Riemann Bilinear Relations Research Articles

Equivariant log-concavity and equivariant Kähler packages

We show that the exterior algebra ΛR[α1,⋯,αn], which is the cohomology of the torus T=(S1)n, and the polynomial ring R[t1,…,tn], which is the cohomology of the classifying space B(S1)n=(CP∞)n, are Sn-equivariantly log-concave. We do so by explicitly giving the Sn-representation maps on the appropriate sequences of tensor products of polynomials or exterior powers and proving that these maps satisfy the hard Lefschetz theorem. Furthermore, we prove that the whole Kähler package, including the Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann bilinear relations, holds on the corresponding sequences in an equivariant setting.

Hodge–Riemann Relations for Schur Classes in the Linear and Kähler Cases

Abstract We prove a version of the Hodge–Riemann bilinear relations for Schur polynomials of Kähler forms and for Schur polynomials of positive forms on a complex vector space.

Goldman form, flat connections and stable vector bundles

We consider the moduli space $\mathscr{N}$ of stable vector bundles of degree 0 over a compact Riemann surface and the affine bundle $\mathscr{A}\to \mathscr{N}$ of flat connections. Following the similarity between the Teichmüller spaces and the moduli of bundles, we introduce the analogue of the quasi-Fuchsian projective connections – local holomorphic sections of $\mathscr{A}$ – that allow to pull back the Liouville symplectic form on $T\*\mathscr{N}$ to $\mathscr{A}$. We prove that the pullback of the Goldman form to $\mathscr{A}$ by the Riemann–Hilbert correspondence coincides with the pullback of the Liouville form. We also include a simple proof, in the spirit of Riemann bilinear relations, of the classic result – the pullback of Goldman symplectic form to $\mathscr{N}$ by the Narasimhan–Seshadri connection is the natural symplectic form on $\mathscr{N}$, introduced by Narasimhan and Atiyah & Bott.

Strict log-concavity of the Kirchhoff polynomial and its applications to the strong Lefschetz property

Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are “strictly” log-concave for simple matroids. In this paper, we show this strictness for simple graphic matroids, that is, we show that Kirchhoff polynomials of simple graphs are strictly log-concave. Our key observation is that the Kirchhoff polynomial of a complete graph can be seen as the (irreducible) relative invariant of a certain prehomogeneous vector space, which may be independently interesting in its own right. Furthermore, we prove that for any ai∈R>0, a1x1+⋯+anxn∈RM1 satisfies the strong Lefschetz property (moreover, Hodge–Riemann bilinear relation) at degree one of the Artinian Gorenstein algebra RM⁎ associated to a graphic matroid M, which is defined by Maeno and Numata for all matroids.

On Kawai theorem for orbifold Riemann surfaces

We prove a generalization of Kawai theorem for the case of orbifold Riemann surface. The computation is based on a formula for the differential of a holomorphic map from the cotangent bundle of the Teichmuller space to the $${\mathrm {PSL}}(2,\mathbb {C})$$ -character variety, which allows to evaluate explicitly the pullback of Goldman symplectic form in the spirit of Riemann bilinear relations. As a corollary, we obtain a generalization of Goldman’s theorem that the pullback of Goldman symplectic form on the $${\mathrm {PSL}}(2,\mathbb {R})$$ -character variety is a symplectic form of the Weil–Petersson metric on the Teichmuller space.

A remark on the Alexandrov–Fenchel inequality

In this article, we give a complex-geometric proof of the Alexandrov–Fenchel inequality without using toric compactifications. The idea is to use the Legendre transform and develop the Brascamp–Lieb proof of the Prékopa theorem. New ingredients in our proof include an integration of Timorin's mixed Hodge–Riemann bilinear relation and a mixed norm version of Hörmander's L2-estimate, which also implies a non-compact version of the Khovanskiĭ–Teissier inequality.

