Quantum Cohomology of Hypertoric Varieties

Abstract

Abstract. We give a complete description of the equivariant quantum cohomology ring of any smooth hypertoric variety, and find a mirror formula for the quantum differential equation.

SYZ Mirror Symmetry for Hypertoric Varieties

“…Closed-string equivariant mirror symmetry for hypertoric manifolds was found by Mcbreen and Shenfeld [MS13]. They derived a presentation of the T dˆCˆequivariant quantum cohomology of a hypertoric manifold and relate it with the Gauss-Manin connection of the mirror moduli.…”

Hypertoric geometry and Gromov–Witten theory

“…The quantum product by divisors is reduced to the calculation of 2-point Gromov-Witten invariants. The computation of quantum product by divisors for smooth hypertoric varieties in [40] uses a result stated as [40, Proposition 2.2], which was proven in [7] in general setting of symplectic resolutions. We prove a similar result for hypertoric Deligne-Mumford stacks.…”

“…In this paper we study the quantum cohomology of hypertoric Deligne-Mumford stacks, and generalize the quantum product formula by a divisor class in [7] to hypertoric cases. Note that for the smooth hypertoric varieties, the quantum cohomology has been studied in [40].…”

Quantizations of multiplicative hypertoric varieties at a root of unity

“…Hypertoric varieties were introduced by Bielawski and Dancer [BD], and are one of the main examples of hyperkähler symplectic manifolds. As such, they have become a central object in modern representation theory ( [BeKu,BLPW12,HS,MS,St]). Multiplicative hypertoric varieties are group-like analogues of (ordinary) hypertoric varieties, and retain many of their nice algebraic and 1 Hence, Oq(G) denotes the ad-equivariant quantized coordinate algebra, as opposed to the bi-equivariant 'FRT' algebra.…”