The Witten equation, mirror symmetry, and quantum singularity theory

Abstract

Abstract. For any non-degenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity A r−1 .We also resolve two outstanding conjectures of Witten. The first conjecture is that ADE-singularities are self-dual; and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.

A Landau–Ginzburg mirror theorem without concavity

“…Polishchuk and Vaintrob have provided in [32] an algebro-geometric counterpart. The compatibility between [17] and [32] is only partly proven, for instance for simple singularities [32,Theorem 7.6.1] or for invariants with so-called narrow entries [3,Theorem 1.2]. In Theorem 3.25, we establish it also for (almost) every invertible polynomials with the maximal group of symmetries.…”

“…As mentioned above, precisely as in Witten [34, §3.1], we look at Calabi-Yau via singularities; namely we pass from a CalabiYau (CY) hypersurface {W = 0} ⊂ P(w) to the Landau-Ginzburg (LG) model W : C N → C whose monodromy around the origin is given by the group µ d = j of order d generated by the grading element This quantum theory of singularities was recently introduced under the name of FJRW theory by Fan, Jarvis, and Ruan [17,18] building upon Witten's initial analytic construction [34]. Polishchuk and Vaintrob have provided in [32] an algebro-geometric counterpart.…”

“…Single non-concave quantum invariants were inferred from concave ones using tautological relation (e.g. using WDVV equation as in [16,17,26]), but so far no systematic approach tackling directly the virtual cycle has been taken. Polishchuk and Vaintrob recently opened the way to an algebraic computation: their construction [32] of a virtual cycle is given by applying the Chern character to a universal object, the fundamental matrix factorization.…”

“…In genus one, this difficulty was overcome by Zinger after a great deal of hard work, but we still lack a comprehensive approach for higher genus. Guided by the frame of ideas of mirror symmetry and the Landau-Ginzburg/Calabi-Yau correspondence, we switch to the quantum theory of singularities introduced by Fan, Jarvis, and Ruan [17,18] based on ideas of Witten [34] (FJRW theory). Single non-concave quantum invariants were inferred from concave ones using tautological relation (e.g.…”

The moduli space in the gauged linear sigma model

“…theory [FJR07,FJR08,FJR12]. This so-called FJRW-theory can be viewed as the Landau-Ginzburg phase of a Calabi-Yau hypersurface X W = {W = 0} ⊂ W P n−1 in weighted projective space.…”