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Register a new userSuper quantum cohomology I: Super stable maps of genus zero with Neveu-Schwarz punctures
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In this article we define stable supercurves and super stable maps of genus zero via labeled trees. We prove that the moduli space of stable supercurves and super stable maps of fixed tree type are quotient superorbifolds. To this end, we prove a slice theorem for the action of super Lie groups on Riemannian supermanifolds and discuss superorbifolds. Furthermore, we propose a Gromov topology on super stable maps such that the restriction to fixed tree type yields the quotient topology from the superorbifolds and the reduction is compact. This would, possibly, lead to the notions of super Gromov-Witten invariants and small super quantum cohomology to be discussed in sequels.
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In this paper, we show that the derivative of the genus-1 Virasoro conjecture for Gromov-Witten invariants along the direction of quantum volume element holds for all smooth projective varieties. This result provides new evidence for the Virasoro conjecture.
In this paper we give some sufficient conditions for the vanishing of the genus-2 G-function, which was introduced by B. Dubrovin, S. Liu and Y. Zhang in [DLZ]. As a corollary we prove their conjecture for the vanishing of the genus-2 G-function for ADE singularities.
In this paper we prove that for Gromov-Witten theory of $P^1$ orbifolds of ADE type the genus-2 G-function introduced by B. Dubrovin, S. Liu, and Y. Zhang vanishes. Together with our results in [LW], this completely solves the main conjecture in their paper [DLZ]. In the process, we also found a sufficient condition for the vanishing of the genus-2 G-function which is weaker than the condition given in our previous paper [LW].
In modern mathematical and theoretical physics various generalizations, in particular supersymmetric or quantum, of Riemann surfaces and complex algebraic curves play a prominent role. We show that such supersymmetric and quantum generalizations can be combined together, and construct supersymmetric quantum curves, or super-quantum curves for short. Our analysis is conducted in the formalism of super-eigenvalue models: we introduce $beta$-deformed version of those models, and derive differential equations for associated $alpha/beta$-deformed super-matrix integrals. We show that for a given model there exists an infinite number of such differential equations, which we identify as super-quantum curves, and which are in one-to-one correspondence with, and have the structure of, super-Virasoro singular vectors. We discuss potential applications of super-quantum curves and prospects of other generalizations.
As we have shown in the previous work, using the formalism of matrix and eigenvalue models, to a given classical algebraic curve one can associate an infinite family of quantum curves, which are in one-to-one correspondence with singular vectors of a certain (e.g. Virasoro or super-Virasoro) underlying algebra. In this paper we reformulate this problem in the language of conformal field theory. Such a reformulation has several advantages: it leads to the identification of quantum curves more efficiently, it proves in full generality that they indeed have the structure of singular vectors, it enables identification of corresponding eigenvalue models. Moreover, this approach can be easily generalized to other underlying algebras. To illustrate these statements we apply the conformal field theory formalism to the case of the Ramond version of the super-Virasoro algebra. We derive two classes of corresponding Ramond super-eigenvalue models, construct Ramond super-quantum curves that have the structure of relevant singular vectors, and identify underlying Ramond super-spectral curves. We also analyze Ramond multi-Penner models and show that they lead to supersymmetric generalizations of BPZ equations.
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