The space of graded traces for holomorphic VOAs of small central charge

Holomorphic integer graded vertex superalgebras

In this note we study holomorphic integer graded vertex superalgebras. We prove that all such vertex superalgebras of central charge 8 and 16 are purely even. For the case of central charge 24 we prove that the weight-one Lie superalgebra is either zero, of superdimension 24, or else is one of an explicit list of 1332 semisimple Lie superalgebras.

$\mathcal{N}=4$ SYM, Argyres-Douglas Theories, and an Exact Graded Vector Space Isomorphism

In this first of two papers, we explain in detail the simplest example of a broader set of relations between apparently very different theories. Our example relates $\mathfrak{su}(2)$ $\mathcal{N}=4$ super Yang-Mills (SYM) to a theory we call "$(3,2)$." This latter theory is an exactly marginal diagonal $SU(2)$ gauging of three $D_3(SU(2))$ Argyres-Douglas (AD) theories. We begin by observing that the Schur indices of these two theories are related by an algebraic transformation that is surprisingly reminiscent of index transformations describing spontaneous symmetry breaking on the Higgs bran…

Orbifold construction and Lorentzian construction of Leech lattice vertex operator algebra

We generalize Conway-Sloane's constructions of the Leech lattice from Niemeier lattices using Lorentzian lattice to holomorphic vertex operator algebras (VOA) of central charge 24. It provides a tool for analyzing the structures and relations among holomorphic VOAs related by orbifold construction. In particular, we are able to get some useful information about certain lattice subVOAs associated with the Cartan subalgebra of the weight one Lie algebra. We also obtain a relatively elementary proof that any strongly regular holomorphic VOA of central charge $24$ with a non-trivial weight one sub…

Holomorphic potentials for graded D-branes

We discuss gauge-fixing, propagators and effective potentials for topological A-brane composites in Calabi-Yau compactifications. This allows for the construction of a holomorphic potential describing the low-energy dynamics of such systems, which generalizes the superpotentials known from the ungraded case. Upon using results of homotopy algebra, we show that the string field and low energy descriptions of the moduli space agree, and that the deformations of such backgrounds are described by a certain extended version of `off-shell Massey products' associated with flat graded superbundles. As…

中心荷不超过24的全纯顶点算子超代数的幺正性与强分次局部性

Unitarity and strong graded locality of holomorphic vertex operator superalgebras with central charge at most 24

本文研究中心荷不超过24的良好全纯VOSA的幺正性与强分次局部性，排除特定假设赝副本与例外情形，证明多数此类VOSA满足相关性质，可生成对应共形网。

L-infinity Algebras and Deformations of Holomorphic Maps

We construct the deformation functor associated with a pair of morphisms of differential graded Lie algebras, and use it to study infinitesimal deformations of holomorphic maps of compact complex manifolds. In particular, using L-infinity structures, we give an explicit description of the differential graded Lie algebra that controls this problem.

微分格架范畴在全纯辛几何中的应用

Differential graded categories in holomorphic symplectic geometry

本文研究全纯辛流形的微分格架范畴及其变形量子化，探讨其形式性条件及Kaledin类对形式性的阻碍作用。

Modular forms and almost linear dependence of graded dimensions

For every positive integral level $k$ we study arithmetic properties of certain holomorphic modular forms associated to modular invariant spaces spanned by graded dimensions of $L_{\hat{sl_2}}(k Λ_0)$-modules. We found a necessary and sufficient condition for their vanishing and showed that these modular forms resemble classical Eisenstein series $E_{2k+2}(τ)$. Furthermore, we derived similar results for $\mathcal{M}(p,p')$ Virasoro minimal models, thus generalizing some results of Mortenson, Ono and the author.

The graded ring of quantum theta functions for noncommutative torus with real multiplication

For quantum torus generated by unitaries $UV = e(θ)VU$ there exist nontrivial strong Morita autoequivalences in case when $θ$ is real quadratic irrationality. A.Polishchuk introduced and studied the graded ring of holomorphic sections of powers of the respective bimodule (depending on the choice of a complex structure). We consider a Segre square of this ring whose graded components are spanned by Rieffel scalar products of Polishchuk's holomorphic vectors. These graded components are linear spaces of quantum theta functions in sense of Yu.Manin.

Differential Graded Cohomology and Lie algebras of Holomorphic Vector Fields

The Dolbeault resolution of the sheaf of holomorphic vector fields $Lie$ on a complex manifold $M$ relates $Lie$ to a sheaf of differential graded Lie algebras, known as the Frölicher-Nijenhuis algebra $g$. We establish - following B. L. Feigin - an isomorphism between the differential graded cohomology of the space of global sections of $g$ and the hypercohomology of the sheaf of continuous cochain complexes of $Lie$. We calculate this cohomology up to the singular cohomology of some mapping space. We use and generalize results of N. Kawazumi on complex Gelfand-Fuks cohomology. Applications a…

中心荷小于25且具有两个原初场的幺正RCFT的分类

Classification of Unitary RCFTs with Two Primaries and Central Charge Less Than 25

本文分类了中心荷小于25且具有两个原初场的二维幺正有理共形场论，发现此类理论或为特定MMS理论，或为全纯VOA的陪集，最终枚举出123个理论并验证了方法的通用性。

Modularity of trace functions in orbifold theory for Z-graded vertex operator superalgebras

We study the trace functions in orbiford theory for Z-graded vertex operator superalgebras and obtain a modular invariance result. More precisely, let V be a C_2-cofinite Z-graded vertex operator superalgebra and G a finite automorphism group of V. Then for any commuting pairs (g,h) in G, the hσ-trace functions associated to the simple g-twisted V-modules are holomorphic in the upper half plane where σis the canonical involution on V coming from the superspace structure of V. If V is further g-rational for every g n G, the trace unctions afford a representation for the full modular group SL(2,…

微分格拉斯曼代数的形性与复拉格朗日子流形

Formality of differential graded algebras and complex Lagrangian submanifolds

本文通过变形量子化证明了复拉格朗日子流形的端射影微分格拉斯曼代数的形性，并推广到拉格朗日子流形对的情况，同时研究了形性在A∞模块族中的行为。

Characterizing Moonshine Functions by Vertex-Operator-Algebraic Conditions

Given a holomorphic $C_2$-cofinite vertex operator algebra $V$ with graded dimension $j-744$, Borcherds's proof of the monstrous moonshine conjecture implies any finite order automorphism of $V$ has graded trace given by a "completely replicable function", and by work of Cummins and Gannon, these functions are principal moduli of genus zero modular groups. The action of the monster simple group on the monster vertex operator algebra produces 171 such functions, known as the monstrous moonshine functions. We show that 154 of the 157 non-monstrous completely replicable functions cannot possibly …

Quasi-modular forms and trace functions associated to free boson and lattice vertex operator algebras

We study graded traces of vectors in free bosonic vertex operator algebras and lattice vertex operator algebras. We show in particular that trace functions in these two theories always have the shape f(q)/η(q)^d where f(q) is quasi-modular in the case of d free bosons, and modular (i.e., a sum of holomorphic modular forms of various weights) in the case of theories based on a lattice L of rank d. We also show how spherical harmonic polynomials with respect to L are related to primary fields in lattice theories.