(4.35)
an insertion of the operator SI
re ection permutation.
Finally, performing the same orbifold construction at the other xed point f g leads to
(
)(~ ) where : f1; : : : ; N g ! fN; : : : ; 1g is the longest or
We now perform the same computation at f+g using a zero mode analysis, highlighting
the additional features that appear compared with a single chiral multiplet. For simplicity,
we restrict ourselves here to the zero
ux sector. The computation in the general
ux
sector is obtained by replacing a !
aj+ = a
orbifold action.
na everywhere with j+ xed under the
2
In absence of the orbifold defect, we can expand the elds X, Y and
around the
1
`=0
point f+g as holomorphic functions of z,
Xai(z) =
X(x`)ai z` ;
Y ia(z) =
eld starts at O(z) in the twisted theory.
we can introduce the deformed elds
Note that since the adjoint chiral multiplet
has U(1)V charge +2, the expansion of this
In the presence of the ZN orbifold with discrete holonomies (4.29) and (4.30) at f+g
1
X(y`)ia z` ;
`=0
1
`=1
ab(z) =
X( `)ab z` :
(4.36)
1
`=0
1
`=1
Xe ai(z) = zIa i X(x`)ai z` ;
Ye ia(z) = zi Ia X(x`)ai z` ;
eab(z) = zIa Ib X( `)ab z` ;
1
`=0
(4.37)
HJEP1(207)5
8 1
> P ( e`)ai z` ; a < b
>
: `=1
> P ( e`)ai z` ; a
>
b
i
>
Ye ia(z) = < `=0 e
1
8 1
> P (y`)ai z` ; Ia
: `=1 e
> P (y`)ai z` ; Ia < i
>
i
;
near f+g.
The orbifold defect has completely changed the zero mode structure at f+g, as one
can see from the above expansions. Note that before orbifolding, each component of X and
Y had a zero mode at f+g, while
some of the uctuations of X and Y at f+g with equivariant weights
vanished there. However, the orbifold has eliminated
k
Y
a=1
Ia 1
Y
i=1
a
mi +
now project onto modes in the expansion that are invariant under ZN . For example, the
invariant modes in the expansion of Xe ai are parametrized by (x`)a i with Ia
for some `~ 2 Z. Replacing z ! z1=N and relabelling the coe cients, the invariant parts of
i + ` = `~N
the expansions are
;
i=Ia+1
Y( a
a<b
b
~) 1
:
Y( a
a<b
b) 1
;
while introducing additional zero modes for
with equivariant weights
Furthermore, the orbifold breaks the gauge symmetry U(k) ! U(1)k at f+g and the
broken generators develop additional zero modes in the defect background parametrizing
the complete ag variety Fk = U(k)=U(1)k. This leads to an additional contribution
corresponding to the equivariant weight at a xed point of Fk. This combines with the
contribution from
to form the equivariant weight of the cotangent bundle T Fk. The
symmetrization over 1; : : : ; k together with these denominator factors can be interpreted
as an equivariant integral over the moduli space T Fk of the defect. The appearance of a
hyperKahler moduli space is expected since in the absence of the mass parameter ~ the
defect preserves a N = (0; 4) supersymmetry. Combining these contributions reproduces
the function SI ( ) obtained in equation (4.34).
4.3
Bethe wavefunctions
The functions SI ( ) are up to normalization the wavefunctions of the o shell Bethe states
for the spin chain in the updown basis, hIj 1; : : : ; ki. The Bethe eigenstates themselves
(4.38)
(4.39)
(4.40)
(4.41)
f ( )
g( )
Bf
Bg
are obtained by evaluating the auxiliary variables on a solution aJ of the Bethe equations,
j J i = j 1J ; : : : ; kJ i. The wavefunctions of the Bethe eigenstates in the updown basis are
then hIj J i = ( 1)jIjN ( J ) 1SI ( J ) where the normalization factor N ( ) is de ned in
equation (2.19).
These wavefunctions arise in the supersymmetric gauge theory from expectation value
of the stable basis elements SI ( ) in the cigar background with vacuum
in equation (3.51). More precisely, we compute the limit
function with SI ( ) inserted at the tip of cigar and vacuum vJ at in nity,
! 0 of a normalized correlation
J at in nity, as
!0
h1ivJ
lim hSI (~ )ivJ = SI ( J ) :
(4.42)
We have already mentioned that this evaluates the function SI ( ) at the solution to Bethe
equations (2.10) associated to the vacuum vJ , with expansion
emphasize that the Bethe wavefunctions SI ( J ) can be found directly from the gauge
aJ = mJa
~2 + O(q). We
theory computation without solving any Bethe equations.
