[Paper Review] W_{N+1}-constraints for singularities of type A_N

Bojko Bakalov, Todor Milanov|ArXiv.org|Nov 12, 2008

Algebraic Geometry and Number Theory24 references3 citations

TL;DR

This paper establishes that the total descendant potential of an $A_N$-singularity satisfies $\mathcal{W}_{N+1}$-constraints by constructing a projective representation of the Lie algebra of differential operators on the circle with central charge $h = N+1$. Using Picard–Lefschetz periods and Givental's symplectic loop space formalism, the authors prove that the potential is a highest weight vector for the $\mathcal{W}_{N+1}$ algebra, providing a complete set of recursion relations for intersection numbers on the moduli space of $h$-spin curves.

ABSTRACT

Using Picard--Lefschetz periods for the singularity of type $A_N$, we construct a projective representation of the Lie algebra of differential operators on the circle with central charge $h:=N+1$. We prove that the total descendant potential $\D_{A_N}$ of $A_N$-singularity is a highest weight vector. It is known that $\D_{A_N}$ can be interpreted as a generating function of a certain class of intersection numbers on the moduli space of $h$-spin curves. In this settings our constraints provide a complete set of recursion relations between the intersection numbers. Our methods are based entirely on the symplectic loop space formalism of A. Givental and therefore they can be applied to the mirror models of symplectic manifolds.

Motivation & Objective

• To determine whether the total descendant potential $\mathcal{D}_{A_N}$ of an $A_N$-singularity satisfies constraints beyond the Virasoro algebra.
• To extend the known Virasoro constraints for the Witten–Kontsevich $\tau$-function to a larger algebraic structure in the context of $A_N$-singularities.
• To establish that $\mathcal{D}_{A_N}$ is a highest weight vector for the $\mathcal{W}_{N+1}$ algebra via a projective representation of differential operators on the circle with central charge $h = N+1$.
• To provide a complete set of recursion relations between intersection numbers on the moduli space of $h$-spin curves using the $\mathcal{W}_{N+1}$-constraints.

Proposed method

• Constructs a projective representation of the Lie algebra of differential operators on the circle with central charge $h = N+1$ using Picard–Lefschetz periods of the $A_N$-singularity.
• Applies Givental's symplectic loop space formalism to analyze the total descendant potential $\mathcal{D}_{A_N}$ as a generating function of intersection numbers on the moduli space of $h$-spin curves.
• Uses the theory of ancestor potentials and monodromy-invariant cycles to analyze the regularity of the potential and its logarithmic coefficients.
• Employs Stokes' theorem and contour integrals around discriminants to show that certain integrals are single-valued and constant, enabling the derivation of constraints.
• Relies on the uniqueness of solutions to the $h$-KdV hierarchy satisfying the string equation, which implies that $\mathcal{D}_{A_N}$ is a highest weight vector for $\mathcal{W}_h$, with $h = N+1$.
• Proves that the $\mathcal{W}_{A_1}$-constraints for the ancestor potential of the $A_1$-singularity imply the $\mathcal{W}_{A_N}$-constraints for the $A_N$-singularity via monodromy and cycle invariance arguments.

Experimental results

Research questions

• RQ1 Can the total descendant potential $\mathcal{D}_{A_N}$ of an $A_N$-singularity be shown to satisfy constraints beyond the Virasoro algebra?
• RQ2 Is $\mathcal{D}_{A_N}$ a highest weight vector for the $\mathcal{W}_{N+1}$ algebra under a suitable representation of the Lie algebra of differential operators on the circle?
• RQ3 Do the $\mathcal{W}_{N+1}$-constraints provide a complete set of recursion relations for intersection numbers on the moduli space of $h$-spin curves with $h = N+1$?
• RQ4 How does the symplectic loop space formalism of Givental enable the derivation of $\mathcal{W}_{N+1}$-constraints for $A_N$-singularities?

Key findings

• The total descendant potential $\mathcal{D}_{A_N}$ of the $A_N$-singularity is a highest weight vector for the $\mathcal{W}_{N+1}$ algebra, as established via a projective representation of the Lie algebra of differential operators on the circle with central charge $h = N+1$.
• The $\mathcal{W}_{N+1}$-constraints provide a complete set of recursion relations between intersection numbers on the moduli space of $h$-spin curves, where $h = N+1$.
• The proof relies on the regularity of logarithmic coefficients of the potential, shown via monodromy invariance and contour integration, particularly using Stokes’ theorem to demonstrate that certain integrals are constant.
• The $\mathcal{W}_{A_1}$-constraints for the ancestor potential of the $A_1$-singularity are sufficient to imply the $\mathcal{W}_{A_N}$-constraints for the $A_N$-singularity, due to cycle invariance and monodromy arguments.
• The authors establish that the logarithmic coefficients of the potential are single-valued near discriminants by showing that their monodromy-induced changes vanish via contour integrals of $\mathcal{W}_{\alpha,\beta}$ forms.
• The key technical step involves proving that $\oint_\gamma \mathcal{W}_{a,a} = 0$ for a specific loop $\gamma$, which follows from the invariance of the cycle $a$ under $\gamma$ and Lemma 4.8, thereby ensuring regularity of the potential.