Symmetry TFTs for 3d QFTs from M-theory Marieke van Beest1 Dewi S.W. Gould2 Sakura Schäfer-Nameki2 Yi-Nan Wang34 1Simons Center for Geometry and Physics SUNY

2025-05-02 0 0 901.35KB 62 页 10玖币

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Symmetry TFTs for 3d QFTs from M-theory

Marieke van Beest 1, Dewi S.W. Gould 2, Sakura Schäfer-Nameki 2, Yi-Nan Wang 3,4

1Simons Center for Geometry and Physics, SUNY,

Stony Brook, NY 11794, USA

2Mathematical Institute, University of Oxford,

Andrew-Wiles Building, Woodstock Road, Oxford, OX2 6GG, UK

3School of Physics,

Peking University, Beijing 100871, China

4Center for High Energy Physics, Peking University,

Beijing 100871, China

We derive the Symmetry Topological Field Theories (SymTFTs) for 3d supersymmetric quan-

tum ﬁeld theories (QFTs) constructed in M-theory either via geometric engineering or holog-

raphy. These 4d SymTFTs encode the symmetry structures of the 3d QFTs, for instance the

generalized global symmetries and their ’t Hooft anomalies. Using diﬀerential cohomology, we

derive the SymTFT by reducing M-theory on a 7-manifold Y7, which either is the link of a

conical Calabi-Yau four-fold or part of an AdS4×Y7holographic solution. In the holographic

setting we ﬁrst consider the 3d N= 6 ABJ(M) theories and derive the BF-couplings, which

allow the identiﬁcation of the global form of the gauge group, as well as 1-form symmetry

anomalies. Secondly, we compute the SymTFT for 3d N= 2 quiver gauge theories whose

holographic duals are based on Sasaki-Einstein 7-manifolds of type Y7=Yp,k(CP2). The

SymTFT encodes 0- and 1-form symmetries, as well as potential ’t Hooft anomalies between

these. Furthermore, by studying the gapped boundary conditions of the SymTFT we constrain

the allowed choices for U(1) Chern-Simons terms in the dual ﬁeld theory.

arXiv:2210.03703v1 [hep-th] 7 Oct 2022

Contents

1 Introduction 3

2 The Symmetry Topological Field Theory 7

3 SymTFT from M-Theory on Y710

3.1 Reduction using the Free Part of Cohomology . . . . . . . . . . . . . . . . . . . 10

3.2 Review of Diﬀerential Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Accounting for Torsion using Diﬀerential Cohomology . . . . . . . . . . . . . . 15

4 SymTFT Coeﬃcients from Geometry 20

4.1 SymTFT Coeﬃcients from Intersection Theory . . . . . . . . . . . . . . . . . . 20

4.2 Intersection Numbers of Toric 4-Folds . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Diﬀerential Cohomology Generators and Toric Divisors . . . . . . . . . . . . . . 23

5 SymTFT for Holography: ABJ(M) 25

5.1 Global Form of the Gauge Group . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2 ’t Hooft Anomaly for the 1-Form Symmetry . . . . . . . . . . . . . . . . . . . . 29

6 SymTFT for Holography: AdS4×Yp,k(CP2)31

6.1 SymTFT for General p, k .............................. 32

6.2 TheBF-Term..................................... 34

6.3 Boundary Conditions and Global Symmetries . . . . . . . . . . . . . . . . . . . 36

6.4 1-Form Symmetry Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7 Comparison to the Field Theory dual to Yp,k(CP2)38

7.1 QuiverGaugeTheories................................ 38

7.2 1-Form Symmetry of the Quivers . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.3 A Check on the Holographic Dictionary . . . . . . . . . . . . . . . . . . . . . . 42

8 Outlook 43

A Gauging Isometries 44

A.1 EquivariantCohomology............................... 45

A.2 SymTFT with Gauged Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . 49

B BF-Terms from Type IIA for Yp,k 49

1 Introduction

Symmetries and their anomalies have proven to be powerful tools in analysing quantum ﬁeld

theories (QFTs). Following the proposal in [1], symmetries are now understood to be the

set of topological operators in a given QFT. This generalization leads to a broadening of the

paradigm of ‘symmetry’, including higher-form symmetries [1], higher-group symmetries [2–7]

and non-invertible symmetries – the most recent progress being in higher-dimensional d≥4

theories [8–32]. Since their recent inception, generalized symmetries in string theory and

related theories have thus been studied extensively1.

