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Mirror symmetry of 3D N = 4 gauge theories and supersymmetric indices

Tadashi Okazaki

Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5

(Received 4 August 2019; published 24 September 2019)

We compute supersymmetric indices to test mirror symmetry of three-dimensional (3D) N ¼ 4 gauge

theories and dualities of half-Bogomol’ny Prasad Sommerfield enriched boundary conditions and

interfaces in four-dimensional N ¼ 4 super Yang-Mills theory. We find the matching of indices as

strong evidences for various dualities of the 3D interfaces conjectured by Gaiotto and Witten under the

action of S-duality in Type IIB string theory.

DOI: 10.1103/PhysRevD.100.066031

I. INTRODUCTION AND CONCLUSIONS

Three-dimensional (3D) N ¼ 4 supersymmetric gauge

theories have a moduli space of supersymmetric vacua

consisting of a Higgs branch M

and a Coulomb branch

which are hyperkähler manifolds. The theories have an

intriguing duality, known as mirror symmetry, which

relates theories with completely different UV descriptions

where the Higgs branch and the Coulomb branch are

exchanged and fayet iliopoulos parameters and mass

parameters are exchanged [1]. This mapping is very non-

trivial. The Higgs branch is a hyperkähler quotient realized

as a zero locus of the D-term constraints divided by the

gauge group. On the Higgs branch, the gauge symmetry is

broken completely and the Higgs branch is not affected by

the quantum correction. On the Coulomb branch, the gauge

group is generically broken to its maximal torus. Unlike the

Higgs branch, the Coulomb branch receives perturbative

and nonperturbative quantum corrections.

The quantum Coulomb branch is studied in terms of

Hilbert series in [2]. The Hilbert series is a generating

function that counts chiral operators on the branches M

H=C

of vacua, graded by their dimensions and quantum numbers

under global symmetry. It encodes the quantum numbers of

generators and relations of the chiral ring C½M

H=C

 of the

corresponding branches M

H=C

of vacua. Subsequently, the

quantum Coulomb branch has been described in [3] in

terms of the Abelianization map that translates the vev of a

monopole operator into a linear combination of Abelian

monopole operator vevs in the low-energy effective theory.

Furthermore, the quantum Coulomb branch has been

mathematically defined in [4–6].

Three-dimensional N ¼ 4 gauge theories can be realized

in Type IIB string theory using the brane configuration [7].

Mirror symmetry can be viewed as arising from S-duality of

Type IIB string theory. Type IIB brane construction is

extended to further study of half-Bogomol’ny Prasad

Sommerfield boundary conditions and interfaces for four-

dimensional N ¼ 4 super Yang-Mills (SYM) theory [8–10],

quarter-BPS corner configuration for four-dimensional

N ¼ 4 SYM theory [11,12], half-BPS boundary conditions

for three-dimensional N ¼ 4 gauge theory [13], and two-

dimensional N ¼ð0; 4Þ supersymmetric gauge theories

[14]. S-duality turns out to give a physical underpinning

to geometric Langlands program [11,15–17] and symplectic

duality [18,19].

In this paper, we compute supersymmetric full- and half-

indices to test mirror symmetry of 3D N ¼ 4 gauge theories

and to extend the analysis in [12] for the dualities of half-BPS

boundary conditions and interfaces in 4D N ¼ 4 SYM

theory.

These dualities were originally conjectured by

Gaiotto and Witten [10]. There have been plenty of works

on the subject of 3D N ¼ 4 full superconformal indices

[21–27]; however, general tests of mirror symmetry for 3D

N ¼ 4 gauge theories have not appeared in the literature

except for the simplest Abelian mirror symmetry [28],in

contrast to 3D N ¼ 2 gauge theories [29–31]. Additionally,

one drawback of the 3D full-indices is that they are

insensitive to the boundary conditions for 4D N ¼ 4 gauge

theory, which involves singular boundary conditions speci-

fied by the Nahm poles [8]. In order to address these open

issues, we evaluate supersymmetric 4D half-indices and 3D

full-indices which count local operators both in four and

three dimensions and present many identities of indices by

[email protected]

Published by the American Physical Society under the terms of

the Creative Commons Attribution 4.0 International license.

