Chiral algebra, localization, modularity, surface defects, and all that

J. Math. Phys. 

61

, 092302 (2020); 

https://doi.org/10.1063/5.0002661

61

, 092302

© 2020 Author(s).

Chiral algebra, localization, modularity,

surface defects, and all that

Cite as: J. Math. Phys. 

61

, 092302 (2020); 

https://doi.org/10.1063/5.0002661

Submitted: 27 January 2020 . Accepted: 17 August 2020 . Published Online: 11 September 2020

Mykola Dedushenko

, and 

Martin Fluder

Journal of

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Chiral algebra, localization, modularity, surface

defects, and all that

Cite as: J. Math. Phys.

, 092302 (2020); doi: 10.1063/5.0002661

Submitted: 27 January 2020

•

Accepted: 17 August 2020

•

Published Online: 11 September 2020

Mykola Dedushenko

1, a)

and Martin Fluder

1,2, b)

AFFILIATIONS

Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA

Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583, Japan

Author to whom correspondence should be addressed:

dedushenko@gmail.com

Electronic mail:

fluderm@gmail.com

ABSTRACT

We study the 2D vertex operator algebra (VOA) construction in 4D

2 superconformal field theories on

, focusing on both old

puzzles and new observations. The VOA lives on a two-torus

⊂

, it is

-graded, and this torus is equipped with the natural choice

of spin structure (1,0) for the

-graded operators, corresponding to the NS sector vacuum character. By analyzing the possible refinements

of the Schur index that preserves the VOA, we find that it admits discrete deformations, which allows access to the remaining spin structures

(1,1), (0,1), and (0,0), of which the latter two involve the inclusion of a particular surface defect. For Lagrangian theories, we perform the

detailed analysis: we describe the natural supersymmetric background, perform localization, and derive the gauged symplectic boson action

on a torus in any spin structure. In the absence of flavor fugacities, the 2D and 4D path integrals precisely match, including the Casimir

factors. We further analyze the 2D theory: we identify its integration cycle and the two-point functions and interpret flavor holonomies as

screening charges in the VOA. Next, we make some observations about modularity; the

-transformation acts on our four partition functions

and lifts to a large diffeomorphism on

. More interestingly, we generalize the four partition functions on the torus to an infinite

family labeled by both the spin structure and the integration cycle inside the complexified maximal torus of the gauge group. Members

of this family transform into one another under the full modular group, and we confirm the recent observation that the

-transform of

the Schur index in Lagrangian theories exhibits logarithmic behavior. Finally, we comment on how locally our background reproduces the

-background.

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I. INTRODUCTION

Accessing the strongly coupled and non-perturbative dynamics of a quantum field theory is hard, and even upon incorporating sim-

plifying assumptions, such as supersymmetry or conformal symmetry, techniques allowing for control in such regimes are rare. Certain

supersymmetric or protected quantities in supersymmetric field theories nevertheless prove amenable to analytic study, and much of the

focus in the last few decades has been on such examples. One prominent class of such theories are four-dimensional

2 quantum field

theories. The infrared (IR) dynamics on the Coulomb branch of such theories has long been understood since the release of the seminal set

of works.

1–4

Incorporating techniques of supersymmetric localization was another major achievement in getting analytical control of such

theories.

The advent of a whole new set of structures in these theories also resulted from the invention of the

-background.

3,4,6–8

Recent years have seen another surge of research activity on four-dimensional

2 theories. More specifically, the discovery and

precise formulation of a connection between vertex operator algebras (VOAs) and four-dimensional superconformal field theories (SCFTs)

in Ref. 9 has sparked a rekindled research interest (see, e.g., Refs. 10–45 for subsequent developments). Such VOAs are rich yet rigid objects

that, quite surprisingly, allow us to gain a non-perturbative access to an algebraically closed sector in the operator product expansion (OPE)

data of the superconformal field theory. The latter fact is completely independent of whether the theory admits a weakly coupled Lagrangian

description or not. This has grown into a subfield by itself, sometimes referred to as the “SCFT/VOA correspondence.”

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In this work, we are set to contribute to this subfield by combining the latest developments for the (older) Lagrangian technique of

supersymmetric localization with the newer ideas on four-dimensional

2 superconformal field theories. The original construction of

Ref. 9 was formulated in the flat Euclidean space

, though the need to also understand it in other backgrounds was repeatedly raised over

the years. Here, we study the chiral algebra construction

on the

geometry, which, incidentally, is also the geometry relevant for

studying the superconformal index, including its Schur limit.