5
Quantum mechanical description
In this section, we provide an alternative viewpoint on Amodel computations on CP1 in
terms of supersymmetric quantum mechanics. We replace CP1 by a long cylinder capped
o
by Atwisted cigars. In the cylindrical region, the theory preserves 2d N = (2; 2)
supersymmetry and we can reduce on S1 to obtain an e ective N = 4 supersymmetric
quantum mechanics. The capped regions become boundary conditions in the
supersymmetric quantum mechanics preserving the supercharges Q ; Q+. This setup is summarized
in gure 13.
In principle, we can nd a nite dimensional N = 4 supersymmetric quantum
mechanics for each
to the zero
ux sector n 2 Z individually. Here we restrict ourselves to a description
ux sector n = 0, or equivalently the limit q ! 0. As we have emphasized
in the introduction, much of the representation theoretic apparatus of the algebraic Bethe
ansatz does not depend on the choice of quasiperiodic boundary condition speci ed by q,
and should therefore have a description in this supersymmetric quantum mechanics.
After a description of the N = 4 supersymmetric quantum mechanics and a general
description of how to translate insertions of twisted chiral operators to boundary
conditions, we will provide two explicit constructions of the boundary conditions that arise from
insertions SI( )(~ ). The rst involves a combination of Neumann or Dirichlet boundary
conditions coupled to boundary degrees of freedom. The second involves the notion of a
thimble boundary condition.
Let us rst consider the N = 4 supersymmetric quantum mechanics obtained from the
ux sector n = 0 of our 2d N = (2; 2) theory, which is obtained by plain dimensional
reduction of section 3.1 on a circle. In the absence of mass deformations, the
supersymmetric quantum mechanics has U(1)V
SU(2)A Rsymmetry with U(1)A
SU(2)A as the
Cartan subalgebra. We denote the euclidean coordinate of the supersymmetric quantum
mechanics by . We refer the reader to appendix A.2 for further details on our conventions
for supersymmetric quantum mechanics.
The U(k) vectormultiplet now contains scalar elds A transforming in a triplet of the
SU(2)A Rsymmetry. The complex combination
=
1 + i 2 is the twodimensional scalar
and 3 arises from the integral of the twodimensional gauge eld around S1. The fermion
now transform as a spinor of SU(2)A. The lagrangian for the vectormultiplet
Lvec =
1
A denote the SU(2)A gamma matrices and we have suppressed contractions over
SU(2)A spinor indices. A chiral multiplet consists of a complex scalar
and fermions
transforming as a spinor of SU(2)A with lagrangian
Lchiral =
jD
2
D
A j2 +
D
+ i
+ i
+ jF j2 :
(5.2)
We have N fundamental chiral multiplets X, N antifundamental chiral multiplets Y , and
an adjoint chiral multiplet . There is also a cubic superpotential W = Tr( XY ).
The supersymmetric quantum mechanics has a vacuum manifold determined by
solutions to the equations
R
A
which coincide with con gurations annihilated by all four supercharges. As in the
twodimensional theory, with r > 0 solutions are forced to have
A = 0 and the vacuum
manifold is V = T G(k; N ).
avour symmetry around the S1.
The supersymmetric quantum mechanics has avour symmetry Gf = PSU(N ) U(1)~.
We can turn on SU(2)A triplets of mass parameters (m1A; : : : ; mAN ; ~A) of mass parameters
by coupling to a background vectormultiplet with a vacuum expectation value for
A in
Tf . The
components of these mass parameters are the complex masses (m1; : : : ; mN ; ~)
introduced in two dimensions. In section 5.4, we will also want to turn on real mass
parameters (m31; : : : ; m3N ) which arise from turning on background holonomy for the PSU(N )
In the presence of such mass parameters, equations (5.5) are deformed to
~
2
a
mj +
~
2
mj +
) ab = 0 :
(5.6)
(5.7)
For generic values of the mass parameters, this lifts the vacuum manifold to the isolated
xed points of the Tf action on V.
Empty boundary condition
Before constructing boundary conditions in the supersymmetric quantum mechanics that
correspond to the inserting SI( )( ) at the tip of the cigar, we rst consider the empty
boundary condition associated to a cigar without any insertion. Our discussion of such
boundary conditions has much in common with the description of Btype boundary
conditions in 2d N = (2; 2) gauge theories [19, 20, 25].
The appropriate boundary condition for the vectormultiplet can be determined as
follows. First, as we are considering the sector with vanishing ux and 3
impose the Dirichlet boundary conditions 3j = 0. The remaining boundary conditions,
RS1 A, we should
A j = 0 ;
3j = 0 ;
D
j = 0 ;
1j = 1j = 0 ;
are determined by preserving the 0d N
Q1 = Q
and Q1 = Q+ at the boundary.
