Composite-order modular moonshine and the surviving open problems of Borcherds-Ryba's Modular moonshine II: Tate cohomology of the moonshine module for composite-order elements, the 4B compressed algebra, normalizer actions, and generalized Kac-Moody algebras mod p

Modular moonshine, initiated by Ryba and developed by Borcherds and Ryba in Modular moonshine II (Duke Math. J. 83 (1996), 435-459), attaches to each element $g$ of prime order $p$ in the monster group a vertex superalgebra ${}^gV$ over $\mathbb{F}_p$ obtained as the Tate cohomology $\hat H^*(g,V^\natural)$ of an integral form of the moonshine vertex operator algebra, with graded Brauer character equal to a McKay-Thompson series. That prime-order theory, its integral-form assumptions, and the pB conjecture are now resolved: Borcherds proved the pB conjecture in Modular moonshine III, and Carnahan (2019) constructed the self-dual $\mathbb{Z}$-form $V^\natural_\mathbb{Z}$ that discharges the remaining assumptions. The still-open core consists of the separately posed open problems of Section 7 of Modular moonshine II (items 3-6, item 3 in its corrected form after Urano's class-8A counterexample refuted the naive conjecture): (1) what happens for elements $g$ of composite order $N$ - the module structure, possible vertex (super)algebra structure, and generalized Brauer characters of $\hat H^*(g,V^\natural_\mathbb{Z})$; (2) the corrected vanishing problem for $\hat H^1$ (Urano's Conjecture 5.11 is open); (3) construction of the compressed 4B modular vertex algebra, conjecturally a reduction modulo 2 or 4 of a vertex algebra for $F_4$; (4) the traces of the normalizer $N_\mathbb{M}(\langle g\rangle)$ on $\hat H^*(g,V)$ beyond the known centralizer action; (5) a structure theory of the Lie superalgebras obtained as Tate cohomology of integral forms of the monster Lie algebra, i.e. generalized Kac-Moody algebras mod p. Each is independently open, and resolving any single one of them counts as resolving the task.

Background

Vertex operator algebras and monstrous moonshine. A vertex operator algebra (VOA) is a graded complex vector space $V=\bigoplus_{n\ge0}V_n$ with a vacuum vector and a conformal vector, equipped with a state-field correspondence $Y$ assigning to each state $v$ a formal field $Y(v,z)=\sum_n v_nz^{-n-1}$ subject to locality and the Jacobi identity; it is the algebraic structure underlying chiral conformal field theory. The moonshine module $V^\natural$ of Frenkel-Lepowsky-Meurman is a holomorphic VOA of central charge 24 whose automorphism group is the monster group $\mathbb{M}$, the largest sporadic finite simple group. Monstrous moonshine (Conway-Norton, proved by Borcherds) asserts that for each $g\in\mathbb{M}$ the graded trace, or McKay-Thompson series, $T_g(q)=\sum_{n\ge1}\mathrm{Tr}(g|V_n)q^{n-1}$ is the normalized generator (Hauptmodul) of the function field of a genus-zero modular curve; an element is called Fricke when $T_g$ is invariant under the Fricke involution $\tau\mapsto-1/(N\tau)$, which holds for 120 of the 171 McKay-Thompson series (the monster has 194 conjugacy classes, but $T_g$ depends only on the cyclic subgroup $\langle g\rangle$ generated by $g$, so these classes merge into 171 series). Borcherds' proof used the monster Lie algebra $\mathfrak{m}$, a $\mathbb{Z}_2$-graded generalization of affine Kac-Moody algebras now called a Borcherds-Kac-Moody or generalized Kac-Moody (GKM) algebra, whose root spaces are the homogeneous pieces of $V^\natural$.

