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Applying the Worldvolume Hybrid Monte Carlo method to lattice gauge theories

Masafumi Fukuma

Abstract

The numerical sign problem remains one of the central challenges in computational physics. The Worldvolume Hybrid Monte Carlo (WV-HMC) method has recently been proposed as a reliable and computationally efficient algorithm that crucially avoids the ergodicity issues inherent in Lefschetz-thimble approaches. In these proceedings, after outlining the key ideas behind WV-HMC, we present its extension to group manifolds. This provides a rigorous framework for applying WV-HMC to lattice gauge theories.

Applying the Worldvolume Hybrid Monte Carlo method to lattice gauge theories

Abstract

The numerical sign problem remains one of the central challenges in computational physics. The Worldvolume Hybrid Monte Carlo (WV-HMC) method has recently been proposed as a reliable and computationally efficient algorithm that crucially avoids the ergodicity issues inherent in Lefschetz-thimble approaches. In these proceedings, after outlining the key ideas behind WV-HMC, we present its extension to group manifolds. This provides a rigorous framework for applying WV-HMC to lattice gauge theories.

Paper Structure

This paper contains 5 sections, 1 theorem, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Complex analysis on complexified groups
  3. WV-HMC for group manifolds
  4. Numerical tests: one-site model
  5. Conclusions and outlook

Key Result

Theorem 1

Let $\mathcal{D}$ be a domain in $G^\mathbb{C}$ and $f(U)$ be a holomorphic function on $\mathcal{D}$. Then, the integral $I_\Sigma$ of $f(U)$ over a real $N$-dimensional oriented submanifold $\Sigma\subset\mathcal{D}$, depends only on the boundary of $\Sigma$.

Figures (2)

  • Figure 1: Deformation of $\Sigma_0 = G$ into a submanifold $\Sigma$ within $G^\mathbb{C}$Fukuma:2025gya. The deformed surface $\Sigma$ approaches a Lefschetz thimble $\mathcal{J}$, which consists of points flowing out from a critical point $U_\ast$.
  • Figure 2: Real and imaginary parts of $\langle e \rangle$ in the one-site model for $\beta \in i\,\mathbb{R}$ with $G = SU(2)$ [top] and $G = SU(3)$ [bottom] Fukuma:2025gya. The dashed lines represent the analytical results (for $G = SU(2)$, $\langle e \rangle = -I_2(\beta)/I_1(\beta)$, for which $\mathrm{Re}\, \langle e \rangle = 0$).

Theorems & Definitions (1)

  • Theorem 1