Topological modularity of Supermoonshine Jan Albert12Justin Kaidi324Ying-Hsuan Lin5 1C. N. Yang Institute for Theoretical Physics

2025-05-06 3 0 664.43KB 46 页 10玖币

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Topological modularity of Supermoonshine

Jan Albert,1,2Justin Kaidi,3,2,4Ying-Hsuan Lin5

1C. N. Yang Institute for Theoretical Physics,

Stony Brook University, Stony Brook, NY 11794-3840, USA

2Simons Center for Geometry and Physics,

Stony Brook University, Stony Brook, NY 11794-3636, USA

3Department of Physics, University of Washington, Seattle, WA, 98195, USA

4Kavli Institute for the Physics and Mathematics of the Universe,

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

5Jeﬀerson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA

jan.albertiglesias@stonybrook.edu, jkaidi@uw.edu, yhlin@fas.harvard.edu

Abstract

The theory of topological modular forms (TMF) predicts that elliptic genera of

physical theories satisfy a certain divisibility property, determined by the theory’s

gravitational anomaly. In this note we verify this prediction in Duncan’s Supermoon-

shine module, as well as in tensor products and orbifolds thereof. Along the way we

develop machinery for computing the elliptic genera of general alternating orbifolds

and discuss the relation of this construction to the elusive “periodicity class” of TMF.

arXiv:2210.14923v2 [hep-th] 27 Mar 2023

Contents

1 Introduction and summary 1

1.1 Periodicityelements ............................... 3

1.2 Organization ................................... 7

2 Alternating orbifolds 8

2.1 Review of symmetric orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Second-quantized formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Discretetorsion.................................. 13

2.4 Heckeformula................................... 15

3 Topological modularity of Supermoonshine 17

3.1 Supermoonshine.................................. 17

3.2 McKay-Thompson data for Supermoonshine . . . . . . . . . . . . . . . . . . 19

3.3 Symmetric orbifolds of Supermoonshine . . . . . . . . . . . . . . . . . . . . . 21

3.4 Alternating orbifolds of Supermoonshine . . . . . . . . . . . . . . . . . . . . 25

3.5 Saturation of divisibility by decomposable theories . . . . . . . . . . . . . . . 28

A Permutation anomaly 29

B Proofs of alternating orbifold formulae 32

B.1 ProofofTheorem1 ............................... 32

B.2 ProofofTheorem2................................ 35

B.3 ProofofTheorem3................................ 37

1 Introduction and summary

Moonshine [1] is one of the most beautiful subjects at the interface of mathematics, physics,

and folklore. What originated from a curious observation about modular forms and the

Monster group has transcended into the ﬁelds of conformal ﬁeld theory [2], string theory [3],

and quantum gravity [4]. Subsequent developments have lead to the discovery of Super-

moonshine [5,6], Mathieu Moonshine [7–12], and Umbral Moonshine [13–18].

In a similar spirit, topological modular forms [19] have begun to make a surprise appear-

ance in physics thanks to a conjecture by Stolz and Teichner [20, 21] based on earlier work

by Segal [22, 23]. The conjecture roughly states the following:

1. Every 2d supersymmetric quantum ﬁeld theory (SQFT) with N= (0,1) supersymme-

try can be associated with a “topological modular form,” or more precisely a class in

TMF,

2. Every class in TMF can be realized by at least one N= (0,1) SQFT,

3. TMF is a complete supersymmetric deformation invariant, i.e. any two SQFTs can be

continuously connected if and only if they are associated with the same topological

modular forms.

Although the map between SQFTs and TMF is not yet fully understood (though see [24,25]

for key progress), in some cases the image of the map is a familiar object: the elliptic genus,

i.e. the torus partition function with Ramond boundary conditions along both space and

time. The non-surjectivity of the map from SQFTs to the space of modular forms implies a

remarkable divisibility property of certain coeﬃcients in the elliptic genus, as will be reviewed

below for the cases of interest to us.1

By now, intricate connections between topological modular forms and Moonshine have

been uncovered [28–31], and this note continues this pursuit. The protagonist of our story is

Duncan’s Supermoonshine module Vf \ [5], a holomorphic N= 1 supersymmetric conformal

ﬁeld theory (SCFT) with central charge c= 12 that enjoys Conway symmetry.23 Its (twisted

and twined) elliptic genera are all constants due to supersymmetry, while its partition func-

tions with other boundary conditions exhibit many of the same extraordinary properties as

their Monster cousins, including the celebrated genus-zero property [5, 6, 32, 33]. A brief

review of the construction and properties of Vf\ is given in Section 3.1.

For a general c= 12nholomorphic N= 1 SCFT whose elliptic genus is a constant and

1The physicist readers are referred to [26, 27] for a friendly introduction to this divisibility.

2This theory actually has N= (1,1) supersymmetry because the anti-holomorphic sector can be equipped

with trivial supersymmetry.

3In the math literature, the notation Vf \ only refers to the supersymmetric vertex operator algebra

(SVOA) of the Neveu-Schwarz sector, whereas the Ramond sector is denoted by Vf \

tw . In this note, we

slightly abuse Vf\ to mean the entire fermionic theory Vf \ ⊕Vf \

tw .

equal to the Witten index I, the aforementioned divisibility property states that4

gcd(24, n)Ior equivalently 24 |nI.(1.4)

The Supermoonshine module has precisely I=−24, saturating divisibility for n= 1. Since

this divisibility is at present still conjectural, it is a valuable exercise to check its validity in

a variety of theories of physical interest. Such checks were performed in [31] in the context

of the Monster module V\as well as its tensor products and various orbifolds. Conversely,

assuming the validity of the conjecture, one can rule out the existence of a number of tentative

VOAs proposed in the literature, including many of the extremal CFTs proposed in [4].