Further distribution of this work must maintain attribution to

the author(s) and the published article’s title, journal citation,

and DOI. Funded by SCOAP

See [20] for further dualities of N ¼ð0; 4Þ half-BPS boun-

daries for 3D N ¼ 4 gauge theories and quarter-BPS corners for

4D N ¼ 4 gauge theories.

PHYSICAL REVIEW D 100, 066031 (2019)

2470-0010=2019=100(6)=066031(42) 066031-1 Published by the American Physical Society

checking several terms of series expansion. As discussed in

[28], after the special limits, the full-indices of 3D N ¼ 4

gauge theories reduce to the Hilbert series. As a result, the

identities of indices provide promotions of identities of

Hilbert series discussed in [2,28]. Furthermore, the full-

indices of 3D N ¼ 4 gauge theories can be used to general-

ize the analysis in [12] of the half-BPS boundary conditions

and interfaces for 4D N ¼ 4 SYM theory. We present

general half-indices for enriched half-BPS boundary con-

ditions and interfaces in 4D N ¼ 4 gauge theory and check

precise matching of the indices for dual pairs.

The organization of this paper is straightforward. In

Sec. II, we briefly review the supersymmetric indices

introduced in [12] and present formulae and some proper-

ties of half-indices for 4D N ¼ 4 gauge theories and full-

indices for 3D N ¼ 4 gauge theories. In Sec. III,we

evaluate full-indices for 3D N ¼ 4 Abelian gauge theories

and check mirror symmetry. In Sec. IV, we further examine

mirror symmetry by computing full-indices for 3D N ¼ 4

non-Abelian gauge theories. We also briefly check in

Sec. V Seiberg-like duality for 3D N ¼ 4 gauge theories

proposed in [10] between ugly theory and good theory in

terms of full-indices. In Sec. VI, we discuss the half-BPS

enriched boundary conditions for 4D N ¼ 4 SYM theory

which involve 3D N ¼ 4 gauge theories. We present strong

evidence for dualities between them conjectured from

string theory by calculating half-indices, which contain

nonregular Nahm pole b.c. Finally, in Sec. VII, we study

the half-BPS interfaces in 4D N ¼ 4 Uð1Þ gauge theory

including 3D N ¼ 4 Abelian gauge theories. We test

dualities between the interfaces by computing half-indices.

In the Appendix A, we show the q-expansions of indices

generated by Mathematica as well as the confirmed orders

which the indices agree up to.

II. INDICES

We begin with a definition of the quarter-index introduced

in [12]. It is a generalization of superconformal index in that it

can count local operators living in different dimensions, i.e.,

in 4D bulk, 3D boundary, and 2D junction. When the

configuration has a trivial junction, it becomes the half-

index that counts boundary local operators, while for the

trivial interface, it becomes the full-index that counts bulk

local operators. The quarter-index can be defined as the trace

over the cohomology of the preserved supercharges

IV ðt; x; qÞ ≔ Tr

ð−1Þ

Jþ

HþC

H−C

: ð2:1Þ

Here F is the Fermion number, J is the generator of the Uð1Þ

rotational symmetry in the space-time on which local

operators are supported. H and C stands for the Cartan

generators of the SUð2Þ

and SUð2Þ

R-symmetry groups,

respectively. f is the Cartan generator of the global sym-

metry. The choice of fugacity in the index (2.1) is fixed in

such a way that the power of q is always strictly positive for a

nontrivial local operator by a unitarity bound. This ensures

the convergence of the index. Consequently, the index can be

a formal power series in q whose coefficients are Laurent

polynomials in the other fugacities.

In this paper, we focus on the configurations of 3D

N ¼ 4 gauge theories which may couple to 4D N ¼ 4

gauge theories so that the indices (2.1) reduce to the full-

indices I of 3D N ¼ 4 gauge theories and/or the half-

indices II of 4D N ¼ 4 gauge theories. One may compute

the indices for appropriate configurations by a localization

procedure. However, we will not pursue that direction in

this paper, instead we will count local operators seriously

from physical consideration.

