Supersymmetric Rényi entropy and defect operators
HJE
Supersymmetric Renyi entropy and defect operators
Tatsuma Nishioka 0 1 3 4
Itamar Yaakov 0 1 2 4
0 Kashiwa , Chiba 2778583 , Japan
1 Bunkyoku, Tokyo 1130033 , Japan
2 Kavli IPMU (WPI), UTIAS, The University of Tokyo
3 Department of Physics, Faculty of Science, The University of Tokyo
4 induces a holonomy for A, around @
We describe the defect operator interpretation of the supersymmetric Renyi entropies of superconformal eld theories in three, four and tors involved are supersymmetric codimensiontwo defects in an auxiliary Zn gauge theory coupled to n copies of the SCFT. We compute the exact expectation values of such operators using localization, and compare the results to the supersymmetric Renyi entropy. The agreement between the two implies a relationship between the partition function on a squashed sphere and the one on a round sphere in the presence of defects.
Nonperturbative E ects; Supersymmetric Gauge Theory; Supersymmetry

and Duality
1 Introduction
Supersymmetric Renyi entropy and defects
Renyi entropy and discrete gauge theories
Supersymmetric Zn gauge theory
Supersymmetric defects from supersymmetric Zn gauge theory
Coupling defects to the ncopy theory
Mass terms and vortices
The matrix models
The scalar moduli space and classical contributions
Perturbative contributions
Nonperturbative contributions  instantons and contactinstantons
Sewing of instantons
Classical and ChernSimons contributions depending on instantons
Fundamental hypermultiplets uctuations
Adjoint hypermultiplet or vector multiplet uctuations
The third point
3.7
n ! 1=n duality
4
Discussion
A Conventions
B Special functions C The instanton partition function
{ i {
Introduction
The nonlocal nature of quantum entanglement is one of the sharpest characteristics by
which quantum physics di erentiates itself from classical physics. Entanglement occupies
a central position in quantum information theory and, increasingly, in various branches of
theoretical physics such as condensed matter and high energy physics. Entangled states
are ubiquitous and of particular interest in manybody quantum systems. Renyi entropy
is a re ned measure of the entanglement a given state possesses when the Hilbert space is
split into states supported on a spatial region
and those supported on its complement.
In a local quantum
amounts to
eld theory, employing the replica trick [1], the nth Renyi entropy
of a (Euclidean) manifold on which the theory is placed. The absolute value
taken in the de nition (1.1) has no e ect in unitary theories, but is necessary to incorporate
the case of a complex partition function which we will deal with when supersymmetry is
implemented on a curved space.
It follows from (1.1) that knowing the partition function Zn is more or less equivalent to
calculating Renyi entropies, and there are a few situations where the exact values are known
(see e.g. [1{3]). A common practice in handling the conical singularity around @ , present
in the calculation of Zn for n > 1, is to smooth out the tip by introducing a regulator,
calculate the partition function on the smoothed space, and take the singular limit [4].
This approach is highly advantageous as it reformulates the problem as a calculation in
quantum
eld theory on a curved space.
Another complementary approach is to represent the partition function Zn as a product
of correlation functions of twist operators that create the proper monodromies around
[1, 5]. Twist operators are codimensiontwo (nonlocal in d > 2 dimensions) objects
that specify the boundary conditions on the entangling surface, and the twisting is done
for an nfold copy of the original theory. In what follows, we will illustrate the interplay
between the two approaches in a particular situation where the exact calculation of Zn
is possible: the supersymmetric Renyi entropy computed using localization [6]. We will
restrict ourselves to a spherical entangling surface @
= S
d 2 in d dimensions, work in the
vacuum state of the SCFT, and examine supersymmetric gauge theories of type 3d N = 2,
4d N = 2, and 5d N = 1. The motivation for doing so is twofold. First, it is interesting to
compare the two di erent looking localization calculations and
nd out how they match.
Second, the microscopic de nition of the defect operators,1 presented in section 2, could be
useful when examining dualities in which the SCFT participates. These could be dualities
between di erent Lagrangian eld theories, or holographic dualities between an SCFT and
string theory on an appropriate background.
1Twist operators are a subclass of defect operators. These terms will be used interchangeably in
this paper.
{ 1 {
The vacuum Renyi entropy is a nonlocal observable which can be de ned for any
ddimensional quantum
eld theory whose Hilbert space can be factorized in a local manner.
For a conformal eld theory, and for integer Renyi parameter n, this observable is equivalent
to two di erent objects, each of which can be de ned by a suitable Euclidean path integral:
I. The partition function on an nfold multiple covering of the dsphere Sd, branched
along @ , with appropriate boundary conditions at the branching loci [7].
II. The partition function of an nfold copy of the theory  henceforth referred to as the
ncopy theory  on Sd with codimensionone defect operators acting between copies
(cf. [8]).
The equivalence between these two objects is tautological. The object in II is simply a
relabeling of the degrees of freedom, one for each of the n sheets of the branched sphere.
The defect operators are de ned to reproduce the boundary conditions implied by the
original geometry.
One may introduce a linear eld rede nition, acting in the ncopy theory, to diagonalize
the action of the defects, introducing defects which act on just one copy at a time. In the
special case of a free theory, the action written using the rede ned elds does not couple
the copies. The computation in this case is equivalent to a third object:
III. The partition function of an nfold copy of the theory on Sd with codimensiontwo
defect operators acting on each copy  equivalently, the kth copy is coupled to a
background connection with holonomy e2 i nk around @ , with k 2 f0; : : : ; n
also [10{18] for further developments).3 It has been observed that the result in this case is
equivalent to yet another object:
IV. The partition function of a single copy of the theory on a squashed Sd, with a suitable
supersymmetry preserving action. The squashing parameter, which determines the
nonround metric, is related to the Renyi parameter n in a simple way, such that
n = 1 corresponds to the round sphere.
2The holonomy prescription could be di erent if the eld in question is a fermion, or carries additional
global symmetry charges [9]. We will make a speci c choice later on.
3Some of the 3d theories we consider are not superconformal. However, deformation invariance of the
Renyi entropy of the SCFT to which the original theory ows [6].
means that the Euclidean action is invariant under some supersymmetry transformation .
It is well known that the path integral in this case is insensitive to exact deformations,
either of the action or in the form of exact insertions.4 The objects I and IV are related
by such a deformation (c.f [19]).
Despite their simplicity, there is something interesting to be said about forms III and
IV of the supersymmetric Renyi entropy. The calculation of the path integrals representing
either III or IV can be performed exactly using localization. This involves splitting the
elds of the theory into an interacting part  the moduli  and a free part  the uctuations.
The latter can be put into the form III. Since the two calculations look quite di erent, and
in some contexts have di erent interpretations, it is interesting and potentially useful to
determine exactly how they give the same result. This will be our primary goal.
1.2
Renyi entropy and discrete gauge theories
The setup described above for calculating the Renyi entropy at integer Renyi parameter
can be alternatively thought of as introducing a defect in a discrete gauge theory coupled
to the ncopy theory.5 For our purposes, it is su cient to consider the gauge group Zn
acting by cyclic permutation on the copies, although the ncopy theory is invariant under
the full permutation group Sn.
6 If we choose to think of Zn as a gauge symmetry, we
can reasonably treat the defects which implement the calculation of the Renyi entropy as
codimensiontwo objects. Only operators charged under the Zn symmetry can detect the
position of the codimensionone defect. Gauging Zn means that all such operators are
projected out.
We would now like to incorporate supersymmetry into the de nition of the Zn gauge
theory and the defect. A simple way of doing so is to write down a version of the discrete
gauge theory which is realized by higher form abelian gauge elds [20]. A BF type theory
with one ordinary gauge eld (A) and one (d 2)form
eld (B) works nicely. The reduction
to Zn gauge symmetry is implemented by using B as a Lagrange multiplier. The action
for the gauge sector is
in Z
2
SBF =
F ^ B ;
where F is the eld strength of A. SBF is invariant under gauge transformations for both
A and B as long as n is an integer.
The codimensiontwo defect could be realized by taking a at connection with
prescribed holonomy and coupling it to the free elds of the ncopy theory. The way to do
this in the BF theory is to insert the operator
I
exp ik
= exp ik
Z
(1.2)
(1.3)
4There are additional restrictions on the deformation. A exact deformation to the action must be
annihilated by
and should have a positive semide nite real part. Speci cally, any deformation must be
such that it does not alter the convergence properties of the path integral.
5The authors are grateful to Daniel Ja eris for suggesting a discrete gauge theory interpretation for
our calculation.
6The symmetry which shifts the copies by one is known as the replica Zn symmetry.
{ 3 {
where [@ ] is the Poincare dual to the cycle @
representing the entangling surface. This
operator is invariant under the B gauge transformations as long as k is an integer.
