In the abelian case (the subject of several beautiful books) fixing some combinatorial structure (so called theta structure of level k) one obtains a special basis in the space of sections of canonical polarization powers over the jacobians. These sections can be presented as holomorphic functions on the "abelian Schottky space". This fact provides various applications of these concrete analytic formulas to the integrable systems, classical mechanics and PDE's. Our practical goal is to do the same in the non abelian case that is to give an answer to the Beauville's question. In future we hope to extend this digest to a mathematical mohograph with title "VBAC".
Geometric Quantization links holomorphic geometry with real geometry, a relation that is a prototype for the modern development of mirror symmetry. We show how to use this treatment to construct a special basis in every space of conformal blocks. This is a direct generalization of the basis of theta functions with characteristics in every complete linear system on an Abelian variety (see Mumford's "Tata lectures on theta" cite(Mumford)). The same construction generalizes the classical theory of theta functions to vector bundles of higher rank on Abelian varieties and K3 surfaces. We also discu…
A linear algebraic group $G$ is represented by the linear space of its algebraic functions $F(G)$ endowed with multiplication and comultiplication which turn it into a Hopf algebra. Supplying $G$ with a Poisson structure, we get a quantized version $F_q(G)$ which has the same linear structure and comultiplication, but deformed multiplication. This paper develops a similar theory for abelian varieties. A description of abelian varieties $A$ in terms of linear algebra data was given by Mumford: $F(G)$ is replaced by the graded ring of theta functions with symmetric automorphy factors, and comult…
Quantum theta functions were introduced by the author in [Ma1]. They are certain elements in the function rings of quantum tori. By definition, they satisfy a version of the classical functional equations involving shifts by the multiplicative periods. This paper shows that for a certain subclass of period lattices (compatible with the quantization form), quantum thetas satisfy an analog of another classical functional equation related to an action of the metaplectic group upon the (half of) the period matrix. In the quantum case, this is replaced by the action of the special orthogonal group …
The paper consists of two sections. The first section provides a new definition of mirror symmetry of abelian varieties making sense also over $p$-adic fields. The second section introduces and studies quantized theta-functions with two-sided multipliers, which are functions on non-commutative tori. This is an extension of an earlier work by the author. In the Introduction and in the Appendix the constructions of this paper are put into a wider context.
A log Calabi--Yau surface $(X,D)$ is given by a smooth projective surface $X$, together with an anti-canonical cycle of rational curves $D \subset X$. The homogeneous coordinate ring of the mirror to such a surface, or to the complement $X\setminus D$, is constructed in the work of Gross-Hacking-Keel, following previous work of Gross-Siebert, using wall structures, and it is generated by theta functions. In our work with Mark Gross we had provided a recipe to concretely compute these theta functions from a combinatorially constructed wall structure in arbitrary dimensions. In this paper, we fi…
We propose a generalization of the classical theta function to higher cohomology of the polarization line bundle on a family of complex tori with positive index. The constructed cocycles vary horizontally with respect to the (projective) flat connection on this family coming from a heat operator. They also possess modular properties similar to the classical ones.
We address quantization of the natural symplectic structure on a moduli space of parabolic vector bundles of parabolic degree zero over $\mathbf{CP}^1$ with four parabolic points and parabolic weights in {0,1/2}. Identifying such parabolic bundles as vector bundles on an elliptic curve, we obtain explicit expressions for the corresponding non-abelian theta functions. These non-abelian theta functions are described in terms of certain naturally defined distributions on the compact group SU(2).
We show that a large class of maximally degenerating families of n-dimensional polarized varieties come with a canonical basis of sections of powers of the ample line bundle. The families considered are obtained by smoothing a reducible union of toric varieties governed by a wall structure on a real n-(pseudo-)manifold. Wall structures have previously been constructed inductively for cases with locally rigid singularities and by Gromov-Witten theory for mirrors of log Calabi-Yau surfaces and K3 surfaces by various combinations of the authors. For trivial wall structures on the n-torus we retri…
We give an explicit description of theta-characteristics on tropical curves and characterize the effective ones. We construct the moduli space for tropical theta-characteristics of genus g as a generalized cone complex. We describe the fibers of the specialization map from the classical moduli space of theta-characteristics on algebraic curves to our moduli space of tropical theta-characteristics.
