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BRST Cohomology is Lie Algebroid Cohomology

Weizhen Jia, Marc S. Klinger, Robert G. Leigh

TL;DR

This work reframes the physicist's BRST complex as the exterior algebra of an Atiyah Lie algebroid, showing that BRST cohomology arises from Lie algebroid cohomology and that isomorphisms between algebroids encode gauge transformations together with diffeomorphisms. By introducing Lie algebroid trivializations and an atlas, the authors connect the BRST differential to a geometric d + s structure on a bi-graded complex, thereby unifying horizontal and vertical gauge data with de Rham data. The paper then develops a Chern–Weil/Chern–Simons formalism in the algebroid setting to derive descent equations and to distinguish consistent and covariant anomalies, confirming this geometry through explicit 2d examples of chiral and Lorentz–Weyl anomalies. Overall, the work provides a geometric framework that not only recasts BRST cohomology in Lie algebroid terms but also yields a natural language for anomaly polynomials and their covariant/consistent forms, with future directions including a configuration algebroid and anomaly inflow.

Abstract

In this paper we demonstrate that the exterior algebra of an Atiyah Lie algebroid generalizes the familiar notions of the physicist's BRST complex. To reach this conclusion, we develop a general picture of Lie algebroid isomorphisms as commutative diagrams between algebroids preserving the geometric structure encoded in their brackets. We illustrate that a necessary and sufficient condition for such a diagram to define a morphism of Lie algebroid brackets is that the two algebroids possess gauge-equivalent connections. This observation indicates that the aforementioned set of Lie algebroid isomorphisms should be regarded as equivalent to the set of local diffeomorphisms and gauge transformations. Moreover, a Lie algebroid isomorphism being a chain map in the exterior algebra sense ensures that isomorphic algebroids are cohomologically equivalent. The Atiyah Lie algebroids derived from principal bundles with common base manifolds and structure groups may therefore be divided into equivalence classes of isomorphic algebroids. Each equivalence class possesses a local representative which we refer to as the trivialized Lie algebroid, and we show that the exterior algebra of the trivialized algebroid gives rise to the BRST complex. We conclude by illustrating the usefulness of Lie algebroid cohomology in computing quantum anomalies, including applications to the chiral and Lorentz-Weyl (LW) anomalies. In particular, we pay close attention to the fact that the geometric intuition afforded by the Lie algebroid (which was absent in the naive BRST complex) provides hints of a deeper picture that simultaneously geometrizes the consistent and covariant forms of the anomaly. In the algebroid construction, the difference between the consistent and covariant anomalies is simply a different choice of basis.

BRST Cohomology is Lie Algebroid Cohomology

TL;DR

This work reframes the physicist's BRST complex as the exterior algebra of an Atiyah Lie algebroid, showing that BRST cohomology arises from Lie algebroid cohomology and that isomorphisms between algebroids encode gauge transformations together with diffeomorphisms. By introducing Lie algebroid trivializations and an atlas, the authors connect the BRST differential to a geometric d + s structure on a bi-graded complex, thereby unifying horizontal and vertical gauge data with de Rham data. The paper then develops a Chern–Weil/Chern–Simons formalism in the algebroid setting to derive descent equations and to distinguish consistent and covariant anomalies, confirming this geometry through explicit 2d examples of chiral and Lorentz–Weyl anomalies. Overall, the work provides a geometric framework that not only recasts BRST cohomology in Lie algebroid terms but also yields a natural language for anomaly polynomials and their covariant/consistent forms, with future directions including a configuration algebroid and anomaly inflow.

Abstract

In this paper we demonstrate that the exterior algebra of an Atiyah Lie algebroid generalizes the familiar notions of the physicist's BRST complex. To reach this conclusion, we develop a general picture of Lie algebroid isomorphisms as commutative diagrams between algebroids preserving the geometric structure encoded in their brackets. We illustrate that a necessary and sufficient condition for such a diagram to define a morphism of Lie algebroid brackets is that the two algebroids possess gauge-equivalent connections. This observation indicates that the aforementioned set of Lie algebroid isomorphisms should be regarded as equivalent to the set of local diffeomorphisms and gauge transformations. Moreover, a Lie algebroid isomorphism being a chain map in the exterior algebra sense ensures that isomorphic algebroids are cohomologically equivalent. The Atiyah Lie algebroids derived from principal bundles with common base manifolds and structure groups may therefore be divided into equivalence classes of isomorphic algebroids. Each equivalence class possesses a local representative which we refer to as the trivialized Lie algebroid, and we show that the exterior algebra of the trivialized algebroid gives rise to the BRST complex. We conclude by illustrating the usefulness of Lie algebroid cohomology in computing quantum anomalies, including applications to the chiral and Lorentz-Weyl (LW) anomalies. In particular, we pay close attention to the fact that the geometric intuition afforded by the Lie algebroid (which was absent in the naive BRST complex) provides hints of a deeper picture that simultaneously geometrizes the consistent and covariant forms of the anomaly. In the algebroid construction, the difference between the consistent and covariant anomalies is simply a different choice of basis.
Paper Structure (18 sections, 97 equations, 1 figure)

This paper contains 18 sections, 97 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on Atiyah Lie Algebroids
  3. The Lie Algebroid Derived from a Principal Bundle
  4. The Exterior Algebra of an Atiyah Lie Algebroid
  5. Lie Algebroid Isomorphisms
  6. A Commutative Diagram
  7. Local Trivialization of an Atiyah Lie Algebroid
  8. The Cohomology of Trivialized Lie Algebroids
  9. Anomalies from Lie Algebroid Cohomology
  10. Characteristic Classes and Lie Algebroid Cohomology
  11. Descent Equations and the Consistent Anomaly
  12. The Horizontal-Vertical Splitting and the Covariant Anomaly
  13. Examples
  14. Chiral Anomaly in $2d$
  15. Lorentz-Weyl Anomaly in $2d$
  16. ...and 3 more sections

Figures (1)

  • Figure 1: A visualization of a Lie algebroid. A connection gives a global split $A=H\oplus V$, which locally can be viewed as determined by a gauge field $b$ defined with respect to "axes" corresponding to sub-bundles $TM$ and $L$.