47,48

Given that the VOA precisely captures the so-called Schur sector of the

theory (i.e., states counted by the Schur index), we take it as no coincidence, but rather as an indication that the

background is in many

ways an inherently natural playground for the study of SCFT/VOA correspondence.

As a courtesy to busy readers, and to avoid getting lost in technicalities, we now provide a somewhat more detailed overview of the main

results presented in this work.

A. Chiral algebra on

The original construction

of the two-dimensional vertex operator algebra as a subsector of the full four-dimensional

2 supercon-

formal field theory relies on a powerful “abstract” operator approach to the four-dimensional theory, which does not require a Lagrangian

description. From such a point of view, any conformal theory can be placed on

via a Weyl transformation (on Weyl invariance of

conformal theories, see, e.g., Ref. 49). If we further close

to a circle, which we label as

throughout this paper, we find that the theory

breaks supersymmetry unless we define it in a twisted sector with respect to the R-symmetry. In this case, one manages to preserve half of the

flat space supercharges, namely, those commuting with

−

, where

is the dilatation generator (that generates translations along

in the

geometry) and

is a Cartan generator of the

(2)

symmetry. The surviving supercharges, together with isometries of

and the

(1)

⊂

(2)

(1)

R-symmetry subgroup, form a supersymmetry algebra,

∣

ℓ

⊕

∣

(1)

which is centrally extended by

−

. This should be regarded as an

2 superconformal symmetry on

. Conformal invariance is

crucial as this algebra contains

(1)

that is only known to be unbroken in conformal theories. A similar algebra without central extension has

previously appeared as three-dimensional

4 supersymmetry on

in Ref. 50. In this case, the algebra admitted central extensions due

to the addition of masses and Fayet–Iliopoulos (FI) parameters in Lagrangian theories. In the present context, this

−

should be regarded

as a mass-like central charge. One can also introduce additional mass-like central charges by turning on flavor symmetry holonomies around

the

circle (corresponding to flavor fugacities in the index), and they can be regarded as lifts of the three-dimensional mass terms. However,

there is no interesting lift of three-dimensional FI parameters because we study four-dimensional superconformal field theories, and even if

they admit Lagrangian descriptions, they do not have any Abelian factors in their gauge groups.

The algebra (1) contains everything required to define the two-dimensional VOA sector. All ingredients of the original construction

from Ref. 9 can be easily translated to this context, and one finds that the VOA is now supported on the torus

, where

⊂

is a great

circle. The labeling of circles stems from the coordinates we use;

is parameterized by

∈

[

ℓ

]

, while

is parameterized, just like in Ref.

50, by the fibration coordinates (

) in which

is a circle parameterized by

that sits at

2, where the

-circle shrinks to a point (see

Sec. II B for details). Since we essentially view

with

shrinking at

∂

, we sometimes refer to

as the “boundary,” even

though

is, of course, closed.

B. Localization on

With the exception of the above generalities, in this paper, we focus on Lagrangian

2 superconformal field theories in four dimen-

sions. In this case, by employing supersymmetric localization on a rigid background of the form

, we explicitly localize a given

Lagrangian superconformal field theory and find that it indeed reproduces the expected two-dimensional VOA on the torus

⊂

described as a gauged symplectic boson. The symplectic boson VOA is also known as a

−

system of weight

, and we interchangeably use

the two terms.

To derive the two-dimensional VOA on the torus, we first define the appropriate rigid supersymmetric

background, reproducing

the superconformal index. In doing so, we analyze the supersymmetry algebra and classify the possible fugacities and their preserved subal-

gebras. In order to retain the VOA construction, the minimal amount of supersymmetry we ought to preserve is

∣

ℓ

⊕

∣

. We find

that we may turn on fugacities preserving an

∣

ℓ

⊕

∣

subalgebra (which can be further broken to the minimal one by defects). Sur-

prisingly, this goes beyond the fugacity in the Schur limit, which is well-known to be relevant for the chiral algebra construction. Specifically,

we are allowed to turn on

discrete

fugacities

∈

, where

corresponds to an insertion of

(

)

in the Schur index, while non-zero

modifies the geometry. For non-zero

, the

is no longer equipped with a product metric, but rather one rotates

as we go around the

. As we shall argue, these deformations

do not

affect the VOA construction but change the complex structure of the

torus and affect the boundary conditions (spin structure) upon going around one of the cycles,

(see below). The inclusion of

actually

does not affect the partition function, but throughout we keep both

and

as generic integers.