Let G2 be the exceptional Lie group of automorphisms of the complex Cayley algebra and C be a smooth, connected, projective curve of genus at least 2. Using the map obtained from extension of structure groups, we prove explicit links between the space of generalized G2-theta functions over C and spaces of generalized theta functions associated to the classical Lie groups SL_2 and SL_3.
Let C be a curve of genus g, and let SU(r) be the moduli space of vector bundles of rank r on C, with trivial determinant. A general E in SU(r) defines a theta divisor in the linear system |r Theta|, where Theta is the canonical theta divisor in Pic^{g-1}(C). This defines a rational map SU(r) - - > |r Theta|, which is the map associated to the determinant bundle on SU(r) (the positive generator of Pic(SU(r)). In this paper we prove that in genus 2 this map is generically finite and dominant. The same method, together with some classical work of Morin, shows that in rank 3 and genus 3 the theta…
From a block-diagonal $(n+1) \times (m+1) \times (m+1)$ tensor symmetric in the last two entries one obtains two varieties: an intersection of symmetric determinantal hypersurfaces $X$ in $n$-dimensional projective space, and an intersection of quadrics $\mathfrak{C}$ in $m$-dimensional projective space. Under mild technical assumptions, we characterize the accidental singularities of $X$ in terms of $\mathfrak{C}$. We apply our methods to algebraic curves and show how to construct theta characteristics of certain canonical curves of genera 3, 4, and 5, generalizing a classical construction of…
We associate to a unimodular lattice L, endowed with an automorphism of square -1, a principally polarized abelian variety A:= L_R/L. We show that the configuration of i-invariant theta divisors of A follows a pattern very similar to the classical theory of theta characteristics; as a consequence we find that A has a high number of vanishing thetanulls. When L = E_8 we recover the 10 vanishing thetanulls of the abelian fourfold discovered by R. Varley.
Abelian Chern-Simons theory relates classical theta functions to the topological quantum field theory of the linking number of knots. In this paper we explain how to derive the constructs of abelian Chern-Simons theory directly from the theory of classical theta functions. It turns out that the theory of classical theta functions, from the representation theoretic point of view of A. Weil, is just an instance of Chern-Simons theory. The group algebra of the finite Heisenberg group is described as an algebra of curves on a surface, and its Schrodinger representation is obtained as an action on …
We give an algebraic analog of the functional equation of Riemann's theta function. More precisely, we define a `theta multiplier' line bundle over the moduli stack of principally polarized abelian schemes with theta characteristic and prove that its dual is isomorphic to the determinant bundle over the moduli stack. We do so by explicitly computing with Picard groups over the moduli stack. This is all done over the ring R=Z[1/2,i]: passing to the complex numbers, we recover the classical functional equation.
This paper outlines an approach to the non-abelian theta functions of the $SU(2)$-Chern-Simons theory with the methods used by A. Weil for studying classical theta functions. First we translate in knot theoretic language classical theta functions, the action of the finite Heisenberg group, and the discrete Fourier transform. Then we explain how the non-abelian counterparts of these arise in the framework of the quantum group quantization of the moduli space of flat $SU(2)$-connections on a surface, in the guise of the non-abelian theta functions, the action of a skein algebra, and the Reshetik…
Let $\X$ be an irreducible, smooth, projective curve of genus $g \geq 2$ defined over the complex field $\C.$ Then there is a covering $π: \X \longrightarrow ¶^1,$ where $¶^1$ denotes the projective line. The problem of expressing branch points of the covering $π$ in terms of the transcendentals (period matrix, thetanulls, e.g.) is classical. It goes back to Riemann, Jacobi, Picard and Rosenhein. Many mathematicians, including Picard and Thomae, have offered partial treatments for this problem. In this work, we address the problem for cyclic curves of genus 2, 3, and 4 and find relations among…
We give a comprehensive treatment of the transformation laws of theta functions from an algebro-geometric perspective, that is, in terms of moduli of abelian schemes. This is accomplished by introducing geometric notions of theta-descent structures, metaplectic stacks, and bundles of half-forms, whose analytic incarnations underlie different aspects of the classical transformation laws. As an application, we lay the foundations for a geometric theory of modular forms of half-integral weight and, more generally, for modular forms taking values in the Weil representation. We discuss further appl…
We give a geometric proof of a formula, due to Segal and Wilson, which describes the order of vanishing of the Riemann theta function in the direction which corresponds to the direction of the tangent space of a Riemann surface at a marked point. While this formula appears in the work of Segal and Wilson as a by-product of some non-trivial constructions from the theory of integrable systems (loop groups, infinite-dimensional Grassmannians, tau functions, Schur polynomials, ...) our proof only uses the classical theory of linear systems on Riemann surfaces.
