Symplectic gauge group on the Lens Space Antonio Amaritiaand Simone Rotaab

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Symplectic gauge group on the Lens

Space

Antonio Amaritiaand Simone Rotaa,b

aINFN, Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy

bDipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16, I-20133 Milano,

Italy

E-mail: antonio.amariti@mi.infn.it, simone.rota@mi.infn.it

Abstract: We compute the Lens space index for 4d supersymmetric gauge theo-

ries involving symplectic gauge groups. This index can distinguish between diﬀerent

gauge groups from a given algebra and it matches across theories related by super-

symmetric dualities. We provide explicit calculations for N= 4 SYM and for classes

of N= 2 and N= 1 Lagrangian quivers related by S-duality. In these cases the

index matches across the S-dual phases, while models in diﬀerent S-duality orbits

have a diﬀerent Lens index. We provide analogous computations for a 4d N= 1

toric quiver gauge theory corresponding to a Z7orbifold of C3. This SU(n)7gauge

theory becomes interesting in the case of n= 2 because it is conformally dual to

other two models, with symplectic and unitary gauge groups, bifundamentals and

antisymmetric tensors. We explicitly check this triality at the level of the Lens space

index.

arXiv:2210.12240v1 [hep-th] 21 Oct 2022

Contents

1 Introduction 2

2 Review 3

2.1 Line operators in gauge theories 3

2.2 Lens space index 4

3N= 4 S-duality with USp(2n)gauge group 7

3.1 SO(5) and USp(4) 8

3.2 SO(7) and USp(6) 10

3.3 SO(9) and USp(8) 12

3.4 SO(11) and USp(10) 13

4N= 2 S-duality of elliptic models with orientifolds 14

4.1 SO(5) ×SU(3) and USp(2) ×SU(4) 16

4.2 SO(7) ×SU(5) and USp(4) ×SU(6) 17

5N= 1 inherited S-duality of elliptic models with orientifolds 18

5.1 SO(5) ×SU(3) and USp(2) ×SU(4) 19

5.2 SO(7) ×SU(5) and USp(4) ×SU(6) 20

6 A conformal triality 21

6.1 Frame A 21

6.2 Frame B 22

6.3 Frame C 24

7 Conclusions 26

A Lens space index for symplectic gauge group 27

A.1 Almost commuting holonomies for USp(2n)27

A.2 Symmetric and Antisymmetric representations for USp(2n)29

A.3 Symmetric and Antisymmetric representations for SU(2n)32

A.4 Bifundamental representations 33

A.5 Almost commuting holonomies for SO(2n+ 1) 36

– 1 –

1 Introduction

Higher forms play a crucial role in the modern formulation of symmetries in QFT,

because they involve extended object and constrain their charges [1]. For example

the extended objects charged under a 1-form symmetry are loop operators. The

constraints on the charge spectrum of such lines, i.e. Wilson and ’t Hooft lines, is

reﬂected in the choice of the gauge group from a given gauge algebra in a gauge

theory [2,3]. Fixing the global properties of a gauge theory can have important

consequences, for example it ﬁxes the periodicity of the theta angle (see for example

[4] for the phenomenological implication in the SM).

Another more formal consequence of having diﬀerent gauge groups for a given

gauge algebra is that this diﬀerence can be observed on the partition function com-

puted in curved space on a spin manifold [3,5]. In general partition functions on

curved space are complicated quantities, but the diﬃculty of such problem is highly

simpliﬁed for some manifolds in supersymmetric gauge theories, thanks to the help

of localization [6].

In general such partition functions can still fail in distinguishing among diﬀerent

global structures because of the presence of further symmetries and dualities relating

theories with diﬀerent gauge groups. This is for example the case of S-duality in

N= 4 SYM. In this case many of the choices of the global structure are related to

each other by the action of the S-duality group on the spectrum of charges of the line

operators. There are nevertheless, depending on the choices of the gauge algebras

and the ranks, cases where multiple orbits of the S-duality group are present. The

partition functions on the curved space can in principle distinguish such orbits. This

picture has been conﬁrmed by explicit calculation from the Lens space index in [5].

This index was originally deﬁned in [7] and it corresponds to the superconformal

index computed on L(r, 1)×S1, where L(r, 1) 'S3/Zris the three-dimensional Lens

space and S1is the Euclidean time.

The ﬁrst explicit calculations of the index on such space have been performed

in [5] for the case of N= 4 SU(n). Furthermore Seiberg duality for SO(n)gauge

theories with vectors has been analyzed as well. Other calculations of the Lens space

index have been performed in [8] for the N= 2 SU(n)quiver corresponding to the

non-chiral orbifolds of C3(see also [9–11] for an analysis for non lagrangian theories).

In all the cases the index has been shown to match for models connected by duality

while it gave diﬀerent results for diﬀerent orbits of the S-duality group.