Let us now consider the boundary conditions for the chiral multiplets. A 1d N = 4
chiral multiplet can be decomposed into a chiral multiplet ( ; 2) and a Fermi multiplet
1 with superpotential E ( ) = D
in terms of the boundary N = (0; 2) supersymmetry
algebra generated by Q
and Q+. A basic boundary condition therefore involves Neumann
for the chiral component and Dirichlet for the Fermi component,
= (0; 2) supersymmetry algebra generated by
D
= 0j ;
1j = 0 ;
D
2j = 0 :
The components ( ; 2) transform as a N = (0; 2) chiral multiplet at the boundary. We
call this a `Neumann' boundary condition. In the presence of a bulk superpotential, this
boundary condition must be supplemented by a choice of matrix factorization.
The boundary condition corresponding to the empty cigar can now be described as
follows. We rst impose Neumann boundary conditions for X, Y and
. We then couple
to a 0d Fermi multiplet
with the same charges as
and boundary superpotentials
E =
j
;
J = X Y j :
This provides a matrix factorization of the bulk superpotential, W j = E
J . Note that the
boundary superpotential E
=
j e ectively modi es the boundary condition for
from
Neumann to Dirichlet. This is compatible with the zero mode analysis in section 4.2. In
particular, the chiral elds X(z) and Y (z) were nonzero at z = 0 corresponding to a
Neumann boundary condition, whereas (0) = 0 reproducing a Dirichlet boundary condition.
(5.8)
(5.9)
(5.10)
The `empty' boundary condition is compatible with any vacuum con guration in V.
It will therefore ow to Neumann boundary conditions in the supersymmetric sigma model
supported on the whole of V. In particular, the partition function on an interval with
the empty boundary condition at each end reproduces the n = 0 contribution to the CP1
partition function,
h1i1d = ( 1)k2 Z d
k
Q ( a
k! k N
Q Q ( a
k
b) Q ( a
a;b=1
mi + ~2 )(
;
(5.11)
provide the factor ( 1)k2 Qa;b( a
following the Je reyKirwan prescription.
Stable boundary conditions
which is the equivariant integral of `1' over V. From this perspective, the denominators
arise from the N = (0; 2) chiral multiplets X; Y . The contributions coming from two Fermi
multiplets
at two boundaries cancel the contribution from the chiral multiplet
and
~) in the numerator. The remaining numerator
b) is the vector multiplet contribution. The contour
is chosen by
We now consider the class of boundary conditions that arise from inserting SI ( ) at the
tip of the cigar. Flowing to a supersymmetric sigma model to the vacuum manifold V,
such boundary conditions are supported on holomorphic lagrangian submanifolds in V that
are xed by Tf . In this section, we provide an elementary description of these boundary
conditions in the supersymmetric gauge theory by coupling the empty boundary condition
to additional boundary degrees of freedom. The construction is similar in spirit to the
exceptional Dirichlet boundary conditions introduced in [26].
Let us rst consider the abelian case. Following the arguments of section 4.1, let us
rst consider the boundary condition obtained by inserting the homogeneous polynomial
mj + ~2 at the tip of the cigar, corresponding to the equivariant weight of the coordinate
Xj . In the vanishing
ux sector, this is a polynomial representative of the equivariant
cohomology class Poincare dual to fXj = 0g
T CPN 1
.
This boundary condition is described in the supersymmetric quantum mechanics by
coupling the empty boundary condition to a 0d N = (0; 2) Fermi multiplet j with the
same charges as Xj with boundary superpotentials
E j = Xj ;
J j = 0 :
The 0d N = (0; 2) chiral multiplet part of Xj will receive a mass from the superpotential
at the boundary modifying the boundary condition for Xj from Neumann to Dirichlet.
This boundary condition is therefore supported on the subspace fXj = 0
g
of the vacuum manifold. In the computation of partition functions, the boundary Fermi
T CPN 1
multiplet j provides an additional contribution
mj +
~
2
;
(5.12)
(5.13)
in the numerator of the integrand. This reproduces the e ect of inserting the twisted chiral
operator f ( ) =
mj + ~2 at the tip of the cigar.
It is now straightforward to write down the boundary condition corresponding to
inserting the twisted chiral operator SI ( ) with I = fig in abelian theories. We deform the
empty boundary condition by introducing (N
with the boundary superpotentials
1) boundary Fermi multiplets j for j 6= i
E j =
(Xj if j < i
Yj
if j > i
;
J i = 0 :
The charges of the Fermi multiplets are uniquely determined by their superpotentials. The
boundary condition is now supported on the subspace
HJEP1(207)5
Xj = 0 for j < I ;
Yj = 0 for j > I ;
of the vacuum manifold V. A partition function with this boundary condition will include
an additional contribution compared to the empty boundary condition,
I 1
Y
i=1
mi +
~
2
N
Y
i=I+1
+ mi +
;
~
2
from the additional contributions of the Fermi multiplets j . Therefore, this boundary
condition reproduced the computation with SI ( ) inserted. Note that due to the symmetry
( j ; Ej ; Jj ) $ ( j ; Jj ; Ej ), we could equivalently have coupled to boundary fermi multiplets
with J type superpotentials.
