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The BV construction for finite spectral triples

Roberta Anna Iseppi

Abstract

This article presents how the BV formalism naturally inserts in the framework of noncommutative geometry for gauge theories induced by finite spectral triples. Reaching this goal entails that not only all the steps of the BV construction, from the introduction of ghost/anti-ghost fields to the construction of the BRST complex, can be expressed using noncommutative geometric objects, but also that the method to go from one step in the construction to the next one has an intrinsically noncommutative geometric nature. Moreover, we prove that both the classical BV and BRST complexes coincide with another cohomological theory, naturally appearing in noncommutative geometry: the Hochschild complex of a coalgebra. The construction is presented in detail for $U(n)$-gauge theories induced by spectral triples on the algebra $M_n(\mathbb{C})$.

The BV construction for finite spectral triples

Abstract

This article presents how the BV formalism naturally inserts in the framework of noncommutative geometry for gauge theories induced by finite spectral triples. Reaching this goal entails that not only all the steps of the BV construction, from the introduction of ghost/anti-ghost fields to the construction of the BRST complex, can be expressed using noncommutative geometric objects, but also that the method to go from one step in the construction to the next one has an intrinsically noncommutative geometric nature. Moreover, we prove that both the classical BV and BRST complexes coincide with another cohomological theory, naturally appearing in noncommutative geometry: the Hochschild complex of a coalgebra. The construction is presented in detail for -gauge theories induced by spectral triples on the algebra .

Paper Structure

This paper contains 15 sections, 13 theorems, 156 equations, 2 figures.

Table of Contents

  1. Why a noncommutative geometric approach to the BV formalism
  2. The BV construction: origin and physical relevance
  3. The BV extension
  4. The BV cohomology complex
  5. Auxiliary fields and the gauge-fixing process
  6. The BRST cohomology complex
  7. Toward the BV quantization
  8. From spectral triples to gauge theories: notions and constructions
  9. A $\mathrm{U}(n)$-model.
  10. The BV extension for finite spectral triples
  11. The BV complex as Hochschild complex
  12. The isomorphism of cohomology complexes
  13. Auxiliary fields and the total spectral triple
  14. The BRST cohomology as a Hochschild complex
  15. Conclusions: the BV construction in the setting of NCG

Key Result

Proposition 3.7

Given $(\mathcal{A}, \mathcal{H}, D)$ a finite spectral triple, let $X_{0}$ denote the space of inner fluctuations for the operator $D$, $X_{0} := [\Omega^{1}_{D}(\mathcal{A})]^{*}$, $\mathcal{G}$ be the space of unitary elements of $\mathcal{A}$ which acts on $X_0$ via the map and, finally, let $S_0$ be the spectral action defined on $X_{0}$ as for any $\varphi\in X_0$ and some $f\in \mathbb{R}

Figures (2)

  • Figure 1:
  • Figure 2: The BV construction for finite spectral triples and its compatibility at the level of the cohomology complexes

Theorems & Definitions (56)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 46 more