Integrating out B in the theory with the action (1.2), and in the presence of this operator,
of strength e2 i nk . After the linear eld rede nition,
A can serve as the at connection for the kth copy for the Renyi parameter n. One may
then extend the elds A; B and the terms in the action to their supersymmetric versions
to create a SUSYBF theory [21].7
One possible obstruction to using ordinary gauge elds is that the original theory may
have its own, possibly nonabelian, gauge elds. It may not be possible to couple the eld
A to these nonabelian gauge
elds in a manner consistent with gauge invariance. Note
HJEP1(207)
that the original description of the codimensionone defect already implies that any gauge
group of the ncopy theory is broken to a diagonal subgroup, acting simultaneously and
identically on the copy at either side, at the defect. This type of the reduction of the gauge
invariance at the position of a defect is quite common (see e.g. [22]). Moreover, gauge elds
often carry supersymmetric moduli which cannot be coupled to the codimensiontwo defect
in the way described above. We will treat all of these problems in an ad hoc manner.
The organization of the paper is as follows. In section 2 we introduce a supersymmetric
version of the Zn gauge theory based on higher form abelian gauge super elds.
Component expressions for the super elds and actions are readily available in the literature. We
then derive the abelian version of the codimensiontwo supersymmetric defect operator
as a classical supersymmetric con guration for a vector supermultiplet on the round
ddimensional sphere. We use this con guration to write down an ordinary supersymmetric
Wilson type codimensiontwo operator for the tensor multiplet, using the supersymmetric
version of (1.3), and perform localization in the presence of this operator and the BF term.
We then derive the resulting e ect on any matrix model to which the original vector
multiplet could potentially be coupled. In section 3, we show that the matrix model for the
ncopy theory on the round Sd, after a speci c modi cation by such abelian defects, is equal
to the matrix model on the squashed Sd, including all classical and oneloop contributions.
In the process, we show how the moduli for the squashed sphere can be thought of as n
sets of moduli for the round sphere, which are sewn together by the defect. We end with
a discussion of possible applications.
2
Supersymmetric defects from supersymmetric Zn gauge theory
In this section, we de ne the supersymmetric Zn gauge theory, and supersymmetric
codimensiontwo defects, which we later use to calculate the supersymmetric Renyi
entropy. We then calculate the expectation values of these defects using localization. We
treat supersymmetric gauge theories of type 3d N = 2, 4d N = 2, and 5d N = 1.
7By SUSYBF, we mean an untwisted supersymmetric version of BF theory of the type considered in [21].
We construct a supersymmetric Zn gauge theory, and a supersymmetric codimensiontwo
defect, by introducing a pair of dynamical supermultiplets: a vector multiplet V and
a (d
2)form multiplet E. The eld strength for the latter sits in the familiar linear
multiplet, denoted by G. We introduce the supersymmetric analogues of the terms used in
the discrete gauge theory
and
The terms on the right hand side are schematic superspace integrals, for which we write
component expressions later. The operator (2.2) can also be written as
ik Z
2
exp
G Vdefect ;
where Vdefect is a background con guration for a vector multiplet invariant under a subset
of the supersymmetries, whose component expression will be worked out below, and the
integral is over the entire superspace. Integrating out G results in a supersymmetric delta
function setting
We call the coe cient the vortex charge.
k
n
V =
Vdefect :
in Z
2
G V ;
I
D ;
2.2
The 3d N = 2 codimensiontwo defect
In three dimensions, both V and E are ordinary vector multiplets. The super eld G is
the eld strength super eld which can be used to write the YangMills term in 3d.8 The
codimensiontwo defect is a vortex loop of the type examined, for instance, in [23]. Its
supersymmetric version was analyzed in [24, 25]. The calculation of the
supersymmetric Renyi entropy in three dimensions using defects was carried out in [6]. We brie y
review it below.
The localization calculation for an N = 2 theory on the round S3 reduces the path
integral to an integration over a single Lie algebra valued scalar: the constant mode
of the real elds
and D appearing in the N = 2 vector multiplet [26, 27]. The resulting
matrix model expression for the partition function will be given in section 3.3. Speci cally,
the uctuation determinant for an abelian vector multiplet is 1.
8This is a real linear super eld, which is sometimes denoted by .
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
where [ ] is the volume form on a maximal S1
S3, and [ ]D is the Poincare dual to
= nk , this con guration can be imposed on a dynamical vector multiplet by
considering an additional vector multiplet with connection B, an o diagonal ChernSimons
SB(3Fd) =
and a supersymmetric abelian Wilson loop of the form
A supersymmetric defect which mimics the e ects of computing the supersymmetric
Renyi entropy can be introduced by considering an additional background abelian vector
multiplet
A ; A; DA; fermions :
The defect con guration is of the form
dA =
Localization reduces the above terms to9
Integrating over the new modulus B then sets
I
W (3d) (k)
exp
SB(3Fd)
W (3d) (k)
exp 2 i n A B ;
exp
2 k B ;
!
!
A =
i :
k
n
{ 6 {
Note that this vev for A is o the original contour of integration and represents an
imaginary mass term for chiral multiplets to which the A vector multiplet is coupled. We will
argue later that an imaginary Higgs type mass for a dynamical vector multiplet can also
be thought of this way. The imaginary mass is the entire e ect of the original defect on the
localization computation. By continuity, the same is true for an arbitrary . We will show
below that this continues to hold, for appropriate defects, in four and in ve dimensions.
2.3
The 4d N = 2 codimensiontwo defect
We consider a 4d N = 2 theory consisting of vector multiplets and hypermultiplets. We
would like to show that the e ect of inserting a codimensiontwo defect is equivalent to the
introduction of an imaginary mass term.
9We universally denote the Lie algebra valued scalar zero mode as .
We begin by reviewing some aspects of the localization calculation for an N = 2 theory
on the round, and on the branched foursphere following [10, 28, 29]. We refer the reader
to these papers for explicit actions and localizing terms. The authors of [10] considered a
smooth resolution of the branched foursphere, the resolved foursphere, which is
deformation equivalent to it, and on which localization computations can be performed. In this
section, we retain an overall scale `, associated with the size of the foursphere, which will
later be set to 1.
We use the following coordinates on an nfold covering of S4 (the branched sphere)
and take a basis for the Cli ord algebra
where i are the Pauli matrices.
to the reality condition [29]
a =
i = i i ;
0
a 0
a
!
;
i
i i ;
4 = 4 = 1 ;
e1 = ` d ;
e2 = n` sin d ;
e3 = ` cos d ;
e4 = ` cos sin d ;
(2.14)
An N = 2 supersymmetry is generated by a foursome of Weyl spinors
A; _ A, subject
A
( A)y =
AB
B ;
A _
_ A y = _ _ AB
_B :
The subscript A is an SU(2)R index. Indices ; _ indicate a spinor transforming as a
doublet under the left and right SU (2) factors of Spin (4). Indices ; _ ; A are raised with
; _ _ ; AB such that 12 = 1 and AB =
AB.
In order to preserve rigid supersymmetry, one must solve the Killing spinor equation.
The relevant Killing spinor equation for the round/branched foursphere is
1
8
1
8
b a) A
iASU(2)R B
iASU(2)R B
A
A
B =
B =
i
i
A0 ;
A0 ;
Here, we have introduced an SU(2)R background connection ASU(2)R , and set all other
supergravity background elds besides the metric to zero. The solution is given by
ASU(2)R =
n
1
2
1 0 !
0
1
d ;
{ 7 {
(2.13)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
0 can be extracted by contracting with
/ . In order to use the transformation generated
by
to perform localization, one should impose an additional constraint to ensure that the
square of the transformation does not contain scale or U(1)R transformations
This constraint has been taken into account above.
As shown in [10, 29], one can introduce supergravity backgrounds for the resolved
foursphere such that the same Killing spinors are preserved. Moreover, one may use the same
localizing term for the vector and hypermultiplets in the presence of defects/squashing as
one does in the round sphere case. These localizing terms, which are described in [28, 29],
imply that all components of a hypermultiplet must vanish on the moduli space. The
contour of integration for vector multiplet scalars compatible with the localizing terms is
and
The localizing terms yield the following moduli space of zero modes for a vector multiplet
y =
;
DAB y =
DAB :
`
2
=
=
i
= 0), and antiinstantons at the south pole (
=
0;
=
=2). The complete matrix model expression for the partition function is given in
section 3.3.
