Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stochastic Calculus and Hochschild Homology

Si Li, Zichang Wang, Peng Yang

TL;DR

The paper develops a probabilistic framework for topological correlations in topological quantum mechanics on $S^1$ by constructing Gaussian free fields and taking a large variance limit, connecting stochastic loop calculus to noncommutative geometry. It shows that loop-based correlations reproduce the Moyal product for Schwartz functions and yields explicit Wick-contraction formulas via a Green’s function on $S^1$, establishing a bridge between stochastic methods and deformation quantization. The work then formulates a quantum HKR map $\sigma^\hbar$ that intertwines Hochschild chains of the Weyl algebra with differential forms under a BV operator $\Delta$, proving a BV-compatible quasi-isomorphism for $\hbar\neq 0$ and relating to the algebraic index theorem in a QFT framework. Overall, the results illuminate deep links between stochastic calculus on loops, Hochschild homology, and BV quantization within topological quantum mechanics.

Abstract

This paper is a case study of probabilistic approach to homological aspects of topological quantum field theory via the example of topological quantum mechanics. We propose topological correlations in terms of large variance limit. An investigation on the relation between probabilistic topological correlations on the circle and Hochschild homology is illustrated.

Stochastic Calculus and Hochschild Homology

TL;DR

The paper develops a probabilistic framework for topological correlations in topological quantum mechanics on by constructing Gaussian free fields and taking a large variance limit, connecting stochastic loop calculus to noncommutative geometry. It shows that loop-based correlations reproduce the Moyal product for Schwartz functions and yields explicit Wick-contraction formulas via a Green’s function on , establishing a bridge between stochastic methods and deformation quantization. The work then formulates a quantum HKR map that intertwines Hochschild chains of the Weyl algebra with differential forms under a BV operator , proving a BV-compatible quasi-isomorphism for and relating to the algebraic index theorem in a QFT framework. Overall, the results illuminate deep links between stochastic calculus on loops, Hochschild homology, and BV quantization within topological quantum mechanics.

Abstract

This paper is a case study of probabilistic approach to homological aspects of topological quantum field theory via the example of topological quantum mechanics. We propose topological correlations in terms of large variance limit. An investigation on the relation between probabilistic topological correlations on the circle and Hochschild homology is illustrated.
Paper Structure (12 sections, 13 theorems, 91 equations, 1 figure)

This paper contains 12 sections, 13 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Stochastic Calculus on Loops
  3. Probabilistic Preliminaries
  4. Gaussian Free Field on $S^1$
  5. Topological Correlations via Large Variance Limit
  6. Polynomials and Wick's Theorem
  7. Schwartz functions and Moyal product
  8. Hochschild Homology
  9. Hochschild Homology
  10. Quantum HKR
  11. $S^1$-product
  12. Quantum HKR

Key Result

Proposition 2.1

If $\{X_m\}_{m\ge 1}$ are Gaussian variables such that then $X$ is also a (possibly degenerate, i.e. variance zero) Gaussian variable, and the convergence holds in $L^p$ for $1\leq p<\infty$.

Figures (1)

  • Figure 1: The value of $G(t,s)$ as a periodic function of $s-t$

Theorems & Definitions (27)

  • Proposition 2.1: JFLGall, Proposition 1.1
  • Proposition 2.2: KaiLaiChung, Theorem 5.3.4
  • Definition 2.3
  • Proposition 2.4: JFLGall, Theorem 2.9, Kolmogorov’s Lemma
  • Proposition 2.5: Gaussian Free Field on $S^1$
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • ...and 17 more