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Spectral action and heat kernel trace for Ricci flat manifolds from stochastic flow over second quantized $L^2$-differential forms

Sita Gakkhar, Matilde Marcolli

Abstract

A quantum stochastic differential equation (qsde) on Fock space over $L^2$ differential 1-forms is given from the small "time" flow of which the trace of the connection Laplacian heat kernel for the spinor endomorphism bundle can be computed over any compact Ricci-flat Riemannian manifold. The existence of the stochastic flow is established by adapting the construction from [14]. When the manifold supports a parallel spinor - Ricci-flatness is a required integrability condition for parallel spinors, the trace of Dirac Laplacian heat kernel of the spinor bundle can be recovered. For 4-manifolds, this corresponds to the spectral action, and realizes Einstein-Hilbert action as a stochastic flow.

Spectral action and heat kernel trace for Ricci flat manifolds from stochastic flow over second quantized $L^2$-differential forms

Abstract

A quantum stochastic differential equation (qsde) on Fock space over differential 1-forms is given from the small "time" flow of which the trace of the connection Laplacian heat kernel for the spinor endomorphism bundle can be computed over any compact Ricci-flat Riemannian manifold. The existence of the stochastic flow is established by adapting the construction from [14]. When the manifold supports a parallel spinor - Ricci-flatness is a required integrability condition for parallel spinors, the trace of Dirac Laplacian heat kernel of the spinor bundle can be recovered. For 4-manifolds, this corresponds to the spectral action, and realizes Einstein-Hilbert action as a stochastic flow.
Paper Structure (13 sections, 16 theorems, 81 equations)

This paper contains 13 sections, 16 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Organization and overview
  3. Spectral action and stochastic flows
  4. Dirac Laplacian, endomorphism connection and parallel spinors
  5. The structure maps for the Laplacian generated flow
  6. The Kolmogorov decomposition
  7. The structure matrix
  8. Growth bounds and continuity
  9. Growth of Laplacian iterates
  10. Picard iterates for the Laplacian generated flow
  11. The extended squareroot trick
  12. Laplacian on products
  13. Appendix: Map-valued Evans-Hudson quantum sde's

Key Result

Proposition 1.2

For $f\in {\mathcal{C}}^\infty(M)$, $\mathop{}\!\mathbin\bigtriangleup^{\mathop{\mathrm{End}}\nolimits(E)}(f\cdot {\textbf{1}}_\mathop{\mathrm{End}}\nolimits) = \mathop{}\!\mathbin\bigtriangleup^M(f)\cdot{\textbf{1}}_\mathop{\mathrm{End}}\nolimits$.

Theorems & Definitions (41)

  • Remark 1.1
  • Proposition 1.2
  • proof
  • Remark 1.3
  • Remark 1.4
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • ...and 31 more