Homogeneity of cohomology classes associated with Koszul matrix factorizations

In this work we prove the so-called dimension property for the cohomological field theory associated with a homogeneous polynomial $W$ with an isolated singularity, in the algebraic framework of [A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1–122]. This amounts to showing that some cohomology classes on the Deligne–Mumford moduli spaces of stable curves, constructed using Fourier–Mukai-type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result established in [A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class, in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (American Mathematical Society, Providence, RI, 2001), 229–249] for certain characteristic classes of Koszul matrix factorizations of $0$. To reduce to this result, we use the theory of Fourier–Mukai-type functors involving matrix factorizations and the natural rational lattices in the relevant Hochschild homology spaces, as well as a version of Hodge–Riemann bilinear relations for Hochschild homology of matrix factorizations. Our approach also gives a proof of the dimension property for the cohomological field theories associated with some quasihomogeneous polynomials with an isolated singularity.

Local Hodge theory of Soergel bimodules

We prove the local hard Lefschetz theorem and local Hodge–Riemann bilinear relations for Soergel bimodules. Using results of Soergel and Kubel, one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction.

Tau-functions for quiver gauge theories

The prepotentials for the quiver supersymmetric gauge theories are defined as quasiclassical tau-functions, depending on two different sets of variables: the parameters of the UV gauge theory or the bare compexified couplings, and the vacuum condensates of the theory in IR. The bare couplings are introduced as periods on the UV base curve, and the consistency of corresponding gradient formulas for the tau-functions is proven using the Riemann bilinear relations. It is shown, that dependence of generalised prepotentials for the quiver gauge theories upon the bare couplings turns to coincide with the corresponding formulas for the derivatives of tau-functions for the isomonodromic deformations. Computations for the SU(2) quiver gauge theories with bi- and tri-fundamental matter are performed explicitly and analysed in the context of 4d/2d correspondence.

Hochster's theta invariant and the Hodge–Riemann bilinear relations

Let R be an isolated hypersurface singularity, and let M and N be finitely generated R-modules. As R is a hypersurface, the torsion modules of M against N are eventually periodic of period two (i.e., Tor i R ( M , N ) ≅ Tor i + 2 R ( M , N ) for i ≫ 0 ). Since R has only an isolated singularity, these torsion modules are of finite length for i ≫ 0 . The theta invariant of the pair ( M , N ) is defined by Hochster to be length ( Tor 2 i R ( M , N ) ) − length ( Tor 2 i + 1 R ( M , N ) ) for i ≫ 0 . H. Dao has conjectured that the theta invariant is zero for all pairs ( M , N ) when R has even dimension and contains a field. This paper proves this conjecture under the additional assumption that R is graded with its irrelevant maximal ideal giving the isolated singularity. We also give a careful analysis of the theta pairing when the dimension of R is odd, and relate it to a classical pairing on the smooth variety Proj ( R ) .

The mixed Hodge–Riemann bilinear relations for compact Kähler manifolds

We prove the Hodge–Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kahler manifolds in the mixed situation.

The Chen-Ruan Cohomology Ring of Mirror Quintic

We compute the Chen-Ruan orbifold cohomology ring of the Batyrev mirror of a smooth quintic hypersurface in ℙ4. We identify the obstruction bundle for this example by using the Riemann bilinear relations for periods. We expect that our method can be used to compute the Chen-Ruan ring for Calabi-Yau hypersurfaces in projective simplicial toric varieties when every obstruction bundle is a direct sum of orbifold line bundles.

BAXTER EQUATIONS AND DEFORMATION OF ABELIAN DIFFERENTIALS

In this paper the proofs are given of important properties of deformed Abelian differentials introduced earlier in connection with quantum integrable systems. The starting point of the construction is the Baxter equation. In particular, we prove the Riemann bilinear relation. The duality plays an important role in our consideration. The classical limit is considered in details.

New formulae for solutions of quantum Knizhnik–Zamolodchikov equations on level-4

We present a new form of solution to the quantum Knizhnik-Zamolodchikov equation on level -4 in a special case corresponding to the Heisenberg XXX spin chain. Our form is equivalent to the integral representation obtained by Jimbo and Miwa in 1996 . An advantage of our form is that it is reduced to the product of single integrals. This fact is deeply related to a cohomological nature of our formulae. Our approach is also based on the deformation of hyper-elliptic integrals and their main property -- deformed Riemann bilinear relation. Jimbo and Miwa also suggested a nice conjecture which relates solution of the qKZ on level -4 to any correlation function of the XXX model. This conjecture together with our form of solution to the qKZ makes it possible to prove a conjecture that any correlation function of the XXX model can be expressed in terms of the Riemann zeta-function at odd arguments and rational coefficients suggested in our previous papers. This issue will be discussed in our next publication.