We perform localization for the

2 vector multiplets; in this case, the two-dimensional theory is determined

indirectly

. As we show,

the Yang–Mills action is

-exact, and thus, the four-dimensional theory solely localizes to a one-loop determinant piece. Nonetheless, the

two-dimensional action can be “bootstrapped” from the knowledge of the partition function and the four-dimensional propagators and is

given by the

small bc

ghost system on the torus.

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Localization for the four-dimensional

2 hypermultiplets more straightforwardly reproduces the two-dimensional theory. Specifi-

cally, the remnant “classical” piece in the localization precisely reduces to the two-dimensional symplectic boson theory on the boundary torus

⊂

. As alluded above, the discrete label

changes the complex structure of the torus (essentially,

≠

0 differs from

0 by

precisely

Dehn twists or a

modular transformation), while

changes the spin structure (periodicity) of symplectic bosons along

The localization from four to two dimensions produces almost no overall one-loop determinant, except for a simple Casimir energy

factor. The part of it that depends on the gauge holonomy around the

cancels between the hypermultiplets and vector multiplets as a

consequence of conformal invariance. If we turn on flavor holonomies, however, there is a simple leftover term that survives and describes

the mismatch of the four- and two-dimensional path integrals. It is given by

−

∑

∈

⟨

⟩

(2)

where the sum goes over all weights in a flavor symmetry representation

in which the matter transforms,

is the background holonomy in

the Cartan of flavor group, and

, where

is a complex structure of the torus. This formula can also be generalized to non-Lagrangian

theories relying on the results of Ref. 13, but this goes beyond the scope of this paper.

C. Spin structures

All operators in the VOA have either integral or half-integral conformal dimensions, as a consequence of the SCFT/VOA correspondence

for general, not necessarily Lagrangian SCFTs. Thus, in order to place the VOA on a torus, one has to pick a particular spin structure, which

amounts to choosing operators of half-integral conformal degree to be either periodic or anti-periodic along the one-cycles of

. [This can

be interpreted as a background connection for the

automorphism acting as (

−

, which always makes sense in the general context of

SCFT/VOA correspondence but can act trivially if all eigenvalues of

are integers.] The canonical choice that follows from how we put a

four-dimensional theory on

is (1, 0), meaning that spinors are anti-periodic (in the NS sector) along

and periodic (in the R sector)

along

We mentioned above that the nontrivial

mod 2 amounts to flipping the spin structure of symplectic bosons along

. This is, in fact,

more general and holds in an arbitrary VOA (arising from a superconformal field theory): starting with the “standard” NS sector vacuum

character, corresponding to (1, 0) spin structure, and turning on

mod 2 in the four-dimensional background, we arrive at the torus partition

function with (1, 1) spin structure. Given our general localization, we immediately get the corresponding four-dimensional result,

(1,1)

, by

setting

1, which, incidentally, is the “modified Schur index” considered by Razamat.

Of course, it precisely agrees with the NS–NS

character. (Since we only work with chiral algebras, we sometimes use the NS–NS and NS–R terminology to denote spin structures along the

pair of one-cycles of

, which should not be confused with the spin structures of left and right movers along the same

, which is a widely

used convention in string theory.)

In two dimensions, we can also change the periodicity of the fields along the other, here

, cycle to obtain the remaining (0, 1) and (0, 0)

spin structures. However, because the

-cycle is contractible in

, we cannot continuously change the periodicity of the four-dimensional

fields along it; thus, we have to make

non-contractible. To do so in a physically sensible way, we have to introduce a surface defect at

i.e., describing

as a circle fibration over the disk

0 is at the origin of

, and the defect extends in the

and

directions.

D. The R-symmetry defect

The surface defect that can switch the

spin structure should act similarly to the

)

monodromy that was able to alter the

spin structure. (Note that

)

1, since

∈

in our conventions.) Specifically, we first define the symmetry interface that does the

transformation

)

on a theory—we call it the canonical R-symmetry interface. Then, we claim that the relevant surface defect should sit

at the boundary of such an interface. If we place the surface defect at

0, with the canonical R-symmetry interface ending on it, we get what

we want: operators that have half-integral dimension in the VOA will obtain the opposite periodicity around

. In this way, we can achieve

the (0, 1) and (0, 0) or NS–R and R–R spin structures from four dimensions.

