125 research outputs found
Affine Macdonald conjectures and special values of Felder–Varchenko functions
No full textWe refine the statement of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof–Kirillov Jr. (Duke Math J 78(2):229–256, 1995) and prove the first non-trivial cases of these conjectures. Our results provide a q-deformation of the computation of genus 1 conformal blocks via elliptic Selberg integrals by Felder–Stevens–Varchenko (Math Res Lett 10(5–6):671–684, 2003). They allow us to give precise formulations for the affine Macdonald conjectures in the general case which are consistent with computer computations. Our method applies recent work of the second named author to relate these conjectures in the case of U_q(sl_2) to evaluations of certain theta hypergeometric integrals defined by Felder–Varchenko (Int Math Res Not 21:1037–1055, 2004). We then evaluate the resulting integrals, which may be of independent interest, by well-chosen applications of the elliptic beta integral introduced by Spiridonov (Uspekhi Mat Nauk 56(1(337)):181–182, 2001)
Fórmulas de Brion y Lawrence-Varchenko
No full textEl objetivo de este trabajo es presentar una demostración asequible de las fórmulas de Brion y Lawrence-Varchenko. Éstas no solo permiten contar la totalidad de puntos con coordenadas enteras dentro de un politopo reticular de Rd, sino también mostrar de cuales se trata. Separadamente Michel Brion y James Lawrence junto con Alexander
Varchenko descubrieron, de forma simultánea, dichas fórmulas de gran sencillez que utilizan las funciones generadoras racionales de los conos tangentes a cada vértice del politopo, siendo la fórmula de Lawrence-Varchenko aplicable solo cuando estos conos son simpliciales. El trabajo abarca ambas demostraciones de forma detallada, así como ejemplos que facilitan su comprensión.Grado en Matemática
Twisted de Rham Complex on Line and Singular Vectors
No full textThis work studies the connection between the representation theory of affine Lie algebra and the relations between the cohomology classes of certain logarithmic differential forms. Following work of V. Schechtman and A. Varchenko we consider two complexes. The first complex is the twisted de Rham complex of scalar meromorphic differential forms on projective line, holomorphic on the complement to a finite set of points. The second complex is the chain complex of the Lie algebra of -valued algebraic functions on the same complement, with coefficients in a tensor product of contragradient Verma modules over affine Lie algebra . In [Rational Differential Forms on Line and Singular Vectors in Verma Modules over , Mosc. Math. J. 17:4 (2017), 782-802] a construction of a monomorphism of the first complex to the second was suggested and it was indicated that under this monomorphism the existence of singular vectors (the Malikov--Feigin--Fuchs singular vectors) is reflected in the relations between the cohomology classes of the twisted de Rham complex. In this work we prove these results.Doctor of Philosoph
Twisted de Rham Complex on Line and Singular Vectors in sl₂ˆ Verma Modules
No full textWe consider two complexes. The first complex is the twisted de Rham complex of scalar meromorphic differential forms on the projective line, holomorphic on the complement to a finite set of points. The second complex is the chain complex of the Lie algebra of sl₂-valued algebraic functions on the same complement, with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra sl₂ˆ. In [Schechtman V., Varchenko A., Mosc. Math. J. 17 (2017), 787-802] a construction of a monomorphism of the first complex to the second was suggested, and it was indicated that under this monomorphism, the existence of singular vectors in the Verma modules (the Malikov-Feigin-Fuchs singular vectors) is reflected in the relations between the cohomology classes of the de Rham complex. In this paper, we prove these results.The authors thank V. Schechtman for useful discussions. The second author was supported in part by NSF grant DMS-1665239
Invariant Curves in Cherkis Bow Varieties
No full textInvariant curves in type A bow varieties, both compact and noncompact, are completely classified. They have many applications, including a proof that the geometric order on fixed points and one of the combinatorial Bruhat orders on (0,1)-matrices coincide, as well as possible implications in equivariant cohomology and stable envelopes. Finally, we discuss some possible constructions for invariant curves in affine type A bow varieties.Doctor of Philosoph
Determinants of the Hypergeometric Period Matrices of an Arrangement and its Dual
No full textWe fix three natural numbers k, n,N, such that n + k + 1 = N, and introduce the
notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement
of N hyperplanes in a k-dimensional affine space, the other is an arrangement of
N hyperplanes in an n-dimensional affine space. We assign weights α1, . . . , αN to the
hyperplanes of the arrangements and for each of the arrangements consider the associated
period matrices. The first is a matrix of k-dimensional hypergeometric integrals
and the second is a matrix of n-dimensional hypergeometric integrals. The size of each
matrix is equal to the number of bounded domains of the corresponding arrangement.
We show that the dual arrangements have the same number of bounded domains and
the product of the determinants of the period matrices is equal to an alternating product
of certain values of Euler’s gamma function multiplied by a product of exponentials
of the weights
Norms of eigenfunctions to trigonometric KZB operators
No full textLet g be a simple Lie algebra and V[0] be the zero weight subspace of a tensor product of g-modules. The trigonometric KZB operators are commuting differential operators acting on V[0]-valued functions on the Cartan subalgebra of g. Meromorphic eigenfunctions to the operators are constructed by the Bethe ansatz. We introduce a scalar product on a suitable space of functions such that the operators become symmetric, and the square of the norm of a Bethe eigenfunction equals the Hessian of the master function at the corresponding critical point
Quantum Integrable Model of an Arrangement of Hyperplanes
No full textThe 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
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