Seiberg-like dualities for 3d N = 2 theories with SU(N) gauge group

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

Abstract

We work out Seiberg-like dualities for 3d ... theories with SU(N) gauge group. We use the ... action on 3d conformal field theories with U(1) global symmetry. One of generator S of ... acts as gauging of the U(1) global symmetry. Utilizing S=S ^sup -1^ up to charge conjugation, we obtain Seiberg-like dual of an SU(N) theory by gauging topological U(1) symmetry of the Seiberg-like dual of the U(N) theory with the same matter content. We work out the Aharony dualities for SU(N) gauge theory with N ^sub f^ fundamental flavors, with/without one adjoint matter with the superpotential. We also work out the Giveon-Kutasov duality for the SU(N) gauge theory with Chern-Simons term and with N ^sub f^ fundamental flavors. For all the proposed dualities, we give various evidences such as chiral ring matching and the superconformal index computation. We find the perfect matching.

Published for SISSA by Springer

Received: July 9, 2013

Accepted: October 3, 2013

Published: October 29, 2013

Jaemo Parka,b and Kyung-Jae Parka

aDepartment of Physics, Postech,

Pohang 790-784, Korea

bPostech Center for Theoretical Physics (PCTP), Postech, Pohang 790-784, Korea

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We work out Seiberg-like dualities for 3d N = 2 theories with SU(N) gauge group. We use the SL(2, Z) action on 3d conformal eld theories with U(1) global symmetry. One of generator S of SL(2, Z) acts as gauging of the U(1) global symmetry. Utilizing S = S1 up to charge conjugation, we obtain Seiberg-like dual of an SU(N) theory by gauging topological U(1) symmetry of the Seiberg-like dual of the U(N) theory with the same matter content. We work out the Aharony dualities for SU(N) gauge theory with Nf fundamental avors, with/without one adjoint matter with the superpotential. We also work out the Giveon-Kutasov duality for the SU(N) gauge theory with Chern-Simons term and with Nf fundamental avors. For all the proposed dualities, we give various evidences such as chiral ring matching and the superconformal index computation. We nd the perfect matching.

Keywords: Supersymmetric gauge theory, Field Theories in Lower Dimensions, Duality in Gauge Field Theories

ArXiv ePrint: 1305.6280

c

Seiberg-like dualities for 3d N = 2 theories with SU(N) gauge group

JHEP10(2013)198

SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP10(2013)198

Web End =10.1007/JHEP10(2013)198

Contents

1 Introduction 1

2 3d superconformal index 2

3 SL(2, Z) action on the 3d CFTs with U(1) symmetry 6

4 Ungauging N = 2 Nf = 1 SQED and its Ahrony dual 94.1 Index of ungauged SQED Nf = 1 9

4.2 Index of ungauged (or gauged) XYZ 10

5 Aharony duality for SU(Nc) gauge theory with Nf fundamental avors 115.1 Index of SU(Nc) theory obtained from ungauging U(Nc) 135.2 Gauging U(1)T of magnetic U(Nf Nc) theory 145.3 Results of indices 15

6 Chern-Simons theory of SU(Nc)k and its dual 17

7 SU(N) theory with Nf fundamental avors and adjoint matter 19

1 Introduction

Recently, there has been renewed interest in nonperturbative dualities of three dimensional theories such as mirror symmetry and Seiberg-like dualities. One of the reasons of such renewed interest is the recent progress on the understanding of the partition function on S3 and the superconformal index [14]. Using these tools, one can give impressive evidences for various 3d dualities. Some of works in this area are [5][25]. One can also obtain the R-charge of the elds by maximizing the free energy of the theory of interest [26].

In this paper we continue this line of research and study Seiberg-like duality [5, 27] for N = 2 d = 3 gauge theories with SU(Nc) group. In 3-d, Seiberg-like dualities are known for U(Nc) theories but the duality for SU(Nc) theories remains elusive. If we knew the dual pair of an SU(Nc) theory, we could obtain the dual pair of the U(Nc) theory by gauging overall U(1) symmetry of the SU(Nc) theory. On the other hand, to obtain the SU(Nc) dual pair from the U(Nc) dual pair, we need ungauging overall U(1) of U(Nc) gauge symmetry, which is not obvious how to do it.1

The purpose of this paper is to propose such ungauging procedure to obtain Seiberg-like dual pair for various SU(Nc) theories. The idea comes from the observation that theres

SL(2, Z) transformation for 3d conformal eld theories with U(1) global symmetry [31]. In particular, one of the generator S of SL(2, Z) involves the gauging of U(1) global symmetry

1This idea was previously put forward by A. Kapustin. See [28].

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and introduces a new U(1) avor symmetry, often called topological symmetry. Applying the S operation to an SU(Nc) theory, we can obtain the U(Nc) theory by gauging the overall U(1) of the SU(Nc) theory. The new avor symmetry of the U(Nc) theory is the topological symmetry U(1)T , whose current is given by dTrA where TrA denotes the overall U(1) gauge eld of the U(Nc) theory. Since S2 = C with C being charge conjugation, by gauging the U(1)T we can obtain the same superconformal eld theory as the original SU(Nc) theory up to charge conjugation.

With respect to U(1)T , monopole operators are charged, which has the nonperturbative origin in the original theory. Thus we had better carry out the 2nd S-operation or gauging U(1)T on the Seiberg-like dual pair of the U(Nc) theory. If the matter content is charge conjugation invariant, we obtain the Seiberg-like dual pair of the original SU(Nc) theory. Thus we can obtain the Seiberg-like dual of any SU(Nc) theory, if the Seiberg-like duality is known for the corresponding U(Nc) theory. All we have to is to gauge U(1)T of Seiberg-like dual of the given U(Nc) theory. We apply this idea to various SU(Nc) theories to obtain Aharony duals [27]. Giveon-Kutasov duality [5] for an SU(Nc) gauge theory with Chern-Simons term can be obtained from the Aharony duality by giving axial mass to some of avors. We subject such candidates of dual pairs to various tests, e.g., computation of the superconformal index. We nd the perfect matching.

The content of the paper is as follows; In section 2, we review the basics of the super-conformal index in 3-dimensions. In section 3, we review the SL(2, Z) transformation of 3d conformal eld theories with U(1) avor symmetry, where gauging U(1) avor symmetry is realized as S operation. In section 4, we consider the simplest example of U(1) gauge theory with one avor and carry out gauging/ungauging operation. In section 5, we work out Aharony duality of the SU(Nc) gauge theory with Nf fundamental and anti-fundamental chiral muiltiplets2 and carry out various tests to nd the nice agreements. In section 6, we obtain Giveon-Kutasov duality for the SU(Nc) gauge theory with Nf fundamental avors and with Chern-Simons term. This duality can be obtained from the Aharony duality of the section 5 by giving axial mass to some avors. In section 7, we obtain the Aharony dual of the SU(Nc) gauge theory with Nf fundamental avors, one adjoint X and with the superpotential W = TrXn+1. Again we put the proposal to various tests and nd the perfect matching.

As this work is nished, we receive the paper [29] which has overlap with ours.

2 3d superconformal index

Let us discuss the superconformal index for N = 2 d = 3 superconformal eld theories (SCFT). The bosonic subgroup of the 3d N = 2 superconformal group is SO(2, 3) SO(2). There are three Cartan elements denoted by , j3 and R which come from three factors

SO(2) SO(3)j3 SO(2)R in the bosonoic subgroup, respectively. The superconformal index for an N = 2 d = 3 SCFT is dened as follows [37]:

I(x, t) = Tr(1)F exp({Q, S})x+j3

YatFaa (2.1)

2Nf fundamental and anti-fundamental chiral muiltiplets will be called Nf fundamental avors for convenience.

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where Q is a supercharge with quantum numbers = 12, j3 = 12 and R = 1, and S = Q. The trace is taken over the Hilbert space in the SCFT on R S2 (or equivalently over the space of local gauge-invariant operators on R3). The operators S and Q satisfy the following anti-commutation relation:

{Q, S} = R j3 := . (2.2)

As usual, only BPS states satisfying the bound = 0 contribute to the index, and therefore the index is independent of the parameter . If we have additional conserved charges Fa commuting with the chosen supercharges (Q, S), we can turn on the associated chemical potentials ta, and then the index counts the number of BPS states weighted by their quantum numbers.

