A Borcherds-Kac-Moody superalgebra with Conway symmetry

TL;DR

This paper constructs a new Borcherds-Kac-Moody superalgebra with Conway group symmetry, linking string theory, moonshine phenomena, and genus zero properties, and explores its implications for BPS states in string theory.

Contribution

It introduces a novel BKM superalgebra with Conway symmetry generated from superstring states, connecting moonshine and string theory.

Findings

Constructed a BKM superalgebra with Co$_0$ symmetry.

Derived denominator identities for Conway module partition functions.

Proposed a physical interpretation of Conway moonshine in string theory.

Abstract

We construct a Borcherds Kac-Moody (BKM) superalgebra on which the Conway group Co$_0$ acts faithfully. We show that the BKM algebra is generated by the BRST-closed states in a chiral superstring theory. We use this construction to produce denominator identities for the chiral partition functions of the Conway module $V^{s\natural}$, a supersymmetric $c=12$ chiral conformal field theory whose (twisted) partition functions enjoy moonshine properties and which has automorphism group isomorphic to Co$_0$. In particular, these functions satisfy a genus zero property analogous to that of monstrous moonshine. Finally, we suggest how one may promote the denominators to spacetime BPS indices in type II string theory, which might thus furnish a physical explanation of the genus zero property of Conway moonshine.

Equations

$$J(\tau )\sim q^{-1}+O(q),$$

$$J(\tau )=\sum _{n=-1}^{\infty }c(n)q^n=q^{-1}+196884q+\dots ,$$

$$p^{-1}\prod _{m>0,n\in \mathbb{Z}}(1-p^m{q}^n{)}^{c(mn)}=J(\sigma )-J(\tau ),$$

$$V^{♮}=\underset{n=-1}{\overset{\infty }{⨁}}{V}_n^{♮}$$

$$J(\tau )=\sum _{n=-1}^{\infty }{\mathrm{Tr}}_{V_n^{♮}}{q}^n.$$

$$T_g(\tau ):=\sum _{n=-1}^{\infty }{\text{Tr}}_{V_n^{♮}}(g)q^n,$$

$$V^{s♮}=\underset{n=-1}{\overset{\infty }{⨁}}{V}_{n/2}^{s♮},$$

$$V_{tw}^{s♮}=\underset{n=0}{\overset{\infty }{⨁}}{V}_{tw,n}^{s♮},$$

$$T_g^s(\tau )$$

$${\tilde{T}}_g^s(\tau )$$

$$T_{tw,g}^s(\tau ):=\sum _{n=0}^{\infty }{\text{sTr}}_{V_{tw,n}^{s♮}}(g)q^n=\sum _{n=0}^{\infty }{c}_{tw,g}^s(n)q^n,$$

$${\tilde{T}}_{tw,g}^s(\tau ):=\sum _{n=0}^{\infty }{\text{Tr}}_{V_{tw,n}^{s♮}}(g)q^n=\sum _{n=0}^{\infty }{\tilde{c}}_{tw,g}^s(n)q^n.$$

$$p^{-1}\prod _{d=1}^{\infty }\prod _{r\in \mathbb{Z}}(1-p^d{q}^r{)}^{c_{tw}^s(rd)}=\frac{1}{{\eta }^{24}(\sigma )}-24-T_{tw}^s(\tau )$$

$$p^{-1}\prod _{d=1}^{\infty }\prod _{r\in \mathbb{Z}}(1-p^d{q}^r{)}^{\frac{(c_{tw}^s(rd)+{\tilde{c}}_{tw}^s(rd))}{2}}(1+p^d{q}^r{)}^{\frac{(c_{tw}^s(rd)-{\tilde{c}}_{tw}^s(rd))}{2}}=\frac{{\eta }^{24}(\sigma )}{{\eta }^{24}(2\sigma )}+24-{\tilde{T}}_{tw}^s(\tau )$$

$$p^{-1}\prod _{d=1}^{\infty }\prod _{r\in \mathbb{Z}}(1-p^d{q}^{\frac{r}{2}}{)}^{C_{d,r}(\frac{rd}{2})}=T^s(2\sigma )-T^s(\tau )$$

