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JHEP07(2018)052

Published for SISSA by Springer

Received: April 24, 2018

Accepted: July 1, 2018

Published: July 9, 2018

An N = 1 3d-3d correspondence

Julius Eckhard,

Sakura Sch¨afer-Nameki

and Jin-Mann Wong

Mathematical Institute, University of Oxford,

Woodstock Road, Oxford, OX2 6GG, U.K.

Kavli Institute for the Physics and Mathematics of the Universe (WPI),

University of Tokyo, Kashiwa, Chiba 277-8583, Japan

Abstract: M5-branes on an associative three-cycle M

in a G

-holonomy manifold give

rise to a 3d N = 1 supersymmetric gauge theory, T

N =1

]. We propose an N = 1

3d-3d correspondence, based on two observables of these theories: the Witten index and

the S

-partition function. The Witten index of a 3d N = 1 theory T

N =1

] is shown to

be computed in terms of the partition function of a topological ﬁeld theory, a super-BF-

model coupled to a spinorial hypermultiplet (BFH), on M

. The BFH-model localizes on

solutions to a generalized set of 3d Seiberg-Witten equations on M

. Evidence to support

this correspondence is provided in the abelian case, as well as in terms of a direct derivation

of the topological ﬁeld theory by twisted dimensional reduction of the 6d (2, 0) theory. We

also consider a correspondence for the S

-partition function of the T

N =1

] theories,

by determining the dimensional reduction of the M5-brane theory on S

. The resulting

topological theory is Chern-Simons-Dirac theory, for a gauge ﬁeld and a twisted harmonic

spinor on M

, whose equations of motion are the generalized 3d Seiberg-Witten equations.

For generic G

-manifolds the theory reduces to real Chern-Simons theory, in which case we

conjecture that the S

-partition function of T

N =1

] is given by the Witten-Reshetikhin-

Turaev invariant of M

Keywords: Duality in Gauge Field Theories, Field Theories in Higher Dimensions, Su-

persymmetric Gauge Theory, Topological Field Theories

ArXiv ePrint: 1804.02368

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP07(2018)052

JHEP07(2018)052

D.2 5d N = 2 SYM on R

1,2

× S

D.3 Generalization to curved M

D.4 Reduction on S

D.4.1 Spherical harmonics and eigenspinors on S

D.4.2 Spectrum of massless ﬁelds 48

D.4.3 Real Chern-Simons theory: v = 0, 1 49

1 Introduction

Starting with the Alday-Gaiotto-Tachikawa (AGT) correspondence [1, 2], a series of

conjectures were put forward, relating d-dimensional N = 2 supersymmetric theories

N =2

6−d

], labeled by (6 − d)-dimensional manifolds M

6−d

, and topological or confor-

mal theories in 6 − d dimensions on the manifold M

6−d

. Each of these correspondences is

established in terms of the agreement between observables, e.g. sphere partition functions

(or indices) of the supersymmetric theories and partition functions/correlation functions

of the topological/conformal theories on M

6−d

One way to motivate these conjectures is to consider M5-branes wrapped on M

6−d

with

a topological twist which preserves N = 2 supersymmetry in d dimensions. For d = 4 it was

shown that the sphere partition function of the class S theories T

N =2

] [3] agree with

correlation functions of 2d Toda theory on the Riemann surface M

[1, 2]. The so-called

3d-3d correspondence [4, 5] similarly relates the S

-partition function of 3d N = 2 theories

N =2

] to the partition function of complex Chern-Simons theory on M

. Finally, for

d = 2 there is a correspondence for 2d (0, 2) theories labeled by four-manifolds M

. The

half-topologically twisted sphere partition function is conjectured to be computed by a

topological sigma-model into the monopole moduli space [6], whereas the Witten index is

identiﬁed with the Vafa-Witten (VW) [7] partition function of 4d N = 4 Super-Yang Mills

theory on M

[8].

Much of the progress in establishing these conjectures relies on the computational ad-

vances that were made for sphere partition functions of N = 2 supersymmetric theories

thanks to localization techniques [9, 10] — for a recent review see [11]. For less supersym-

metry, many of these tools are not quite as well developed thus far. One may hope that

formulating similar correspondences for N = 1 theories could give further insight into their

structure. An initial step towards developing an N = 1 version of the AGT correspondence

has been made in [12], with the goal to relate the sphere partition function of the class S

theories [13, 14] to 2d conformal blocks.

The goal of this paper is to develop an N = 1 version of the 3d-3d correspondence,

motivating it from ﬁrst principles by starting with the 6d (2, 0) theory. As is well-known the

N =2

] are obtained by wrapping M5-branes on special Lagrangian (sLag) three-cycles

– 1 –

JHEP07(2018)052

in Calabi-Yau three-folds. To retain N = 1 in 3d, we will show that the natural setup for

N =1

] is to realize M

as an associative three-cycle in a G

-holonomy manifold.

A priori we do not know what the topological theories are, which would complement the

3d N = 1 theories in such a 3d-3d correspondence. To determine these, it is useful to recall

the approach applied in the N = 2 setting: the topological theory, whose partition function

computes the sphere partition function of the T

N =2

6−d

] theories can be determined

from the sphere reduction of the 6d (2, 0) in an N = 2 preserving conformal supergravity

background [6, 15, 16]. We will employ this approach in the following to determine the

topological theories, which compute the following two observables of T

N =1

]: the T

partition function, i.e. Witten index, and the S

-partition function.