The superconformal index is exactly calculable using the localization technique [3, 4]. It can be written in the following form:

I(x, t) =

XmZ

Z

da 1

|Wm|(sgn)eS

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(0)

CS (a,m)eib0(a,m)

"

Xn=11nftot(eina, tn, xn)#. (2.3)

The origin of this formula is as follows. To compute the trace over the Hilbert space on S2 R, we use path-integral on S2 S1 with suitable boundary conditions on the elds. The path-integral is evaluated using localization, which means that we have to sum or integrate over all BPS saddle points. The saddle points are spherically symmetric congurations on S2 S1 which are labeled by magnetic uxes on S2 and holonomy along S1. The magnetic uxes are denoted by {m} and take values in the cocharacter lattice of G (i.e. in Hom(U(1), T ), where T is the maximal torus of G), while the eigenvalues of the holonomy are denoted {a} and take values in T . S(0)CS(a, m) is the classical action for the (monopole+holonomy) conguration on S2 S1, 0(m) is the Casmir energy of the vacuum state on S2 with magnetic ux m, q0a(m) is the Fa-charge of the vacuum state, and b0(a, m) represents the contribution coming from the electric charge of the vacuum state. The last factor comes from taking the trace over a Fock space built on a particular vacuum state. |Wm| is the order of the Weyl group of the part of G which is left unbroken by the magnetic uxes m . These ingredients in the formula for the index are given by the following explicit expressions:

S(0)CS(a, m) = i

XRFk(m)(a), (2.4)

b0(a, m) = 1

2

X XR |(m)|(a),

q0a(m) = 1

2

X XR |(m)|Fa( ),

0(m) = 1

2

X (1 ) XR |(m)| 1 2

XG |(m)|,

Yatq0a(m)ax0(m) exp

3

ftot(x, t, eia) = fvector(x, eia) + fchiral(x, t, eia),

fvector(x, eia) =

XG ei(a)x|(m)|,

fchiral(x, t, eia) =

X XR

"ei(a)

YatFaa x|(m)|+ 1 x2 ei(a)

YatFaa x|(m)|+2 1 x2

#

PG represent summations over all fundamental weights of G, all chiral multiplets, all weights of the representation R , and all roots of G, respectively. There are additional sign factors sgn in the formula (2.3) depending on the monopole sectors, which will be explained shortly.

We will nd it convenient to rewrite the integrand in (2.3) as a product of contributions from the di erent multiplets. First, note that the single particle index f enters via the so-called Plethystic exponential:

P.E. f(x, t, z = eia, m)

exp

where

PRF ,

P ,

PR and

Xn=11nf(xn, tn, zn = eina, m)!(2.5)

while we dene zj = eiaj. Specically, consider a single chiral eld , whose single particle index is given by:3

XR

ei(a)tafa( ) x|(m)|+ 1 x2 ei(a)tafa( )

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x|(m)|+2

1 x2

!. (2.6)

Then the index contribution is simply

YR

P.E. ei(a)tafa( ) x|(m)|+

1 x2 ei(a)tafa( )

x|(m)|+2

1 x2

!

(2.7)

The full index will involve a product of such factors over all the chiral elds in the theory, as well as the contribution from the gauge multiplet. It is given by:

I(x, t) =

XmZ

I Y

j

dzj 2izj

1 |Wm|(sgn)eSCS(m,a)Z

gauge(x, z, m)

Y

Z (x, t, z, m) (2.8)

where:

Zgauge(x, z = eia, m) =

Yad(G) x|(m)|

1 ei(a)x2|(m)|

,

|(m)|/2

Z (x, t, z, m) =

YR

x(1 )ei(a)

Ya tafa( )

(2.9)

P.E. ei(a)tafa( ) x|(m)|+

1 x2 ei(a)tafa( )

x|(m)|+2

1 x2

!

3Note that a in (a) and the subscript a in ta or fa denotes the di erent object.

4

Note that by shifting ta taxca, one can change the value of the R-charge . Hence remains the free parameter for generic cases.

The above is the ordinary superconformal index. We need two more generalizations for later purposes. The rst one is the notion of the generalized index. When we turn on the chemical potential ta, which can be regarded as turning on a Wilson line for a xed background gauge eld. The generalized index is obtained when we turn on the nontrivial magnetic ux na for the corresponding background gauge eld. Only the contribution to the chiral multiplets are changed and this is given by the replacement (m) (m) +

Pa fa( )na

Z (x, t, z = eia, m) (2.10)

=

YR

x(1 )ei(a)

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Ya tafa( )

|(m)|/2+[summationtext]a fa( )na/2

P.E. ei(a)tafa( ) x|(m)+

[summationtext]a fa( )na|+

1 x2 ei(a)tafa( )

x|(m)+[summationtext]a fa( )na|+2

1 x2

!

Here na should take integer value as does mj.

For every U(N) gauge group, we can dene another abelian symmetry U(1)T whose conserved current is F of overall U(1) factor. To couple this topological current to background gauge eld we introduce BF term

R

ABG trdA + and terms needed for super-

symmetric completion. This introduces to the index

znw[summationtext]j mj (2.11)

where n is the new discrete parameter representing the topological charge while w is the chemical potential for U(1)T .

Finally let us discuss the sign factor in the index formula. This subtlety was initially explained by Dimofte, Gaiotto, Gukov (DGG) [33]. This subtle sign factor arises since DGG takes (1)F = (1)2j3 in the denition of the index. In the initial derviation [3, 4] (1)F = (1)2jf+em is used, where e is the electric charge of the eld and m is the magnetic ux. Because of this, if we consider the free chiral multiplet with the background CS level 1/2, we have

I (m; x, t) = (x)

1

2 (m+|m|)t

1

2 (m+|m|)

Yk=01 t1x|m|+2+2k1 tx|m|+2k . (2.12)

so that we have additional sign (1)

m+|m|

2 . In DGG, this relative phase factor has been extensively checked. This sign cannot be derived from the direct localization computation but we will assume such relative phase in the index formulae. Such additional sign was also taken account in the factorization of the superconformal index [38]. For example if we consider the U(1) theory with CS level with Nf fundamental chiral and anti-chiral

5

multiplets , the avor symmetry is U(1)A SU(Nf) SU(Nf). The index is given by:

I(x, t, ~t, w, ) =

XmZ

I

dz2iz wmxNf|m|(z)mNf|m| (2.13)

Yk=0

Nf

Ya=11 z1t1a1x|m|+2+2k 1 ztax|m|+2k

Nf

Ya=11 z~t1a1x|m|+2+2k 1 z1~tax|m|+2k

where , ta, ~ta are the fugacities for U(1)A, Cartans of SU(Nf), SU(Nf) respectively. Note that we include the additional phase (1)m to the original index. Similar factor should be included for non-abelian cases as well for later applications, which have obvious generalizations. Such phase was crucial for the factorization of the index [38]. It turns out that incorporation of the phase gives the desired result for the Seiberg-like dualities. One can also nd detailed discussions and the explicit formluae in [29]

3 SL(2, Z) action on the 3d CFTs with U(1) symmetry

Gauging and ungauging of U(1) factor we adpoted in this paper is closely related to the S-operation for the 3d conformal eld theories (CFTs) with U(1) avor symmetry. It was found that there is an SL(2, Z) action on the space of 3-dimensional conformal eld theories with U(1) avor symmetry. This action was rst described in [31] as a way to understand the meaning of di erent choices of boundary conditions for an abelian gauge eld in AdS4 in the context of AdS4/CF T3. And in [32, 33] such SL(2, Z) action on 3d abelian gauge theories with U(1) avor symmetry was considered. We closely follow their explanation in the below.

SL(2, Z) acts on the space of 3d theories equipped with a specic way to couple a U(1) avor symmetry to a background U(1) gauge eld. The SL(2, Z) action has two generators S, T satisfying

S4 = (ST )3 = I (3.1)

The T -opration on the 3d conformal theories only modies the prescription of how to couple the theory to the background gauge eld A, by adding F = dA to the conserved current for the background U(1) symmetry. At the Lagrangian level, this means adding a background Chern-Simons interaction at level k = 1,

T : L L + 1

2A dA . (3.2)

On the other hand, the S-operation changes the structure of the 3d theory by making the background gauge eld A dynamical.4 Once the old U(1) avor symmetry turns into gauge eld, it has the new U(1) avor current given by F of the gauged U(1). At the Lagrangian level

S : L L + Anew dA (A dynamical) . (3.3)

4One can add a Yang-Mills kinetic term at intermediate stages in the calculation. But for S to have the correct properties, one must ow to the IR at the end, and then gYM ! 1 and this term is removed.

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Monopole operators for A are charged under the new U(1) avor symmetry, hence this U(1) is sometimes called topological. Since S2 = C, if one carries out the gauging twice following the prescription eq. (3.3) , the resulting theory is equivalent to the original theory up to charge conjugation. One can also say that gauging U(1) (S) is equivalent to ungauging U(1) up to charge conjugation (S1C). When we work out gauging/ungauging U(1), we are always intended to apply S / S1 operation.

From the denitions of S and T , one can prove that the relations S2 = C and (ST )3 = I hold, where the transformation C (charge conjugation) just inverts the sign of the background gauge eld.

This has a suitable N = 2 generalization. Suppose that we have a theory with U(1) global symmetry coupled to a background vector multiplet V . V has a real scalar , two Majorana fermions and the gauge eld A. This can be dualized to a linear multiplet

V = DDV (3.4)

with the lowest component of being . Now in order to supersymmetrize SL(2, Z) action, we simply have to substitute V for A dA. In particular, S-operation is given by

L(V ) L(V ) + Z

d4 newV (3.5)

Now we can apply this idea to obtain the Aharony dual of an SU(N) gauge theory. The basic idea is that we start from an SU(N) gauge theory with matters which are charge-conjugation invariant so that if we apply S-operation twice we are back to the original theory. Given SU(N) gauge group we have obvious global U(1) symmetry and if we apply the S-operation we introduces the gauging of U(1) theory with the BF type coupling to the background gauge eld. Thus we now have the U(N) gauge theory with the usual U(1)T topological symmetry for which the monopole operators are charged. This is the typical setting where Aharony duality of U(N) gauge theory is discussed. If we apply S-operation again, then we gauge topological symmetry and introduce another U(1) avor symmetry. By this procedure we are back to our original theory of SU(N) theory assuming the matter contents is charge-conjugation invariant. Thus gauging topological U(1) corresponds to ungauging overall U(1) gauge symmetry. On the other hand, by applying the same S-operation to the Aharony dual of the U(N) theory, we obtain the Aharony-dual of the SU(N) theory.

