[Paper Review] Effective Batalin-Vilkovisky quantization and geometric applications

Si Li|arXiv (Cornell University)|Sep 3, 2017

Black Holes and Theoretical Physics6 references3 citations

TL;DR

This paper develops an effective Batalin-Vilkovisky (BV) quantization framework to connect quantum field theory with deep geometric structures. It demonstrates that BV quantization of BCOV theory on Calabi-Yau manifolds compactified over $X \times \mathbb{C}$ yields an effective 2d chiral theory on $\mathbb{C}$, whose symmetries generate an integrable hierarchy; this construction provides a geometric realization of Dubrovin's classical and quantum integrable hierarchies via topological string theory.

ABSTRACT

We explain the effective renormalization method of quantum field theory in the Batalin-Vilkovisky formalism and illustrate its mathematical applications by three geometric examples: (1) Topological quantum mechanics and algebraic index, (2) Elliptic curve and higher genus mirror symmetry, (3) Calabi-Yau geometry and integrable hierarchy. This note is an expansion of author's talk at ICCM 2016.

Motivation & Objective

• To establish a rigorous mathematical framework for effective renormalization in quantum field theory using the Batalin-Vilkovisky formalism.
• To connect quantum field theoretic constructions—specifically BV quantization—with classical geometric invariants such as the algebraic index and higher genus mirror symmetry.
• To provide a geometric interpretation of the emergence of integrable hierarchies in topological string theory on Calabi-Yau manifolds.
• To generalize known results on elliptic curve mirror symmetry and the algebraic index theorem using BV quantization and effective field theory techniques.
• To propose a uniform mechanism for the appearance of integrable systems in topological string theory via compactification of BCOV theory on $X \times \mathbb{C}$.

Proposed method

• Utilizes the Batalin-Vilkovisky formalism to handle gauge symmetries and anomalies in quantum field theories with infinite-dimensional field spaces.
• Applies effective field theory techniques to integrate out massive modes on $X$ in BCOV theory on $X \times \mathbb{C}$, resulting in an effective 2d chiral theory on $\mathbb{C}$.
• Employs homological perturbation theory with a choice of Hodge filtration splitting to derive the effective field space $\operatorname{PV}(\mathbb{C}) \otimes_{\mathbb{C}} H_X[[t]]$.
• Identifies an abelian subalgebra of background symmetries $\partial_z \otimes H_X[[t]]$ whose Noether currents generate commuting operators $\{I_{\alpha}^{\hbar,\epsilon}\}$.
• Derives a Poisson bracket structure on the classical limit $\hbar, \epsilon \to 0$, yielding $\{a(z), b(w)\} = \langle a,b \rangle \partial_z \delta(z-w)$.
• Constructs the generating function $F_0^X$ for genus-zero amplitudes and uses descendant insertions $t^k\mu$ to define $I_{k,\mu} = \oint dz J_{k,\mu}(b(z))$, which commute under the Poisson bracket.

Experimental results

Research questions

• RQ1 How can effective BV quantization be used to derive geometric invariants such as the algebraic index from topological quantum mechanics?
• RQ2 What is the role of BV quantization in realizing higher genus mirror symmetry for elliptic curves?
• RQ3 How does compactification of BCOV theory on $X \times \mathbb{C}$ lead to an effective 2d chiral theory on $\mathbb{C}$?
• RQ4 In what way does the effective theory on $\mathbb{C}$ realize Dubrovin's classical and quantum integrable hierarchies?
• RQ5 What is the geometric origin of integrable hierarchies in topological string theory on Calabi-Yau manifolds?

Key findings

• The effective 2d chiral theory on $\mathbb{C}$ arising from BCOV theory on $X \times \mathbb{C}$ carries an infinite-dimensional abelian symmetry algebra, whose Noether currents generate commuting operators $I_{\alpha}^{\hbar,\epsilon}$.
• In the classical limit $\hbar, \epsilon \to 0$, the Poisson bracket $\{a(z), b(w)\} = \langle a,b \rangle \partial_z \delta(z-w)$ is realized, matching the classical integrable hierarchy of Dubrovin on a Frobenius manifold.
• The limit $\lim_{\epsilon \to 0} I^{\hbar,\epsilon}_{\alpha}$ yields a dispersion deformation of the classical hierarchy, consistent with Dubrovin-Zhang theory for semi-simple Frobenius manifolds.
• The generating function $F_0^X$ for genus-zero amplitudes on $H_X[[t]]$ encodes the classical integrable hierarchy via $I_{k,\mu} = \oint dz J_{k,\mu}(b(z))$ with $J_{k,\mu} = \partial_{t^{k}\mu} F_0^X|_{\mathbb{H}_X}$.
• The construction provides a full generalization of Dijkgraaf's elliptic curve result to higher genus, with the Gromov-Witten invariants matching the vertex operator algebra construction via boson-fermion correspondence.
• The framework suggests a natural path to quantum BCOV theory on $X \times \mathbb{C}$, which would yield a full quantum integrable hierarchy in general Calabi-Yau geometries.