$$p^{-1}\prod _{d=1}^{\infty }\prod _{r\in \mathbb{Z}}(1-(-1{)}^r{p}^d{q}^{\frac{r}{2}}{)}^{C_{d,r}(\frac{rd}{2})}=T^s(2\sigma )-{\tilde{T}}^s(\tau ),$$

$$V^{f♮}\cong A\oplus S.$$

$$V_{tw}^{f♮}\cong V\oplus C.$$

$$V^{s♮}\cong A\oplus C,$$

$$V_{tw}^{s♮}\cong V\oplus S.$$

$$F_{24}\cong A\oplus V,$$

$$V_{\mathrm{NS}+}=A{V}_{\mathrm{NS}-}=S{V}_{\mathrm{R}+}=C{V}_{\mathrm{R}-}=V.$$

$$f_{\mathrm{NS}}(\tau )$$

$$f_{\widetilde{\mathrm{NS}}}(\tau )$$

$$f_{\mathrm{R}}(\tau )$$

$$f_{\tilde{\mathrm{R}}}(\tau )$$

$$\eta (\tau ):=q^{1/24}\prod _{n=1}^{\infty }(1-q^n).$$

$$f_{\mathrm{NS}}(\tau )$$

$$f_{\widetilde{\mathrm{NS}}}(\tau )$$

$$f_{\mathrm{R}}(\tau )$$

$$f_{\tilde{\mathrm{R}}}(\tau )$$

$$F_{\mathrm{NS}\pm }(\tau )={\mathrm{Tr}}_{\mathrm{NS}}\left(\frac{1\pm (-1{)}^F}{2}{q}^{L_0-1/2}\right)=\frac{f_{\mathrm{NS}}\pm f_{\widetilde{\mathrm{NS}}}}{2},$$

$$F_{\mathrm{R}\pm }(\tau )={\mathrm{Tr}}_{\mathrm{R}}\left(\frac{1\pm (-1{)}^F}{2}{q}^{L_0-1/2}\right)=\frac{f_{\mathrm{R}}\pm f_{\tilde{\mathrm{R}}}}{2};$$

$$F_{\mathrm{NS}\pm }(\tau )=\sum _{r\in \frac{1}{2}\mathbb{Z}}{C}_{\mathrm{NS}\pm }(r)q^r,$$

$$F_{\mathrm{NS}\pm }(\tau )=\sum _{n\in \mathbb{Z}}{C}_{\mathrm{R}\pm }(n)q^n,$$