The realization of QFTs within string theory has most of its utility when studying strongly

coupled regimes of theories, either in geometric engineering or from a dual holographic per-

spective. In some instances this provides otherwise inaccessible information about strongly

coupled QFTs, and in particularly favorable circumstances even a framework for classiﬁcation

of particular types of (supersymmetric) QFTs. The string theoretic realization has to capture

some of the salient physical properties of the QFTs, in particular the generalized symme-

tries and their ’t Hooft anomalies, which are robust under RG-ﬂow. The symmetry structure

of a QFT can be encoded in the so-called Symmetry Topological Field Theory (SymTFT or

Symmetry TFT) [75–77], see [28,29,31,78–80] for recent applications.

In a nutshell, the SymTFT is a (d+ 1)-dimensional topological ﬁeld theory, which upon

reduction on an interval with topological boundary conditions on one side, and physical (non-

topological) boundary conditions on the other, gives rise to the physical theory (and its

anomaly theory). The SymTFT contains, for example, the BF-couplings of the background

ﬁelds for global symmetries and the couplings that give rise to ’t Hooft anomalies. We will

shortly give a more thorough introduction in section 2.

The main observation in [77], is that for QFTs that have a realization in string theory, the

SymTFT can be derived from a supergravity approach. For a geometric engineering setup

that corresponds to a dimensional reduction on a (non-compact) space X, the SymTFT is

obtained by a suitable dimensional reduction on ∂X – and thus is naturally one dimension

higher than the QFT that is being engineered. In order to capture subtle aspects such as ﬁnite

group (higher-form) symmetries, the dimensional reduction is not a standard KK-reduction in

supergravity, but we are required to utilize diﬀerential cohomology to capture background ﬁelds

of ﬁnite group symmetries. Prior applications of diﬀerential cohomology to string/M-theory

have appeared in [77, 81–85], and for a mathematical review see [86].

Closely related to this is that of holography, where the strongly-coupled regime of a su-

1For a sample list of references see [33–74].

perconformal ﬁeld theory is realized in terms of string/M-theory on AdSd+1 ×Xspacetime.

In this case, the SymTFT can be interpreted as the topological couplings in the bulk super-

gravity on AdSd+1 (or in more general holographic setups). The most well-studied example

of AdS5×S5has the bulk coupling NRAdS5B2∧dC2, which is precisely an example of such

a BF-coupling for the 1-form symmetries of the dual 4d gauge theories (with gauge algebra

su(N)) [87]. More precisely, the SymTFT in holography lives in the near-boundary region

of the bulk and models the choice of global forms of the gauge group and the singleton sec-

tor [87, 88]. In terms of the formulation as generalized symmetries and SymTFTs, there has

been much recent interest in the holographic literature [24,28,84,87, 89–95], in particular for

AdS4/CFT3in [91] for 3d N= 6 SCFTs of ABJM type [96].

The goal of this paper is to determine the SymTFT for 3d QFTs which either have a

realization as geometric engineering in M-theory on an 8-manifold, or in terms of AdS4/CFT3

holographic setups in M-theory. These two constructions are closely related and we provide

a systematic computational approach to determining the SymTFT in both cases. The main

focus will be on conical 8-manifolds (with special holonomy) X8=C(Y7)in setups with and

without branes. Using diﬀerential cohomology in the supergravity reduction allows us to take

into account the eﬀects of torsion in the homology of Y7, which is associated with a new set

of background ﬁelds for ﬁnite higher-form symmetries.

For Y7a Sasakian 7-manifold we provide a prescription for computing the SymTFT coeﬃ-

cients explicitly, which correspond to secondary invariants in diﬀerential cohomology, from the

intersection theory in the non-compact complex 4-fold X82. We give detailed examples when

the cone X8is toric, in particular for X8=C4/Zkand X8=C(Yp,k(CP2)), where combinato-

rial formulas for intersection numbers can be explicitly computed. As such, we explain how

physical anomaly coeﬃcients and BF-terms are encoded in the geometric information of the

toric diagram. In summary, we will derive the SymTFT and give a procedure for computing

the coeﬃcients for

1. Geometric engineering: M-theory on a singular, non-compact Calabi-Yau 4-fold X8=

C(Y7), i.e. Y7is a Sasaki-Einstein 7-manifold.