Tate cohomology and modular moonshine. For a finite cyclic group $\langle g\rangle$ acting on an abelian group $A$, write $N_g=\sum_{h\in\langle g\rangle}h$ for the norm and $I$ for the augmentation ideal of $\mathbb{Z}[\langle g\rangle]$; Tate cohomology is the 2-periodic family $\hat H^0(g,A)=A^{\langle g\rangle}/N_g(A)$ and $\hat H^1(g,A)=\ker(N_g)/IA$, and the cup product makes $\hat H^*(g,A)=\hat H^0\oplus\hat H^1$ a superalgebra (even and odd parts). If $A$ is torsion-free then $\hat H^*(g,A)$ is annihilated by $|\langle g\rangle|$. Ryba conjectured, and Borcherds-Ryba proved in Modular moonshine II, that for $g$ of prime order $p$ the Tate cohomology of a suitable integral form of $V^\natural$ (an integral form being a $\mathbb{Z}$-submodule closed under the vertex-algebra operations whose tensor with $\mathbb{C}$ recovers $V^\natural$) is a vertex superalgebra ${}^gV$ over $\mathbb{F}_p$ on which the centralizer $C_\mathbb{M}(g)$ acts by automorphisms through its quotient $C_\mathbb{M}(g)/\langle g\rangle$ (Modular moonshine III, Theorem 6.1, shows that for the classes 3B, 5B, 7B and 13B the largest normal $p$-subgroup $O_p$ of $C_\mathbb{M}(g)$ acts trivially, so there the action factors through $C_\mathbb{M}(g)/O_p$), with graded Brauer character equal to a McKay-Thompson series; a Brauer character of a finite-dimensional representation over $\mathbb{F}_p$ is the complex-valued class function on the $p$-regular elements (the elements of order prime to $p$) obtained by summing the Teichmuller lifts of the eigenvalues. Concretely, for an element $g$ of type pA the odd part vanishes ($\hat H^1=0$) and ${}^gV=\hat H^0$ is an ordinary vertex algebra acted on by $C_\mathbb{M}(g)/\langle g\rangle$, the Brauer trace of any $p$-regular element $h$ being the Hauptmodul $T_{gh}$ (Borcherds-Ryba, Corollary 4.8, for the classes 3A, 3C, 5A, 7A, 11A and 23A; Borcherds, Modular moonshine III, Theorem 4.1, for class 13A and all $p\ge17$; the class 2A in Theorem 5.2, under Assumption 5.1), while for the elements of type pB (classes 2B, 3B, 5B, 7B, 13B, the prime-order classes whose McKay-Thompson series have negative coefficients and whose groups are $\Gamma_0(p)$ rather than $\Gamma_0(p)^+$) both parts are nonzero, and for 3B, 5B, 7B and 13B the central involution of $C_\mathbb{M}(g)/O_p$ acts as $+1$ on $\hat H^0$ and $-1$ on $\hat H^1$ (the pB conjecture, proved in Modular moonshine III, Theorem 6.1). For composite order $N$ the coefficient ring $\mathbb{Z}/N\mathbb{Z}$ is not a field and ordinary Brauer characters are unavailable; Urano introduced generalized Brauer characters for finite-length modules over a discrete valuation ring that repair this.

Integral forms. The original proofs rested on integral structures: the $\mathbb{Z}[1/2]$-form of $V^\natural$ from the Frenkel-Lepowsky-Meurman construction, Assumption 5.1 of Modular moonshine II providing a compatible $\mathbb{Z}[1/3]$-form, and self-duality hypotheses on integral forms of the monster Lie algebra. The Lie-algebra hypotheses were proved by Borcherds in The fake monster formal group via an integral no-ghost theorem (the Goddard-Thorn positivity result from string theory, whose integral version shows that the degree pieces of the $\mathbb{Z}_p$-form of the monster Lie algebra are self-dual lattices). The remaining assumptions, needed for $p=2$, were discharged by Carnahan, who constructed a unique self-dual $\mathbb{Z}$-form $V^\natural_\mathbb{Z}$ with monster symmetry (using cyclic orbifold constructions, which reconstruct a VOA from the fixed points of a finite cyclic group of automorphisms together with its twisted sectors, made rigorous by Abe-Lam-Yamada, and faithfully flat descent), which makes modular moonshine for prime-order elements unconditional in a slightly weakened form: his published Corollary 3.10 (weak modular moonshine) yields, for each prime $p$ and each element of class pA, a vertex algebra over $\mathbb{F}_{p^n}$ (for some $n$) with centralizer action whose graded Brauer characters are the McKay-Thompson series, the case $p=2$ running over $\mathbb{F}_{2^n}$ for some $n\le4$.