Such surface defects should exist in general superconformal field theories, but they are by no means unique. In fact, they correspond

to the Ramond sector modules of the VOA, and there might be many of them. For Lagrangian theories, we define in the main text and

in Appendix D the simplest possible pair of such defects. Because

)

acts trivially on the vector multiplets, we define defects that only

directly couple to hypermultiplets by changing their asymptotic behavior in the vicinity of the defect. This is closely related to the monodromy

defect considered in Ref. 23, and we elaborate more on their relation in Sec. II F. We further generalize the localization answer to include this

defect at

0. In this way, we end up with four partition functions,

(1,0)

(1,1)

(0,1)

(0,0)

, labeled by the spin structure, where the first one

is the usual Schur index and the second one is, of course, the “modified Schur index” of Razamat.

E. The two-dimensional theory

Once we have the two-dimensional action of gauged symplectic bosons, it is straightforward to study its various properties, such as the

proper integration cycle and the two-point functions in all spin structures. One interesting observation that we make is that in the presence

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of flavor holonomies—which appear as mass-like central charges in the supersymmetry algebra—vertex operators charged under the flavor

symmetries fail to remain holomorphic. The sector that remains holomorphic is formed by flavor-neutral operators. At the level of the VOA,

it corresponds to the well-known operation of screening. Somewhat formally, it expands the class of vertex algebras we might study. One

simple example we describe is a free hypermultiplet: turning on its

(1)

⊂

(2)

flavor holonomy screens the free symplectic boson VOA

to a slightly simpler VOA described by a pair of chiral bosons.

F. Modularity of the Schur index

It has long been suggested that the four-dimensional Schur index observes modular properties (see, for example, Refs. 51 and 52).

Naively, one expects that the modular properties are related to an exchange of the “thermal cycle” with the Hopf fiber of the index. Of course,

with the advent of Ref. 9, the identification of the Schur index with some vacuum character of a two-dimensional chiral algebra immediately

suggests that the Schur index ought to be part of a modular vector. Indeed, in Ref. 26, differential equations observing modular properties

were derived, and it was suggested that the vacuum character—or the Schur index—is a solution of them.

Nevertheless, a four-dimensional understanding of the modular properties of the two-dimensional character remains an unresolved

problem. For Lagrangian theories, our discussion of the spin structures of the torus partition function and its relation to the four-dimensional

Schur index gives us immediately the

-modular transformation of the latter. In particular, we explicitly find the following general result:

(1,0)

(

+ 1,

)

∝

(1,1)

(

(1,1)

(

+ 1,

)

∝

(1,0)

(

(0,0)

(

+ 1,

)

∝

(0,0)

(

(0,1)

(

+ 1,

)

∝

(0,1)

(

(3)

In order to shed light on the action of the modular

-transformation, we introduce novel (formal) partition functions labeled by two

additional indices,

∈

and

(

)

(

)

. They are defined as the partition function in the given spin structure (

), but now with the

modified contour

(

)

of the holonomy integral in the localization formula, labeled by the integers

and

. The upshot of introducing this

extended, infinite set of partition functions is that they exhibit a simple behavior under modular transformations.

For instance, we find that

(

)

(

)

(

−

)

∝

(

)

(

−

)

(

)

(

)

(

)

(

+ 1,

)

∝

(

)

(

)

(

)

(4)

Thus, we suggest that the objects

(

)

(

)

furnish an (infinite-dimensional projective) representation of

(2,

), and given the relation to two-

dimensional chiral algebras, we expect it to truncate, i.e., there are relations among them, and a—possibly finite—set of them transform as

some modular vector.

Physically, the remanent independent objects

(

)

(

)

are expected to correspond to partition functions in the presence

of general defects (corresponding to the non-trivial modules of the two-dimensional chiral algebra

9,18,23,54

Finally, we discuss two simple examples and explicitly observe that the

-transformation of the Schur index exhibits logarithmic behavior,

which is in accordance with the expected solutions to the modular differential equations derived in Ref. 26.