2. Holography: AdS4×Y7solutions of M-theory, which are dual to M2-branes probing

X8=C(Y7), where Y7is a Sasakian 7-manifold (Sasaki-Einstein when X8is a Calabi-

Yau 4-fold).

For concrete applications, we will mostly focus on the holographic setups, leaving the explo-

ration of geometrically engineered 3d QFTs for future work. We ﬁrst compute the SymTFT

2We assume that X8has a resolution, so that we can rely on a smooth model and intersection theory therein.

in the M-theory models dual to ABJM and ABJ theories. This relatively simple holographic

setup is well-suited to demonstrate these new reﬁned geometric methods while, at the same

time, allowing for a match with known results from type IIA [91] in the case where discrete

background torsional ﬂux is turned oﬀ. Finally, we apply this machinery in a much more

subtle (and not completely ﬁxed) duality of 3d N= 2 theories realized on M2-branes probing

C(Yp,k(CP2)) [97–99]. By computing the SymTFT from the geometry, we obtain previously

unknown anomalies for these theories. Furthermore, we will see that analysing consistent

gapped boundaries of the SymTFT provides some further checks and balances to the pro-

posed dictionary, coming from the spectrum of extended operators.

Generalized symmetries and their ’t Hooft anomalies have a rich structure that has been

studied ﬁeld-theoretically from various angles in e.g. in [59, 100–103]. Some of these results

will be used later on to cross-check against our string theoretic results.

Let us summarize some of the main results.

ABJ(M). The N= 6 ABJM theories were conjectured in [96] as a class of U(N)k×U(N)−k

Chern-Simons matter theories with bifundamental matter, realized on NM2-branes probing

C(S7/Zk) = C4/Zk. The addition of bfractional M2-branes takes us to the ABJ variant [104]

with U(N+b)k×U(N)−kgauge group. The theories are conjectured to be holographically

dual to M-theory on AdS4×S7/Zkwith Nunits of 4-form ﬂux over the external space. It

was argued in [104] that the presence of fractional branes gives rise to an additional bunits of

torsional G4ﬂux in the near-horizon limit.

In [91], generalized symmetry methods were used to derive a suite of gauge theories within

the framework without fractional branes by considering diﬀerent boundary conditions of a

topological ﬁeld theory in one dimension higher. In particular, these gauge theories have the

same Lie algebra as the U(N)×U(N)ABJM theory, but diﬀerent global forms of the gauge

group.

In this work we use diﬀerential cohomology tools in M-theory to determine the BF-term

Skin

2π=ZAdS4

kB2∧dB1+NB2∧F , (1.1)

which matches that found from IIA in [91]. We emphasize that whilst the BF-term is familiar,

we must employ recent technology [77] to understand the geometric origin of the 1-form

symmetry background. In this work, we make this link precise by reducing 11-dimensional

supergravity on the torsional components of the M-theory geometry. We identify the 1-form

symmetry background as the reduction of ˘

G4, a diﬀerential cohomology reﬁnement of G4, on

the generator of H2(S7/Zk,Z) = Zk. Furthermore, we give a geometric derivation of a 1-form

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SymmetryTFTsfor3dQFTsfromM-theoryMariekevanBeest1,DewiS.W.Gould2,SakuraSchäfer-Nameki2,Yi-NanWang3;41SimonsCenterforGeometryandPhysics,SUNY,StonyBrook,NY11794,USA2MathematicalInstitute,UniversityofOxford,Andrew-WilesBuilding,WoodstockRoad,Oxford,OX26GG,UK3SchoolofPhysics,PekingUniversity,Beijing1008...

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Symmetry TFTs for 3d QFTs from M-theory Marieke van Beest1 Dewi S.W. Gould2 Sakura Schäfer-Nameki2 Yi-Nan Wang34 1Simons Center for Geometry and Physics SUNY

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