2.3.2
Introducing the defect
We would like to introduce a surface defect into the computation on the round sphere. The
data for a surface defect can be embedded in a background N = 2 abelian vector multiplet
A ; ; ; DAB; A; A ;
where A is the gauge eld,
and
are complex scalar elds whose relationship depends
on the contour of integration, DAB is an SU (2)R triplet of auxiliary scalars, and
A; A
independent Weyl fermions which are SU (2)R doublets. We want a supersymmetric defect,
on the round foursphere, supported only on the twosphere at
= 0. Such a defect would
have a eld strength
Fdefect =
( ) d ^ d :
{ 8 {
In order to regularize the above con guration, we introduce a background
such that10
g ( ) is a smooth function satisfying
g ( )
;
<
The supersymmetry variation of the gaugino in an abelian vector multiplet is [29]
A =
A =
1
2
1
2
F
F
A + 2D=
A + 2D=
A + DAB
A + DAB
B + 2 D= A ;
B + 2 D= A :
We can complete (2.27) to a supersymmetric con guration by introducing additional
bosonic backgrounds
i
In this background, the gaugino variations vanish with an arbitrary spinor from (2.20). In
the limit
! 0 we get
g ( ) ! 1 ;
Note that both the smooth and singular con gurations are o the original contour of
integration for the vector multiplet (2.22) (D12 is imaginary).
2.3.3
The supersymmetric Lagrange multiplier
Following the logic of the Zn gauge theory, we will impose the con guration for the
background vector multiplet using a supersymmetric Lagrange multiplier. We make the vector
multiplet dynamical and introduce an N = 2 abelian tensor multiplet E
E ; G; G; LAB; fermions ;
whose eld strength sits in the linear multiplet G [30].11
We couple E to the vector multiplet (actually its eld strength) using a supersymmetric BF term [30]
SB(4Fd) =
2
1
2
1
4
"
G +
G
DABLAB +
E
F
+ fermions :
(2.35)
11We use the letters E and G to indicate both the supermultiplets and some of their components. We
hope this does not cause too much confusion.
(2.26)
(2.27)
(2.28)
(2.29)
(2.30)
(2.31)
(2.32)
(2.33)
(2.34)
We then introduce a Wilson surface operator for the tensor multiplet by adding an
additional copy of (2.35), but replacing all elds in the vector multiplet with their values
from (2.32) and (2.33), and with
exp
Integrating out the tensor multiplet would result in a supersymmetric delta function setting
all elements of the vector multiplet to their values for the surface defect. Instead of doing
this, we rst perform the localization.
2.3.4
Localization of the tensor multiplet
The localizing term for the abelian vector multiplet can be taken from [29]. For the tensor
multiplet, we must consider the transformation of the fermions in this multiplet, denoted
'A; 'A.12 The variations are
(2.36)
(2.37)
(2.39)
(2.40)
(2.41)
(2.42)
(2.43)
(2.38)
12We have decomposed each spinor 'i from [30] as a doublet 'A; 'A. Note that the de nition of the
LeviCivita symbol in [30] contains an extra factor of \i" compared to the usual de nition: "1234 = 1.
Examination of the gravitino transformation in [30] yields the following dictionary in relation to [29]
i
!
A!
A ;
i
! 2
1
=
D
A!
A
= 2i
0A!
0A :
Superscripts on both sides indicate SU(2)R transformations. Note the factor of 2 di erence with the used
in e.g. [31, 32].
'A =
Stensor localizing term =
h
( ')yA 'A + ( ')yA 'Ai :
Z
S4
1
2
"
G =
Gy ;
LAB y =
LAB ;
L11 y =
L22 ; Im L12 = 0 :
H
G = Gr + iGi :
We take the localizing term multiplet elds (2.35) which yields
We also de ne
We will use reality conditions for G; G; LAB appropriate for the coupling to the vector
H
= 0 ;
L11 = L22 = Gr = 0 ;
Gi =
= G0 :
(2.46)
with G0 a real constant.
2.3.5
Modi cation of the matrix model
We will couple the vector multiplet representing the surface defect to the physical vector
multiplets and hypermultiplets of the N
= 2 SCFT as detailed in section 3.1.
After
localization, the matrix model representing the S4 partition function is modi ed by the
presence of the defect. We would like to show that this modi cation amounts to giving
the hypermultiplets or the vector multiplets an imaginary mass, given by
discuss all contributions of V and E to the matrix model.
n
i k . Below we
Classical contribution.
The multiplets V and E have a classical action given by (2.35)
and (2.36). After localization, these terms give insertions in the matrix model. Using the
vevs for the scalar elds
and
we get
=
=
i ;
2
D12 =
G = G = i G0 ;
L12 =
`
;
`
2
G0 ;
exp
SB(4Fd)
Wsurface (k)
!
!
exp
2 i`4 n
G0 ;
exp 2 `3 k G0 :
i
`
The bosonic part of the resulting localizing term is
Z
S4
"
+2
The vector eld v is such that
The localization locus is thus
1
The quadratic approximation to the localizing term (2.44)
for the tensor multiplet around the locus (2.46) is independent of G0. The oneloop
determinant is thus a G0 independent number, which is furthermore equal to the partition
function of a free tensor multiplet on the round foursphere. This multiplet can be dualized
into an uncharged massless hypermultiplet whose partition function can be deduced from
the expressions in 3.5. The oneloop determinant for A is trivial.
H = ?dE = 0 ;
implies that we can set
up to tensor gauge transformations. When there are nontrivial twocycles, the form E is
closed and the tensor gauge transformations imply that E is gauge equivalent to 0 whenever
it represents an integral class. The remaining moduli of E can be identi ed with the values
of the possible Wilson surfaces. The gauge
eld A also has moduli in this situation. A
smooth instanton con guration for A may exist, where A is a nontrivial connection with
integral ux on the twocycle.
The analysis above applies only to smooth con gurations for A; E and their gauge
transformation parameters. Experience shows that we should allow singular con gurations
for A, as we did for the nonabelian gauge elds. The coupling of E to the eld strength
FA means that we should consider allowing singular con gurations for E as well, at least
at the poles. In fact, the con guration (2.32) which we are trying to reproduce has a
singularity on the entire maximal twosphere. If we allow E to have the singularities at
the same position as A, then the coupling
(2.50)
(2.51)
(2.52)
(2.53)
(2.54)
(2.55)
(2.56)
Nonperturbative contributions. In the absence of twocycles, the equation
I
I
E =
H = 0 ;
exp in
Z
F Ainst
^ [ ] ;
induces a nonvanishing classical contribution upon localization
in Z
2
in Z
2
FA ^ E
FA ^ E
!
in Z
2
V G ;
in Z
2
F Ainst
^ Eclassical :
the form
trivial
The form E is again closed and the tensor gauge transformations again imply that E is
gauge equivalent to 0 whenever it represents an integral class. The remaining moduli of
E, parameterizing singular closed twoforms localized at the poles modulo integral forms,
are angular parameters. Upon localization, there is a contribution to the matrix model of
n
Z
FA ^ E
!
n
Z
F Ainst
^ [ ] ;
A
where F inst is the eld strength of some singular instanton con guration at the poles and
[ ] is a representative of any singular twoform at the poles, which is de ned only up to
integral classes. The part of the electric defect containing an integral over E is, however,
because E is being paired with a trivial cycle represented by @ . Integration of
A
over all
then restricts F inst to vanish. This means that we only need to work in the
instanton number 0 sector for A.
Integrating out. In addition to the above, there is a remaining integration over G0.
This integral, and the one over , can be done explicitly. The result is simply to set
or in the units of the rest of the sections
This was what we set out to show.
The parameter entering the oneloop and instanton contributions in the matrix model,
described in section 3, is im, where m is identi ed with the
of a background vector
multiplet. Hence, our surface operator is equivalent to an imaginary mass
HJEP1(207)
equivalently, a shift of the parameters by
2.4
The 5d N = 1 codimensiontwo defect
We describe the implementation of codimensiontwo defect operators in vedimensional
N = 1 theories. Most of the analysis is similar to the fourdimensional case, so we will be
brief. Conventions and notations are the same as in [14].
2.4.1
Killing spinors on the vesphere
Supersymmetric eld theories on curved spaces are systematically obtained in the rigid
limit of the N = 1 supergravity in ve dimensions which has an SU(2)R symmetry whose
indices are denoted by A; B as in the 4d case. There are SU(2)R gauge eld V AB, SU(2)R
triplet scalar eld tAB and the other elds which are irrelevant to the following discussion
in the Weyl multiplet [33]. The variations of the fermions in the multiplet have to
vanish for preserving supersymmetries on given background
elds and the solutions are the
Killing spinors. We set the radius of the vesphere to one from the beginning to simplify
the discussion.