This boundary condition can be easily generalized to nonabelian theories in a manner
consistent with the zero mode analysis of section 4.2. The boundary condition for SI (~ )
with I = fI1; : : : ; Ikg should be supported on the subspace of V de ned by
Xaj = 0 for j < Ia ;
Y j a = 0 for j > Ia :
These constraints can be implemented by introducing k(N
1) Fermi multiplets denoted
by aj for j < Ia and j a for j > Ia with superpotentials
(E )aj = Xaj for j < Ia ;
(E )ja = Y j a for j > Ia ;
together with J = 0.
Recalling that Ia
Ib if and only if a
b, these constraints imply that the complex
moment map for the gauge symmetry is upper triangular at the boundary,
(X Y )abj = <
8
>
>:0 ;
X
Ia<j<Ib
Xaj Y j bj ; if a
b ;
if a > b :
Therefore the matrix factorization requires that we introduce only k(k+1) Fermi
multiplets at the boundary, with components
ab for a
2
b, with boundary superpotentials
(5.14)
(5.15)
(5.16)
(5.17)
(5.18)
(5.19)
SI ( ) =
k
Q
Q ( a
i=1
mi + ~2 ) IaQ+1(
N
!
Here, the numerator factors are the contributions from the boundary Fermi multiplets a
and
j a. The denominator factors arise from additional bosonic zero modes of
a < b and from the breaking of the gauge symmetry. As mentioned in section 4.2, the
denominator factors combine to form an equivariant weight of the tangent bundle at a
xed point of T Fk, and symmetrizing over 1; : : : ; k, the result can be interpreted as an
equivariant integral over this boundary moduli space.
Thimble boundary conditions
In this section, we turn on real mass parameters (m31; : : : ; m3N ) valued in the Cartan
subalgebra of the PSU(N ) avour symmetry. These parameters arise from a background avour
holonomy around S1 in the twodimensional setup. In the presence of real mass parameters,
there is a natural class of boundary conditions for the supersymmetric quantum mechanics
preserving Q , Q+, which are analogous to thimble boundary conditions in 2d N = (2; 2)
theories [27{29]. We will need a slight generalization of the standard notion appropriate
for theories with multiple isolated vacua connected by gradient ows [26]. We will adapt
this construction here to the context of 1d N = 4 supersymmetric quantum mechanics. We
expect this to provide a physical counterpart to the construction of stable envelopes [11].
In the presence of real mass parameters fm31; : : : ; m3N g, the con gurations of the
supersymmetric quantum mechanics preserving Q , Q+ are
(J )ab = X Y j and (E )ab =
boundary from the components
ab with a < b.
abj. This leads to additional bosonic uctuations at the
In addition, the boundary condition breaks complex gauge transformations GC = GL(k)
to the Borel subgroup B of upper triangular transformations. There are therefore additional
bosonic uctuations at the boundary parametrizing Fk = GC=B, which is complex version
of the breaking of U(k) to its maximal torus described in section 4.2. As mentioned there,
these
uctuations combine with those of
a
bj for a < b to form the cotangent bundle
T Fk, whose hyperKahler structure is a re ection of the fact that the boundary condition
preserves a N = (0; 4) supersymmetry in the limit ~ ! 0.
In the computation of interval partition functions, this boundary condition leads to an
additional contribution compared to the empty boundary condition,
R
r1 =
D X =
Xy
D X =
3 X + X
m3 ;
D Y = Y
=
These equations can be reformulated as the gradient ow equations
(5.20)
j
a
bj for
(5.21)
(5.22)
where
h =
R;m ;
(5.23)
and R and R;m denote the moment maps for the gauge and avour symmetry respectively.
The notation X refers to one of the elds fX; Y; ; 3
g and gXXy is the inverse metric on
the space of elds inherited from the Lagrangian.
Note that inside of gauge invariant combinations of the elds X and Y , we have
(5.24)
and therefore gauge invariant combinations of X and Y will grow or decay along the
direction according to their charge under the
avour transformation generated by m3.
This is gradient ow on the vacuum manifold V for the Morse function hm = m3
A preliminary de nition of a left thimble boundary condition BI on
be given as follows: it is a boundary that is equivalent for the computation of correlation
functions preserving Q , Q+ to the placing theory on
1 with a
xed isolated
vacuum vI at
vacuum vI at
the submanifold of points in V that can be reached by an in nite gradient ow from the
parameters. Suppose that the real masses are ordered as m3(1) <
< m3(N) for some
1. This submanifold clearly depend on the ordering of the real mass
1. The support of such a boundary condition is in the rst instance
0 can now
permutation , then we denote this submanifold by V
1 to
However, in passing from
= 0, we will need to allow for a sequence of
domain walls preserving Q , Q+ that interpolate between di erent isolated vacua connected
by gradient ows. In order to formalize this notion, we can introduce a partial ordering on
the set of isolated vacua fvI g depending on the permutation , by the requirement
(I ).
there exists a gradient ow vI ! vJ
vI < vJ ;
and extending transitively, namely if vI < vJ and vJ < vK then also vI < vK . Allowing
for sequences of domain walls interpolating between vacua, the support of a left thimble
boundary condition BI is
V(BI) =
[
vI vJ
V
(J ) :
We claim that the thimble boundary condition BI in the presence of mass parameters
corresponds to boundary condition constructed in section 5.3
ordered by the permutation
associated to inserting SI( )( ).