For example, if we start from U(N) gauge theory with Nf fundamental avors, Aharony dual is given by U(N Nf) gauge theory with Nf avors with the following superpotentials

W = v+V + vV+ +

Nf

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Xa,b=1Mabqa~qb (3.6)

where M, v is the singlet for U(N Nf) corresponding to the mesons and the monopole operators for the U(N) gauge theory.5 By applying the S-operation we obtain U(1)T

5In Mab, the upper indices denote the fundamental representation of SU(Nf), while the lower indices denote the antifundamental representation. We will later write [summationtext]Nfa,b=1 Mabqa~qb Mq~q.

7

U(N Nf) gauge theory with the above superpotential and the monopole operators v, V is charged under U(1)T . Furthermore, we have the additional BF type coupling

AT dTrA (3.7)

where TrA denotes the overall U(1) gauge eld of U(N Nf). Following the above logic this should be the Aharony dual of SU(N) with Nf avors. We subject this claim to the various tests in the next section.

Furthermore this logic applies to any SU(N) gauge theory with charge-conjugation invaraint matter contents to obtain Aharony dual if the corresponding Aharony dual of U(N) theory is known. Thus we also consider the theory of the U(N) theory with Nf avors and adjoint matter X with the superpotential W = TrXn+1 and work out its Aharony dual. One should note that starting from SU(N) theory one can generate whole classes of SCFTs by SL(2, Z) transformation. We are currently working out the details such SCFTs [36].

For later purpose we need to work out how the index would transform under the S-transformation. Suppose that(z, s) denotes the generalized index with U(1) global symmetry and z, s are respectively chemical potential and magnetic ux associated with the U(1) global symmetry. Let us denote the generalized index of S-transformed theory by I(u, m) with u, m are respectively chemical potential and magnetic ux associated with new U(1) global symmetry. Then I is given by

I(w, m) =

XsZ

I

dz2iz wszm

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(z, s) (3.8)

The relation between the index and the S-transformed index is well known. Its convenient to use the charge basis [33]

(z, s) =

XeZ(e, s)ze, I(w, m) =

XeZI(e, m)we (3.9)

Then the right hand side of (3.8) becomes

XsZ

Z

dz2iz wszm

(z, s) =

XsZ

Z

dz2iz wszm+e

(e, s) =

XeZws(m, s) (3.10)

XeZ

PeZ I(e, m)we. Thus we have

I(e, m) =(m, e). (3.11)

Note that in the charge basis S-operation takes the simple form

e m

!

0 1

1 0

which is equal to I(w, m) =

!

e m

!

(3.12)

Thus we have

(z, s) =

XeZze

Z

dw2iws+1 I(w, e). (3.13)

8

We regard this formula as ungauging U(1). We also have the inverse relation

I(u, m) =

XeZue

Z

dz 2izm+1

(z, e) (3.14)

Lets denote the generalized index of the theory obtained from S-operation on the theory with index being I(u, m) by I(z, s). One can easily check that(z, s) = I(z1, s). Or in charge basis(e, s) = I(e, s), which is the consequence of S2 = C. If we consider U(N) theory with Nf avors, the theory obtained by S-transformation di ers from SU(N) theory with Nf avors by the overall sign of the charges of the matter. Thus the role of chiral multiplet and the anti chiral multiplet is exchanged. But since we are dealing with the same number of chiral multiplets and the anti chiral multiplets, we obtain the same theory. However when we interpret the index result, we should keep in mind of such sign ipping.

4 Ungauging N = 2 Nf = 1 SQED and its Ahrony dual

Lets consider the simplest example of N = 2 Aharony dual pair. N = 2 SQED with Nf = 1 avor. Its dual is given by XYZ model. In using the convention of the section 3, this is the theory with no gauge group with the superpotential

W = v+vM (4.1)

where v is charged under U(1)T . If we ungauge U(1) gauge group for SQED, we are left with the free theory with Nf = 1 avor, since this corresponds to S1-operation.

This is the N = 4 theory with one free hypermultiplet. On the other hand, if we gauge U(1)T of XYZ model, we obtain U(1) theory with Nf = 1 avor v with additional neutral chiralmultiplet M whose superpotential is given by eq. (4.1). This is N = 4 SQED with one hypermultiplet. Thus N = 4 theory with one free hypermultiplet and N = 4 SQED with one hypermultiplet are related by S2 = C. But since the matter content is C-invariant, we obtain the equivalent theory. This is nothing but the simplest mirror pair. Thus we can regard this mirror pair as a special case of Aharony dual pair for the SU(N) theory with Nf avors with N = Nf = 1.

This simple example also shows that why S-operation involves the duality transformation. Starting from N = 4 one free hypermultiplet, we obtain SQED with Nf = 1. Under the U(1)T symmetry, monopole operators of SQED are charged. Thus in order to carry out the gauging of U(1)T we had better go to the frame where the monopole operators are elementary elds. This is possible if we work in the Aharony dual of SQED with Nf = 1.

This is nothing but the XYZ model and U(1)T is mapped to the usual U(1) global symmetry. Hence gauging U(1)T is straightforward and we obtain N = 4 SQED with one hypermultiplet. Following this example, we carry out gauging of U(1)T for the Seiberg-like dual of an original U(Nc) theory to obtain Seiberg-like dual of an SU(Nc) theory.

4.1 Index of ungauged SQED Nf = 1

As a warmup exercise of the index gymnastics, we consider the ungauging of SQED with one avor. Since our major concern is the gauge symmetry and the topological U(1)T symmetry,

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we will turn o the chemical potential for U(1) axial symmetry. Similar manipulation will be used for handling U(N) theory with Nf avors. The index formula of the SQED with one avor is given by .

I(w, mw; x) =

Xm1Z

I

dz12iz1 wm1zmw1ZQZ

~Q

=

Xm1Z

I

dz12iz1 wm1zmw1Z (z1, m1; x) (4.2)

where z1 is the holonomy of U(1)T and w is the chemical potential for the background gauge eld coupled to U(1)T . ZQ and Z ~Q is some function of x, z1 and m1. If we ungauge

U(1), we expect to have the free theory with Nf = 1. Using the formula eq. (3.13), the ungauged index is

(z, s; x) =

XmwZ zmw

I

dw2iws+1 I(w, mw; x)

=

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XmwZ zmw

I

dw 2iws+1

X

I

dz12iz1 wm1zmw1Z (z1, m1; x)

(4.3)

m1Z

We can expand the Z (z1, m1; x) in the integer power of z1, such as Z (z1, m1; x) =

Pz1Z zn1 (n, m1; x). So eq. (4.3) is equal to

(z, s; x) =

XmwZ zmw

I

dw 2iws+1

X

I

dz12iz1 wm1zmw1

XnZzn1 (n, m1; x)

. (4.4)

The above integral has very simple dependence on w and z1. Integration over z1 gives simply the restriction that n = mw and the integration over w gives m1 = s. So(z, s) becomes

(z, s; x) =

XmwZzmw (mw, s; x) = Z (z, s; x) (4.5)

Explicit form of this ungauged index is(z, s; x) = ZQZ ~Q, where ZQ and Z ~Q is

ZQ = x1rz1

|s|/2 P.E.

m1Z

zxr z1x2r1 x2 x|s|

Z ~Q = x1rz

z1xr zx2r1 x2 x|s| . (4.6)

The resulting index is that of the free theory with a chiral and an anti-chiral eld as expected. The chemical potential of the U(1) global symmetry is z.

4.2 Index of ungauged (or gauged) XYZ

Lets do the same process to the dual XYZ theory. Then the index of ungauged XYZ theory becomes

(z, s; x) =

XmwZ zmw

I

dw2iws+1 I(w, mw; x)

=

XmwZ

|s|/2 P.E.

I

dw2iw wszmwZMZv+Zv (4.7)

10

Fields U(1)R U(1)A SU(Nf) SU(Nf)

Q r 1 Nf 1

~Q r 1 1 Nf

M 2r 2 Nf Nf

Y 2Nf(1 r) 2Nc + 2 2Nf 1 1

q 1 r 1 Nf 1

~q 1 r 1 1 Nf v Nf(1 r) Nc + 1 Nf 1 1 V Nf(r 1) + Nc + 1 Nf 1 1 u Nf(r 1) + Nc Nf 1 1

Table 1. The global symmetry charges of the chiral elds.

where

Zv+ =

x(1r)w1

|mw|/2 P.E.

wxr w1x(2r)1 x2 x|mw|

Zv =

JHEP10(2013)198

x(1r)w

|mw|/2 P.E.

w1xr wx(2r)1 x2 x|mw|

ZM = x12rP.E.

x2r x22r 1 x2

. (4.8)

The R-charge of the v elds r = Nf(1 r) Nc + 1 = 1 r. Eq. (4.7) is the generalized index of U(1) theory with matter M, v where v has charge 1 under U(1). And the original U(1) gauge symmetry becomes U(1) topological symmetry of this new theory. If we directly gauge XYZ model, the generalized index has ws instead of ws in (4.7), which is again due to S2 = C.