The round unit vesphere allows the Killing spinor in the coordinates
ds2S5 = d 2 + sin2 d 2 + cos2 ds2S3 ;
when the background elds are set to [14, 34]
1
2
tAB =
( 3)AB ;
V AB =
( 3)AB d ;
others = 0 :
With the hermitian gamma matrices in tensor product forms
1 =
1
(2.57)
(2.58)
(2.59)
(2.60)
(2.61)
(2.62)
(2.63)
;
=
i
2 i
i
eld describing the defect is
whose eld strength becomes
We want a codimensiontwo surface defect at
= 0 on the round sphere. Introducing
a smoothing function g ( ) as in the fourdimensional case (2.28), the background gauge
a spinor A in ve dimensions is also written as tensor products of spinors A and
two and three dimensions
A =
A
A :
The Killing spinor on the round sphere (2.61) is given by
(2.64)
(2.65)
(2.66)
(2.67)
(2.68)
(2.69)
A
defect and
(2.70)
(2.71)
where 1;2 are constant spinors and
are the Killing spinors on a unit threesphere
Codimensiontwo defects
N = 1 abelian vector multiplet
A surface defect in a avor symmetry is speci ed as a singular con guration in a background
a real scalar, Y AB an SU(2)R triplet scalar and
A an
SU(2)R
This con guration is supersymmetric if the real scalar defect, the gaugino
the triplet scalar YdAefBect take the following forms
(Ydefect)AB =
g0 ( )( 3)AB ;
defect =
A
defect = 0 :
i
2
1
4
This con guration is invariant under the supersymmetric transformation, especially
one sees
0 =
A
defect =
A(Fdefect)
+ (Ydefect)AB
B :
One can replace g0 ( ) with the delta function ( ) in the singular limit
! 0.
fLAB; E ; N; 'Ag ;
consisting of an SU(2)R triplet scalar LAB, an antisymmetric tensor gauge eld E , a real
scalar N and an SU(2)RMajorana fermion 'A, through a supersymmetric BF coupling [33]
We will implement defects in a avor symmetry by coupling the abelian vector multiplet
to the linear multiplet
It is gauge invariant for an integer n as E
transforms as gaugeE
supersymmetric transformation laws of the linear multiplet with only the backgrounds tAB
and V AB turned on are given by
10i A'BtAB ;
4tBC LCB A
6t(AC LB)C B +
ik Z pg
2
ik
Z
S3=0
i
2
(Y defect)BA LAB +
(L11
L22) !3 +
4
1 F defectE
1
2
E
#
;
In addition, we introduce a surface operator for the linear multiplet with the BF
coupling to the defect abelian vector multiplet
where !3 is the volume form of the unit threesphere and
is the Hodge operator in ve
dimensions. Choosing
= 2 the defect operator is gauge invariant for an integer k.
The linear multiplet plays a role of the Lagrange multiplier and integrating it out in
the path integral with the BF term and the Wilson surface results in setting the vector
Instead of doing so, we localize the tensor multiplet on the round sphere with the
LAB = 2i (A'B) ;
E
= 2i
N =
2
' ;
(D ')
'A = i(D LAB)
Wsurface(k) = exp
1
2
where we de ned L^AB
multiplet to the defect con guration.
localizing term
whose bosonic part is written as
Vt(elnosco)r =
i(D LAB) B
4tBC LCB A
6t(AC LB)C B
) A
+
N A
(D LAB)(D LBA) +
(N
8L^AA)2 + 18 L^(AB)L^(AB) ;
= exp
B
"
1
2
i
2
i
2
1
4
tAC LCB.
1
4
SB(5Fd) =
In localizing the tensor multiplet, we impose the reality condition for LAB
that is equivalent to
(LAB)y = LBA ;
(L^AA)y = L^AA ;
(L^(AB))y = L(AB) :
This choice makes the bosonic part of the localizing term be semipositive de nite and the
tensor multiplet localizes to
D LAB = 0 ;
which on the round sphere yields
Localizing the vector multiplet to the xed locus
N = 4(L11
the BF term (2.73) and the surface operator (2.75) end up with
exp
SB(5Fd)
Wsurface(k)
!
!
=
exp
exp
i :
k
n
i 2
2
2
2
n N0 ;
kN0 :
(2.78)
(2.79)
(2.80)
(2.81)
(2.82)
(2.83)
(2.84)
(3.1)
(3.2)
the integration over N0 sets
we will con rm in section 3.
3
Squashing from defects
This value is consistent with the mass shift by the parameter (2.60) as in the 4d case as
We are now in a position to demonstrate the relationship between the supersymmetric
Renyi entropy and supersymmetric codimensiontwo defects. In the original setup, the
supersymmetric Renyi entropy is de ned as
Snsusy
1
1
n
log
Znsusy
(Z1) n ;
where Zn is the appropriate supersymmetry preserving partition function on the branched
dsphere, or the squashed dsphere. The schematic form of the localization calculation for
Zn is
Zn =
X Z
moduli
hZnclassical (moduli) Znpert (moduli)i ;
(3.3)
(3.4)
where the sum/integral is over the moduli space of supersymmetric zero modes, Znclassical
is the exponential of minus the Euclidean action evaluated on the moduli space, and Znpert
is the oneloop determinant obtained by evaluating the path integral over nonzero modes
in the quadratic approximation around the moduli space.
The partition function for the ncopy theory on the round sphere is simply
We would like to demonstrate that Zn can be computed using Z1 and the insertion of
Where the right hand side is the partition function of the ncopy theory on the round sphere
in the presence of a speci c codimensiontwo defect described in the next subsection. After
the linear eld rede nition, the result is
Zndefect
X Z
n moduli
"
The outer sum/integral is over n copies of the moduli. The symbol moduli is a placeholder
for the sewing operation which identi es how the squashed sphere moduli space
fractionalizes into n copies. Zclassical is the classical contribution of a single copy of the theory
1
evaluated on the round sphere. Zpertdefect (k; moduli) is the perturbative contribution, in
1
the quadratic approximation around the moduli space, in the presence of the appropriate
defect for the kth copy.
3.1
Coupling defects to the ncopy theory
The super eld V , which carries the information about the supersymmetric defect, must
be coupled to the physical elds of the ncopy theory. For matter multiplets, this is the
usual minimal coupling of V to chiral multiplets or hypermultiplets. We need not consider
the e ect of the defect on the other terms in the action involving matter elds, since these
vanish at the level of the quadratic approximation around the localization locus for any of
the setups we consider. The coupling of a at connection carried by V to physical gauge
elds can be accomplished by formally performing the eld rede nition in the introduction.
Since nonabelian gauge elds are not free elds at nite gauge coupling, the resulting action
would inevitably look like a gauge noninvariant mess. However, gauge invariance, with
the caveats already mentioned, is guaranteed by the ability to undo the eld rede nition.
If the gauge group is SU(N ), there is a physical procedure which implements the right
defect and makes clear the form of the coupling to a background vector, at the level of the
quadratic approximation to the moduli space. We use the language of 4d N = 2, but the
same applies to any of the theories under consideration.
First, consider the enlarged gauge group U(nN ), where the gauge group of the ncopy
theory, SU(N )n, is embedded as a block diagonal subgroup
The required codimensiontwo defect, before the eld rede nition, can be viewed as a at
connection, on Sdn@ , which is represented by a oneform with holonomy
data [35, 36].
vector multiplet
once in a chosen direction. To preserve supersymmetry,
we consider a GukovWitten type surface defect with the holonomy (3.7) specifying the
To go back to the ncopy theory, while keeping the defect, one should
rst Higgs
U(nN ) down to SU(N )n by giving an appropriate large vev to the adjoint scalar in the
0 SU(N )0
B
B
B
B
.
.
.
0
SU(N )1
0
.
.
.
0
.
.
.
0
0
.
.
.
SU(N )n 1
1
C
CC :
C
A
BBB ... ...
B 0 0
B
1 0
0
1
0
0
.
. . ... CCC ;
XHiggs = BBBB 0... t...1 .
0 0
.
. .
0
0
.
.
tn 1
1
C
CC :
C
A
(3.6)
(3.7)
(3.8)
Upon taking ti ! 1, all modes not coming from the original ncopy theory are in nitely
massive and do not contribute to the computation. XHiggs should really be considered
only up to permutations of the ti, which are a part of the Weyl group of the theory. In
fact, the holonomy (3.7) acting on XHiggs produces such a permutation. Although we do
not show this explicitly, we take this to mean that the GukovWitten operator with this
data preserves the same supersymmetry as XHiggs. After a change of variables, which is
in this case a constant SU(nN ) gauge transformation, the elds in the vector multiplet for
SU(N )k acquire a monodromy exp (2 ik=n) around @ .13
The physical e ect of the defect on vector multiplets can now be examined more
carefully at the level of the quadratic approximation to the moduli space of the remaining
light elds. Since the action in this approximation is quadratic, the monodromy can be
traded for a coupling to a background vector multiplet with a speci c pro le. This pro le
is singular, and determined by the value of the monodromy and by supersymmetry. It
is this multiplet, denoted by V , which arises in our realization of the supersymmetric Zn
gauge theory.