Let us illustrate this construction with an abelian example: k = 1 and N = 2 with
V = T CP1  see table 1. Introducing a real mass parameter m3 for the U(1)m
symmetry, we have
hm = m3( jX1j2 + jX2j2 + jY1j2
jY2j2) :
Note that in the graphical representation of V = T CP1 in gure 10, the Morse function is
proportional to the coordinate along the horizontal axis. The gradient ows are therefore
straightforward to understand in this graphical representation.
(5.25)
(5.26)
avour
(5.27)
V{B22,1}
(5.28)
(5.29)
HJEP1(207)5
= f1; 2g corresponds to m3 < 0 and the permutation
= f2; 1g
corresponds to m3 > 0. In the graphical representation, the direction of ow for increasing
is from left to right for
= f1; 2g and right to left for
= f2; 1g. The vacua are
therefore ordered such that v1 < v2 for
straightforward to see that
= f1; 2g and v2 < v1 for
= f2; 1g. First, it is
Vf11;2g = CP1
Vf21;2g = F2 ;
fv2g ;
Vf12;1g = F1 ;
Vf22;1g = CP1
fv1g ;
where Fi denotes the ber of V = T CP1 at the xed point vi. We therefore generate
thimble boundary conditions with support
VfB11;2g = Vf11;2g
[ Vf21;2g = CP1 [ F2 ;
VfB21;1g = Vf22;1g = F1 ;
VfB12;2g = Vf21;2g = F2 ;
VfB22;1g = Vf22;1g
[ Vf12;1g = CP1 [ F1 ;
= f1; 2g, and Sf2;1g( ) and Sf2;1g( ) for
1 2
= f2; 1g.
which are illustrated in
gure 14. The supports of these boundary conditions clearly
coincide with those obtained from the cigar with insertions of S1f1;2g( ) and Sf1;2g( ) for
2
Finally, let us attempt to make a general statement. We expect that the left thimble
boundary condition BI generated by real mass parameters m3(1) <
m3(N) is equivalent
for the purpose of computing correlation functions preserving Q , Q+ to the boundary
condition in section 5.3 corresponding to S
ow equations in the same way as mj3 !
mj3, the right thimble boundary condition for the
same mass parameters reproduces the function SI
( ), where : f1; : : : ; N g ! fN; : : : ; 1g
( )
I( )( ). Since
!
transforms the gradient
is the longest permutation.
In this section we return to studying the Heisenberg spin chain and the question of how the
Rmatrix arises in the study of supersymmetric gauge theory. In particular, we will examine
the twopoint functions of stable basis elements S
I( )(~ ) in the Atwisted supersymmetric
gauge theory on the sphere. These correlation functions are in fact independent of q and can
therefore be interpreted as the partition function of a supersymmetric quantum mechanics
of section 5 on a interval with thimble boundary conditions at either end.
Orthonormality of stable basis
and S
Let us rst consider the twopoint correlation functions of basis elements SI( )(~ ) at f+g
( )(~ ) at f g where : f1; : : : ; N g ! fN; : : : ; 1g is the longest permutation. This
J
correlation functions are independent of q and evaluate to
hSI( )(~ )SJ( )(~ )iS2 = I;J :
The appearance of the re ection
here is natural from the orbifold construction. The
orbifold construction at f+g producing SI( )(~ ) has avour holonomy (gF )ij = ! (i) 1 ij .
This is compatible with turning on
form ei i ij such that
(1) <
avour holonomy in a neighbourhood of f+g of the
(N). Translating to f g without allowing these
holonomy eigenvalues to cross, the compatible orbifold construction leads to functions SI
Similarly in the supersymmetric quantum mechanics, introducing constant real masses
ordered such that m3(1) <
< m3(N) leads to left thimble boundary conditions generating
( )(~ ).
SI( )(~ ) and right thimble boundary conditions generating S
I
( )(~ ).
This observation motivates to de ne an inner product,
(6.1)
(6.2)
(6.3)
(6.4)
where the conjugation sends a !
a; mi !
therefore the stable basis elements for a given permutation
are orthonormal,
mi. In particular, SI( )(~ ) = S
I
( )(~ ) and
hf (~ ); g(~ )i = hf (~ ); g(~ )iS2 ;
hSI( )(~ ); SJ( )(~ )i = I;J :
By construction, this inner product depends only on the functions f (~ ) and g(~ ) modulo
the twisted chiral ring relations. Under the correspondence outlined in the introduction 1,
this corresponds to the inner product on the spin chain Hilbert space,3
V
( ) = Cm (1)
2
2
Cm (2)
: : :
2
Cm (N) ;
with sites ordered according to the permutation . In particular, equation (6.3) corresponds
to the orthonormality of the updown basis of the spin chain, hIjJ i = I;J .