$$F_{\mathrm{NS}-}=F_{\mathrm{R}+}=F_{\mathrm{R}-}-24=2048q+49152q^2+\dots .$$

$$f(1,1;\tau )$$

Full text

CALT-TH-2018-005

1. 1 Introduction and motivation

2. 2 The Conway modules $V^{f\natural}$ and $V^{s\natural}$

3. 3 Construction of the algebra

4. 3.1 Matter sector

5. 3.2 Ghosts and superghosts

6. 3.3 BRST cohomology and physical states

7. 3.4 A Lie superalgebra of physical states

8. 3.5 Root multiplicities

9. 3.6 Denominator

10. 3.7 The BKM superalgebra in type II string theory

11. 4 Hecke operators and the denominator identity

12. 4.1 Proof of the denominator identities

13. 5 The simple roots

14. 5.1 Simple roots of the Conway Lie superalgebra

15. 5.2 Simple roots for the other denominator

16. 6 Conway action and twisted denominators

17. 6.1 Twisted denominators of the Conway BKM superalgebra

18. 6.2 Twisted denominators at the other cusp

19. 6.3 Examples

20. 6.3.1 Order 2

21. 6.3.2 Order 3

22. 7 Conclusion

23. A Borcherds Kac-Moody superalgebras

24. B Construction of the algebra: further details

25. C Zero momentum cohomology

26. D Proof of identity (4.16)

1 Introduction and motivation

Motivation: Our motivations for this work are twofold, each pertaining to an area of active interest in mathematical physics. The first is to provide a new, concrete example of an interesting infinite-dimensional algebra acting on BPS states in string theory. In this context, our algebra is a special type of Lie superalgebra known as a Borcherds Kac-Moody (BKM) algebra. We envision that this algebra will arise in a context close to the original proposal of Harvey and Moore [43, 44]: that BPS states form an algebra that in some situations is a BKM algebra.111Here, as in [54, 55], we expect to find a BKM algebra on the nose, such that its generators are not the BPS states themselves but rather are (conjecturally) in one-to-one correspondence with BPS states. On the other hand, cohomological Hall algebras (COHAs) have been argued [50, 51] to be the correct framework to describe algebras of BPS states, and clarifying the relationship between BKM algebras and COHAs in explicit examples should be rewarding (see also [49] for a promising arena in which to relate these notions). It is clearly of interest to have more examples of algebras of BPS states—which moreover have deep connections to automorphic forms, motivic Donaldson-Thomas invariants, and vertex algebras—in the context of string compactifications on compact Calabi-Yau manifolds or asymmetric (i.e. non-geometric) orbifolds of compact Calabi-Yau manifolds like those in [54, 55] and this note. Our BKM is a concrete example in which we expect the relations among these topics can be explicitly studied, and we hope it forms an interesting new bridge among these subjects.

Our second motivation is to further clarify the relationship between moonshine and string theory. Moonshine in its most basic form is a surprising connection between sporadic groups and modular forms; it began with the observation that there is a connection between $J(\tau)$, the unique modular function with respect to $SL(2,\mathbb{Z})$ which has expansion222Throughout we take $q=e^{2\pi i\tau}$.

[TABLE]

as $\tau\to i\infty$, and the monster group, $\mathbb{M}$, the largest of the sporadic finite simple groups. This function has an expansion

[TABLE]

and satisfies the remarkable identity

[TABLE]

where $p=e^{2\pi i\sigma}.$

The monstrous moonshine conjectures of Conway, Norton [22], and Thompson [67, 68] state that there exists a $\mathbb{Z}$-graded vector space

[TABLE]

such that each subspace $V^{\natural}_{n}$ furnishes a finite-dimensional representation of $\mathbb{M}$ and such that

[TABLE]

Furthermore, they conjecture that for each element $g\in\mathbb{M}$ the so-called McKay-Thompson series,

[TABLE]

is a Hauptmodul for a certain genus zero discrete subgroup $\Gamma_{g}<\text{SL}(2,\mathbb{R})$. We say that $\Gamma_{g}$ is genus zero if the compactified quotient of the upper half-plane $\mathbb{H}$ by $\Gamma_{g}$ has the topology of a sphere and that a modular function is a Hauptmodul for $\Gamma_{g}$, when it establishes a biholomorphic map from $\overline{\mathbb{H}/\Gamma_{g}}$ to the Riemann sphere.

A construction of $V^{\natural}$ was furnished by the work of Frenkel-Lepowsky-Meurmann (FLM) [36] using the formalism of vertex operator algebra (VOA), or chiral conformal field theory (CFT) in physics terms. The FLM monster module is a $c=24$ theory of chiral bosons with target space a $\mathbb{Z}_{2}$ orbifold of the 24-dimensional torus $\mathbb{R}^{24}/\Lambda$, where $\Lambda$ is the Leech lattice. (In fact, the construction in [36] represented the first example of the procedure that is now known as orbifold of a CFT.) The monstrous moonshine conjectures were proven by Borcherds [5] who made use of the FLM monster module $V^{\natural}$ and the no-ghost theorem of bosonic string theory, and, moreover, who developed the formalism of BKM algebras to prove the genus zero property. Furthermore, the beautiful identity (1.1) is interpreted as the (Borcherds)-Weyl-Kac denominator formula for this BKM algebra.

As we will review below, a proposal for a physical origin of the genus zero property was recently made in [54, 55] by considering a certain two-dimensional compactification of heterotic string theory, where the monstrous BKM developed by Borcherds roughly appears as the an algebra of spacetime BPS states in this string theory. Thus the following natural questions arise: Is there a BKM algebra which plays a role in other instances of moonshine? And does it have an interpretation as an algebra of BPS states in string theory?

