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Twists and Gorms and Antifields, oh my!

Alex S. Arvanitakis

Abstract

I define topological twists of supersymmetric field theories in the case when the supercharges involved obey an ``open'' algebra. Using the Batalin-Vilkovisky field-antifield formalism, I construct twisted theories algorithmically from the supersymmetry data, and explain supersymmetric localisation in terms of anticanonical transformations. I also treat equivariant topological twists and explain how BV observables contain the equivariant cohomology of the space of histories. Some results are generalised to theories with two topological supercharges -- such as the ``balanced'' topological field theories of Dijkgraaf and Moore -- using the geometry of ``differential gorms'' of Kochan and Ševera. Finally, I exhibit examples of these constructions, including a $\mathrm{U}(1)$-equivariant topological B-model.

Twists and Gorms and Antifields, oh my!

Abstract

I define topological twists of supersymmetric field theories in the case when the supercharges involved obey an ``open'' algebra. Using the Batalin-Vilkovisky field-antifield formalism, I construct twisted theories algorithmically from the supersymmetry data, and explain supersymmetric localisation in terms of anticanonical transformations. I also treat equivariant topological twists and explain how BV observables contain the equivariant cohomology of the space of histories. Some results are generalised to theories with two topological supercharges -- such as the ``balanced'' topological field theories of Dijkgraaf and Moore -- using the geometry of ``differential gorms'' of Kochan and Ševera. Finally, I exhibit examples of these constructions, including a -equivariant topological B-model.

Paper Structure

This paper contains 38 sections, 1 theorem, 123 equations.

Table of Contents

  1. Introduction and summary
  2. A homological perturbation lemma
  3. A single supercharge
  4. Topological twists
  5. Equivariant topological twists of type DM
  6. The Weil algebra and the Cartan calculus.
  7. A BV construction for the "equivariant twist".
  8. The equivariant quantum master equation. Integration.
  9. Remark.
  10. Homotopies between solutions.
  11. On equivariant observables and a "Cartan-style" cohomology
  12. On localisation
  13. Gauge symmetries and equivariant topological twists of type BT
  14. Topological twists and gauge symmetry
  15. BT-type twists
  16. ...and 23 more sections

Key Result

Theorem 1

Let $\delta$ and $\mathcal{D}$ be two odd left derivations acting on some graded-commutative algebra $A$. The algebra is bigraded with $\mathbb Z$-gradings $\mathop{\mathrm{ghdeg}}\nolimits$ and $\mathop{\mathrm{Rdeg}}\nolimits$ such that $\mathop{\mathrm{ghdeg}}\nolimits+\mathop{\mathrm{Rdeg}}\noli for $s_1$ some odd derivation of appropriate degrees. Finally, $\delta$ is required to be acyclic:

Theorems & Definitions (2)

  • Theorem 1: 8.3 henneaux1992quantization
  • Definition 1