5 Aharony duality for SU(Nc) gauge theory with Nf fundamental avors

Lets consider the Aharony duality for SU(Nc) gauge theory with Nf avors. Following the procedure of the previous section, we propose the following;

Electric theory: SU(Nc) gauge theory(without Chern-Simons term), Nf pairs of fundamental/anti-fundamental chiral superelds Qa, ~Qb(where a, b denote avor indices).

Magnetic theory: U(1) U(Nf Nc) gauge theory with the BF coupling

AU(1) dTrAU(NfNc) , (5.1)

with Nf pairs of fundamental/anti-fundamental chiral superelds qa, ~qa of U(Nf Nc), Nf Nf singlet superelds Mab. We have v, V charged under U(1) with

11

charge 1. The superpotential is given by

W = v+V + vV+ + Mq~q (5.2)

where u is the monopole operator of U(1).

Note that the gauged elds v do not have the usual U(1)R charge compared with the elementary elds Q, q since they have the nonperturbative origin. This will lead to interesting dynamics such as the nonperturbatuve truncation of the chiral ring.

Lets compare the chiral ring elements of the both sides. For SU(Nc) gauge theory with no superpontial, there are mesons Mab = Qa ~Qb, a monopole operator Y which parametrizes the Coulomb branch, and baryons of the form

Ba1aNfNc = a1aNfNc b1bNc i1iN

c Qb1i1 QbNciNc . (5.3) And, similary, there are ~B ~QNc in the chiral ring. Where a, b and i are the avor and gauge indices a, b = 1, , Nf and i = 1, , Nc. In the magnetic theory Y is mapped to v+v, mesons are mapped to singlet elds M. Baryon elds B, ~B are mapped to the monopole operators ~b u+~qa1 ~qa ~Nc and b uqa1 qa ~

Nc wherec = Nf Nc and the avor and gauge indices are totally anti-symmetric. The baryon operators have the charged matters of SU(Nf Nc) coupled to the basic monopole operator u. This structure is required due to the BF coupling. We can view the monopole operators as states on S2 R by operator-state correspondence of conformal eld theories. When we turn on unit monopole of AU(1), due to the BF coupling (5.1),c = Nf Nc matters should couple.6

Due to the residual gauge invariance of SU(Nf Nc) the allowed operator should have the form u+~qa1 ~qac and uqa1 qac . Note that number of the baryons of both sides are the same Nf CNc =Nf CNfNc =

Nf !(Nf Nc)!Nc! . u+u is Q-exact since it carries no charge so it is not a BPS state. It is truncated nonperturbatively.7

Its worthwhile to consider the special cases. When Nf = Nc 6= 1 the dual is simply given by U(1) gauge theory with singlets M, unit charged matter v with the superpotential

W = v+vdetM (5.4)

Note that the superpotential is inherited from the Aharony dual of U(Nc) theory with Nf = Nc. The electric theory has two baryon operators B, ~B. The corresponding operator in the magnetic theory is given by u+, u. For Nf = Nc 6= 1, the theory can also be described by mesons M, baryon elds B, ~B and monopole eld Y with the superpotential [30]

W = Y (detM B ~B) (5.5)

One can check that both theories have the same chiral ring structure and the same superconformal index. Note that for the U(1) theory with the superpotential (5.4), the corresponding chiral ring relation detM u+u = 0 should be generated by the U(1)

gauge dynamics.8

6In our convention, each qa has the charge 1~

Nc withc = Nf Nc under the overall U(1) of U(

7This is the special for U(1) theory. For U(Nc > 1) theory, we can take u+ (1, 0, ), u (0, 1, )

where we denote the monopole charge for Cartans U(1)Nc of U(Nc). u+u is a BPS state.

8This was pointed out by the authors of [29] after the rst version of the draft was distributed. We thank them for making this point clear.

12

JHEP10(2013)198

c).

When Nf = Nc = 1 the dual is U(1) gauge theory with singlet M, charged matter v with the superpotenial

W = v+vM (5.6)

This was already discussed in the previous section.

Also interesting case is Nc = 1 with arbitrary Nf > 1. The electric theory is free theory with Q, ~Q. The magnetic theory is given by U(1) U(Nf 1) theory. The interesting thing is that we have 2Nf free elds Q, ~Q. These are matched by u+~qa1 ~qaNf1 and uqa1 qaNf1 . Apparently the chiral ring element v+v exists in the magnetic side, which has no counterpart in the electric side. The truncation of this element occurs nonperturbatively. This is similar to what happens to N = 2 U(1) theory with Nf > 1 avors.

The monopole operator v+, v in this theory has the same quantum number as the above v+, v. Since the product of v+v has carries no charge to protect, it is truncated nonperturbatively. We think the similar thing happens in the case at hand as well. Indeed one can see this operator is canceled by a suitable fermionic operator in the index computation.

Of course when Nf = 1, v+, v has the quantum number of elementary elds. In this case due to the usual superpotential W = v+vM, v+v is Q-exact.

5.1 Index of SU(Nc) theory obtained from ungauging U(Nc)

Now lets consider the index of the electric U(Nc) theory with Nf avors.

I(w, mw; x) =

XmiZ1 |Wm|

I

Nc

JHEP10(2013)198

Yi=1dzi2izi wm1+...+mNc (z1 . . . zNc)mwZgaugeZNfQ ZNf~Q . (5.7)

where w is the fugacity for U(1)T and zi denotes the holonomy variable of the Cartans of U(N). z = z1z2 zNc is the holonomy variables of the overall U(1), ZQ and Z ~Q is given by

ZQ =

Nc

Yi=1

x1rz1i

|mi|/2 P.E.

zixr z1ix2r1 x2 x|mi|

Z ~Q =

Nc

Yi=1

x1rzi

|mi|/2 P.E.

z1ixr zix2r1 x2 x|mi| . (5.8)

And Zgauge is

Zgauge = x[summationtext]Nci<j |mimj|

Nc

Yi<j(1 ziz1jx|mimj|)(1 z1izjx|mjmi|) (5.9)

Equivalently this can be viewed as gauging overall U(1) global symmetry of SU(Nc)

with additional BF term Anew dAU(1) where U(1) symmetry acts as Qi eiQi. Thus quark has charge 1 under this U(1) symmetry. Thus the same index can be written as

I(w, mw; x) =

XmZ

I

dz2iz wszmISU(Nc)(z, s) (5.10)

13

where ISU(Nc)(z, s) is the generalized index of SU(Nc) theory. Comparing it with eq. (5.7) we nd that

z = z1z2 zNc, s = m1 + m2 + mNc. (5.11)

Lets just denote that 1

|Wm| ZgaugeZNfQ ZNf~Q = Z (m1, z1, m2, z2, , mNc, zNc; x) for simplicity. Concentrating on an arbitrary zi out of z1, , zNc, say zi = zNc, we can expand Z =

PnZ znNc ( , mNc, n). Then by rewriting (5.7) using this expansion of zNc, and ungauing U(1) of U(Nc), it becomes

(z, s) =

XmwZ zmw

I

dw2iws+1 I(w, mw; x)

=

JHEP10(2013)198

I

dw 2iw

Nc

Yi=1dzi 2izi

!wm1+...+mNc s

z1 . . . zNc z

mw

XnZznNc .

Xmw,miZ

By integrating w and zNc, we obtain

(z, s) =

I N

Yi=1dzi 2izi

!

c1

XmiZ

XmwZ

zz1 . . . zNc1

mw ( , s m1 mNc1, mw).

This means that the zNc is changed by the

z z1...zNc1

and the mNc is changed by the s m1 mNc1 from the origianl U(Nc) gauge theory, which we already nd at eq. (5.11). Thus we recover the generalized index of SU(Nc) gauge theory with the global U(1) with fugacity z and charge s, as expected. By setting z = 1 and s = 0, we obtain the ungauged index(1, 0), which is equal to the index of usual SU(Nc) theory with Nf avors.

5.2 Gauging U(1)T of magnetic U(Nf Nc) theory

Index of magnetic thoery with U(Nf Nc =c) gauge theory is

I =

XmiZ1 |Wm|

I

c

Yi=1dzi2izi wm1+...+mNc (z1 . . . zNc)mwZgaugeZNfq ZNf~q ZN2fM Zv+Zv.(5.12)

Where

Zq =

c

Yi=1

x(1r)z1i

|mi|/2 P.E. zixr z1ix(2r)

1 x2 x|mi|!

Z~q =

|mi|/2 P.E. z1ixr zix(2r)

1 x2 x|mi|!

