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Register a new userThe $mathbb F_p$-Selberg Integral
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We prove an $mathbb F_p$-Selberg integral formula, in which the $mathbb F_p$-Selberg integral is an element of the finite field $mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$.
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We consider the KZ differential equations over $mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $mathbb F_p$. We study the polynomial solutions of these differential equations over $mathbb F_p$, constructed in a previous work joint with V.,Schechtman and called the $mathbb F_p$-hypergeometric solutions. The dimension of the space of $mathbb F_p$-hypergeometric solutions depends on the prime number $p$. We say that the KZ equations have ample reduction for a prime $p$, if the dimension of the space of $mathbb F_p$-hypergeometric solutions is maximal possible, that is, equal to the dimension of the space of solutions of the corresponding KZ equations over $mathbb C$. Under the assumption of ample reduction, we prove a determinant formula for the matrix of coordinates of basis $mathbb F_p$-hypergeometric solutions. The formula is analogous to the corresponding formula for the determinant of the matrix of coordinates of basis complex hypergeometric solutions, in which binomials $(z_i-z_j)^{M_i+M_j}$ are replaced with $(z_i-z_j)^{M_i+M_j-p}$ and the Euler gamma function $Gamma(x)$ is replaced with a suitable $mathbb F_p$-analog $Gamma_{mathbb F_p}(x)$ defined on $mathbb F_p$.
We propose a conjectural explicit formula of generating series of a new type for Gromov--Witten invariants of $mathbb{P}^1$ of all degrees in full genera.
In the previous paper (J. Combin. Theory Ser. A, 120, 2013, 1263--1284) H. Tagawa and the two authors proposed an algebraic method to compute certain Pfaffians whose form resemble to Hankel determinants associated with moment sequences of the classic
al orthogonal polynomials. At the end of the paper they offered several conjectures. In this work we employ a completely different method to evaluate this type of Pfaffians. The idea is to apply certain de Bruijn type formula and convert the evaluation of the Pfaffians to the certain Selberg type integrals. This approach works not only for Pfaffians but also for hyperpfaffians. Hence it enables us to establish much more generalized identities than those conjectured in the previous paper. We also attempt q-analogues.
The relations in the tautological ring of the moduli space M_g of nonsingular curves conjectured by Faber-Zagier in 2000 and extended to the moduli space of stable pointed curves by Pixton in 2012 are based upon two hypergeometric series A and B. The question of the geometric origins of these series has been solved in at least two ways (via the Frobenius structures associated to 3-spin curves and to CP1). The series A and B also appear in the study of descendent integration on the moduli spaces of open and closed curves. We survey here the various occurrences of A and B starting from their appearance in the asymptotic expansion of the Airy function (calculated by Stokes in the 19th century). Several open questions are proposed.
The Doran-Harder-Thompson gluing/splitting conjecture unifies mirror symmetry conjectures for Calabi-Yau and Fano varieties, relating fibration structures on Calabi-Yau varieties to the existence of certain types of degenerations on their mirrors. This was studied for the case of Calabi-Yau complete intersections in toric varieties by Doran, Kostiuk and You for the Hori-Vafa mirror construction. In this paper we prove one direction of the conjecture using a modified version of the Gross-Siebert program. This involves a careful study of the implications within tropical geometry and applying modern deformation theory for singular Calabi-Yau varieties.
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