13Note that the diagonal elements of SU(N )k also acquire this monodromy. Had we tried to implement
the defect using a GukovWitten type surface defect in each SU(N )k, this would not have been so.
3.3
We collect the expressions for the matrix models associated to the squashed sphere partition
functions in three, four and
ve dimensions. Our conventions for integration over the Lie
algebra are in appendix A. Special functions in the oneloop determinants and instanton
contributions are de ned in appendices B and C, respectively.
The matrix model for a 3d N = 2 theory on the squashed sphere is [41, 42]
Zsusy 3d =
1 Z rankG
Y
jW j
i=1
d i
p!1!2
e !1i!2 Tr( 2)
Y
2 +
# of chirals
Y
I=1
S2 (i ( )j!) S2 ( i ( )j!)
Y
2RI
S2 i ( )+imI + j!j
I !
2
+ are the positive roots of g.
to the Ith chiral multiplet.
jW j is the size of the Weyl group.
I is the Rcharge of the Ith chiral multiplet.
is the ChernSimons level.
! = (!1; !2) are squashing parameters, and
i runs over the Cartan of the Lie algebra g of the group G, which we have assumed
is U(N ). Our conventions are such that i are real (see appendix A).
denotes a weight in the representation RI associated
j!j
X !i = j!1 + !2j :
The supersymmetric Renyi entropy is computed using ! = 1; n1 .
We will set the mass parameters mI to zero.
The matrix model for a 4d N = 2 theory on the squashed sphere is [29]
Zsusy 4d =
jW j
Y
2 +
1 Z rank G
Y
i=1
d i
p!1!2
e !1!82g2Y2 M
Tr( 2)
(i ( )j!) ( i ( )j!)
Y
2R
Zi(n4sdt) q(4d); i ; mf0 ; !1; !2
i ( ) + im + j!j !
2
1
2
;
gYM is the YangMills coupling. gYM and YM appear also in the instanton part of
the matrix model. q(4d) is de ned in appendix C.
R is now the total representation of the hypermultiplets.
We have set all hypermultiplet masses to a common value: m.
(3.19)
1
;
(3.20)
(3.21)
The supersymmetric Renyi entropy is computed using ! =
parameter m for all hypermultiplets set to a common value
1; n1 , with the mass
The result for a 5d N = 1 theory on the squashed sphere is still conjectural [34, 43{49].
(3.22)
(3.23)
(3.24)
(3.25)
(3.26)
2
Zinst q; i ; mf0 ; !1; !2; !3
Zinst q; i ; mf0 ; !3; !1; !2
Zinst q; i ; mf0 ; !2; !3; !1
;
2
is the ChernSimons coupling.
and gYM appear also in the instanton part. q is
de ned in appendix C.
into [51]
The derivative of the triple sine function in the integral measure can be written
S30(0j!) =
3(!1; !2; !3)2 1(!1) 1(!2) 1(!3) ;
2(!1; !2) 2(!2; !3) 2(!3; !1)
where r(!) is the Stirling modular form
Note that 1(!) = p2 =!.
metric Renyi entropy is computed using ! = 1; 1; n1 , and setting
! = (!1; !2; !3) are squashing parameters, and j!j = j!1 + !2 + !3j. The
supersym1 Z rank G
Y
i=1
jW j
Y
2 +
We follow the form in [50]
Zsusy 5d =
z!0 z r(zj!)
j!j
2
:
:
3.4
The scalar moduli space and classical contributions
All of the theories we consider have a moduli space which is partially given by the vev of a
Lie algebra valued scalar. The scalar
has eigenvalues ~ . Integration over
is what makes
the result of the localization procedure into a matrix model. In our setup for the ncopy
theory, there is one such
k and one integration for each copy. There are also classical
contributions to the matrix model which depend on .
The mode which
parametrizes is a part of the original theory which cannot be treated
as free, even after localization. Therefore, the values of k are subject to the boundary
conditions implied by the original codimensionone de nition of the replica defect. It is
Starting from the ncopy theory on the round sphere, after using the delta functions to set
all the k equal, we recover the factor of n.
Perturbative contributions
The defect operator interpretation of the supersymmetric Renyi entropy was originally
observed in [6] by rewriting the perturbative partition function on the nfold cover as
ncopies of the partition functions on a round threesphere with vortex loops inserted on each
copy. (See also [16, 17] for related works in two dimensions). We extend this interpretation
to higher dimensions and show the perturbative parts of the partition functions in the 4d
and 5d supersymmetric Renyi entropies also have similar structures. The nonperturbative
contributions arise in higher dimensions will be discussed separately in section 3.6.
First, let us review the story in three dimensions [6]. There are no nonperturbative
contributions in the matrix model (3.19) and we only need to deal with the oneloop
partition functions appearing as the double sine functions S2 from the vector and matter
multiplets. The identity (B.7) for the double sine function yields that the oneloop partition
function of a multiplet with Rcharge
I can be decomposed as a product of those in the
presence of a supersymmetric abelian vortex loop [25]
S2 i ( ) + j!j
2
I 1;
1
n
Y S2 i ( ) +
I
2
1
n
k
n
1 +
+
1; 1
1
=
=
n 1
k=0
n 1
k=0
Y S2 i ( ) + qk3d vortex +
I 1; 1
1
;
1
;
ec(Qi !i 1)Tr( p)
;
Y ! 1 = n;
Y ! 1 = 1:
i
i
(3.27)
(3.28)
(3.29)
(3.30)
(3.31)
(3.32)
trivial to see that this implies that all k are equal.16 Equivalently, the sewing operation
for this set of moduli consists of a set of delta functions, in the matrix model for the ncopy
theory in the presence of the defect, which enforce this equality
mscoadlaurli =
n 2 rank G
Y
Y
k=0 i=1
(( k)i
( k+1)i) :
The classical contributions depending only on
are products of expressions of the form
for some constant c. In the theory considered on the squashed sphere, we have
HJEP1(207)
while for the round sphere
where qk3d vortex is introduced to be
q3d vortex =
k
I
2
1
n
1
+
k
n
:
16This only makes sense if one identi es the gauge transformations at the interface. This also implies the
identi cation of the residual gauge transformations acting in the di erent matrix models.
Comparing it with the shift of the modulus
(2.12), we interpret the decomposition as
a manifestation of the introduction of a supersymmetric abelian vortex loop, described in
section 2.2, of charge qkvortex supported on the entangling surface ( = 0) on each copy of a
round sphere. Note that qkvortex di ers from (2.12) by a term proportional to the Rcharge
I for matter multiplets. This means that the vortex loops for the supersymmetric Renyi
entropy are dressed by the Rsymmetry ux.
One can work out a similar decomposition for the perturbative part in the 4d N = 2
matrix model (3.21) with a slight modi cation. The matter oneloop partition function is
represented by the
function, which enjoys decomposition into the product of the ncopies
with the help of the identity (B.12):
i (a) + im + j!j 1;
2
1
1
Y
k=0
i (a) + 1 + qkvortex 1; 1
1
;
where qkvortex is to be interpreted as the charge of a codimensiontwo surface defect
k
n
;
(3.33)
(3.34)
under the choice of the shifted mass (3.14) and the relation (A.5). It agrees with the shift of
the modulus (2.60) induced by the insertion of a supersymmetric abelian surface operator
of charge k in section 2.3.
Repeating the analogous procedure to the 5d N
oneloop partition function with the identity (B.7):
= 1 theory one
nds the matter
S3 i ( ) + im + j!j 1; 1;
2
1
n
1
=
Y S3 i ( ) + 1 + qkvortex 1; 1; 1
1
;
(3.35)
where we introduce the charges qkvortex by (3.34) and the mass shift (3.16). Once again it can
be interpreted as a supersymmetric codimensiontwo surface defect of charge k described
in section 2.4.
It is straightforward to apply the same argument to vector multiplets in any dimensions
to read o the surface charges and the results (3.32) and (3.34) still hold with
I = 0.
3.6
Nonperturbative contributions  instantons and contactinstantons
The partition functions on the foursphere and the vesphere receive nonperturbative
contributions from instantons and contactinstantons, respectively. These are supersymmetric
con gurations localized at xed loci of the equivariant action generated by the square of
the supersymmetry. For the foursphere, the xed points are at the north and south pole
and give rise to instanton and antiinstanton contributions, respectively. The
partition function includes contributions from contact instantons [43], which are extended
along the ber of
(3.36)
S
1
! S
5
! CP 2 ;
Yli
k = 1
Ylp
k = 0
k = 2
HJEP1(207)
and localized at three points on the base.17 The supersymmetric instanton contributions
are computed by the Nekrasov partition function [52, 53]. We nd it convenient to express
both types of contributions in terms of the 5d, or qdeformed, version of the Nekrasov
partition function, which we review in appendix C. The 4d undeformed partition function
can be recovered by taking an appropriate limit.
