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The tensor multiplet in loop space

Dongsu Bak, Andreas Gustavsson

TL;DR

This work develops a loop-space formulation of the six-dimensional abelian tensor multiplet by recasting fields in a cohomological form and transgressing them to loop space using two commuting lightlike conformal Killing vectors $U$ and $V$. It introduces an integrated transgression map and, crucially, an unintegrated map along with weak $(-1)$-forms to manage information flow and algebraic structures on loop space, enabling a consistent abelian theory and infrastructure for nonabelian generalizations. A nonabelian transgression map based on loop algebras is constructed, with Bianchi identities and supersymmetry variations extended to loop space, yielding on-shell closed SUSY and loop-space equations of motion. The framework is connected to five-dimensional super Yang-Mills through dimensional reduction along a chosen vector, with explicit mappings of the six-dimensional tensor data to five-dimensional fields and a conjectured general nonabelian loop-space reduction, indicating a route to a local, loop-algebraic description of nonabelian tensor multiplets. The paper also discusses multiple transgression schemes and future directions for achieving a fully consistent, supersymmetric nonabelian theory in loop space, potentially illuminating new geometric perspectives on Wilson surfaces and higher gauge structures.

Abstract

We reformulate the abelian tensor multiplet on a curved spacetime with at least two supercharges in a cohomological form where all the bosonic and fermionic fields become tensor fields. These tensor fields are rewritten as fields in loop space by a transgression map. There are two lightlike conformal Killing vectors. By decomposing the spacetime tensor fields in transverse and parallel components to these Killing vectors, we obtain the equations of motion in loop space by closing the supersymmetry variations on-shell. We generalize to nonabelian gauge groups. By closing supersymmetry variations we obtain nonabelian fermionic equations of motion in loop space.

The tensor multiplet in loop space

TL;DR

This work develops a loop-space formulation of the six-dimensional abelian tensor multiplet by recasting fields in a cohomological form and transgressing them to loop space using two commuting lightlike conformal Killing vectors and . It introduces an integrated transgression map and, crucially, an unintegrated map along with weak -forms to manage information flow and algebraic structures on loop space, enabling a consistent abelian theory and infrastructure for nonabelian generalizations. A nonabelian transgression map based on loop algebras is constructed, with Bianchi identities and supersymmetry variations extended to loop space, yielding on-shell closed SUSY and loop-space equations of motion. The framework is connected to five-dimensional super Yang-Mills through dimensional reduction along a chosen vector, with explicit mappings of the six-dimensional tensor data to five-dimensional fields and a conjectured general nonabelian loop-space reduction, indicating a route to a local, loop-algebraic description of nonabelian tensor multiplets. The paper also discusses multiple transgression schemes and future directions for achieving a fully consistent, supersymmetric nonabelian theory in loop space, potentially illuminating new geometric perspectives on Wilson surfaces and higher gauge structures.

Abstract

We reformulate the abelian tensor multiplet on a curved spacetime with at least two supercharges in a cohomological form where all the bosonic and fermionic fields become tensor fields. These tensor fields are rewritten as fields in loop space by a transgression map. There are two lightlike conformal Killing vectors. By decomposing the spacetime tensor fields in transverse and parallel components to these Killing vectors, we obtain the equations of motion in loop space by closing the supersymmetry variations on-shell. We generalize to nonabelian gauge groups. By closing supersymmetry variations we obtain nonabelian fermionic equations of motion in loop space.

Paper Structure

This paper contains 22 sections, 437 equations.

Table of Contents

  1. Introduction
  2. The abelian tensor multiplet
  3. The abelian transgression map
  4. Review of the integrated transgression map
  5. The unintegrated transgression map for $p \geq 0$
  6. The introduction of weak $(-1)$-forms
  7. Inner product
  8. Two geometric vector fields
  9. Bianchi identities in loop space for abelian gauge group
  10. Supersymmetry in loop space for abelian gauge group
  11. Equations of motion in loop space for abelian gauge group
  12. The nonabelian transgression map
  13. Bianchi identities in loop space with nonabelian gauge group
  14. Supersymmetry in loop space with nonabelian gauge group
  15. Five-dimensional super Yang-Mills
  16. ...and 7 more sections