In the description of indices, we use the following

notation by defining q-shifted factorial:

ða; qÞ

≔ 1; ða; qÞ

≔

n−1

k¼0

ð1 − aq

Þ;

ðqÞ

≔

k¼1

ð1 − q

Þ;n≥ 1;

ða; qÞ

∞

≔

∞

k¼0

ð1 − aq

Þ; ðqÞ

∞

≔

∞

k¼1

ð1 − q

Þ;

ða



; qÞ

∞

≔ ða; qÞ

∞

ða

−1

; qÞ

∞

; ð2:2Þ

where a and q are complex variables with jqj < 1.

We compute the indices to test the 3D dualities and

dualities of the half-BPS boundary conditions/interfaces

which are conjectured from string theory [10]. We consider

five types of branes in Type IIB string theory whose world-

volumes span the following directions:

(i) D3-branes extended along x

(ii) NS5-branes extended along x

(iii) D5-branes extended along x

(iv) NS5

-branes extended along x

(v) D5

-branes extended along x

In other words, the brane configuration is summarized as

0123456789

D3 ∘∘∘−−− ∘ −−−

NS5 ∘∘∘∘∘∘−−−−

D5 ∘∘∘−−−− ∘∘∘

NS5

∘∘−−−− ∘∘∘∘

∘∘− ∘∘∘∘−−−

ð2:3Þ

The 3D N ¼ 4 gauge theories are realized by considering

D3-branes which are finite segments in the x

direction

between NS5-branes and may intersect with D5-branes [7].

The half-BPS boundaries and interfaces in 4D N ¼ 4

gauge theories are realized by considering D3-branes

which are (semi-)infinite D3-branes which end on or pass

through a sequence of NS5- and D5-branes [8].

TADASHI OKAZAKI PHYS. REV. D 100, 066031 (2019)

066031-2

Such brane setup is a nice tool for finding mirror pairs of

3D N ¼ 4 gauge theories and dual pairs of half-BPS

boundary conditions and interfaces in 4D N ¼ 4 SYM

theory by studying the action of S-duality [10].

A. Indices of 4D N = 4 Sym Theory

1. Full-indices

Four-dimensional N ¼ 4 SYM theory has SUð4Þ

R-symmetry. It contains the adjoint scalar fields transforming

as 6 under the SUð4Þ

.LetX and Y be the scalar fields

transforming as ð1; 3Þ and ð3; 1Þ under the SUð2Þ

SUð2Þ

⊂ SUð4Þ

. In the brane construction (2.3),the

scalar fields X and Y describe the positions of D3-branes

along the (x

, x

) directions and (x

, x

) directions,

respectively. The theory also has the 4D gauginos λ trans-

forming as ð2; 2Þ under the SUð2Þ

× SUð2Þ

The local operators in 4D N ¼ 4 gauge theory of gauge

group G which contribute to index have charges

∂

X ∂

Y ∂

λ ∂

G Adj Adj Adj Adj

Uð1Þ

nnnþ

n þ

Uð1Þ

02þþ

Uð1Þ

20þþ

fugacity

nþ

−2

−q

nþ1

−q

nþ1

ð2:4Þ

From (2.4), one can express the index for 4D N ¼ 4

UðNÞ gauge theory as

4D UðNÞ

ðt;qÞ¼

ðqÞ

∞

ðq

;qÞ

∞

ðq

−2

;qÞ

∞

i¼1

2πis

i≠j

;qÞ

∞

ðq

;qÞ

∞

ðq

;qÞ

∞

ðq

−2

;qÞ

∞

ð2:5Þ

Here the denominator comes from the scalar fields X and Y,

while the numerator captures the 4D gauginos. The integra-

tion contour for gauge fugacities s

is taken as a unit torus T

2. Half-indices

Four-dimensional N ¼ 4 SYM theory admits half-BPS

boundary conditions which preserve three-dimensional

N ¼ 4 supersymmetry with the R-symmetry group

SUð4Þ

broken down to SUð2Þ

× SUð2Þ

. In the brane

setup (2.3), they arise when parallel D3-branes end on a

single five-brane. There are two types of three-dimensional

boundaries/interfaces at x

¼ 0 realized by NS5

- and D5

branes and those at x

¼ 0 realized by NS5- and D5-branes.