3In the supersymmetric gauge theory, we have allowed the parameters ( 1; : : : ; k), and (m1; : : : ; mN ; ~)
to be complex. In order to recover the honest inner product on the spin chain Hilbert space, we would need
to specify certain reality conditions on these parameters such that they map exactly onto the corresponding
spin chain parameters.
6.2
The natural next step is to consider the inner product of stable basis elements for di
erent permutations
and 0. These correlation functions are again independent of q. Let
us therefore consider this problem from the perspective of the supersymmetric quantum
mechanics of section 5.
(m31; : : : ; m3N ) vary as a function of
Consistency of such a correlation function requires that the mass parameters
across an interval, such that they are ordered by
the permutation
at the left boundary and by 0 at the right boundary. This is an exact
deformation and therefore correlation functions do not depend on the particular pro le
HJEP1(207)5
of this variation. In particular, we can say that such a correlation function requires the
presence of a `Janus interface' J ; 0 for the real mass parameters. The inner product
hSI( )(~ ); SJ( 0)(~ )i ;
One can immediately recognize this table as the matrix elements of the spin chain
R2
matrix (2.13) acting on Cm1
sets of correlation functions
C2m2 , up to a sign. Summarizing this example, there are two
which can be represented graphically as in gure 15. The rst line consists of correlation
functions consistent with constant real masses ordered by the permutation f1; 2g, whereas
the second line contains correlation functions consistent with the presence of a Janus
interface Jf2;1g;f1;2g.
is computing the partition function of the supersymmetric quantum mechanics on a interval
with the interface J ; 0 between thimble boundary conditions generated by the vacua vI
on the left and vJ on the right.
In order to develop the connection between such correlation functions and spin chain
quantities, let us consider the simplest example corresponding to a spin chain of length
N = 2. In that case we have two distinct permutations, f1; 2g and f2; 1g, and three
supersymmetric theories with k = 0, 1 and 2. The correlation functions
hSIf2;1g(~ ); SJf1;2g(~ )i ;
are straightforward to evaluate explicitly. The result is summarized in the following table
Sf2;1g
fg
Sff12g;1g
Sff22g;1g
Sff12;;21gg
Sf1;2g
fg
Sff11g;2g
Sff21g;2g
Sff11;;22gg
1
0
0
0
m1
m1 + ~
m1 + ~
0
m1 + ~
m2
m1 + ~
0
0
0
1
hSIf1;2g(~ ); SJf1;2g(~ )i = IJ ;
hSIf2;1g(~ ); SJf1;2g(~ )i
RIJ (m1
m2) ;
(6.5)
(6.6)
(6.8)
:
(6.7)
1
2
1
2
a spin chain of length N = 2.
In order to extend this correspondence to supersymmetric gauge theories with N > 2,
we will need to introduce the notion of a Weyl Rmatrix depending on a pair of permutations
,
sitions.
functions
where
In order to de ne it, let decompose the +
1 = (i1 j1) (i2 j2) : : : (iL jL) :
R( +; )(m1; : : : ; mN ) := R
(i1) (j1) m
: : :
As a consequence of the YangBaxter equation (2.12) and the unitarity condition (2.14)
the Weyl Rmatrix is independent of the choice of decomposition into elementary
transpoWe have found that the components of the Weyl Rmatrix coincide with the correlation
RI(J+; )(m1; : : : ; mN ) = NIJ
( +; )
h I
S( +)(~ ); SJ( )(~ )i ;
NIJ
( +; ) = ( 1)j +1(I)j+j 1(J)j ;
is a sign. We have performed extensive checks of this relation in numerous examples. In the
supersymmetric quantum mechanics setup of section 5, the matrix elements of the Weyl
Rmatrix are therefore identi ed with the partition function of a Janus interface J +;
between thimble boundary conditions generated by the vacua vI and vJ .
We end this section with an example of a Weyl Rmatrix. Let us x the spin chain
length N = 5 and choose permutations
decompose the permutation
1 = (25) (34) (24) (23) and therefore de ne
+ = f1; 4; 3; 5; 2g and
= f1; 2; 3; 4; 5g. We can
R(f1;4;3;5;2g;f1;2;3;4;5g)(m1; : : : ; m5) = R25(m52)R34(m43)R24(m42)R23(m32) ;
(6.14)
which can be straightforwardly computed from the matrix elements of the elementary
Rmatrices, Rij (mji). This Rmatrix can be depicted as in gure 16. Equivalently, the same
matrix elements can be computed from the correlation functions
RI(fJ1;4;3;5;2g;f1;2;3;4;5g)(m1; : : : ; m5) = NIJ hSIf1;4;3;5;2g(~ ); SJf1;2;3;4;5g(~ )i ;
(6.15)
1 as a convolution of elementary transpositions
: : : R
(iL) (jL) m
(6.9)
(6.10)
(6.12)
(6.13)
1
2
5
Weyl Rmatrix is independent of the decomposition as a consequence of the YangBaxter equation
( gure 5a) and unitarity ( gure 5b).
which can be evaluated by directly performing the contour integral in (3.16). Both of these
computations give the same result.