Prepotential and instanton corrections in Script N = 2 supersymmetric SU(N1) × SU(N2) Yang Mills theories

In this paper we analyse the non-hyperelliptic Seiberg-Witten curves derived from M-theory that encode the low energy solution of N=2 supersymmetric theories with product gauge groups. We consider the case of a SU(N_1)xSU(N_2) gauge theory with a hypermultiplet in the bifundamental representation together with matter in the fundamental representations of SU(N_1) and SU(N_2). By means of the Riemann bilinear relations that hold on the Riemann surface defined by the Seiberg--Witten curve, we compute the logarithmic derivative of the prepotential with respect to the quantum scales of both gauge groups. As an application we develop a method to compute recursively the instanton corrections to the prepotential in a straightforward way. We present explicit formulas for up to third order on both quantum scales. Furthermore, we extend those results to SU(N) gauge theories with a matter hypermultiplet in the symmetric and antisymmetric representation. We also present some non-trivial checks of our results.

The Hard Lefschetz Theorem and the topology of semismall maps

A line bundle on a complex projective manifold is said to be lef if one of its powers is globally generated and defines a semismall map in the sense of Goresky–MacPherson. As in the case of ample bundles the first Chern class of lef line bundles satisfies the Hard Lefschetz Theorem and the Hodge–Riemann Bilinear Relations. As a consequence, we prove a generalization of the Grauert contractibility criterion: the Hodge Index Theorem for semismall maps, Theorem 2.4.1. For these maps the Decomposition Theorem of Beilinson, Bernstein and Deligne is equivalent to the non-degeneracy of certain intersection forms associated with a stratification. This observation, joint with the Hodge Index Theorem for semismall maps gives a new proof of the Decomposition Theorem for the direct image of the constant sheaf. A new feature uncovered by our proof is that the intersection forms involved are definite.

Schottky uniformization¶and vector bundles over Riemann surfaces

We study a natural map from representations of a free group of rank g in GL(n,C), to holomorphic vector bundles of degree 0 over a compact Riemann surface X of genus g, associated with a Schottky uniformization of X. Maximally unstable flat bundles are shown to arise in this way. We give a necessary and sufficient condition for this map to be a submersion, when restricted to representations producing stable bundles. Using a generalized version of Riemann's bilinear relations, this condition is shown to be true on the subspace of unitary Schottky representations.

Bilinear identity for q-hypergeometric integrals

We describe a bilinear identity satisfied by certain multidimensional q-hypergeometric integrals. The identity can be considered as a deformation of the Riemann bilinear relation for the twisted de Rham (co)homologies. The identity also gives an explicit formula for scalar products of solutions of the qKZ and the dual qKZ equations, therefore providing a deformation of Gaudin-Korepin's formula for norms of the Bethe vectors.

Riemann bilinear relations on minimal surfaces

From a physical point of view, minimal surfaces in R are objects submitted to a balanced force system, consisting in the forces associated to non zero onedimensional homology classes in the surface. Several geometric properties can be studied in terms of those forces, as embeddedness, symmetries and deformations, leading up to uniqueness and non existence results, see [11–13]. More precisely, each closed curve γ in a minimal surface S ⊂ R carries a force that expresses the stress produced by an unit conormal vector field η along γ on the whole surface. The action of this conormal field provides a tendency of translation, or linear momentum F , and another one of rotation around an axis, or angular momentum M . The first one is given by the force vector ∫

Exact solution for superfluid film vortices on a torus.

The problem of describing the velocity field of quantized vortices on a torus is solved exactly by use of Riemann's bilinear relations of 1857. The solution is used to discuss the behavior of superfluids on porous media. The close connection between topology and quantization of circulation is emphasized as well as its implications for a successful theory of the experiments.