Instanton and contactinstanton contributions introduce new moduli, classical
contributions, and perturbative contributions into the calculation. We begin by discussing how
the moduli of the replicas in the ncopy theory are sewn up to produce those of the original
theory. We assert a speci c pattern for the fractionalization of a particular instanton
conguration, given by a vector of Young diagrams. We then show that, given this pattern,
the classical and perturbative contributions recombine to yield (3.4).
Similar decompositions of partitions appear in the context of instantons on ALE spaces
(cf. [54]). A relationship between the instanton partition function on a Zn orbifold of
C2, acting on just one C factor, and surface operators appears in [55, 56] (see also the
review [57]). It is possible that the results we need for the covering space associated
to the ncopy theory can be recovered from the latter papers. Speci cally, this seems
plausible given the connection between the supersymmetric Renyi entropy at n and at 1=n.
However, we need 5d results and a very speci c surface operator, so we derive the necessary
fractionalization and relationships between the determinants in this context.
3.6.1
Sewing of instantons
After employing equivariant localization, the instanton moduli space localizes to a set of
points, each given by a vector of partitions [52, 53]. The partitions describing the instanton
moduli space of the theory on the branched sphere fractionalize to yield partitions for each
17To the best of our understanding, the precise form of these contributions is still conjectural. We will
use the form considered in e.g. [50] and nd that it works well.
of the replicas in the ncopy theory. This happens by splitting one set of Young diagrams,
yielding a partition vector Y~, into n Young diagrams, yielding partitions n Y~ (k)on 1
,
having the same combined number of boxes. This process is illustrated in gure 1. The
diagrams can be split along the vertical or along the horizontal, depending on which
deformation parameter i is taken to be n times smaller. Since the instanton partition function
is invariant under the simultaneous transposition of the diagrams and 1 $ 2, it su ces
k=0
to consider the situation in gure 1.
For the contribution of a hypermultiplet in the fundamental representation, each
element of the uctuation determinant around an instanton can be associated to a box in the
Young diagram, and
gure 1 describes the decomposition into replicas. The contribution
from a vector multiplet, or from an adjoint hypermultiplet, involves pairs of partitions in
the vector Y~. We nd that the decomposition into replicas follows the pattern
HJEP1(207)
n 1
Y
k1;2=0
(Y1; Y2)
(Y1 (k1) ; Y2 (k2)) jk1 k2=k mod n :
(3.37)
ZY5~d;CS ~a; ; 1; 2
1
2
= exp 42 i
2
n 1
k=0
= Y exp 42 i
X
X
a
l
X
X
l (p;t)2Yl(k)
s 1
n
a
l
k
n
The moduli space of contactinstantons can also fractionalize along the additional S1
direction. This process is simpler, amounting to decomposing the KaluzaKlein (KK)
momentum, and is described in section 3.6.5. It does not involve the partitions.
Classical and ChernSimons contributions depending on instantons
The classical weight in the instanton partition function of a con guration with instanton
number j Y~j in a 5d SCFT is
The number of boxes in the Young diagrams on the right pane of gure 1 sum to the
number of boxes on the left, and therefore the combined weight from each of the replicas
matches that of the original theory
qj Y~j :
n 1
k=0
~
qjYj =
Y qj Y~(k)j :
In the presence of a 5d ChernSimons term, we also need to split the contribution
2
ZY5~d;CS (~a; 1; 2; ) = exp 4i
X
X
(al
(s
1) 1
We do this rst for the case where 1 = 1=n by setting s = np
(3.38)
(3.39)
(3.40)
(t
k
3
(t 1) 5 ;
3
where in the second line we have reparametrized
(3.42)
(3.43)
(3.44)
(3.45)
(3.46)
(3.47)
The expression on the second line can be thought of as the contributions from the nreplicas,
where the additional shift
is the e ect of the monodromy brought on by the defect. Since we associate this shift with
k ! n
1
k :
l ! a
qsurface =
k
n
;
n
;
we see that the contribution from the Higgs type mass enters the ChernSimons term in
the same way as it enters the uctuation determinant for the fundamental hypermultiplet
considered below.
The contribution from the kth replica is sensitive only to the partition represented by
the Young diagrams Y~ (k). It might seem strange to see the mass shift appear at all in a
classical contribution. Note, however, that the combination appearing in the exponential
comes from evaluating the classical ChernSimons term at the positions of the poles for
the integral over the scalar associated to the auxiliary U jYj symmetry [58{60]. This
~
position is shifted by the Higgs type mass term.
Fundamental hypermultiplets
uctuations
We now demonstrate the relationship between the contribution of a hypermultiplet on the
squashed sphere, with deformation parameters ( 1; 2; ) =
n1 ; 1; 2 , and n
hypermultiplets in the presence of defects with deformation parameters ( 1; 2; ) = (1; 1; 2 ). At the
Qm10 = t :
f
Expressing the squashed sphere contribution using the round sphere values for t; q etc.,
conformal point
we get
=
=
4
4
n 1 2 N
Y
Y Y
1
n 1 2 N
Y
Y Y
1
k=0 l=1 p=1
l=1 j=1 Ql qYlj+1 t n ; q
k=0 l=1 j=1 Ql qYlj+1 t n ; q
Ql q t n ; q
j 1
j 1 1
;
Ql q t n ; q
j 1
1
j 1 1
1
3
5 j=np k ;
Qm1f (k) Ql q t (p 1); q
Qm1f (k) Ql qYl;np k+1 t (p 1); q
1
1
3
5 ;
Qm1f (k)
t nk 1+ n1 :
We now identify the terms in the square parentheses with the contribution of
uctuations of a hypermultiplet in the presence of a codimensiontwo defect
~
Y
Zhyperdefect (~a; 1; 1; 2 ; k)
2 N
4
Y Y
1
l=1 p=1
Qm1f (k) Ql q t (p+1); q
Qm1f (k) Ql qYl;np+k+1 t (p+1); q
1
1
3
5 ;
horizontal position given by
~
Y
The expression for Zhyperdefect (k) di ers from the expression Zhyper in two ways
~
Y
1. The fugacity, or mass parameter, involving the background vector is shifted in the
We ascribe this to the e ect of the codimensiontwo defect on the uctuations.
2. The kth such contribution is sensitive only to the boxes of the Young diagram with
We ascribe this to the fractionalization of the instanton moduli corresponding to
the partition.
These are the same e ects visible for the ChernSimons contribution. We conclude that
n 1
k=0
~
Y
=
Y Zhyperdefect (~a; 1; 1; 2 ; k) :
3.6.4
Adjoint hypermultiplet or vector multiplet
uctuations
An adjoint hypermultiplet contributes to the uctuation determinant around an instanton
con guration as
(3.48)
(3.49)
(3.50)
(3.51)
(3.52)
(3.53)
(3.54)
Reparametrizing the product over k as k ! n
k, we get
implying
n
qvortex =
k
:
i = k mod n :
Y
Zadjoint hyper ~a; mf0 ; 1; 2;
~
Y
(l;i)6=(m;j)
Qm0 Ql Qm1 qYli Ymj tj i+1; q
f
Qm10 Ql Qm1 qYli Ymj tj i; q
f
1
1
This involves pairs of partitions Yl and Ym. The virtue of the form of the
uctuation
determinant written above is that t appears raised only to a power corresponding the
Y
= Y
k=0
= Y
k=0
4
4
n 1 2i j=n(p q)+k
column indices i; j. As such, the determinant can be decomposed in a way similar to the
fundamental hypermultiplet
Zadjoint hyper ~a; n1 ; 1; 1; 2
~
Y
Ql Qm1 qYli Ymj t n1 (j i); q
(l;i);(m;j) Ql Qm1 qYli Ymj t n1 (j i+1); q
n 1 2j i=n(p q)+k
1
1
Ql Qm1 t n1 (j i+1); q
Ql Qm1 t n1 (j i); q
1 ;
1
(3.55)
(3.56)
1 ;
1
(3.57)
(3.58)
(3.59)
1 :
1
such that
Zadjoint hyper ~a; mf0 ; n1 ; 1; 2
~
Y
n 1
k=0
~
Y
=
Y Zadjointdefect (~a; 1; 1; 2 ; k) :
We now consider the contribution of a vector multiplet. Its uctuation determinant
is inverse to that of an adjoint hypermultiplet with zero mass, as was the case for the
perturbative contribution. As we did there, we keep a \mass fugacity" to keep track of the
deformation brought on by the defect. Since this mass is now associated to the Higgs vev,
it is not shifted from mf = 0 by the OkudaPestun prescription. Repeating the calculation
for the adjoint hypermultiplet using the same manipulations, we get
Zv~ector ~a; mf0 ; n1 ; 1; 2
Y
=
Y Zv~ectordefect (~a; 1; 1; 2 ; k) ;
Y
n 1
k=0
Ql Qm1 qYli Ymj tp q+ nk ; q
Ql Qm1 qYmj Yli tp q+1 nk ; q
Ql Qm1 tp q+1 nk ; q
Ql Qm1 tp q+ nk ; q
1
1
3
1 5 ;
1
Qmf (k) Ql Qm1 qYli Ymj tp q; q
Qm1f (k) Ql Qm1 qYmj Yli tp q+1; q
1
1
Qm1f (k) Ql Qm1 tp q+1; q
Qmf (k) Ql Qm1 tp q; q
3
1 5 :
1
In the second line, we have replaced in two of the factors k ! n
1
k. If we de ne
~
Y
Zadjoint defect (~a; 1; 1; 2 ; k)
i j=n(p q)+k
Y
Qm1f (k) Ql Qm1qYmj Yli tp q+1; q
1
1
Qm1f (k) Ql Qm1 tp q+1; q
Qmf (k) Ql Qm1 tp q; q
with
Zv~ectordefect (~a; 1; 1; 2 ; k)
Y
i j=n(p q)+k
Y
(l;i)6=(m;j)
Qmf (k) Ql Qm1 qYli Ymj tp q; q
Qm1f (k) Ql Qm1 qYmj Yli tp q+1; q
1
1
Qm1f (k) Ql Qm1 tp q+1; q
Qmf (k) Ql Qm1 tp q; q
Since the original product is over pairs of Young diagrams corresponding to l and m,
the decomposition of an adjoint hypermultiplet or a vector multiplet into n parts is not as
simple as in gure 1. Instead, each pair of diagrams of the theory on the squashed sphere
splits into n pairs for each of the n copies.
