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Statistical de Rham Hodge operators, spectral Einstein functionals and the noncommutative residue

Yuchen Yang, Yong Wang

TL;DR

This work extends the noncommutative residue framework to spectral functionals associated with statistical de Rham Hodge operators with torsion on compact manifolds with boundary. By deriving Lichnerowicz-type formulas and defining Einstein functionals via $Wres$, the authors obtain explicit bulk and boundary contributions that encode geometric data such as the Ricci tensor, scalar curvature, and torsion terms. In particular, they establish Dabrowski-Sitarz-Zalecki-type theorems for these operators, providing concrete 4D formulas for both interior and boundary terms and illuminating the interplay between spectral invariants and boundary geometry in noncommutative settings.

Abstract

Inspired by statistical de Rham Hodge operators and the spectral functionals, we carry on some promotion to spectral functionals to noncommutative fields, and associate them with the noncommutative residue on manifolds with boundary. We prove the Dabrowski-Sitarz-Zalecki type theorem for statistical de Rham Hodge operators on manifolds with boundary.

Statistical de Rham Hodge operators, spectral Einstein functionals and the noncommutative residue

TL;DR

This work extends the noncommutative residue framework to spectral functionals associated with statistical de Rham Hodge operators with torsion on compact manifolds with boundary. By deriving Lichnerowicz-type formulas and defining Einstein functionals via , the authors obtain explicit bulk and boundary contributions that encode geometric data such as the Ricci tensor, scalar curvature, and torsion terms. In particular, they establish Dabrowski-Sitarz-Zalecki-type theorems for these operators, providing concrete 4D formulas for both interior and boundary terms and illuminating the interplay between spectral invariants and boundary geometry in noncommutative settings.

Abstract

Inspired by statistical de Rham Hodge operators and the spectral functionals, we carry on some promotion to spectral functionals to noncommutative fields, and associate them with the noncommutative residue on manifolds with boundary. We prove the Dabrowski-Sitarz-Zalecki type theorem for statistical de Rham Hodge operators on manifolds with boundary.
Paper Structure (4 sections, 12 theorems, 85 equations)

This paper contains 4 sections, 12 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Spectral functionals for the statistical de Rham Hodge operator with Torsion
  3. Residue for Statistical de Rham Hodge Operators $\widetilde{\nabla}_{U}\widetilde{\nabla}_{V}T^{-2}$ and $T^{-2}$
  4. Residue for statistical de Rham Hodge operators with torsion $\widetilde{\nabla}_{U}\widetilde{\nabla}_{V}T^{-1}$ and $T^{-3}$

Key Result

Lemma 2.1

PS When $\lambda_1=1$, for the statistical de Rham Hodge operator $T$, we have where $s$ is the scalar curvature.

Theorems & Definitions (14)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 4 more