Let us consider the half-BPS boundary conditions for 4D

N ¼ 4 Uð1Þ gauge theory. When a single D3-brane ends

on the NS5-brane and D5-brane, one finds the Neumann

b.c. N and Dirichlet b.c. D at x

¼ 0 for Uð1Þ gauge

theory, respectively,

N ∶ F

6μ

∂

¼ 0; ∂

∂

¼ 0; ∂

∂

¼ 0

D∶ F

μν

∂

¼ 0; ∂

∂

¼ 0; ∂

∂

¼ 0

μ;ν ¼ 0; 1;2:

ð2:6Þ

On the other hand, when the NS5

-brane and D5

-brane end

on a single D3-brane, one obtains the Neumann b.c. N

and Dirichlet b.c. D

at x

¼ 0 for Uð1Þ gauge theory,

respectively,

∶ F

2μ

∂

¼ 0; ∂

∂

¼ 0; ∂

∂

¼ 0

∶ F

μν

∂

¼ 0; ∂

∂

¼ 0; ∂

∂

¼ 0

μ; ν ¼ 0; 1;6:

ð2:7Þ

The half-indices of the Neumann b.c. N and Dirichlet

b.c. D

for 4D N ¼ 4 Uð1Þ gauge theory take the form

4D Uð1Þ

ðt; qÞ¼II

4D Uð1Þ

ðt; qÞ¼

ðqÞ

∞

ðq

−2

; qÞ

∞

: ð2:8Þ

The denominator is associated to the scalar fields Y charged

under Uð1Þ

, while the numerator correspond to a half of

the 4D gauginos. Likewise, the half-indices of Neumann

b.c. N

and Dirichlet b.c. D are

4D Uð1Þ

ðt; qÞ¼II

4D Uð1Þ

ðt; qÞ¼

ðqÞ

∞

ðq

; qÞ

∞

: ð2:9Þ

The denominator captures the scalar fields X charged under

Uð1Þ

, whereas the numerator is associated to a half of the

4D gauginos.

The half-BPS boundary conditions corresponding to N

D3-branes ending on a single NS5-brane (or NS5’) are also

Neumann b.c. for the UðNÞ gauge theory. We can denote

them as N and N

as in the Abelian case. By contrast,

when N multiple D3-branes end on a single D5-brane, one

finds a singular boundary condition associated to a regular

Nahm pole [8,32]. A single D5-brane or D5

-brane on

which N D3-branes end give rise to the Nahm or Nahm

pole boundary conditions,

Nahm∶ F

μν

∂

¼ 0;D



X þ



X ×



∂

¼ 0;



∂

¼ 0 μ; ν ¼ 0; 1; 2

Nahm

∶ F

μν

∂

¼ 0;D



∂

¼ 0;



Y þ



Y ×



∂

¼ 0 μ; ν ¼ 0; 1; 6; ð2:10Þ

MIRROR SYMMETRY OF 3D N ¼ 4 GAUGE … PHYS. REV. D 100, 066031 (2019)

066031-3

where we denote the scalar fields by



X and



Y as they are the

SUð2Þ

triplet and the SUð2Þ

triplet, respectively. The

Nahm equations for the scalar fields



X and



Y have singular

solutions



Xðx

Þ¼



;



Yðx

Þ¼



; ð2:11Þ

where



t ¼ðt

; t

Þ is a triplet of elements of the Lie

algebra g ¼ uðNÞ obeying the commutation relation

½t

; t

¼t

and cyclic permutation thereof. The choice



t specifies a homomorphism of Lie algebras ρ∶ suð2Þ →

g which maps the fundamental representation of UðNÞ to

the dimension N irreducible representation of suð2Þ. When

N D3-branes end on multiple D5-branes, one finds other

Nahm poles, including the Dirichlet b.c. as the trivial Nahm

pole corresponding to the case with N D5-branes.