G. Relation to the (flat)

-background

⊕

For a while, it has been suggested

(see also Refs. 40 and 56) that the two-dimensional chiral algebra can be obtained by an

-deformation

3,4

of the holomorphic-topological twist, introduced by Kapustin in Ref. 57 (see Refs. 44 and 45 for recent results to this

end). Indeed, our analysis of the

background, with the two-dimensional theory arising on the torus,

⊂

, suggests that the (flat

space)

-background is hidden as some “local” (and decompactified) version near the torus.

In Sec. VI, we comment on this connection

in four dimensions, determining an expansion of the

background in the vicinity of the torus and thereby explicitly obtaining the flat

-background

⊕

. The theory then effectively localizes to the tip of the

-background,

, giving the symplectic boson action in the

remaining (flat) two-dimensional space. We remark here that our background seems to be in accordance with the recent results in Refs. 44

and 45.

The remainder of this paper is organized as follows: We start in Sec. II by setting the stage and define the supersymmetric background

. We further discuss the possible fugacities including their preserved subalgebra and argue that we may slightly deform away from

the Schur limit while still preserving the VOA construction. Furthermore, we introduce the notion of spin structures on the torus to the game

and discuss how they are realized from a four-dimensional point of view, which marks the inception of the canonical R-symmetry interface

and surface defect. In Sec. III, we explicitly localize the

partition function onto the gauged symplectic bosons in two dimensions.

This includes the results for the (four) different spin structures and the R-symmetry surface defect. We point out various subtleties along

the way. In Sec. IV, we consequently discuss the various aspects of the resulting two-dimensional theory. This includes the determination of

its integration cycle, the propagators, and the interpretation of flavor fugacities as screening charges in two dimensions. In Sec. V, we then

discuss some implications of our results toward understanding the modular properties of the Schur index. Finally, in Sec. VI, we mention the

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connection to the flat

-background. We close in Sec. VII with a brief discussion and mention some work for the future. We collect some

more technical details in Appendixes A–E.

II. THE INDEX, THE BACKGROUND, AND THE ALGEBRA

In this section, we provide the necessary details of the constructions explored in the remainder of this paper. We start by recalling the

definition of the superconformal index of four-dimensional

2 superconformal field theories and choose a particular representation of it

convenient for our purposes. We then proceed to describe the rigid background in four-dimensional

2 conformal supergravity, which

is appropriate for studying the index. Such a background, of course, ought to have the topology of

(we denote the Euclidean time

direction by

). However, we may also replace

by an interval or

. In the latter case, it can be obtained by a Weyl transformation from flat

space.

We further explicitly describe the most general fugacities in the superconformal index compatible with the supercharge

, which is

required to be preserved for the vertex operator algebra (VOA) construction, and explain in detail how the VOA construction works on

. In particular, we obtain a VOA on the torus

⊂

. Interestingly, such a specialization of fugacities goes slightly beyond the

well-known Schur limit of the superconformal index. Specifically, we add a discrete parameter that switches the spin structure of the torus

(along

) at the level of the VOA. Furthermore, we introduce a surface defect that switches the spin structure along the other cycle of the

torus—such defects correspond to the Ramond sector modules for the VOA.

A. Different representations of the superconformal index

The superconformal index in four-dimensional

2 superconformal field theories was introduced in Refs. 59–61. In its most basic

form, and given a choice of the supercharge

, it is defined as

(

)

(

−

(5)

where the trace is taken over the Hilbert space of radial quantization

and

is the fermion number, and we introduced the definition

≡

{

†

}

(6)

Furthermore,

is a maximal set of mutually commuting generators that necessarily also commute with

. Mirroring the choice in Ref. 48,

we pick the particular supercharge

−

to define the index. Thus, we fix

−

(7)

and the maximal set of commuting generators consists of the following anti-commutators:

−

(8)

+ 2

−

(9)

+ 2

(10)

A key property of the superconformal index is its independence of

. This is due to the pairwise cancellations of non-zero modes of

−

Thus, we may shift

, as long as we are preserving the convergence of (5). In particular, shifting

by various linear combinations of chemical

potentials

is equivalent to redefining

→

, where

is some number. Because such redefinitions do not affect the answer, we may

write the following equivalent formulas for the index:

(

)

Tr (

−

(

−

)

(

−

)

−

(11)

(

)

Tr (

−

(

−

)

(

−

)

−

(12)

(

)

Tr (

−

(13)

Passing from (11) to (12) involves the change of variables

τσ

τρ

, and

and a shift of

, while passing from (12) to (13) is

accomplished solely by shifting

Note that even though the answer is

-independent,

also plays the role of a regulator, regularizing the trace over an infinite-dimensional

Hilbert space. The importance of this factor differs in the alternative representations of the index. For example, since the factor (

ρστ

)

ensures

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convergence, as long as

∣

ρστ

∣

1 (with some extra assumptions), we may safely put

0 in (11). Similarly, we can set

0 in (12) as long as

∣

1. However, we

cannot

put

0 in (13), as none of the other factors can obviously serve as a regulator.

