Genus Two Modular Bootstrap

We study the Virasoro conformal block decomposition of the genus two partition function of a two-dimensional CFT by expanding around a Z3-invariant Riemann surface that is a three-fold cover of the Riemann sphere branched at four points, and explore constraints from genus two modular invariance and unitarity. In particular, we find 'critical surfaces' that constrain the structure constants of a CFT beyond what is accessible via the crossing equation on the sphere.Comment: 23 pages, 6 figures. v2: updated references, typos corrected. v3-v5: minor typos correcte

On the existence of generically smooth components for moduli spaces of rank 2 stable reflexive sheaves on P3

AbstractThe goal of this work is to prove that for almost all possible triples (c1, c2, c3) ϵ Z3 the moduli scheme M(2; c1, c2, c3), which parametrizes isomorphism classes of rank 2 stable reflexive sheaves on P3 with Chern classes c1, c2 and c3, has a generically smooth component. In order to obtain these results we construct a wide range of non-obstructed, m-normal curves with suitable degree and genus. We conclude this paper by adding some examples and remarks

A spectral curve approach to Lawson symmetric CMC surfaces of genus 2

Minimal and CMC surfaces in $S^3$ can be treated via their associated family of flat \SL(2,\C)-connections. In this the paper we parametrize the moduli space of flat \SL(2,\C)-connections on the Lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of Lawson's geometric construction. The parametrization uses Hitchin's abelianization procedure to write such connections explicitly in terms of flat line bundles on a complex 1-dimensional torus. This description is used to develop a spectral curve theory for the Lawson surface. This theory applies as well to other CMC and minimal surfaces with the same holomorphic symmetries as the Lawson surface but different Riemann surface structure. Additionally, we study the space of isospectral deformations of compact minimal surface of genus $g\geq2$ and prove that it is generated by simple factor dressing.Comment: 39 pages; sections about isospectral deformations and about CMC surfaces have been added; the theorems on the reconstruction of surfaces out of spectral data have been improved; 1 figure adde

Rings whose total graphs have genus at most one

Let $R$ be a commutative ring with $\Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by \T(\Gamma(R)). It is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x, y\in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y\in\Z(R)$. We investigate properties of the total graph of $R$ and determine all isomorphism classes of finite commutative rings whose total graph has genus at most one (i.e., a planar or toroidal graph). In addition, it is shown that, given a positive integer $g$, there are only finitely many finite rings whose total graph has genus $g$.Comment: 7 pages. To appear in Rocky Mountain Journal of Mathematic