YangBaxter equation
Janus interface J +;
the real masses mi3 and m3,
j
As explained in the previous section, we have performed extensive checks that the Weyl
Rmatrix RIJ
( +;
)(m1; : : : ; mN ) corresponds to matrix elements of a Janus interface J +;
in the supersymmetric quantum mechanics setup of section 5.
In the same manner that the Weyl Rmatrix RIJ
elementary Rmatrices according to a decomposition
( +;
)(m1; : : : ; mN ) is constructed from
1 = (i1 j1) : : : (iL jL), the
is a composition of elementary Janus interfaces Jij that interchanges
J +;
= Ji1j1 : : : JiLjL :
(6.16)
This can be understood since deformations of the pro le m31( ); : : : ; m3N ( ) for the real
mass parameters are exact in Q , Q+. We are therefore free to choose a pro le consisting
of a sequence of `jumps' where pairs of mass parameters mi3 and mj3 are interchanged.
Each of these jumps can be regarded as an elementary Janus interface Jij . Inserting the
complete set of states provided by the stable basis SI( )(~ ) in between each elementary Janus
interface then reproduces the decomposition (6.11) of the Weyl Rmatrix. Equation (6.16)
can therefore be understood as a basisindependent statement of this decomposition.
The fact that the Weyl Rmatrix is independent of the choice of decomposition into
elementary transpositions followed from the YangBaxter equation (2.12) and the
unitarity condition (2.14). From the perspective of supersymmetric quantum mechanics, this
property is guaranteed since di erent decompositions of a pro le m31( ); : : : ; m3N ( ) into
elementary jumps are related by exact deformations. In particular, we have
and
where I is an identity interface preserving the order of the real mass parameters. This is
a basisindependent statement of the YangBaxter equation and unitarity relation. The
standard equations for Rmatrices are recovered by inserting the complete set of states
provided by the stable basis SI( )(~ ) in between each elementary Janus interface.
Jij Jik Jjk = Jjk Jik Jij ;
Jij Jji = I ;
(6.17)
(6.18)
Discussion
In this paper, we have investigated aspects of the correspondence between XXX 1
Heisenberg spin chains and 2d N
= (2; 2) supersymmetric gauge theories. We have focussed
2
on reproducing components of the algebraic Bethe ansatz for spin chains from correlation
functions in Atwisted supersymmetric gauge theory and their reduction to partition
functions in N = 4 supersymmetric quantum mechanics. In particular, we have provided a
concrete construction of the wavefunctions of o shell Bethe states as orbifold defects in
Atwisted supersymmetric gauge theory, and as thimble boundary conditions in
supersymmetric quantum mechanics. We have also developed a new interpretation of the spin chain
Rmatrix as the matrix elements of Janus interfaces for mass parameters, leading to a novel
and basisindependent presentation of the YangBaxter equations.
Let us conclude with some directions for further research:
First, there are some important components of the algebraic Bethe ansatz that we
have omitted from our presentation.
One example is the generators of the
Yangian symmetry of the spin chain. Unlike the Rmatrix, Yangian generators have
nonvanishing matrix elements between spin chain states with di erent number of
excitations.
On the supersymmetric side of the correspondence, this will
correspond to correlation functions of interfaces that change the rank of the gauge group,
U(k) ! U(k0). It is straightforward to construct such interfaces in the
supersymmetric quantum mechanics description of section 5, following methods introduced
in [30]. However, we expect a complete discussion of Yangian representation theory
and the algebraic Bethe ansatz will arise from `tripartite' interfaces in
supersymmetric quantum mechanics relating theories (k; N ), (k0; N 0) and (k00; N 00) with di erent
gauge and avour symmetries [31].
Secondly, in this paper we have considered only su(2) spin chains with the
fundamental representation at each site. It would be interesting to extend the results
presented here to more general groups and representations, by studying more general
supersymmetric quiver gauge theories.
Finally, it would be interesting to extend our results to trigonometric or elliptic spin
chains, corresponding to three and four dimensional supersymmetric gauge theories.
The corresponding localization techniques for correlation functions of twisted theories
on S2
S1 or S2
T 2 have been developed in [7, 32, 33].