In the following, we will set

0 and work with the representation of the superconformal index given in (12). We further perform the

following change of variables:

(

)

−

(14)

βζ

(15)

(

)

βγ

(16)

upon which the index takes the following form:

(

)

(

−

βζ

βγ

(

)

(17)

Indeed, this representation is the most convenient for the path integral interpretation. We put the theory on a space

, and the factor

−

suggests that we have to evolve it for the Euclidean time

. The additional factors

βζ

βγ

(

)

further imply that afterward we have to

perform various symmetry transformations on the system, and the trace means that we close the time direction into a circle

. Finally, (

−

implies that fermions ought to have periodic boundary conditions (up to the twists introduced by

βζ

βγ

(

)

Thus, we have to put the theory on a Euclidean space given by

, defined in a twisted sector, where upon going once around

, we

perform the R-symmetry transformations

βγ

(

)

and rotate

βζ

. The latter means that the geometry can be constructed by first

taking

with the product metric, where

is an interval, and then identifying the two boundary three-spheres by a rotation generated

βζ

. It is straightforward to describe a background satisfying these criteria, and we do so in Subsection II B. Quite importantly, this

background should preserve both

(1)

⊂

(2)

and

(1)

R-symmetries, simply because we work in a twisted sector with respect to both

and

On the contrary, the background does not have to preserve the supercharge

−

used in the original definition of the index (5). Once

we have arrived at expression (17) that concerns a particular way to count states in the radial quantization Hilbert space

, it no longer

matters what supercharge we started with in (5). In what follows, we will investigate what supersymmetry is preserved by such a background

for various values of the fugacities in (17).

B. The background

Using off-shell supergravity to construct rigid supersymmetric backgrounds has become a standard technique in supersymmetry litera-

ture.

(This technique goes back to as early as Refs. 151–153 in the context of topologically twisted theories. Of course, it is also possible to

define (rigid) supersymmetry on the

background using other methods.

154

Nevertheless, we are going to follow the general approach

of Ref. 62.) The relevant supergravity theory for our problem is given by the four-dimensional

2 off-shell conformal supergravity of

Refs. 63–65 based on the standard Weyl multiplet. (General rigid supersymmetric backgrounds have been analyzed in Ref. 155 within the

setting of four-dimensional conformal supergravity.) It has been successfully used to put general four-dimensional

2 gauge theories

on the ellipsoid in Refs. 66 and 67 in which case the theory does not have to be conformal. We follow the conventions of Ref. 67 (see

also Appendix A). As we will see, theories that are not conformal quantum-mechanically appear to break supersymmetry in our

background, and therefore, for the scope of this paper, we focus solely on conformal theories.

We parameterize the space

by variables (

), where

∈

[

]

∈

[

−

]

, and

∈

[

0, 2

]

are “fibration” coordinates for

the three-sphere

of radius

ℓ

and

∈

[

ℓ

]

parameterizes the circle

of circumference

ℓ

. The corresponding metric then reads

ℓ

[

+ sin

(

ℓ

)

+ cos

(

ℓ

)

]

(18)

where we further introduced the additional deformation

, which is related to the “standard” fugacities

, and

via equations (14)–(16).

Note that for

0, this is just a product of a round

with

, while non-zero

introduces a twist, i.e., as we go around the

, the three-

sphere is rotated by

βζ

. Alternatively, we could write our metric in terms of the coordinates

ℓ

and

ℓ

in which

it would simply become a product metric, where gluing a patch

∈

(0,

ℓ

) into a circle, i.e., identifying

0 with

ℓ

, would involve a

rotation (“twist”) by

βζ

. This rotation is precisely generated by

, which appears in the definition of the superconformal index in

equations (11)–(13). We prefer to use the coordinates (

), in which the metric takes the form (18), and up to the R-symmetry twists

J. Math. Phys.

, 092302 (2020); doi: 10.1063/5.0002661

, 092302-6

Published under license by AIP Publishing