The half-index of Neumann b.c. N for 4D N ¼ 4 UðNÞ

gauge theory takes the form

4DUðNÞ

ðt;qÞ¼

ðqÞ

∞

ðq

−2

;qÞ

∞

i¼1

2πis

i≠j

;qÞ

∞

ðq

−2

;qÞ

∞

ð2:12Þ

Again the integration contour for gauge fugacities s

is a

unit torus T

. The half-index of Dirichlet b.c. D for 4D

N ¼ 4 UðNÞ gauge theory is given by

4D UðNÞ

ðt; z

; qÞ¼

ðqÞ

∞

ðq

; qÞ

∞

i≠j

ðq

; qÞ

∞

ðq

; qÞ

∞

; ð2:13Þ

where z

is the fugacities associated to the boundary UðNÞ

global symmetry. The half-index for Nahm pole boundary

conditions in 4D N ¼ 4 UðNÞ gauge theory is

4D UðNÞ

Nahm

ðt; qÞ¼

k¼1

ðq

kþ1

2ðk−1Þ

; qÞ

∞

ðq

; qÞ

∞

: ð2:14Þ

As discussed in [12], the duality between the Neumann b.c.

and the regular Nahm pole b.c. implies equality between the

half-indices (2.12) and (2.14). In Sec. VI, we discuss more

general dual descriptions of the half-indices, including half-

index (2.13) for Dirichlet b.c.

We can get similar expressions for the mirror boundary

conditions, i.e., N

, D

, and Nahm

by setting t → t

−1

B. Indices of 3D N = 4 gauge theory

The 3D N ¼ 4 hypermultiplet consists of a pair of

complex scalars H,

H forming a doublet of SUð2Þ

and a

pair of complex fermions ψ

, ψ

forming a doublet of

SUð2Þ

. The charges of 3D N ¼ 4 hypermultiplet is

H ψ

−

Uð1Þ

00 −−þþ

Uð1Þ

þþ 0000

ð2:15Þ

The 3D N ¼ 4 Abelian vector multiplet consists of a 3D

Uð1Þ gauge field A

, three scalars, which we denote by

real and complex scalars σ, φ forming the SUð2Þ

triplet,

and two complex fermions ðλ

; η

Þ. The charges of 3D

N ¼ 4 vector multiplet is

σφλ



Uð1Þ

002þ − þ −

Uð1Þ

000þ −−þ

ð2:16Þ

The 3D N ¼ 4 superalgebra has an outer automorphism

that interchanges SUð2Þ

and SUð2Þ

. This automorphism

makes the ordinary supermultiplets into twisted super-

multiplets. The twisted hyper and vector multiplets can

be obtained by exchanging the Uð1Þ

and Uð1Þ

charges

of the hypermultiplet and vector multiplet, respectively.

In three dimensions, a photon is electric magnetic dual to

a scalar field, which we call dual photon. The dual photon is

periodic when the gauge group is compact and the shift

symmetry of the dual photon is a classical topological

symmetry whose conserved current is F where  is the

Hodge star. The conservation of F follows from the

Bianchi identity dF ¼ 0.

1. Matter multiplets

The index of a 3D N ¼ 2 chiral multiplet of charge þ1

under a U ð1Þ

flavor symmetry with fugacity x is

3DCM

ðx; qÞ¼

ðqx

−1

; qÞ

∞

ðx; qÞ

∞

: ð2:17Þ

Its denominator counts complex scalar and its ∂ derivatives,

while its numerator counts fermions and its ∂ derivatives

included in the 3D N ¼ 2 chiral multiplet.

The 3D N ¼ 4 hypermultiplet has the following oper-

ators which contribute to index:

∂

H ∂

∂

−

∂

−

Uð1Þ

þ − þ −

Uð1Þ

nn n þ

n þ

Uð1Þ

00 þþ

Uð1Þ

þþ 00

Fugacity

nþ

tx q

nþ

−1

−q

nþ

−1

x −q

nþ

−1

ð2:18Þ

The index for 3D N ¼ 4 hypermultiplet is

TADASHI OKAZAKI PHYS. REV. D 100, 066031 (2019)