The 4d Lagrangian SCFTs zoology admits however many other possibile behav-

iors that have not yet be studied in terms of the Lens space index and that require

an investigation. For example models with symplectic gauge groups have not been

analyzed so far. This comprises the case of N= 4 where depending on the parity of

the gauge rank we have diﬀerent structure of the S-duality orbits, involving orthogo-

nal groups as well [3]. Furthermore symplectic, orthogonal and unitary gauge groups

– 2 –

are all involved in the examples of S-duality for N= 2 quivers originally found in

[12] that can be studied in terms of the Lens space index. Such S-duality has been

recently shown in [13] to persist when breaking N= 2 to N= 1. In this paper we

study the Lens space index for these models, showing that they match among the

theories in the same S-duality orbit, while they diﬀer for choices of the gauge group

in a diﬀerent orbit.

We conclude our analysis by studying a triality found in [14] that relates three

models with either unitary or symplectic gauge groups. It was pointed out in [14]

that there are in these cases diﬀerent choices of the gauge group for each phase and

we observe that all these choices give rise to the same Lens space index. On one hand

this corroborates the validity of the claim about the triality among these models. On

the other hand we explain the absence of multiple orbits by discussing some general

expectations from the holographic dual description of one of these three phases in

Type IIB string theory.

2 Review

Extended operators, such as Wilson and t’Hooft lines, play an important role in the

study of Quantum Field Theories. When the spacetime manifold is R4the extended

operators of the theory do not aﬀect the correlation function of local operators. Nev-

ertheless, two theories that only diﬀer by their extended operators are still distin-

guished by the correlation functions that involve the extended operators themselves.

When the spacetime manifold is non-trivial the extended operators can have a wider

impact on the physics. Extended operators can be wrapped on non-trivial cycles

of the spacetime providing diﬀerent backgrounds for the local physics. Furthermore

when the theory is compactiﬁed to a lower dimension the presence of extended opera-

tors can change the spectrum of local operators on the lower-dimensional theory. For

example when the spacetime is R3×S1a Wilson line wrapped around S1becomes

a local operator in the eﬀective 3-dimensional theory on R3.

2.1 Line operators in gauge theories

In gauge theories the spectrum of extended operators is closely related to the global

structure of the gauge group. The spectrum and correlation functions of local op-

erators only depend on the gauge algebra gassociated to the gauge group Gwhile

the spectrum of lines depends on the gauge group Gand on discrete theta-like pa-

rameters. In this paper we will only consider compact gauge groups, therefore we

have G=˜

G/H where ˜

Gis the compact simply connected group with associated Lie

algebra gand H∈ Z(˜

G)is a subgroup of the center of ˜

G. The lines can be organized

by their electric and magnetic charges (ne, nm)∈ Z(˜

G)× Z(˜

G). We always have

Wilson lines in every representation of Gthat belong to the classes (ne,0) with ne

– 3 –

Figure 1: The three possible choices of line charges for gauge theories with gauge

algebra usp(2n). The orange dots represent the charges of the line operators that

are included in the corresponding theory.

invariant under the action of H. These are completely determined by the choice

of gauge group G. In addition to the Wilson lines the theory includes t’Hooft and

dyonic lines. Any two lines of the theory must satisfy a Dirac pairing condition, for

example when Z(G) = Zkthe condition reads:

nen0

m−n0

enm= 0 mod k(2.1)

The spectrum of lines is determined by a complete and maximal set of charges

(ne, nm)satisfying (2.1). It turns out that given a choice of gauge group Gthere

still can be diﬀerent choices for the spectrum of lines. These choices are associated

to discrete theta-like parameters that can be introduced in the theory. For example

when g=usp(2n)there are three possible choices for the line spectrum, they are

depicted in Figure 1.

2.2 Lens space index

The Lens space index is a powerful tool for studying the global structure of supersym-

metric gauge theories [5,8,11,15–17] . Unlike the supersymmetric index on S3×S1,

the Lens space index is sensible to the extended (1 + 1)-dimensional operators of the

theory (i.e. Wilson, t’Hooft and dyonic lines) and can be able to distinguish between

theories with the same gauge algebra and matter content but with diﬀerent gauge

group. The index is an RG invariant and is expected to match between dual theories

as well as being stable under exactly marginal deformations. The Lens space index of

a theory can be computed as a supersymmetric partition function on the (Euclidean)

spacetime manifold L(r, 1) ×S1. Here L(r, 1) 'S3/Zris the three-dimensional Lens

space and S1is the Euclidean time. The integer rparametrizes diﬀerent spacetime

– 4 –

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摘要：

SymplecticgaugegroupontheLensSpaceAntonioAmaritiaandSimoneRotaa;baINFN,SezionediMilano,ViaCeloria16,I-20133Milano,ItalybDipartimentodiFisica,UniversitàdegliStudidiMilano,ViaCeloria16,I-20133Milano,ItalyE-mail:antonio.amariti@mi.infn.it,simone.rota@mi.infn.itAbstract:WecomputetheLensspaceindexfor4dsu...

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