1 ei (`Ym (s) 1 (aYl (s)+1) 2+al am)i Y h
1 ei ( (`Yl (t)+1) 1+aYm (t) 2+al am)i ;
the defect decomposition, in this case, follows simply from the identity
n 1
Y h
k=0
1
e2 i( + nk )i = 1
e2 in :
A similar expression exists for the hypermultiplet contribution. Its decomposition follows
from the same method.18
To relate this decomposition to the KK decomposition, we use the regularized in nite
The 5d squashed sphere has one more contribution, not of the type above. The third
point contributes a sum over contactinstantons with deformation parameters ( 1; 2; ) =
(1; 1; 2 n), i.e.
is n times as large as it would be on the round sphere. In order to
decompose this contribution, it is useful to write the determinant part of the qdeformed
instanton partition function as a product over Kaluza Klein modes coming from the extra
circle. Starting from the expressions in [61]
(3.60)
HJEP1(207)
where
NlY;m = Y h
~
s2Yl
product
the form
form
Nc
l;m
t2Ym
Z3d or 4d
!1
!2
:
(3.61)
(3.62)
(3.63)
(3.64)
(3.65)
(3.66)
1
Y
m= 1
(m + a) = 1
e2 ia ;
Im (a) > 0 :
vev of the scalar modulus.
3.7
The defect decomposition can now be thought of as writing the quantum number m in
m = np + k :
The partitions of the various copies are simply identi ed in this case, in analogy with the
The supersymmetric Renyi entropy in three and four dimensions satis es an interesting
property stemming from the fact that for a superconformal theory
Zn3d or 4d = Z13=dnor 4d :
This follows simply from two facts:
1. Conformal invariance implies a dependence on the squashing parameters !1=2 of the
18The mass shift given by mf0 is immaterial in this case.
(3.67)
(3.68)
(3.69)
!1 $ !2 :
1 = 1=n ;
2 = 1 ;
1 = n ;
2 = 1 ;
Taking !1 = 1=n and !2 = 1 yields (3.65). The same trick does not work in ve dimensions.
This relationship can be thought of as an interacting supersymmetric version of the
BoseFermi duality in three dimensions [62], which does not hold for Renyi entropies in higher
dimensions without introducing supersymmetry.
One can calculate the nth supersymmetric Renyi entropy from the partition function
on the branched sphere. From the point of view of the instanton partition function, taking
corresponds to counting instantons on a space which is branched over a codimensiontwo
surface. Taking
2. There is a trivial change of coordinates which exchanges
on the other hand, corresponds to counting instantons on an orbifold. It is interesting that
the two counts are related.
4
We have shown that the supersymmetric Renyi entropy (SRE) can be computed using
supersymmetric codimensiontwo defects. After giving a microscopic de nition of the
defect operators, we computed the expectation values of these defects using localization.19
We showed that the e ect of such defects on the matrix models calculating the partition
function on the round sphere amounted to imaginary mass terms. We made a conjecture
regarding the details of the sewing operation needed to complete the picture for the
moduli, scalar vevs, instantons and contactinstantons, encountered in localization. We then
showed the equality with the squashed sphere partition function.
Although we explicitly only showed agreement of the partition functions representing
the SRE, the decomposition into defects seems to work at the level of the matrix model
ingredients, and for any deformation parameters !i. It is reasonable to conjecture that it
works at the level of the 5d holomorphic blocks and gluing [50, 63, 64]. If this is the case,
a relationship similar to the one described here should hold for the partition functions on
fourmanifolds and
vemanifolds of the type described in e.g. [65, 66].
In the context of holographic duality, the RyuTakanayagi prescription [67, 68] allows
us to compute the entanglement entropy in a CFT, in a particular limit corresponding to
classical gravity in the bulk, using a minimal area surface in AdS which is homologous to
a given entangling region
. Corrections to this computation have recently been
conjectured in [69, 70]. A variant for the Renyi entropy was put forth in [71]. Somewhat similar
19The supersymmetric codimensiontwo defects coincide, in three and in four dimensions, with speci c
versions of the operators de ned in [24] and [35] respectively. Fivedimensional versions were considered in
e.g. [40].
prescriptions are used to compute the expectation values of supersymmetric nonlocal
operators (see e.g. [72{78]). The authors, and others, have long suspected that there is a
relationship between these computations. We do not, however, know of a concrete example
of such a relationship. We hope that the de nition of the supersymmetric defect operator
version of the SRE calculation can be used to
nd one. This may involve going, rst, to
a dual picture in the SCFT. For instance, the codimensiontwo defects realizing the SRE
in a 3d N = 2 theory are vortex loops, which, in certain situations, are dual to a Wilson
loop under 3d mirror symmetry [24].
Acknowledgments
We would like to thank S. Hellerman, C. Herzog, K. Hosomichi, D. Ja eris, R. Myers,
T. Okuda, A. Sheshmani, Y. Tachikawa and B. Willett for valuable discussions. The work
of T.N. was supported in part by JSPS GrantinAid for Young Scientists (B) No. 15K17628
and JSPS GrantinAid for Scienti c Research (A) No.16H02182. The work of I.Y. was
supported by World Premier International Research Center Initiative (WPI), MEXT, Japan.
A
Conventions
We summarize our conventions for gauge theories and the matrix models resulting from
the localization procedure in three, four, and
ve dimensions. To begin with, we set an
overall scale associated with the size of S3;4;5
HJEP1(207)
Dimensionful parameters such as 1;2; !1;2;3; a; m; etc. are expressed using this scale.
We use physics conventions for the gauge and avor symmetry groups. The generators
of the Lie algebra u (N ) are taken to be Hermitian matrices, and factors of i appear in
appropriate places in the eld strength. Consequently, integration over the Cartan
subalgebra means an N dimensional real integral over variables denoted ~ , which are the
eigenvalues of a matrix . In 5d, ~ is related to the scalar vev as
In 4d we have
In 3d, where the real adjoint scalar in the vector multiplet is also denoted , we have
This convention extends to mass parameters, which are vevs for scalars in background
vector multiplets. The physical mass of a chiral multiplet or hypermultiplet is a real
number m, which bears the same relation to the background vev as
does to the dynamical
` = 1 :
h i =
=
i :
2
h i =
:
(A.1)
(A.2)
(A.3)
(A.4)
We summarize the de nitions and identities for the special functions appear in the text.
Multiple gamma function. For ! = (!1;
; !r)
zeta function is de ned by
0 and z 2 C, the multiple Hurwitz
where n = (n1;
; nr)
0. The integral representation is
r(s; z; !) := X(n ! + z) s ;
n 0
r(s; z; !) =
e zt
e !it) ts 1dt :
r s; z; !1;
; !r 1; !Nr
r s; z +
N
k!r ; ! :
For an integer N , one can prove the identity
The Barnes multiple gamma function r(zj!) is de ned by
vev. The deformation parameters a and mf , which are used when discussing the instanton
contributions to the partition function in 4d and 5d, are set to
a = i ;
mf = i m :
Our conventions for spinors and supersymmetry transformations are di erent in
different dimensions. However, supersymmetry transformation parameters are always taken
to be commuting spinors.