ZM = x12r P.E. x2r x22r

1 x2

!

14

c

Yi=1

x(1r)zi

Zv+ =

x(1r)aw1

|mw|/2 P.E. wxr w1x(2r)

1 x2 x|mw|!

Zv =

x(1r)w

|mw|/2 P.E. w1xr wx(2r)1 x2 x|mw|!

. (5.13)

The R-charge of chiral eld q is r = 1 r and R-charge of v is r = Nf(1 r) Nc + 1.Lets ungauge U(1)T of the magnetic theory

(z, s) =

XmwZ zmw

I

dw2iws+1 I(w, mw; x)

=

XmwZ

JHEP10(2013)198

I

dw2iw wm1+...+mN

c s

z1 . . . zc z

mwZv+Zv ( ), (5.14)

where the ( ) term is independent of w and mw. Hence we can nd v are gauged in the index expression. Terms denoted above are just the form of the U(1) gauge theory with fugacity w, charge mw. Also from the functional form of w, z one can see that there is a BF coupling between gauged U(1)T and overall U(1) factor of U(Nf Nc).

5.3 Results of indices

We can check the dualities by using superconformal index. We expand the index in x upto some orders and compared the dual pairs for 0 < Nc,c < 3. We turned o the ungauged global U(1) symmetry by setting fugacity variable z = 1 and charge s = 0.

Its worthwhile to examine the chiral ring structure of the theories and to work out the gauge invariant operators of the rst few lower orders of the indices. For the case of the electric theory, SU(Nc) gauge theory with no superpotential, there are mesons Mab = Qa ~Qb, a monopole operator Y , and baryons of the form B QNc , ~B ~QNc with a, b the avor indices a, b = 1, , Nf.9

The lowest components of meson superelds give N2fx2r to the index. From the bosonic components B and ~B, we pick Nc of chiral elds among Nf so that it gives Nf CNcxNcr from each B and ~B.10 For example, for Nc = 2, indices have the contribution

I = + N2f + Nf(Nf 1)

x2r + (5.15)

9In principle, there could be boson-fermion cancelation so that the index does not show the full operators. However for the chiral ring elements we consider, such cancelation does not occur. One can argue that there is no fermionic state that would be paired up with the chiral ring generators. For example, lets consider mesons Q ~Q whose energy is 2r. By the Z-minimization procedure similar to [14], one nds r < 12 . Any

fermionic state that has the energy + j3 = 2r and zero topological charge (and appropriate other global charges) would cancel the index contribution of mesons. However, there is no such fermionic state. Recall that possible fermionic excitations are in fundamantal/anti-fundamental representation and gaugino , which have + j3 = 2 r and 2 respectively. Both values are larger than 2r. Thus any fermionic state

with zero topological charge must have + j3 > 2r. Similar arguments can be made to the other chiral ring elements.

10Nf CNc = Nf

Nc

[parenrightbig] =

Nf !

Nc!(Nf Nc)!

15

Electric Magnetic(Nc, Nf) SU(Nc) U(1) U(Nf Nc) Index (r is the IR R-charge of Q.)

(1, 1)

free

U(1)

1 4x2 5x4 + x42r 2x2r + 2xr + 3x2r + 4x3r + 5x4r 4x2+r 4x2+2r +

(1, 2)

free

U(1) U(1)

1 16x2 + 6x42r 4x2r + 4xr + 10x2r + 20x3r 36x2+r +

(1, 3)

free

U(1) U(2)

1 36x2 + 15x42r 6x2r + 6xr + 21x2r + 56x3r

120x2+r +

(1, 4)

free

U(1) U(3)

1 64x2 +28x42r 8x2r +8xr +36x2r +120x3r

280x2+r +

(2, 2)

SU(2)

U(1)

1 16x2 + 88x4 + x48r + x24r + 26x42r + 6x2r +

20x4r + 50x6r + 105x8r 64x2+2r 160x2+4r +

136x2+558x4+x812r +x46r +21x42r +15x2r +

105x4r + 490x6r 384x2+2r +

1 64x2 + 1888x4 + x1216r + x68r + 36x42r +

28x2r + 336x4r 1280x2+2r +

1 100x2 + 4750x4 + x1620r + x810r + 55x42r +

45x2r + 825x4r 3200x2+2r +

1 + 82x2 + x26r + 9x24r + 36x22r + 9x2r + 2x3r + 45x4r + 18x5r + 167x6r + 90x7r + 513x8r + 332x9r

18x2+r + 81x2+2r 162x2+3r +

132x2+x48r+16x46r+100x44r+16x2r+8x3r+

136x4r + 120x5r + 836x6r 48x2+r 480x2+2r +

150x2+x610r+25x68r+25x2r+20x3r+325x4r+

450x5r 100x2+r +

1 72x2 + x812r + 36x810r + 36x42r + 36x2r +

40x3r + 666x4r 180x2+r +

Table 2. The results of the superconformal index computation.

and, for Nc = 3, indices have

I = + N2fx2r + 13Nf(Nf 1)(Nf 2)x3r + . (5.16)

There is a monopole operator Y which corresponds to v+v of the dual theory. Sinces its R charge is just the twice of v, it gives x2Nf(1r)+22Nc to the indices. For instance, when Nc = 2, Nf = 3 there is a term x46r for Y , x812r for Y 2.

As pointed out in [30] for the case Nf = Nc, the theory is described in terms of M, Y, B, ~B with the superpotential

W = Y (detM B ~B) (5.17)

which gives constraint that detM B ~B = 0. One state corresponding to them becomes Q-exact so it doesnt appear in the index result. We can check it in the index expansion. From detM we have N2fHNf x2Nfr with nHm =n+m1 Cm = (n+m1)!(n1)!m!, and from B2, ~B2,B ~B,

16

JHEP10(2013)198

(2, 3)

SU(2)

U(1) U(1)

(2, 4)

SU(2)

U(1) U(2)

(2, 5)

SU(2)

U(1) U(3)

(3, 3)

U(1)

SU(3)

(3, 4)

SU(3)

U(1) U(1)

(3, 5)

SU(3)

U(1) U(2)

(3, 6)

SU(3)

U(1) U(3)

MB and M ~B we have (3 + 2N2f)x2Nfr. But actually the coe cient of the term is smaller by 1 than the naive counting.

There are terms like x2. This is due to the mixed contribution of bosonic and fermionic operators. For example consider Nc = 1 case. Because these are free chiral theories so just

Q and ~Q are chiral ring elements. Q and ~Q contribute 2Nfxr to the index. Fermion components of chiral elds give 2Nfx2r, and to the next order, two chiral elds give

2Nf H2x2r. When fermions coupled to Q, ~Q they gives (2Nf)2 copies of xr x2r so there are (2Nf)2x2.

As alluded before, the chiral ring elements of magentic theory corresponding to B and ~B are ~b u+~qa1 ~qac and b uqa1 qac where the avor and gauge indices are

totally anti-symmetric. Since u and q, ~q have R-charge ru = Nf(1 r) Nc and 1 r respectively, b and ~b have R-charge Nf(1 r) Nc +c(1 r) = Ncr. Since we pick c = Nf Nc of q and ~q out of Nf, b and ~b together give 2 Nf

xNcr to the index.

For the case of Nc = 1, on the other hand, there is no monopole operator in electric theories because they are just free chiral eld theories. But there are still v exist in the dual theories which possibly can make the chiral ring element v+v. We propose that this chiral ring element is truncated nonperturbatively. vv+ gives xNf(22r) but it is cancled by the contribution of fermionic partner of detM which gives xNf(22r).

When Nf = Nc = 1, the magnetic theory is U(1) theory with 3 elds v, M with superpotential W = v+vM. v give 2xr to the index. Due to the superpotential W = v+vM, v+v are Q-exact. So the singlet M gives x2r to the index and monopoles from charge sector of 2 and 2, u2+ and u2 respectively, give 2x2r to the index. In sum, we have 3x2r. One can go to higher orders if one wishes.

6 Chern-Simons theory of SU(Nc)k and its dual

The duality of Chern-Simons theory of SU(Nc) gauge theory can be obtained from the Aharony duality of SU(Nc) gauge theory. By giving some of the avors axial mass, one can generate Chern-Simons term. We start with SU(Nc) theory with Nf + k avors. Matters have axial charge +1 and integrating k matters gives CS level k > 0. The dual theory starts with U(1) U(Nf + k Nc) gauge theory with BF term AU(1) dTrA, where TrA denotes the overall U(1) of U(Nf + k Nc). Integrating k of q, ~q gives CS level k of U(Nf + k Nc) , since q, ~q have axial charge 1. Integrating out v gives CS level 1 for U(1) gauge eld. Thus the dual theory is given by U(1)1 U(Nf + k Nc)k with Nf avors charged only under U(Nf + k Nc) and the BF term AU(1) dTrA. Subscript of the gauge group denotes the Chern-Simons level. Note that the above theory is charge-conjugation invariant. Now no eld is charged under U(1)1 so it can be integrated out.

Thus we obtain the following duality;

Electric theory: SU(Nc)k gauge theory, Nf pairs of fundamental/ anti-fundamental chiral superelds Qa, ~Qb with Chern-Simons level k.