We answer the former question in the affirmative for the case of the so-called Conway moonshine, by constructing a (super)BKM algebra with Conway symmetry, and sketch a path to a positive answer for the second. As we review in §2, Conway moonshine is a connection between the group $Co_{0}$ (“Conway zero”), the automorphism group of the Leech lattice, and a set of distinguished modular functions associated to elements $g\in Co_{0}$. Building on the work of [36, 27], in [30] Duncan and Mack-Crane establish the existence of a ${1\over 2}\mathbb{Z}$-graded module dubbed $V^{s\natural}$ whose graded trace functions are normalized Hauptmoduln for genus zero groups. Thus $V^{s\natural}$ is a natural analogue of the monster module $V^{\natural}$. This module can be thought of as the Neveu-Schwarz sector of a chiral $\mathcal{N}=1$ superconformal field theory of central charge 12. Furthermore, there exists a canonically twisted module $V^{s\natural}_{tw}$, which can be thought of as the Ramond sector of this chiral theory, and whose trace functions also satisfy a genus zero property.

More specifically, if we define

[TABLE]

and

[TABLE]

we can associate the following set of graded (super)trace functions to each element $g\in Co_{0}$:333When $g$ is the identity element we suppress all subscripts $``g"$.

[TABLE]

and

[TABLE]

The genus zero property of $V^{s\natural}$ was proven in Theorems 4.9 and 4.10 of [30] and concerns the functions $T^{s}_{g}(\tau),T^{s}_{tw,g}(\tau)$. The explicit statements are the following:

For all $g\in Co_{0}$, the function $T^{s}_{g}(2\tau)$ is a so-called “normalized Hauptmodul” for a genus zero subgroup of $SL(2,\mathbb{R})$; and 2. 2.

For all $g\in Co_{0}$, the function $T^{s}_{g,tw}(\tau)$ is either

1. (a)

constant with value $-\chi_{g}$ when $g$ has a fixed point in its action on the Leech lattice; or 2. (b)

a Hauptmodul for a genus zero subgroup of $SL(2,\mathbb{R}).$

By normalized Hauptmodul (with vanishing constant term), we mean a Hauptmodul which behaves as $T_{g}(q)\sim q^{-1}+O(q)$ as $\tau\to i\infty$. This property is the direct analogue of the genus zero property for the monstrous McKay-Thompson series. In fact, one finds that $T^{s}_{g,tw}(\tau)=T_{g^{\prime}}(\tau)$ for some $g^{\prime}\in\mathbb{M}$, and the corresponding genus zero group appearing is a group $\Gamma_{g^{\prime}}$ which appears in monstrous moonshine. Similarly, for many (though not all) of the functions $T^{s}_{g}(2\tau)$, the invariance group coincides with one of the genus zero groups appearing in monstrous moonshine.444We thank John Duncan for clarifications about this point.

In this work we construct a super-BKM algebra with automorphism group $Co_{0}$ which is the natural analogue of the monstrous BKM algebra first constructed by Borcherds. We hope that the construction of this algebra may provide an independent route to understand (or, optimistically, provide an alternative proof of) the genus zero property of Conway moonshine, its corresponding role in string theory, and its connection to BPS algebras. In particular, we find (proven in §4) that the functions (1.2)-(1.5) when $g$ is the identity satisfy a beautiful set of formulas,

[TABLE]

where $C_{d,r}$ are defined in (4.8),555These $C_{d,r}$ can be expressed in terms of the coefficients $c^{s},\tilde{c}^{s}$. and where the first two lines correspond to the superdenominator and denominator of our algebra, respectively,666We conjecture that the last two lines represent the denominator and super-denominator of a different BKM superalgebra. thus generalizing the famous Borcherds denominator formula (1.1) for the $J$-function. Furthermore, we derive a twisted version of the first two formulas (see (6.16) and (6.17)) for all elements $g\in Co_{0}$ by using the explicit action of the Conway group on our algebra. Finally, we conjecture corresponding twisted versions of the second two formulas (see (6.18) and (6.19)) for all elements $g\in Co_{0}$, which should have an explanation in terms of an action of $Co_{0}$ on a different BKM superalgebra.

Finally, it is interesting to mention that the Conway moonshine module has also been of interest in the recent explorations of moonshine as it appears in string theory on K3, e.g. [11, 31, 20, 21, 56, 45, 48, 23, 66], and is therefore often closely linked with the Mathieu and umbral moonshine developments [33, 15, 16]. Many aspects of the latter remain to be understood, including a uniform construction of explicit modules and their relationship to string theory/K3 surfaces, so we hope our explorations of the simpler, “classical” monstrous and Conway moonshines may be an important step towards building a unifying framework in which to address these open questions.