B
Special functions
(A.5)
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
HJEP1(207)
Qir=1(1
N 1
X
k=0
Multiple sine function. One can de ne the rple sine function Sr(zj!) by
with j!j = Pir=1 !i. It satis es the following identities [79]:
Sr(zj!) := r(zj!) 1 r(j!j
zj!)( 1)r ;
Sr(j!j
zj!) = Sr(zj!)( 1)r 1
Sr(N zj!) =
N =
Y
0 ki N 1
Y
0 ki N 1;k6=0
Sr z +
Sr
k !
N
k !
N
! ;
! ;
Sr(czjc!) = Sr(zj!) ; for c > 0 :
; !i 1; !i+1;
The formula (B.3) yields an additional identity
Sr z !1;
k=0
Y Sr z +
k!r !
N
:
This is the generalization of the identity for the hyperbolic gamma function found in [6].
The
function.
The double gamma function is used to de ne the
function [29, 80]
(zj!1; !2) :=
2
2 j!j !1; !2 ( 2(zj!1; !2) 2(j!j
2
zj!1; !2)) 1 :
satis es several identities
where (z) := (z)= (1
z) and the scaling law
Some literatures including [29, 80] use
(z + !1j!1; !2) = ! !2z2 1
2
(z + !2j!1; !2) = ! !2z1 1
1
(z=!2) (zj!1; !2) ;
(z=!1) (zj!1; !2) ;
(j!j 2z)2
(czjc!1; c!2) = c 4!1!2
(zj!1; !2) :
b(z) :=
(zjb; 1=b) :
which is sometimes denoted
(z) without the subscript.
The formula (B.3) yields
z !1; !N2
N 1
Y
k=0
z +
N
k!2 !1; !2 :
(B.7)
(B.8)
(B.9)
(B.10)
(B.11)
(B.12)
(C.1)
C
The instanton partition function
Nekrasov's instanton partition function, [52, 53], is the equivariant volume of the instanton
moduli space with respect to the action of
U(1)a
U(1) 1
U(1) 2 :
The three factors correspond to (constant) gauge transformations and to rotations in two
orthogonal twoplanes inside R4, respectively. The qdeformed version of the partition
function counts instantons extended along an additional S1 factor in the geometry of
circumference . The undeformed partition function can be recovered by letting the size of
this S1 shrink to 0. Our expressions for the instanton partition function are taken from [61].
We use a 5d parameter
which can be used to take the 4d limit, and our conventions di er
from those in [61] by the substitutions
1 ! i
1 ;
2 ! i
2 ;
a ! i a :
(C.2)
The qdeformed version of the instanton partition function for G = U(N ) and in the
presence of hypermultiplets can be expressed as follows [61, 81]
Zinst (q;~a; m~f ; 1; 2; ) = X qj Y~jZY5~d;CS (~a; 1; 2; ) Z Y~ (~a; m~f ; 1; 2; ) ;
(C.3)
~
Y
Y
Y
N 1
s2Yl
1
Y h
1
t2Ym
Zfund hyper (~a; mf ; 1; 2; ) = Y Y
~
Y
Qm1f Ql q t j; q
Qm1f Ql qYlj+1 t j; q
1
1
:
Another expression for the vector contribution is
Zv~ector (~a; 1; 2; ) = Y
Y
~
NlY;m =
ei [`Ym (s) 1 (aYl (s)+1) 2+al am]i
ei [ (`Yl (t)+1) 1+aYm (t) 2+al am]i :
l=1
Zv~ector (~a; 1; 2; ) =
Y
Zadjoint hyper (~a; mf ; 1; 2; ) =
~
Y
Ql Qm1 qYli Ymj tj i; q
Ql Qm1 qYli Ymj tj i+1; q
Ql Qm1 tj i+1; q
Ql Qm1 tj i; q
1
1
1 ; (C.4)
Qmf Ql Qm1 qYli Ymj tj i+1; q
Qm1f Ql Qm1 qYli Ymj tj i; q
1
1
Qm1f Ql Qm1 tj i; q
Qmf Ql Qm1 tj i+1; q
The symbols above are de ned as follows:
Y~ is an N vector of partitions Yl. A partition is a nonincreasing sequence of
nonnegative integers which stabilizes at zero
Yl = fYl 1
Yl 2
: : :
Yl nl+1 = 0 = Yl nl+2 = Yl nl+3 = : : :g :
2
X Yl2i ;
i
X Yli :
l;i
The sum in (C.3) is over all such partitions. A partition Yl can be identi ed with
a Young diagram whose ith column is of height Yli. We denote the partition
corresponding to the transposed Young diagram as Yt.
l
For a box s 2 Yl with coordinates s = (i; j), we de ne the leg length and arm length
`Yl (s)
Yltj
i ;
aYl (s)
Yli
j :
(C.10)
1 ;
1
(C.5)
(C.6)
(C.7)
(C.8)
(C.9)
~a is an N vector of deformation parameters corresponding to the equivariant action
of the gauge group on the instanton moduli space. In the partition functions we
compute, they are integrated over the imaginary axis and identi ed with the vev of
a scalar eld in the vector multiplet.
m~f is an Nf dimensional vector of mass deformation parameters associated to
hypermultiplets. When all of the hypermultiplets are in the fundamental representation
of the gauge group, m~f transforms as a fundamental of the avor symmetry group
SU(Nf ). Mass deformations should be viewed as coming from a vev for a background
vector multiplet. Physical masses are the imaginary part of this vev.
q
e
e i 1 ;
Ql
ei al ;
Qmf
e
i mf :
(C.11)
This de nition di ers from [61] by (C.2).
The qPochhammer symbol is de ned as
q is a classical contribution equal to20
{ in 5d we have
{ In a 4d calculation one uses
where e2 i is minus the exponential of the one instanton action of the conformal
theory with coupling constant
If the theory is not conformal, then21
20Later versions of [61] include a di erent convention for the counting parameter, essentially rede ning
There is a similar factor included in the ChernSimons contribution. We will not use these rede nitions.
21We follow [57]. h_ (G) is the dual Coxeter number and k (R) is the quadratic Casimir, normalized such
that k (adjoint) = 2h_. For SU(N ), we have h_ = N and k (fund) = 1. The combination 2h_ (G)
k (R)
is the coe cient of the oneloop beta function for the 4d N = 2 theory with hypermultiplets in the
representation R.
(x; q)1
1
Y (1
p=0
x qp) :
8 2
gY2M :
q(4d)
e2 i ;
=
2
YM +
4 i
2
gYM
2h_(G) k(R) ;
q ! q e 1+ 2
2 :
(C.12)
(C.14)
(C.15)
(C.16)
(C.17)
(C.13)
In the presence of a 5d ChernSimons term with parameter , we have [81]
l
:
An alternative version is [61, 82]
2
ZY5~d;CS (~a; 1; 2; ) = exp 4i
3
1) 1
(C.21)
where
is the holomorphic dynamical scale. According to [57], the relationship
between the 5d and 4d partition functions is
q = q(4d) ( i )2h_(G) k(R) ;
and
Zi(n4sdt) q(4d); ~a; m~f ; 1; 2
1 = !1 ;
2 = !2 ;
1 = !2 ;
2 = !1 ;
!1 = 1 ;
!2 =
(!1; !2; !3) =
1; 1;
{ 37 {
= li!m0 Zinst q(4d) ( i )2h_(G) k(R) ; ~a; m~f ; 1; 2;
N
l=1
X
X
l (s;t)2Yl
1
n
n
(C.18)
:
(C.19)
(C.20)
(C.22)
(C.23)
(C.24)
(C.25)
(C.26)
The parameters 1;2 are associated with the deformation in the 4d theory. They take the
following values:
in the derivation of the prepotential for the SeibergWitten solution one takes
1 =
2
i~ ;
! 0 ;
eventually extracting the leading piece at ~ ! 0.
in the computation on the squashed foursphere, with squashing parameters !1 =
` 1; !2 = `~ 1, the instanton contribution from the north pole involves
and for antiinstantons from the south pole
and we take the limit
symmetric Renyi entropy, we choose
! 0. For the squashed S4 corresponding to the nth
superin the computation on the vesphere,
is the circumference of the circle ber and
there are three xed points, the values at which are given in table 1. These correspond
to the values given in [50, 63].
For the squashed S
5 corresponding to the nth supersymmetric Renyi entropy,
#
1
2
3
2
2
2
e
e
e
2 i !!31
2 i !!12
2 i !!23
e 2 i !!21
e 2 i !!32
e 2 i !!13
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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