We study a discrete analogue of the classical multivariate Gaussian distribution. It is supported on the integer lattice and is parametrized by the Riemann theta function. Over the reals, the discrete Gaussian is characterized by the property of maximizing entropy, just as its continuous counterpart. We capitalize on the theta function representation to derive statistical properties. Throughout, we exhibit strong connections to the study of abelian varieties in algebraic geometry.
In this paper, we give an extension of the classical story of the elliptic modular function to the Hilbert modular case for $\mathbb{Q}(\sqrt{5})$. We construct the period mapping for a family $\mathcal{F}=\{S(X,Y)\}$ of $K3 $ surfaces with $2$ complex parameters $X$ and $Y$. The inverse correspondence of the period mapping gives a system of generators of Hilbert modular functions for $\mathbb{Q}(\sqrt{5})$. Moreover, we show an explicit expression of this inverse correspondence by theta constants.
A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that theta surfaces are precisely the surfaces of double translation, i.e. obtained as the Minkowski sum of two space curves in two different ways. These curves are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their th…
In this paper we establish the relationships between theta functions of arbitrary order and their derivatives. We generalize our previous work math.AG/0310085 and prove that for any n>1 the map sending an abelian variety to the set of Gauss images of its points of order 2n is an embedding into an appropriate Grassmannian (note that for n=1, i.e. points of order 2, we only get generic injectivity). We further discuss the generalizations of Jacobi's derivative formula for any dimension and any order. The series of embedding of the level covers of moduli spaces of principally polarized abelian va…
We construct "quantum theta bases," extending the set of quantum cluster monomials, for various versions of skew-symmetric quantum cluster algebras. These bases consist precisely of the indecomposable universally positive elements of the algebras they generate, and the structure constants for their multiplication are Laurent polynomials in the quantum parameter with non-negative integer coefficients, proving the quantum strong cluster positivity conjecture for these algebras. The classical limits recover the theta bases considered by Gross-Hacking-Keel-Kontsevich. Our approach combines the sca…
Mess showed that the genus 2 Torelli group $T_2$ is isomorphic to a free group of countably infinite rank by showing that genus 2 Torelli space is homotopy equivalent to an infinite wedge of circles. As an application of his computation, we compute the homotopy type of the zero locus of any classical genus 2 theta function in $\mathfrak{h}_2 \times \mathbb{C}^2$, where $\mathfrak{h}_2$ denotes rank 2 Siegel space. Specifically, we show that the zero locus of any such function is homotopy equivalent to an infinite wedge of 2-spheres.
The classical Schur polynomials form a natural basis for the ring of symmetric polynomials, and have geometric significance since they represent the Schubert classes in the cohomology ring of Grassmannians. Moreover, these polynomials enjoy rich combinatorial properties. In the last decade, an exact analogue of this picture has emerged in the symplectic and orthogonal Lie types, with the Schur polynomials replaced by the theta and eta polynomials of Buch, Kresch, and the author. This expository paper gives an overview of what is known to date about this correspondence, with examples.
We discuss the relation between quantum curves (defined as solutions of equation $[P,Q]=\hbar$, where $P,Q$ are ordinary differential operators) and classical curves. We illustrate this relation for the case of quantum curve that corresponds to the $(p,q)$-minimal model coupled to 2D gravity.