Magnetic theory: U(Nf + k Nc) gauge theory, Nf pairs of fundamental/anti-fundamental chiral superelds qa, ~qa of U(Nf + k Nc), Nf Nf singlet superelds

17

Nf Nc

JHEP10(2013)198

Mab. The superpotential is given by

with the Chern-Simons term

d k

where AN~c is the U(c) = U(Nf + k Nc) gauge eld and = TrAc is the overall U(1) gauge eld.

Again we can discuss the chiral ring elements. In the elctric side we have mesons M, and baryons B, ~B. In the magnetic side, mesons are trivially matched. The baryon operators are mapped to the monopole operators of the form u+~qa1 ~qaNfNc and uqa1 qaNfNc .

If we turn on the unit ux of the overall U(1) of U(Nf + k Nc), Gauss constraint dictates that we should turn on Nf Nc matters. Using the residual gauge invariance, one can choose the color index running from 1 to Nf Nc.

The index of Chern-Simons SU(Nc) theory is

I =

XmiZ1Sym

I

c

where the same expressions are adopted from (5.13).

Here the N~c is the rank of the gauge group N~c = Nf + k Nc. Note that theres additional sign factor (1)kmi+mi at eq. (6.4). Such sign factor (1)km is observed for every U(1) factor with CS level k where m is the magnetic ux associated with U(1) gauge group. The origin of such sign factor is explained at section 2.

We compute the index of the above CS theory for some cases and nd the perfect matching. The result is exhibited at table 3.

We examine how gauge invariant BPS operators appear in the index expression. On the electric theory side, since these are Chern-Simons SU(N) gauge theories, chiral ring elements consist of mesons M and baryons B, ~B. Mesons M = Q ~Q contribute N2fx2r to the

index and baryons B QNc, ~B ~QNc give 2 Nf

Nc

xNcr to the index. But when Nf < Nc there are no baryons. It is easily seen from the index computation. For example, when Nc = 2, we should look for the coe cient of x2r = xNcr term. If Nf = 2 there is the baryon contribution I = + (22 + 2 22

)x2r + but if Nf = 1 there is no such contribution. Turning to the magnetic theory, mesons are trivially mapped to the siglet operators M. As explained before, baryon operators are matched by the monopole operators u+~qaNfNc

and uqa1 qaNfNc . They give

Nf

Nf Nc

W = Mq~q. (6.1)

Ac dAc 2i3 Ac Ac Ac

(6.2)

Yi=1dzi2izi (zi)kmiZgaugeZNfQ ZNf~Q , (6.3)

where ZQ, Z ~Q, Zgauge are given by eq. (5.8), (5.9). Here zNc = (z1 zNc1)1 and mNc = (m1 + + mNc1).

The index of dual Chern-Simons U(Nf + k Nc) is

I =

XmiZ1Sym

I

JHEP10(2013)198

Nc

Yi=1(1)kmi+miz kmi+m1++mc i

dzi2izi ZgaugeZNfq ZNf~q ZN2fM , (6.4)

for each 1 monopole ux, which contribute the

18

term 2 Nf

Nf Nc

xNcr.

Electric Magnetic(Nc, Nf, k) SU(Nc) U(Nf + k Nc) Index (r is the IR R-charge of Q.)

(2, 2, 1)

SU(2)

U(1)

1 16x2 + 88x4 x44r + 20x42r + 6x2r + 20x4r

64x2+2r +

(2, 1, 2)

SU(2)

U(1)

1 4x2 5x4 + 4x6 + 3x42r + 4x62r + x2r + x4r + x6r 4x4+2r +

(2, 3, 1)

SU(2)

U(2)

136x2 +558x4 x66r +21x42r +15x2r +105x4r +

490x6r 384x2+2r +

1 16x2 + 88x4 + 10x42r + 6x2r + 20x4r + 50x6r +

105x8r 64x2+2r 160x2+4r +

(2, 4, 1) SU(2) U(3) 1 64x2 + 36x42r + 28x2r + 336x4r + (2, 3, 2) SU(2) U(3) 1 36x2 + 15x2r + 105x4r +

(3, 3, 1)

SU(3)

U(1)

118x2+9x2r+2x3r+45x4r+18x5r+167x6r+90x7r+

513x8r +332x9r 18x2+r 144x2+2r 162x2+3r +

1 8x2 + 28x4 x44r + 8x42r + 16x4r + 4x2r +

10x4r 4x2+r 24x2+2r +

1 32x2 + 16x2r + 8x3r + 136x4r + 120x5r + 836x6r

48x2+r 480x2+2r +

118x2+9x2r+2x3r+45x4r+18x5r+167x6r+90x7r+

513x8r +332x9r 18x2+r 144x2+2r 162x2+3r + (3, 2, 3) SU(3) U(2) 1 8x2 + 4x42r + 4x2r 4x2+r +

Table 3. The results of the superconformal index comutation of Chern-Simons theory.

SU(Nf) SU(Nf) U(1)A U(1)T U(1)R Q Nf 1 1 0 r

~Q 1 Nf 1 0 r

X, ~X 1 1 0 0 2

n+1

Mj Nf Nf 2 0 2r + 2j

n+1

vj, 1 1 Nf 1 Nfr + Nf 2

n+1 (Nc 1) +

2j n+1

n+1 (Nc 1) +

7 SU(N) theory with Nf fundamental avors and adjoint matter

One can apply the same procedure to obtain Aharony duality for N = 2 SU(Nc) gauge theory with Nf fundamental and anti-fundamental avors and with one adjoint matter.

The proposed duality is as follows;

19

(2, 2, 2)

SU(2)

U(2)

JHEP10(2013)198

(3, 2, 2)

SU(3)

U(1)

(3, 4, 1)

SU(3)

U(2)

(3, 3, 2)

SU(3)

U(2)

2j n+1

q Nf 1 -1 0 r + 2

n+1

~q 1 Nf -1 0 r + 2

n+1

~vj, 1 1 Nf 1 Nfr Nf + 2

n+1 (Nc + 1) +

u 1 1 Nf 0 nNf(1 r) + 2n

2n n+1

Table 4. Quantum numbers of various elds.

Electric theory: SU(Nc) gauge theory(without Chern-Simons term), Nf pairs of fundamental/ anti-fundamental chiral superelds Qa, ~Qb(where a, b denote avor indices), an adjoint supereld X, and the superpotential We = Tr Xn+1.

Magnetic theory: U(1) U(nNf Nc) gauge theory with BF coupling,

AU(1) dTrAU(nNfNc) (7.1)

Nf pairs of fundamental/anti-fundamental chiral superelds qa, ~qa of U(nNf Nc), Nf Nf singlet superelds (Mj)ab, j = 0, . . . , n 1, an adjoint supereld ~X of

U(nNf Nc) , 2n superelds v0,,. . . ,vn1, charged under U(1), 2n superelds ~v0,,. . . ,~vn1, charged under U(1) and a superpotential

Wm = Tr ~Xn+1 +

The chiral superelds of the theory have charges under various symmetries as we specied at the table above.v0, and ~v0, are minimal bare monopoles of electric theory and magnetic theory, respectively. Those correspond to excitation of magnetic ux (1, 0, . . . , 0). For the description of the monopole operators we had better use the operator state correspondence to describe the operator as the corresponding state on RS2. When magnetic ux (1, 0, . . . , 0) is excited the gauge group U(Nc) is broken to U(1) U(Nc 1). Then X takes the form

X = X11 0

0 X

!

(7.3)

where X is an adjoint eld of U(Nc 1) unbroken gauge group. We denote the dressed monopole operator vi, Tr(v0,Xi), i = 1, . . . , n 1 with the trace taken over U(1). For example v1, = X11| 1, 0, . . .i. The details can be found in [34].

Lets consider the chiral ring elements and how they are mapped . For the adjoint and mesons, we have the following correspondence

TrXi Tr ~Xi

QaXj ~Qb (Mj)ab. (7.4)

We have n independent monopole operators Yi with i = 0 n 1, which are mapped to vi,+vi,. Baryons can be constructed not only from Q, ~Q but also from some combinations of X and Q or X and ~Q. For example XjiQi or XjiXikQk can replace one or many Qs in the baryon operators, where i, j, k are gauge indicies. Thus B QNc1(XQ) could be a chiral ring element.

On the dual side, baryon like operators b, ~b which is a coupled state of the monopole operators u of the gauged U(1) and (anti)fundamental and adjoint matters elds. Where baryons could be b uqc or b uqc1( ~Xq) wherec = nNf Nc. Anit-baryons ~b could be dened similarly, e.g., ~b u+~qc or ~b u+~qc1( ~Xq). The detailed matching of

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JHEP10(2013)198

n1

Xj=0

Mj ~q ~Xn1jq +

n1

Xi=0(vi,+~vn1i, + vi,~vn1i,+). (7.2)

the chiral ring is quite delicate. For the Aharony duality of U(Nc) gauge theory with an adjoint X, the chiral rings are constrained by characteristic equations of adjoint X and ~X. Classically, there are Nc independent operators Tr Xi, i = 1, . . . , Nc due to characteristic equation of X which is in U(Nc) adjoint representation. With a superpotential W = Tr Xn+1 there are l independent operators Tr Xi, i = 0, . . . , l where l = min(n 1, Nc). As explained in [24, 34], if the gauge group of electric side is smaller than that of magnetic side, Nc Nc, the number of (classical) chiral ring generators of electric side is less than magnetic side. The redundant chiral ring generators of the magnetic theory are cancelled by some monopole operators. On the other hand, if Nc > Nc, the electric theory seems to have more chiral ring generators than magnetic theory. But some non-trivial relation of monopole operators reduce the number of state so the chiral ring is again the same.

