Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher Dimensional Analogues of Donaldson-Witten Theory

B. S. Acharya, M. O'Loughlin, B. Spence

TL;DR

This work constructs higher-dimensional analogues of Donaldson–Witten theory on manifolds with reduced holonomy, notably Spin(7) in eight dimensions, by formulating BRST-invariant actions whose stress tensor is BRST exact for holonomy-preserving metric deformations. It derives the higher-dimensional instanton equations tied to a covariantly constant 4-form $\phi$ and shows how the theories arise naturally as supersymmetric Yang–Mills theories on curved manifolds without twisting, with BRST observables yielding invariants and hints of Floer-type structures in higher dimensions. The paper provides explicit field content and Lagrangians for 8D Spin(7) and 7D G2 cases, analyzes metric variations of the holonomy structure, and constructs a set of BRST-invariant observables $W_k$, leading to metric-independent quantities $I(\gamma)$. It also discusses a Floer formulation, brane/string interpretations, and connections to heterotic and Type I string theory, highlighting the deep link between calibrated geometries, topological field theory, and nonperturbative dualities.

Abstract

We present a Donaldson-Witten type field theory in eight dimensions on manifolds with $Spin(7)$ holonomy. We prove that the stress tensor is BRST exact for metric variations preserving the holonomy and we give the invariants for this class of variations. In six and seven dimensions we propose similar theories on Calabi-Yau threefolds and manifolds of $G_2$ holonomy respectively. We point out that these theories arise by considering supersymmetric Yang-Mills theory defined on such manifolds. The theories are invariant under metric variations preserving the holonomy structure without the need for twisting. This statement is a higher dimensional analogue of the fact that Donaldson-Witten field theory on hyper-Kähler 4-manifolds is topological without twisting. Higher dimensional analogues of Floer cohomology are briefly outlined. All of these theories arise naturally within the context of string theory.

Higher Dimensional Analogues of Donaldson-Witten Theory

TL;DR

This work constructs higher-dimensional analogues of Donaldson–Witten theory on manifolds with reduced holonomy, notably Spin(7) in eight dimensions, by formulating BRST-invariant actions whose stress tensor is BRST exact for holonomy-preserving metric deformations. It derives the higher-dimensional instanton equations tied to a covariantly constant 4-form and shows how the theories arise naturally as supersymmetric Yang–Mills theories on curved manifolds without twisting, with BRST observables yielding invariants and hints of Floer-type structures in higher dimensions. The paper provides explicit field content and Lagrangians for 8D Spin(7) and 7D G2 cases, analyzes metric variations of the holonomy structure, and constructs a set of BRST-invariant observables , leading to metric-independent quantities . It also discusses a Floer formulation, brane/string interpretations, and connections to heterotic and Type I string theory, highlighting the deep link between calibrated geometries, topological field theory, and nonperturbative dualities.

Abstract

We present a Donaldson-Witten type field theory in eight dimensions on manifolds with holonomy. We prove that the stress tensor is BRST exact for metric variations preserving the holonomy and we give the invariants for this class of variations. In six and seven dimensions we propose similar theories on Calabi-Yau threefolds and manifolds of holonomy respectively. We point out that these theories arise by considering supersymmetric Yang-Mills theory defined on such manifolds. The theories are invariant under metric variations preserving the holonomy structure without the need for twisting. This statement is a higher dimensional analogue of the fact that Donaldson-Witten field theory on hyper-Kähler 4-manifolds is topological without twisting. Higher dimensional analogues of Floer cohomology are briefly outlined. All of these theories arise naturally within the context of string theory.

Paper Structure

This paper contains 13 sections, 70 equations.

Table of Contents

  1. Introduction.
  2. Instanton Equations in $D > 4$.
  3. D= 7,6
  4. The Field Theories
  5. On the Metric Dependence of the Theories.
  6. Metric Variation of $\chi$.
  7. Invariants
  8. Floer Formulation
  9. Discussion and Conclusion.
  10. Field Theories From Super Yang-Mills Theory
  11. D = 7, 6
  12. Relation to Heterotic and Type I String Theory.
  13. A Further Relation With Donaldson-Witten Theory.