066031-4

3DHM

ðt; x; qÞ¼

ðq

−1

x; qÞ

∞

ðq

−1

; qÞ

∞

ðq

tx; qÞ

∞

ðq

−1

; qÞ

∞

¼ I

3DCM





× I

3DCM



−1



: ð2:19Þ

It can be expanded as

3DHM

ðt; x; qÞ¼

∞

n¼0

k¼0

ðq

−2

; qÞ

ðq

−2

; qÞ

n−k

ðqÞ

n−k

n−2k

ð2:20Þ

The free 3D N ¼ 4 hypermultiplet has no Coulomb

branch local operators surviving in the H-twist. Therefore,

in the H-twist limit t → q

, the indices (2.19) and (2.20)

become trivial,

3DHM

ðt ¼ q

;x; qÞ¼1: ð2:21Þ

On the other hand, in the C-twist limit t → q

−

, the indices

reduce to

3DHM

ðt ¼ q

−

;x; qÞ¼

ð1 − xÞð1 − x

−1

: ð2:22Þ

This counts two bosonic generators in the algebra of Higgs

branch local operators.

The operators in 3D N ¼ 4 twisted hypermultiplet

which contribute to index are as follows:

∂

T ∂

∂

−

∂

−

Uð1Þ

þ − þ −

Uð1Þ

nnn þ

n þ

Uð1Þ

þþ 00

Uð1Þ

00 þþ

Fugacity

nþ

−1

nþ

−1

−q

nþ

tx −q

nþ

−1

ð2:23Þ

The index for 3D N ¼ 4 twisted hypermultiplet can be

obtained from the index (2.19) by setting t → t

−1

3D tHM

ðt;x;qÞ¼

ðq

tx;qÞ

∞

ðq

−1

;qÞ

∞

ðq

−1

x;qÞ

∞

ðq

−1

;qÞ

∞

¼ I

3DCM

ðq

−1

xÞ· I

3DCM

ðq

−1

Þ: ð2:24Þ

Again, it has an expansion

3D tHM

ðt; x; qÞ¼

∞

n¼0

k¼0

ðq

; qÞ

∞

ðq

; qÞ

∞

ðqÞ

n−k

n−2k

−n

ð2:25Þ

2. Gauge multiplets

While in four-dimensional case, the index only involves

integration over the gauge group [33,34], and in three-

dimensional case the index would have nonperturbative

contributions of monopole operators and contain the sum

over the magnetic fluxes of monopole operators for all

backgrounds [21,29].

Let us first consider the perturbative contributions to the

index. The charges of operators in 3D N ¼ 4 vector

multiplet contributing to the index are

ðσ þ iρÞ D

φ D

−

G Adj Adj Adj Adj

Uð1Þ

nnn þ

n þ

Uð1Þ

02−−

Uð1Þ

00− þ

Fugacity

nþ

−2

−q

nþ

ð2:26Þ

The perturbative index contributed from the local oper-

ators in (2.26) of 3D N ¼ 4 Uð1Þ vector multiplet takes the

form

3D pert Uð1Þ

ðt; qÞ¼

ðq

; qÞ

∞

ðq

−2

; qÞ

∞

2πis

; ð2:27Þ

where the integration contour of gauge fugacity s is

a unit circle. Similarly, the perturbative index for 3D

N ¼ 4 UðNÞ vector multiplet takes the form

3D pert UðNÞ

ðt; qÞ

ðq

; qÞ

∞

ðq

−2

; qÞ

∞

i¼1

2πis

i≠j



1 −



ðq

−2

;

ð2:28Þ

where the integration contour of gauge fugacities s

is a unit

torus T

Likewise, charges of operators in 3D N ¼ 4 twisted

vector multiplet are

ð˜σ þ i

ρÞ D

−

G Adj Adj Adj Adj

Uð1Þ

nnn þ

n þ

Uð1Þ

00þþ

Uð1Þ

02þ −

Fugacity

nþ

−q

nþ

−2

ð2:29Þ

MIRROR SYMMETRY OF 3D N ¼ 4 GAUGE … PHYS. REV. D 100, 066031 (2019)

066031-5

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3D N = 4规范理论和超对称指数的镜像对称

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