Acknowledgments
We would like to thank Stefano Cremonesi for useful discussions, and the organizers of the
Pollica summer workshop for kind hospitality while this paper was being completed. MB
and TL are supported by ERC STG grant 306260. HK would like to thank Mathematical
Institute at University of Oxford and the 2017 Summer Workshop at the Simons Center
for Geometry and Physics for their hospitality and support during di erent stages of this
work. The research of HK is supported in part by NSF grant PHY1067976.
Az =
Az =
2
2
i
i( +
i( +
D = +Dz +
i
2 +[ ;
+);
Dz
2
1
2
1
2
1
2
1
2
+ = i +
D
2iFzz + [ ; ] + 2
Dz ;
= i
D + 2iFzz
[ ; ]
2 +Dz ;
+ =
i +
D + 2iFzz + [ ; ] + 2
Dz ;
D
2iFzz
[ ; ]
2 +Dz ;
Conventions
2d N = (2; 2) supersymmetry
We consider twodimensional N = (2; 2) supersymmetric theory on a at space with
Euclidean coordinates (x1; x2). We will also introduce a complex coordinate z = x1 + ix2.
Our conventions are taken directly from appendix A of [25] with x0 =
The supersymmetry transformations of a vectormultiplet are
(A.2)
(A.4)
(A.6)
(A.7)
(A.8)
(A.10)
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
(A.17)
+Dz + +
Dz
2 +[ ;
= +
+ ;
+ ;
+ = 2i
Dz + +F
= 2i +Dz +
F +
Dz + +F
2i +Dz +
F +
;
;
;
;
+ =
F =
F =
2i
2i
+ ( +
2i
( +
Dz
Dz
+ 2i +Dz +
+ 2i +Dz +
+) + i(
+ ) + i (
) ;
) ;
For a chiral multiplet transforming in a unitary representation of the gauge group G, the
supersymmetry transformations are
where D
= @ + iA and it is understood that vectormultiplet elds act in the appropriate
representation of G.
LV = Tr
d
Z
1
4
and FI term
1
2e2
The chiral multiplet Lagrangian is
L = Tr
Z
d
+ total derivative
= Tr
2Dz Dz
2Dz Dz +i
and superpotential term
2Dz Dz
2Dz Dz +2i
Dz
2i +Dz + +4Fz2z +D2
(A.18)
LF I = Re
Z
d2 ~( t ) =
rD
2 Fzz :
HJEP1(207)5
i
+ +i
+i +
+ +
[ ; ]
i +Dz + + D +jF j2
j j
2
1
2
Dz
Z
LW = Re
d
2 W ( ) :
(A.19)
(A.21)
(A.22)
(A.23)
(A.24)
(A.25)
(A.26)
Writing a general supersymmetry transformation as
we nd that
= i( +Q
Q+
fQ ; Q g = 2iDz
fQ ; Q+g =
fQ+; Q g = 0
Q2 = 0
fQ+; Q+g =
fQ+; Q g =
fQ+; Q g = 0
Q
2 = 0 :
2iDz
The charges of the supersymmetry generators under U(1)J rotations and the axial U(1)A
and vector U(1)V Rsymmetries are shown below:
Q+
Q+
Q
Q
1
1
+1
+1
1
1
+1
+1
U(1)V
U(1)A
1
+1
+1
1
U(1)0J
2
0
0
+2
In order to write the N = (2; 2) supersymmetry algebra as a 1d N = 4 supersymmetry
algebra, we compactify a spatial direction on a circle x1
x1 + 2 R and rename
= x2.
We can then organize the supercharges into spinors
Q
Q+
Q
Q
Q+
Q
combining supercharges of U(1)V charge
1 and +1 respectively. Note that the top (resp.
bottom) components of both spinors have charge +1 (resp. 1) under U(1)A. With this
notation, the supersymmetry algebra with Z = 0 can be reexpressed as follows
where ( I )
mation is given by
are the Pauli matrices. In terms of these elds, the supersymmetry
transforwhere
fQ ; Q g =
fQ ; Q g = 0 ;
D + Z
0
1 0
1
Z
U(1)V
This takes the form of an N = 4 supersymmetric quantum mechanics with Rsymmetry
SU(2)A Rsymmetry. States with KK momentum in the x1 direction clearly break
SU(2)A to the U(1)A axial Rsymmetry in two dimensions.
We now write the supersymmetry transformations of the elds in SU(2)A covariant
notation. We need to choose a convention for raising and lowering indices and will choose
12 =
21 = 1 with
transformations in SU(2)A covariant notation as
and
. We rst write the supersymmetry
= i(
Q
Q ) :
For the vectormultiplet we write
I ( I )
A =
1
2
i
2
i D
= i D
D =
D
A1
i
2
i
2
) ;
[
[
D
A1!
D ;
= i (
) + i (
) ;
[
(A.28)
(A.29)
(A.30)
(A.31)
(A.32)
For the chiral multiplet we have the supersymmetry transformation as
F =
F =
D
D
D
D
F +
F +
(A.34)
HJEP1(207)5
where
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