We expect similar mechanism works here. Especially interesting case is baryon operators. For the electric case, such baryon operators exist for Nf Nc. However, for the magnetic side, the similar condition is Nf c = nNf Nc. Unless n = 2 and Nf = Nc two conditions are incompatible. Thus we expect the nonperturbative truncation of the chiral ring occurs. Here we examine simple cases of n = 2 and Nf = Nc in the below and nd the perfect matching. We will explore more general cases in the future work [36].

We compute the indices for some values of n, Nc and Nf. Results are listed in the table below.

Lets work out the operator contents for the few lower orders of the index results. Consider n = Nc = Nf = 2 case. The x2/3 comes from X of the electric side and ~X of the magnetic theory. In the electric theory, 4x2r term comes from the meson contribution 2x2r and from baryon contribution Q1Q2, ~Q1 ~Q2. On the dual side the singlets M0 give 4x2r and uq1q2, u+~q1~q2 give 2x2r.

We check another term 16x2r+2/3. On the electric side, TrXQ ~Q counts four, QX ~Q counts four. Another four come from Qa(XQb) and one from QQTrX but they are not completely independent of each other. They satises Q1(XQ2)Q2(XQ1)Q1Q2TrX = 0.

So the baryon like operators give 4, and the similar anti-baryons operators (e.g., ~Qa(X ~Qb))

give 4, summing up to 16x2r+2/3. On the dual magnetic theory, the counting is basically the same as the electric case, except the mesons are mapped to singlet Mj and baryons B

and ~B are mapped to b, ~b. From Y , there are x8/34r and the same term comes from v0,+v0, of the magnetic theory. There are also x44r which comes from v1,+v1,.

We consider one more example. For the case of Nc = Nf = 1 and n = 2, the electric theory is a free theory with matter elds Q, ~Q and X. The dual magnetic theory has gauge group U(1) U(1). The chiral ring elements of the electric theory is simply the free chiral matter elds Q, ~Q and X. We can read o it from the index expression. For example, adjoint eld X gives x2/3. Each Q and ~Q gives xr, leading to 2xr in the index. The term 3x2r comes from QQ, ~Q ~Q, Q ~Q, the term 3x2r+2/3 comes from XQQ, X ~Q ~Q, XQ ~Q and so on.

On the dual magnetic theory, the adjoint matter eld ~X correspond to the X of electric theory and it gives x2/3 as expected. The monopole operators u of the gauged U(1) are coupled to the matter elds q, ~q, so u+~q and uq correspond to the chiral elds Q, ~Q. We also have a singlet eld M0, (u+~q)2, (uq)2 giving 3x2r which corresponds to Q ~Q, ~Q ~Q, QQ

21

JHEP10(2013)198

Electric Magnetic(n, Nc, Nf) SU(Nc) U(1) U(nNf Nc) Index (r is the IR R-charge of Q.)

(2, 1, 1)

free

U(1) U(1)

1+x2/34x23x8/3+(3+3x2/3)x2r+(4+4x2/3)x3r+

5x4r + xr(2 + 2x2/3 4x2) + xr(2x2 2x8/3) +

3 10r + (1 +

4x2/3)x4r + x6r + x2r(1 + 4x2/3 + x4/3) + x2r(x2/3 + 2x4/3 + x2) + x4r(x4/3 + 2x2 + x8/3) + x6r(x2 +

2x8/3) + x8r(x8/3 + 2x10/3) +

1 + x2/3 + x4/3 16x2 31x8/3 + x 16

1 + x2/3 + 10x4/3 + 20x2 + x412r + 4x 83 6r + (10 + 26x2/3)x4r+20x6r+4x 23 +3r +x2r(4+8x2/3+24x4/3)+ x2r(4x4/3 + 12x2) + x4r(x4/3 + 3x2 + 12x8/3) + x8r(x8/3 + 3x10/3) +

1+px+2x+x3/2 3x2 +x48r +x4r +x2r(1+4px+ 5x)+x2r(x+2x3/2+3x2+2x5/2)+x4r(x2+2x5/2+

3x3) + x6r(x3 + 2x7/2) +

1 + px + 12x + 47x3/2 + 154x2 + x416r + (1 + 3px)x312r + 4x310r + (4 + 16px)x26r + (20 + 60px)x6r + 35x8r + x4r(10 + 26px + 99x) + x2r(4 + 8px + (36 + 116px)x) + x2r((4 + 16px)x + 60x2) + x4r((1+3px)x+(17+55px)x2)+x8r((1+3px)x2+ 19x3)

1+x2/5+2x4/5+2x6/5+2x8/53x2+(1+4x2/5)x4r + x2r(1 + 4x2/5 + 5x4/5 + 8x6/5) + x2r(x6/5 + 2x8/5 + 3x2 + 4x12/5) + x4r(x12/5 + 2x14/5) +

1 + 2x2/5 + 6x4/5 + x210r + x 85 8r + (1 + 3x2/5)x2r +

x4r+x4r(x4/5+4x6/5)+x2r(x2/5+4x4/5+10x6/5)+ x6r(x6/5 + 4x8/5) +

Table 5. Indices of SU(Nc) gauge theory with adjoint matter and its dual theory.

of the electric theory. The index doesnt have the term x22r which may correspond to v0,+v0, becaue the superpotential W = M0 ~Xq~q + M1q~q + M0v0,+v0, makes it Q-exact.

The last term of the superpotential or similar terms can be generated for special values of (n, Nc, Nf) [34].

Lets work out one more example where nonperturbative truncation of the chiral ring occurs. For terms in x2r+2/3, naively there are four chiral ring elements in the magnetic theory which are M1, ~XM0, ~X(u+~q)2 and ~X(uq)2. The existence of M1 could be a problem here because the electric theory is a free theory, the corresponding operator QX ~Q is not di erent from Q ~QX = XQ ~Q. This means that M1 should not be a chiral ring element and should be canceled by some other terms. If we consider R-charges of the possible states, there are four candidates canceling M1, (~v0,+~v0,, v1,+~v0,+, v1,~v0,, v1,+v1,).

But normally they are canceled against each other due to the superpotential v1,~v0,.

But the state ~v0,+~v0, does not exist for the case of U(1) gauge theory because ~v0,+ and ~v0, have opposite charges for the same Cartan sector, so they are paired up. Rest of

22

(2, 2, 1)

(2, 2, 2)

SU(2)

SU(2)

U(1) U(0)

1 + x2/3 + x4/3 4x2 + x412r + x 10

3 8r + 20x4r + x2r(6+16x2/3+6x4/3)+x4r(x8/3+2x10/3+x4)+

(2, 3, 2)

U(1) U(2)

JHEP10(2013)198

SU(3)

U(1) U(1)

(3, 2, 1)

SU(2)

U(1) U(1)

(3, 4, 2)

SU(4)

U(1) U(2)

U(1) U(2)

(4, 2, 1)

SU(2)

(4, 3, 1)

SU(2)

U(1) U(1)

the candidate states have the same energy states, two of them fermionic and one of them bosonic. Thus M1 truncated nonperturbatively. Therefore we have the right value 3x2r from ~XM0, ~X(u+~q)2 and ~X(uq)2.

Acknowledgments

We thank Dongmin Gang, Eunkyung Koh for the collaboration at the initial stage of the project and thank for Chiung Hwang, Hyungchul Kim for the discussions. We also thank Ofer Aharony, Shlomo S. Razamat, Nathan Seiberg and Brian Willett for the correspondences. J.P. was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) with the Grants No. 2012-009117, 2012-046278 and 2005-0049409 through the Center for Quantum Spacetime (CQUeST) of Sogang University. JP also appreciates APCTP for its stimulating environment for research.

References

[1] A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, http://dx.doi.org/10.1007/JHEP03(2010)089

Web End =JHEP 03 (2010) 089 [arXiv:0909.4559] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0909.4559

Web End =INSPIRE ].

[2] N. Hama, K. Hosomichi and S. Lee, SUSY gauge theories on squashed three-spheres, http://dx.doi.org/10.1007/JHEP05(2011)014

Web End =JHEP 05 (2011) 014 [arXiv:1102.4716] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1102.4716

Web End =INSPIRE ].

[3] S. Kim, The complete superconformal index for N = 6 Chern-Simons theory,http://dx.doi.org/10.1016/j.nuclphysb.2012.07.015

Web End =Nucl. Phys. B 821 (2009) 241 [Erratum ibid. B 864 (2012) 884] [arXiv:0903.4172] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0903.4172

Web End =INSPIRE ].

