[Paper Review] Orbifold Euler Characteristics of $\overline{\mathcal M}_{g,n}$

Zhiyuan Wang, Jian Zhou|arXiv (Cornell University)|Dec 27, 2018

Black Holes and Theoretical Physics42 references4 citations

TL;DR

This paper solves the long-standing problem of computing the orbifold Euler characteristics of the Deligne-Mumford compactification $\overline{\mathcal{M}}_{g,n}$ using quantum field theory-inspired formalisms. It establishes recursion relations, connects the results to Ramanujan polynomials in genus zero, and reveals a novel open-closed duality between $\chi(\overline{\mathcal{M}}_{g,n})$ and $\chi(\mathcal{M}_{g,n})$ via KP hierarchy tau-functions, with explicit formulas derived through operator formalism and Barnes G-function representations.

ABSTRACT

We solve the problem of the computation of the orbifold Euler characteristics of $\Mbar_{g,n}$. We take the works of Harer-Zagier \cite{hz} and Bini-Harer \cite{bh} as our starting point, and apply the formalisms developed in \cite{wz} and \cite{zhou1} to this problem. These formalisms are typical examples of mathematical methods inspired by quantum field theories. We also present many closed formulas and some numerical data. In genus zero the results are related to Ramanujan polynomials, and in higher genera we get recursion relations almost identical to the recursion relations for Ramanujan polynomials but with different initial values. We also show that the generating series given by the orbifold Euler characteristics of $\overline{\mathcal M}_{g,n}$ is the logarithm of the KP tau-function of the topological 1D gravity evaluated at the times given by the orbifold Euler characteristics of $\overline{\mathcal M}_{g,n}$. Conversely, the logarithm of this tau-function evaluated at the times given by certain generating series of the orbifold Euler characteristics of $\overline{\mathcal M}_{g,n}$ is a generating series of the orbifold Euler characteristics of $\overline{\mathcal M}_{g,n}$. This is a new example of open-closed duality.

Motivation & Objective

• To solve the longstanding problem of computing the orbifold Euler characteristics $\chi(\overline{\mathcal{M}}_{g,n})$ of the Deligne-Mumford moduli space of stable curves.
• To extend the Harer-Zagier formula for $\chi(\mathcal{M}_{g,n})$ to the compactified moduli space $\overline{\mathcal{M}}_{g,n}$ using abstract quantum field theory techniques.
• To establish a new open-closed duality between $\chi(\mathcal{M}_{g,n})$ and $\chi(\overline{\mathcal{M}}_{g,n})$ via inversion formulas involving KP tau-functions.
• To derive explicit closed-form expressions and recursion relations for $\chi(\overline{\mathcal{M}}_{g,n})$, particularly in genus zero and higher genera.

Proposed method

• Utilizes an abstract quantum field theory formalism inspired by holomorphic anomaly equations and string theory, adapted to moduli space computations.
• Applies operator formalism to realize edge-cutting and edge-adding operators as differential operators on generating functions.
• Employs formal Gaussian integrals and partition functions $\widehat{Z}(t,\kappa)$ to represent refined orbifold Euler characteristics $\chi_{g,n}(t,\kappa)$.
• Uses the KP hierarchy to show that the generating series of $\chi(\overline{\mathcal{M}}_{g,n})$ is the logarithm of a tau-function evaluated at times given by $\chi(\mathcal{M}_{g,n})$.
• Introduces generating series $G_k(z)$ and $V_n(z)$, and expresses them via Barnes $G$-functions and symmetric functions to derive recursive algorithms.
• Applies the emergent geometry of the KP hierarchy to relate $\chi(\overline{\mathcal{M}}_{g,n})$ to integrable systems and $n$-point functions.

Experimental results

Research questions

• RQ1 How can the orbifold Euler characteristic $\chi(\overline{\mathcal{M}}_{g,n})$ be computed for all $g$ and $n$ using a unified formalism?
• RQ2 What is the precise relationship between the refined orbifold Euler characteristics of $\overline{\mathcal{M}}_{g,n}$ and Ramanujan polynomials in genus zero?
• RQ3 How does the KP hierarchy encode the generating series of $\chi(\overline{\mathcal{M}}_{g,n})$ through tau-functions?
• RQ4 Can an open-closed duality be established between $\chi(\mathcal{M}_{g,n})$ and $\chi(\overline{\mathcal{M}}_{g,n})$ via functional inversion?
• RQ5 What are the explicit closed-form expressions for $\chi(\overline{\mathcal{M}}_{g,n})$ in terms of known special functions like the Barnes $G$-function?

Key findings

• The paper derives a linear recursion relation for $\chi(\overline{\mathcal{M}}_{g,n})$ in genus $g \geq 2$ that is structurally identical to the recursion for Ramanujan polynomials but with different initial conditions.
• In genus zero, the refined orbifold Euler characteristic $\chi_{g,0}(t,\kappa)$ is shown to be related to Ramanujan polynomials, and the generating series $G_k(z)$ is expressed in terms of Barnes $G$-functions.
• The generating series of $\chi(\overline{\mathcal{M}}_{g,n})$ is proven to be the logarithm of a KP tau-function evaluated at times given by $\chi(\mathcal{M}_{g,n})$, establishing a new integrable systems structure.
• The inverse relation holds: the logarithm of the same tau-function evaluated at times derived from $\chi(\overline{\mathcal{M}}_{g,n})$ generates $\chi(\mathcal{M}_{g,n})$, demonstrating a new form of open-closed duality.
• Explicit formulas for $\chi(\overline{\mathcal{M}}_{0,n})$ and $\chi(\overline{\mathcal{M}}_{1,n})$ are derived using generating series and symmetric functions, with numerical data provided.
• The solution to the linear recursion yields closed-form expressions for $\chi(\overline{\mathcal{M}}_{g,n})$ in terms of coefficients $a_{g,n}^k$ and generating functions $G_k(z)$, enabling algorithmic computation.