Compression and the composite-order frontier. Modular moonshine exhibits a striking compression phenomenon: the regraded $\hat H^0(3C,V[1/2])$ (the piece of degree $3n-1$ regraded to $n$, with fields rescaled accordingly) is probably isomorphic to $V_{E_8}/3V_{E_8}$, the mod-3 reduction of the lattice VOA of the $E_8$ root lattice, which is acted on by the finite group $E_8(\mathbb{F}_3)$, although the centralizer quotient $C_\mathbb{M}(3C)/\langle g\rangle$ is the Thompson sporadic simple group, whose connection to $E_8$ has no a priori explanation. Borcherds-Ryba predicted an analogous compression for the class 4B (centralizer $(4\times F_4(2)).2$, McKay-Thompson series $q^{-1}+52q+834q^3+4760q^5+\cdots$): the compressed $\hat H^0(4B,V)$ should be the reduction modulo 2 (or modulo 4) of a vertex algebra for $F_4$ over $\mathbb{Z}$. Beyond prime order the theory is thin: the only composite classes for which vanishing $\hat H^1=0$ is known are 15A and 21A (via the permutation-module argument of Theorem 4.7, completed by Urano); Urano computed a generalized Brauer character formula (a rational linear combination of the McKay-Thompson series $T_{g^dh}$ over the proper divisors $d$ of the order) and refuted the naive composite-order vanishing conjecture using the Fricke class 8A, whose McKay-Thompson series $q^{-1}+36q+128q^2+386q^3+\cdots$ has nonnegative coefficients yet $\hat H^1\neq0$; Urano's weaker Conjecture 5.11 (vanishing when $g$ and all its powers are Fricke) is open. Two satellite problems from Section 7 of Modular moonshine II also remain untouched: the traces of the elements of the normalizer $N_\mathbb{M}(\langle g\rangle)$ outside the centralizer on $\hat H^*(g,V)$ (the normalizer acts on the modules, but the trace values are unknown, and when $\hat H^1\neq0$ the normalizer need not preserve the vertex-superalgebra structure; the source remarks that these traces might be reachable by extending proposition 2.2 of Modular moonshine II, proved for a central element $g$ of prime order, to the case when $\langle g\rangle$ is only normal and not central), and a structure theory for the Lie superalgebras $\hat H^*(g,\mathfrak{m}_\mathbb{Z})$ obtained as Tate cohomology of integral forms of the monster Lie algebra, a GKM-algebras-mod-p program with no known analogue of the characteristic-zero denominator identity (the product-sum formula from which the root multiplicities of a GKM algebra are computed; the source suggests the Lie algebra homology groups of the positive degree subalgebras as a concrete instance of this program, which in characteristic 0 is equivalent to asking for the simple roots).

Problem Statement

Let $V^\natural_\mathbb{Z}$ be the self-dual $\mathbb{Z}$-form of the moonshine module with monster symmetry constructed by Carnahan (SIGMA 15 (2019), 030, Theorem 3.22, doi:10.3842/SIGMA.2019.030), and for $g\in\mathbb{M}$ let $\hat H^*(g,V^\natural_\mathbb{Z})=\hat H^0\oplus\hat H^1$ be the 2-periodic Tate cohomology of the cyclic group $\langle g\rangle$ with coefficients in $V^\natural_\mathbb{Z}$, a superalgebra that is annihilated by $|\langle g\rangle|$ and so is a module over $\mathbb{Z}/N\mathbb{Z}$ for $N=|\langle g\rangle|$. The following open problems, posed separately in Section 7 (Open problems and conjectures) of Borcherds-Ryba, Modular moonshine II (items 3-6, item 3 replaced by its corrected form, Urano's Conjecture 5.11, after the class-8A counterexample), extend the prime-order modular moonshine structure theory (now proved, in a slightly weakened form over $\mathbb{F}_{p^n}$, by Carnahan's Corollary 3.10) to composite order and to the associated modular Lie algebras. Parts (a)-(e) below correspond to source items 3-6; (a) and (c) are two independent aspects of source item 4 (the general composite-order question and the specific 4B compression) rather than two separate source problems. Each part is independently open, and resolving any single one of them counts as resolving the task:

(a) Composite order (item 4). What happens if $g$ is an element of composite order $N$? It suffices to resolve any one of the following: the $\mathbb{Z}/N\mathbb{Z}$-module structure of $\hat H^0(g,V^\natural_\mathbb{Z})$ and $\hat H^1(g,V^\natural_\mathbb{Z})$ as $C_\mathbb{M}(g)$-modules, whether $\hat H^0$ (or the correction of $\hat H^*(g,V^\natural_\mathbb{Z})$ when $\hat H^1\neq0$) carries a natural vertex (super)algebra structure over $\mathbb{Z}/N\mathbb{Z}$ or over $\mathbb{Z}/p^k\mathbb{Z}$ for the prime powers $p^k$ dividing $N$, extending the Borcherds-Ryba construction, and an explicit formula for the generalized Brauer character separating the contributions of $\hat H^0$ and $\hat H^1$ (Urano's Corollary 5.8 gives only the supertrace, as a linear combination of $T_{g^dh}$).

(b) Vanishing (item 3, corrected form). Prove or disprove Urano's Conjecture 5.11 (stated for self-dual integral forms: if $g$ is Fricke and every positive power of $g$ is Fricke, then $\hat H^1=0$). The original form of item 3 - whether $\hat H^1(g,V)$ vanishes whenever the McKay-Thompson series of $g$ has no negative coefficients - was refuted by Urano's counterexample at the Fricke class 8A, whose series has nonnegative coefficients yet $\hat H^1\neq0$.

(c) The 4B compressed algebra (item 4, the 4B case). Construct the modular vertex algebra corresponding to elements of type 4B: identify $\hat H^0(4B,V^\natural_\mathbb{Z})$ or its relevant reduction with the reduction modulo 2 (or modulo 4) of a vertex algebra for $F_4$ defined over $\mathbb{Z}$, in the same way that the compression of the vertex algebra for elements of type 3C is conjecturally the reduction $V_{E_8}/3V_{E_8}$ of an algebra for $E_8$, acted on by $E_8(\mathbb{F}_3)$. (Separately, in the final questions of Modular moonshine III, Borcherds asked whether $C_\mathbb{M}(g)/O_p$ is the full automorphism group of ${}^gV$, noting that for 3C the full automorphism group may be $E_8(\mathbb{F}_3)$.)

(d) Normalizer action (item 6). The modules $\hat H^*(g,V[1/2])$ are acted on not just by the centralizer $C_\mathbb{M}(g)$ but by the normalizer $N_\mathbb{M}(\langle g\rangle)$ of $\langle g\rangle$: what are the traces of the elements of $N_\mathbb{M}(\langle g\rangle)$ that are not in the centralizer? The source notes that when $\hat H^1\neq0$ the elements of the normalizer do not necessarily preserve the algebra structure on $\hat H^*$.

(e) GKM algebras mod p (item 5). Develop some sort of theory of "generalized Kac-Moody algebras mod p" applicable to the study of the Lie superalgebras obtained as Tate cohomology of the $\mathbb{Z}_p$-forms of the monster Lie algebra (whose degree-piece self-duality is proved by the integral no-ghost theorem of The fake monster formal group). The source also observes that many of these modular Lie algebras cannot be obtained by reducing some integral form of a generalized Kac-Moody algebra mod p, because the denominator formula shows that some of the simple roots would then have negative multiplicity.

References

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