Alexander Alexandrov, Todor Milanov
We construct a Hermitian matrix model for the total descendant potential of a simple singularity of type D similar to the Kontsevich matrix model for the generating function of intersection numbers on the Deligne--Mumford moduli spaces $\overline{\mathcal{M}}_{g,n}$.
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Simple, or Kleinian, singularities are classified by Dynkin diagrams of type ADE. Let g be the corresponding finite-dimensional Lie algebra, and W its Weyl group. The set of g-invariants in the basic representation of the affine Kac-Moody algebra g^ is known as a W-algebra and is a subalgebra of the Heisenberg vertex algebra F. Using period integrals, we construct an analytic continuation of the twisted representation of F. Our construction yields a global object, which may be called a W-twisted representation of F. Our main result is that the total descendant potential of the singularity, int…
We derive a Bouchard--Eynard type topological recursion for the total descendant potential of $A_N$-singularity. Our argument relies on a certain twisted representation of a Heisenberg Vertex Operator Algebra (VOA) constructed via the periods of $A_N$-singularity. In particular, our approach allows us to prove that the topological recursion for the total descendant potential is equivalent to a certain generating set of $\mathcal{W}$-algebra constraints.
We present a construction of an open analogue of total descendant and total ancestor potentials via an "open version" of Givental's action. Our construction gives a genus expansion for an arbitrary solution to the open WDVV equations satisfying a semisimplicity condition and admitting a unit. We show that the open total descendant potentials we define satisfy the open topological recursion relations in genus 0 and 1, the open string and open dilaton equations. We finish the paper with a computation of the simplest nontrivial open correlator in genus 1 using our construction.
It was proved in 2010 that the principal Kac--Wakimoto hierarchy of type $D$ is a reduction of the 2-component BKP hierarchy. On the other hand, it is known that the total descendant potential of a singularity of type $D$ is a tau-function of the principal Kac--Wakimoto hierarchy. We find explicitly the point in the Grassmanian of the 2-component BKP hierarchy (in the sense of Shiota) that corresponds to the total descendant potential. We also prove that the space of tau-functions of Gaussian type is parametrized by the base of the miniversal unfolding of the simple singularity of type $D$.
Fano orbifold lines are classified by the Dynkin diagrams of type $A,D,$ and $E$. It is known that the corresponding total descendant potential is a tau-function of an appropriate Kac--Wakimoto hierarchy. It is also known that in the A-case the Kac--Wakimoto hierarchies admit an extension and that the total descendant potential is a tau-function of an extended Kac--Wakimoto hierarchy. The goal of this paper is to prove that in the D-case the total descendent potential is also a tau-function of an extended Kac--Wakimoto hierarchy.
In a recent paper [3], Bakalov and Milanov proved that the total descendant potential of a simple singularity satisfies the W-constraints, which come from the W-algebra of the lattice vertex algebra associated to the root lattice of this singularity and a twisted module of this vertex algebra. In the present paper, we prove that the solution of these W-constraints is unique up to a constant factor, as conjectured by Bakalov and Milanov in their paper.
K. Saito's theory of primitive forms gives a natural semi-simple Frobenius manifold structure on the space of miniversal deformations of an isolated singularity. On the other hand, Givental introduced the notion of a total ancestor potential for every semi-simple point of a Frobenius manifold and conjectured that in the settings of singularity theory his definition extends analytically to non-semisimple points as well. In this paper we prove Givental's conjecture by using the Eynard--Orantin recursion.
We give a global, intrinsic, and co-ordinate-free quantization formalism for Gromov-Witten invariants and their B-model counterparts, which simultaneously generalizes the quantization formalisms described by Witten, Givental, and Aganagic-Bouchard-Klemm. Descendant potentials live in a Fock sheaf, consisting of local functions on Givental's Lagrangian cone that satisfy the (3g-2)-jet condition of Eguchi-Xiong; they also satisfy a certain anomaly equation, which generalizes the Holomorphic Anomaly Equation of Bershadsky-Cecotti-Ooguri-Vafa. We interpret Givental's formula for the higher-genus p…
We prove that the total descendant potential functions of the theory of Fan-Jarvis-Ruan-Witten for D_4 with symmetry group <J> and D_4^T with symmetry group G_{max}, respectively, are both tau-functions of the D_4 Kac-Wakimoto/Drinfeld-Sokolov hierarchy. This completes the proof, begun in [FJR], of the Witten Integrable Hierarchies Conjecture for all simple (ADE) singularities.
The total spine of the Milnor fibration of a plane curve singularity
本文提出了一种方法，通过构造脊柱来实现平面曲线奇点的Milnor纤维的消失拓扑，并在商空间上建立了等价的光滑纤维结构。
It was proved recently that the correlation functions of a semi-simple cohomological field theory satisfy the so called Eynard--Orantin topological recursion. We prove that in the settings of singularity theory, the relations can be expressed in terms of periods integrals and the so called phase forms. In particular, we prove that the Eynard-Orantin recursion is equivalent to $N$ copies of Virasoro constraints for the ancestor potential, which follow easily from the definition of the potential.
The paper math.AG/0108100 gives a construction of the total descendent potential corresponding to a semisimple Frobenius manifold. In math.AG/0209205, it is proved that the total descendent potential corresponding to K. Saito's Frobenius structure on the parameter space of the miniversal deformation of the A_{n-1}-singularity satisfies the modulo-n reduction of the KP-hierarchy. In this paper, we identify the hierarchy satisfied by the total descendent potential of a simple singularity of the A,D,E-type. Our description of the hierarchy is parallel to the vertex operator construction of Kac --…
Orbifold Saito theory of A and D type singularities
本文为A型和D型Landau-Ginzburg轨道折叠构造Saito理论，包括Brieskorn格和Gauss-Manin联络的轨道折叠版本，并显式计算了五族对称群情形。
Consider a normal complex analytic surface singularity. It is called Gorenstein if the canonical line bundle is holomorphically trivial in some punctured neighborhood of the singular point and is called numerically Gorenstein if this line bundle is topologically trivial. The second notion depends only on the topological type of the singularity. Laufer proved in 1977 that, given a numerically Gorenstein topological type of singularity, every analytical realization of it is Gorenstein if and only if one has either a Kleinian or a minimally elliptic topological type. The question to know if any n…
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