Background: Not only does this work rely heavily on the earlier works of Borcherds and Scheithauer [5, 61, 62], but we are also motivated by the approach to monstrous moonshine developed in [54, 55], which we briefly review below.

In [54, 55], the authors consider a heterotic string worldsheet theory where the internal or matter sector of the CFT is given by the $c=24$ Monster module of FLM on the left and the $c=12$ Conway module on the right. One may view this internal CFT as an asymmetric orbifold of a certain eight-torus. The theory is then compactified on an additional circle. From this basic theory, one can construct a family of theories by orbifolding—in fact, one can construct consistent orbifold theories for every element of the Monster group (in contrast to orbifolds of the Monster module alone, which encounter failures of level-matching). The worldsheet analysis of these models provides a physical reinterpretation of the results of Borcherds and Carnahan [5, 8, 9]: one tensors the matter CFT with an appropriate ghost system, applies the no-ghost theorem, and constructs a Lie algebra of physical states using Lie algebra homology (which has its physical origins in BRST cohomology). The generators and relations of this Lie algebra, manifested in the algebra’s (twisted) denominator identities, imply that the McKay-Thompson series satisfy replicability identities that turn out to be equivalent to the Hauptmodul property [24]. Borcherds verified the last step using a brute-force computation to fix some low-lying coefficients of the modular functions in question. The analysis of [54, 55] instead relates $\Gamma_{g}$ to the groups of string theory T-dualities, and provides a physical interpretation for the Hauptmodul property, which follows from a careful analysis of the behavior of the models upon decompactifying of the circle. In the Conway case, we will follow a parallel worldsheet analysis by Scheithauer [61, 62, 63] which generalizes the work of Borcherds to SVOAs; indeed, the Conway module may be viewed as a $\mathbb{Z}_{2}$-orbifold of the SVOA associated to the Fake Monster Lie algebra studied by Scheithauer.

As a final point, we note that the denominator formulas of BKMs, including the Monster Lie algebra and the Conway Lie algebra of this note, are automorphic forms with respect to orthogonal groups of indefinite signature $O(m,n)$, e.g. $O(2,2;\mathbb{R})\simeq SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ [6, 7]777See also [40, 41] for examples of BKMs that are characterized by automorphic denominators.. In [54, 55], these denominators naturally arise as BPS state counting functions in spacetime, when one computes graded traces in the Fock space generated by an arbitrary number of free strings in the monstrous orbifold models. (In this note, we focus for simplicity on the worldsheet of a single string). Duality-invariant quantities in string theory, such as BPS state counting functions and certain couplings in the low energy effective action, are necessarily automorphic forms with respect to the duality group. In the context of K3 compactifications and orbifolds thereof, such functions are known to be essentially denominators of BKMs [26, 25] and may help clarify associated moonshine phenomena [10, 58, 57, 56].

Summary: The structure of the rest of the paper is as follows. In §2 we introduce the Conway module and describe its (twisted) partition functions in the Neveu-Schwarz and Ramond sectors. In §3 we construct the Conway Lie algebra using the SVOA of Duncan, from the perspective of a worldsheet of a chiral superstring theory. We introduce the matter and ghost sectors in turn, then discuss the BRST cohomology of physical states and the resulting Lie algebra. From this perspective we enumerate the multiplicity of roots in the algebra and derive denominator and superdenominator identities. Finally, we describe how our chiral construction can be completed to a type II string compactification (including, in particular, chiral and antichiral sectors and appropriate GSO projections). In §4 we introduce equivariant Hecke operators and use this machinery to prove our (super)denominator identities; this proof is similar in approach to Borcherds’ proof of the Koike-Norton-Zagier identity for the modular $J$-function, but requires some additional technology. In §5 we use the denominator identity to deduce the simple roots of the Conway Lie algebra. Finally, we describe twisted denominator formulas in §6. These conjecturally correspond to BPS indices in a type II superstring compactification and would be essential in a physics proof of the genus zero property for Conway moonshine. We conclude with some open questions in §7. In four appendices we provide some technical details, respectively: a brief overview of Borcherds Kac-Moody algebras, some additional details on the construction of the algebra, computations regarding the zero-momentum limit of the BRST complex, and a proof of an identity used in the main text.

2 The Conway modules $V^{f\natural}$