113 research outputs found

### Quantum Integrable Model of an Arrangement of Hyperplanes

The goal of this paper is to give a geometric construction of the Bethe
algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra.
More precisely, in this paper a quantum integrable model is assigned to a
weighted arrangement of affine hyperplanes. We show (under certain assumptions)
that the algebra of Hamiltonians of the model is isomorphic to the algebra of
functions on the critical set of the corresponding master function. For a
discriminantal arrangement we show (under certain assumptions) that the
symmetric part of the algebra of Hamiltonians is isomorphic to the Bethe
algebra of the corresponding Gaudin model. It is expected that this
correspondence holds in general (without the assumptions). As a byproduct of
constructions we show that in a Gaudin model (associated to an arbitrary simple
Lie algebra), the Bethe vector, corresponding to an isolated critical point of
the master function, is nonzero

### Solutions modulo $p$ of Gauss-Manin differential equations for multidimensional hypergeometric integrals and associated Bethe ansatz

We consider the Gauss-Manin differential equations for hypergeometric
integrals associated with a family of weighted arrangements of hyperplanes
moving parallelly to themselves. We reduce these equations modulo a prime
integer $p$ and construct polynomial solutions of the new differential
equations as $p$-analogs of the initial hypergeometric integrals.
In some cases we interpret the $p$-analogs of the hypergeometric integrals as
sums over points of hypersurfaces defined over the finite field $F_p$. That
interpretation is similar to the interpretation by Yu.I. Manin in [Ma] of the
number of point on an elliptic curve depending on a parameter as a solution of
a classical hypergeometric differential equation.
We discuss the associated Bethe ansatz.Comment: Latex, 19 pages, v2: misprints correcte

### Characteristic variety of the Gauss-Manin differential equations of a generic parallelly translated arrangement

We consider a weighted family of $n$ generic parallelly translated
hyperplanes in \C^k and describe the characteristic variety of the
Gauss-Manin differential equations for associated hypergeometric integrals. The
characteristic variety is given as the zero set of Laurent polynomials, whose
coefficients are determined by weights and the Plucker coordinates of the
associated point in the Grassmannian Gr(k,n). The Laurent polynomials are in
involution.Comment: Latex, 13 page

### Critical set of the master function and characteristic variety of the associated Gauss-Manin differential equations

We consider a weighted family of $n$ parallelly transported hyperplanes in a
$k$-dimensioinal affine space and describe the characteristic variety of the
Gauss-Manin differential equations for associated hypergeometric integrals. The
characteristic variety is given as the zero set of Laurent polynomials, whose
coefficients are determined by weights and the associated point in the
Grassmannian Gr$(k,n)$. The Laurent polynomials are in involution.
An intermediate object between the differential equations and the
characteristic variety is the algebra of functions on the critical set of the
associated master function. We construct a linear isomorphism between the
vector space of the Gauss-Manin differential equations and the algebra of
functions. The isomorphism allows us to describe the characteristic variety. It
also allowed us to define an integral structure on the vector space of the
algebra and the associated (combinatorial) connection on the family of such
algebras.Comment: Latex, 24 pages, v2: references added, misprints corrected; v3:
misprint correc

- …