[4] Y. Imamura and S. Yokoyama, Index for three dimensional superconformal eld theories with general R-charge assignments, http://dx.doi.org/10.1007/JHEP04(2011)007

Web End =JHEP 04 (2011) 007 [arXiv:1101.0557] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1101.0557

Web End =INSPIRE ].

[5] A. Giveon and D. Kutasov, Seiberg duality in Chern-Simons theory,

http://dx.doi.org/10.1016/j.nuclphysb.2008.09.045

Web End =Nucl. Phys. B 812 (2009) 1 [arXiv:0808.0360] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0808.0360

Web End =INSPIRE ].

[6] V. Niarchos, Seiberg duality in Chern-Simons theories with fundamental and adjoint matter, http://dx.doi.org/10.1088/1126-6708/2008/11/001

Web End =JHEP 11 (2008) 001 [arXiv:0808.2771] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0808.2771

Web End =INSPIRE ].

[7] V. Niarchos, R-charges, chiral rings and RG ows in supersymmetric Chern-Simons-matter theories, http://dx.doi.org/10.1088/1126-6708/2009/05/054

Web End =JHEP 05 (2009) 054 [arXiv:0903.0435] [http://inspirehep.net/search?p=find+J+JHEP,0905,054

Web End =INSPIRE ].

[8] A. Kapustin, B. Willett and I. Yaakov, Nonperturbative tests of three-dimensional dualities, http://dx.doi.org/10.1007/JHEP10(2010)013

Web End =JHEP 10 (2010) 013 [arXiv:1003.5694] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1003.5694

Web End =INSPIRE ].

[9] A. Kapustin, B. Willett and I. Yaakov, Tests of Seiberg-like duality in three dimensions, arXiv:1012.4021 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.4021

Web End =INSPIRE ].

[10] D. Bashkirov and A. Kapustin, Dualities between N = 8 superconformal eld theories in three dimensions, http://dx.doi.org/10.1007/JHEP05(2011)074

Web End =JHEP 05 (2011) 074 [arXiv:1103.3548] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1103.3548

Web End =INSPIRE ].

[11] C. Krattenthaler, V. Spiridonov and G. Vartanov, Superconformal indices of three-dimensional theories related by mirror symmetry, http://dx.doi.org/10.1007/JHEP06(2011)008

Web End =JHEP 06 (2011) 008 [arXiv:1103.4075] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1103.4075

Web End =INSPIRE ].

[12] D. Ja eris and X. Yin, A duality appetizer, arXiv:1103.5700 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1103.5700

Web End =INSPIRE ].

[13] A. Kapustin, Seiberg-like duality in three dimensions for orthogonal gauge groups, arXiv:1104.0466 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1104.0466

Web End =INSPIRE ].

23

JHEP10(2013)198

[14] B. Willett and I. Yaakov, N = 2 dualities and Z extremization in three dimensions, arXiv:1104.0487 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1104.0487

Web End =INSPIRE ].

[15] A. Kapustin and B. Willett, Generalized superconformal index for three dimensional eld theories, arXiv:1106.2484 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1106.2484

Web End =INSPIRE ].

[16] D. Bashkirov, Aharony duality and monopole operators in three dimensions, arXiv:1106.4110 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1106.4110

Web End =INSPIRE ].

[17] C. Hwang, H. Kim, K.-J. Park and J. Park, Index computation for 3d Chern-Simons matter theory: test of Seiberg-like duality, http://dx.doi.org/10.1007/JHEP09(2011)037

Web End =JHEP 09 (2011) 037 [arXiv:1107.4942] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1107.4942

Web End =INSPIRE ].

[18] D. Gang, E. Koh, K. Lee and J. Park, ABCD of 3d N = 8 and 4 superconformal eld theories, arXiv:1108.3647 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.3647

Web End =INSPIRE ].

[19] D. Berenstein and M. Romo, Monopole operators, moduli spaces and dualities, http://dx.doi.org/10.1007/JHEP03(2012)063

Web End =JHEP 03 (2012) 063 [arXiv:1108.4013] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.4013

Web End =INSPIRE ].

[20] T. Morita and V. Niarchos, F-theorem, duality and SUSY breaking in one-adjoint Chern-Simons-Matter theories, http://dx.doi.org/10.1016/j.nuclphysb.2012.01.003

Web End =Nucl. Phys. B 858 (2012) 84 [arXiv:1108.4963] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.4963

Web End =INSPIRE ].

[21] F. Benini, C. Closset and S. Cremonesi, Comments on 3D Seiberg-like dualities, http://dx.doi.org/10.1007/JHEP10(2011)075

Web End =JHEP 10 (2011) 075 [arXiv:1108.5373] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.5373

Web End =INSPIRE ].

[22] C. Hwang, K.-J. Park and J. Park, Evidence for Aharony duality for orthogonal gauge groups, http://dx.doi.org/10.1007/JHEP11(2011)011

Web End =JHEP 11 (2011) 011 [arXiv:1109.2828] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1109.2828

Web End =INSPIRE ].

[23] O. Aharony and I. Shamir, On O(Nc) D = 3 N = 2 supersymmetric QCD theories, http://dx.doi.org/10.1007/JHEP12(2011)043

Web End =JHEP 12 (2011) 043 [arXiv:1109.5081] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1109.5081

Web End =INSPIRE ].

[24] A. Kapustin, H. Kim and J. Park, Dualities for 3D theories with tensor matter, http://dx.doi.org/10.1007/JHEP12(2011)087

Web End =JHEP 12 (2011) 087 [arXiv:1110.2547] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1110.2547

Web End =INSPIRE ].

[25] K. Intriligator and N. Seiberg, Aspects of 3D N = 2 Chern-Simons-matter theories, http://dx.doi.org/10.1007/JHEP07(2013)079

Web End =JHEP 07 (2013) 079 [arXiv:1305.1633] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1305.1633

Web End =INSPIRE ].

[26] D.L. Ja eris, The exact superconformal R-symmetry extremizes Z, http://dx.doi.org/10.1007/JHEP05(2012)159

Web End =JHEP 05 (2012) 159 [arXiv:1012.3210] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.3210

Web End =INSPIRE ].

[27] O. Aharony, IR duality in D = 3 N = 2 supersymmetric USp(2Nc) and U(Nc) gauge theories, http://dx.doi.org/10.1016/S0370-2693(97)00530-3

Web End =Phys. Lett. B 404 (1997) 71 [http://arxiv.org/abs/hep-th/9703215

Web End =hep-th/9703215 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9703215

Web End =INSPIRE ].

[28] A. Kapustin, Three-dimensional avatars of Seiberg duality, talk given at Simons Summer Workshop in Mathematics and Physics 2011, July 25Agugust 19, C.N. Yang Institute, Stony Brook, U.S.A. (2011); see http://media.scgp.stonybrook.edu/video/video.php?f=201108101qtp.mp4

Web End =http://media.scgp.stonybrook.edu/video/ http://media.scgp.stonybrook.edu/video/video.php?f=201108101qtp.mp4

Web End =video.php?f=201108101qtp.mp4 around minute 73.

[29] O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3D dualities from 4D dualities, http://dx.doi.org/10.1007/JHEP07(2013)149

Web End =JHEP 07 (2013) 149 [arXiv:1305.3924] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1305.3924

Web End =INSPIRE ].

[30] O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, http://dx.doi.org/10.1016/S0550-3213(97)00323-4

Web End =Nucl. Phys. B 499 (1997) 67 [http://arxiv.org/abs/hep-th/9703110

Web End =hep-th/9703110 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9703110

Web End =INSPIRE ].

[31] E. Witten, SL(2, Z) action on three-dimensional conformal eld theories with Abelian symmetry, http://arxiv.org/abs/hep-th/0307041

Web End =hep-th/0307041 [http://inspirehep.net/search?p=find+EPRINT+hep-th/0307041

Web End =INSPIRE ].

[32] T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, arXiv:1108.4389 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.4389

Web End =INSPIRE ].

24

JHEP10(2013)198

[33] T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3D indices, arXiv:1112.5179 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1112.5179

Web End =INSPIRE ].

[34] H. Kim and J. Park, Aharony dualities for 3D theories with adjoint matter, http://dx.doi.org/10.1007/JHEP06(2013)106

Web End =JHEP 06 (2013) 106 [arXiv:1302.3645] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1302.3645

Web End =INSPIRE ].

[35] D. Gang, E. Koh, S. Lee and J. Park, Superconformal index and 3D-3D correspondence for mapping cylinder/torus, arXiv:1305.0937 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1305.0937

Web End =INSPIRE ].

[36] H. Kim, J. Park and K. Park, work in progress.

[37] J. Bhattacharya and S. Minwalla, Superconformal indices for N = 6 Chern-Simons theories, http://dx.doi.org/10.1088/1126-6708/2009/01/014

Web End =JHEP 01 (2009) 014 [arXiv:0806.3251] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0806.3251

Web End =INSPIRE ].

[38] C. Hwang, H.-C. Kim and J. Park, Factorization of the 3D superconformal index, arXiv:1211.6023 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1211.6023

Web End =INSPIRE ].

JHEP10(2013)198

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