In the current work, we perform a similar exercise for the Supermoonshine module Vf\.

In particular, we check the validity of the divisibility criterion for tensor products of Vf\,

together with orbifolds by Snand Anpermutation symmetries. We also allow for orbifolds by

non-anomalous cyclic subgroups of the diagonal Co0symmetry. While primarily serving as a

check of the Stolz-Teichner conjecture, this exercise has an important secondary motivation:

namely, to develop the tools necessary for realizing a special class in TMF containing the

“periodicity elements,” whose deﬁnition we now review.

1.1 Periodicity elements

TMF is a generalized cohomology ring graded by an integer ν, where the multiplication and

addition operations correspond physically to taking the tensor product and direct sum of

SQFTs and νcharacterizes the gravitational anomaly.5For SCFTs, the quantity νis related

to the chiral central charge by ν= 2(cR−cL), so that in particular a holomorphic SCFT with

(cL, cR)=(c, 0) has ν=−2c. Intuitively, the group TMFνcaptures how much data beyond

the gravitational anomaly νis necessary to specify the deformation class of an SQFT. The

notion of deformation class here includes, but is not necessarily limited to, the identiﬁcation

4Let the prime factorization of a triple of natural numbers D, n, a be

D=Y

pδi

i, n =Y

pνi

i, a =Y

pαi

i, δi, νi, αi∈Z≥0.(1.1)

Then D

gcd(D, n)a⇔δi≤min(δi, νi) + αi∀i , (1.2)

and

D|n a ⇔δi≤νi+αi∀i . (1.3)

If δi≤νi, then both conditions are obviously true; if δi≥νi, then the two conditions become identical. In

(1.4), D= 24 and a=I.

5The gravitational anomaly νis conventionally normalized such that a chiral fermion has ν= 1.

of all theories connected by RG ﬂows (induced by either relevant deformations or vacuum

expectation values) as well as theories connected by marginal deformations [24].

It is known that the cohomology ring has periodicity ν∼ν+ 576. This is a rather

remarkable property: it means that the set of deformation classes of SQFTs with gravita-

tional anomaly νis identical to that of SQFTs with gravitational anomaly ν+ 576. Indeed,

there exists a special class in TMF−576 called the “periodicity class” such that every class of

TMFν−576 is obtained from a unique class in TMFνby taking the product with the period-

icity class. An SQFT with ν=−576 realizes an element in the periodicity class if and only

if its elliptic genus is a constant with value ±1.6

Because a constant elliptic genus is a highly non-generic feature in systems without

supersymmetry, a natural starting point for realizing an element in the periodicity class is

to consider holomorphic N= 1 SCFTs with central charge c= 288. This leads us to the

study of theories constructed from Vf\. Indeed, according to [34, Example 2.4.1], the ’t

Hooft anomaly of the Co1symmetry of Vf\ realizes the generator of SH3(Co1) = Z24.7

Hence, the diagonal Co1symmetry of (Vf\)⊗nis non-anomalous when 24 |n. For n= 24,

the chiral central charge c= 12 ×24 = 288 gives precisely the amount of gravitational

anomaly needed for the periodicity class of TMF, and one may then hope (on the basis of

aesthetics alone) that one periodicity element is realized by the theory Vf \⊗24/Co1.

Unfortunately, with present technology, it is not possible to conclusively refute or conﬁrm

this guess. Indeed, ignorance of the generalized McKay-Thompson data for Vf\ prevents one

from computing the full Witten index of the Co1orbifold. Nevertheless, there is evidence

to suggest that this ﬁrst guess is incorrect. In particular, the Witten index of Vf\⊗24 is

2424, whereas Co1“only” has 4,157,776,806,543,360,000 elements, 15 orders of magnitude

smaller. It thus seems extremely unlikely—though not strictly speaking impossible—for the

Co1orbifold to have Witten index ±1. One is led to consider alternative constructions.

One closely related construction is to allow for permutation orbifolds, e.g. Vf \⊗24/S24.

Indeed, permutation orbifolds are well-known to give rise to massive reductions in the index

or the degeneracies in the light spectra [38,39], and thus seem well-suited for the current task.

Lo and behold, allowing for permutation orbifolds does enable one to identify a periodicity

6One can also realize the periodicity class in TMF576, but this is less interesting since it can be realized

by an N= (0,1) sigma model, as described brieﬂy later. Furthermore, given a holomorphic N= 1 theory

realizing ν=−576 (which will be our focus below), we can get another ν= +576 theory by exchanging left-

and right-movers.

7We remind the reader that the supercohomology group SHd(G) comprises the ﬁrst three layers of the

Atiyah–Hirzebruch spectral sequence for the spin bordism group ΩSpin

d(BG) [35,36]. As a set, this is equiva-

lent to Hd(G, U(1))⊕Hd−1(G, Z2)⊕Hd−2(G, Z2)— referred to as the bosonic, Gu-Wen, and Majorana layers,

respectively—on the E2page, and is generally reduced by non-trivial diﬀerentials on the higher pages. For

d= 3 the group SH3(G) is identical to ΩSpin

3(G) [37], but for larger dit captures less information.

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TopologicalmodularityofSupermoonshineJanAlbert,1;2JustinKaidi,3;2;4Ying-HsuanLin51C.N.YangInstituteforTheoreticalPhysics,StonyBrookUniversity,StonyBrook,NY11794-3840,USA2SimonsCenterforGeometryandPhysics,StonyBrookUniversity,StonyBrook,NY11794-3636,USA3DepartmentofPhysics,UniversityofWashington,Seat...

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