A Generalized Landau-Ginzburg/Gromov-Witten Correspondence.

[摘要] The celebrated Landau-Ginzburg/Calabi-Yau correspondence asserts that the Gromov-Witten theory of a Calabi-Yau hypersurface in weighted projective space is equivalent to its corresponding FJRW-theory via analytic continuation. It is well known that this correspondence fails in non-Calabi-Yau cases. The main obstruction is a collapsing or dimensional reduction of the state space of the Landau-Ginzburg model in the Fano case, and a similar collapsing of the state space of Gromov-Witten theory in the general type case. We state and prove a modified version of the cohomological correspondence that describes this collapsing phenomenon at the level of state spaces. This result confirms a physical conjecture of Witten-Hori-Vafa. The main purpose of this thesis is to provide a quantum explanation for the collapsing phenomenon. A key observation is that the corresponding Picard-Fuchs equation develops irregular singularities precisely at the points where the collapsing occurs. Our main idea is to replace analytic continuation with asymptotic expansion in this non-Calabi-Yau setting. The main result of this article is that the reduction in rank of the Gromov-Witten I-function due to power series asymptotic expansions matches precisely the dimensional reduction of the corresponding state space. Furthermore, asymptotic expansion under a different asymptotic sequence yields a different I-function which can be considered as the mathematical counterpart to the additional ;;massive vacua;; of physics.

[发布机构] University of Michigan

[效力级别] LG/CY correspondence

[关键词] Gromov-Witten theory;LG/CY correspondence;FJRW theory;Mathematics;Science;Mathematics