For 3d N = 1 the Witten index [17] is a well explored observable, much more so

than the S

-partition function. For this reason we will focus much of our attention on

this observable and provide non-trivial checks of the proposed correspondence. We will

derive the ‘dual’ topological theory by considering the 6d (2, 0) theory ﬁrst on T

, which

gives 3d N = 8 SYM, which we then topologically twist along M

, while preserving two

topological supercharges. This twist corresponds to the embedding of M

as an associative

cycle in a G

-manifold. In summary, the theory, whose partition function computes the

Witten index of the T

N =1

] will be shown to be a supersymmetric BF-theory coupled

to a spinorial hypermultiplet (BFH), which is a section of the normal bundle N

of M

inside the G

-manifold.

We should at this point elaborate brieﬂy on the geometry of associative three-cycles

in G

-holonomy manifolds [18, 19], which will play an important role in the behavior of

the Witten index. The normal bundle of an associative three-cycle is N

= S ⊗ V ,

where S is the spin bundle and V an SU(2)-bundle, in particular sections of the normal

bundle are twisted harmonic spinors on M

, satisfying a twisted Dirac equation. On an

odd-dimensional manifold the Dirac operator has vanishing index, which implies that the

dimension of the kernel (inﬁnitesimal deformations) equals that of its co-kernel (obstruc-

tions to these deformations), however the index does not reveal any information about the

non-triviality of each of these spaces. For non-generic choices of G

-structure, there can

be twisted harmonic spinors, which are accompanied with non-trivial obstructions of the

deformations, which they parametrize. This fact will reﬂect itself in the discontinuity/wall-

crossing of the Witten index of T

N =1

The 3d-3d correspondence that we propose for N = 1 results in an identiﬁcation of

the partition function of the BFH-model on M

with the Witten index of T

N =1

]. To

compute the partition function of the BFH-model, we show that its action is minimized on

solutions to a non-abelian generalization of the 3d Seiberg-Witten (gSW) equations. These

diﬀer from the standard SW equations in that the spinor transforms in the adjoint of a

gauge group G as well as under an additional SU(2)-bundle V (that appears in the normal

bundle of the associative cycle). The partition function of the BFH-model is computed by

-holonomy manifolds have two sets of supersymmetric, i.e. calibrated cycles: associative three-cycles

calibrated with the G

-three-form Φ and co-associative four-cycles, calibrated with ?Φ. M5-branes wrapping

co-associative four-cycles results in the VW twist along the four-manifold, i.e. 4d-2d duality studied in [6, 8].

– 2 –

JHEP07(2018)052

the Euler characteristic of the moduli space M

gSW

of solutions to the gSW equations.

particular for gauge group G = U(1) we derive the partition function explicitly and match

it with the index of T

N =1

, U(1)].

We ﬁnd that already in the abelian case the index is discontinuous under metric de-

formations, and jumps depending on the existence of twisted harmonic spinors. The fact

that the partition function of the BFH-model is only topological up to wall-crossing can, as

noted earlier, be traced back to the deformation theory of associative three-cycles within

-manifolds. At the location of the walls the normal deformations, appearing in the

gSW equations, are obstructed. This means that M

gSW

can become singular and its Euler

characteristic can jump.

A second less-explored observable for 3d N = 1 theories is the S

-partition function

(for a discussion of this observable for SCFTs see [20]). Whereas the N = 2 3d-3d cor-

respondence is studied for the S

-partition function, and many computational results are

available thanks to localization methods [21] (and see [11] for a recent review), the situ-

ation for N = 1 is much less explored. In particular localization will not be applicable

for computing the sphere partition functions with 3d N = 1 supersymmetry. Here we will

nevertheless determine what the ‘dual’ topological ﬁeld theory is, whose partition function

on M

would provide a conjecture for the S

-partition function of T

N =1

]. To do so, we

determine the conformal supergravity background similar to [6, 15, 16] and perform the

reduction of the 6d (2, 0) theory on a three-sphere, ﬁrst to 5d SYM and then on an S

3d, whilst preserving N = 1 supersymmetry. The resulting topological theory is shown to

be real Chern-Simons gauge theory on M

coupled to a twisted harmonic spinor φ, i.e. a

Chern-Simons-Dirac theory whose equations of motion are the generalized Seiberg-Witten

equations (for a review see [22]). For generic associatives in G

-manifolds there will be

no twisted harmonic spinors, and the theory reduces to real Chern-Simons theory. In this

case the topological partition function is given in terms of the Witten-Reshetikhin-Turaev

invariant [23, 24], which we conclude must compute the S

-partition function of T

N =1

The most interesting physical application arises when viewing the M5-branes as domain

walls in the 4d N = 1 theory obtained by M-theory on the G

-holonomy manifold. For Lens

spaces this case has been studied in [25] and we will connect these results, when discussing

concrete examples. This may in particular be of interest in recent constructions of new

-holonomy manifolds in [26–28] and singular limits thereof [29] that realize non-abelian

gauge groups.

Finally, we should remark that in the case that there is a non-trivial IR ﬁxed point, the

M5-branes on associatives have a holographic dual description in terms of AdS

-solutions,

where M

is a hyperbolic three-manifold [30]. This means the metric has constant sectional

curvature −1, and by Schur’s lemma the metric is Einstein. Examples of such associatives

exist in the Bryant-Salamon G

-manifolds [31], which are the total space of the spin bundle

over M

, with M

of constant sectional curvature ±1. For metrics on M

with negative

scalar curvature the associatives can indeed have obstructions, which are determined by

zero modes of the Dirac operator. It would be interesting to explore this from a holographic

point of view.

The analog for N = 2 is the moduli space of complex ﬂat connections.

– 3 –

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