# Holomorphic boundary conditions for topological field theories via branes in twisted supergravity

**Authors:** Ilka Brunner, Ioannis Lavdas, Ingmar Saberi  
**arXiv:** [2110.15257](https://arxiv.org/abs/2110.15257)  
**Date:** 2021-10-28  
**Summary page:** https://www.summarizepaper.com/en/paper/2110.15257v1/

## Key points

- The paper explores the relationship between holomorphically twisted field theories and topological twists.
- The authors focus on three-dimensional $\mathcal{N}=4$ supersymmetric field theories with chiral half-BPS boundary conditions preserving $\mathcal{N}=(0,4)$ supersymmetry.
- These boundary conditions are not compatible with topological twists, so the authors revisit Costello and Gaiotto's deformations to define boundary conditions for topological theories.
- They work directly in the setting of the holomorphically twisted theory while viewing topological twists as further deformations.
- Their approach combines intersecting brane configurations of Hanany-Witten with recent work of Costello and Li on twisted supergravity.
- Not all $\mathcal{N}=(0,4)$ boundary conditions admit such deformations, which the authors explore by studying how to realize holomorphically and topologically twisted field theories directly as worldvolume theories in deformed supergravity backgrounds using their brane engineering approach.
- This paper provides new insights into reconciling holomorphically twisted field theories with topological twists through careful analysis of boundary conditions and their associated deformations.

## Summary

The paper titled "Holomorphic boundary conditions for topological field theories via branes in twisted supergravity" by Ilka Brunner, Ioannis Lavdas, and Ingmar Saberi explores the relationship between holomorphically twisted field theories and topological twists. The authors focus on three-dimensional $\mathcal{N}=4$ supersymmetric field theories that have a natural class of chiral half-BPS boundary conditions preserving $\mathcal{N}=(0,4)$ supersymmetry. However, these boundary conditions are not compatible with topological twists. To address this issue, Costello and Gaiotto introduced deformations that define boundary conditions for the topological theories. The authors revisit this construction and work directly in the setting of the holomorphically twisted theory while viewing the topological twists as further deformations. They explain properties of the construction both purely in the context of holomorphic field theory and also by engineering the holomorphic theory on the worldvolume of a D-brane. Their approach combines intersecting brane configurations of Hanany-Witten with recent work of Costello and Li on twisted supergravity. The paper highlights that not all $\mathcal{N}=(0,4)$ boundary conditions admit such deformations. The authors explore this issue by studying how to realize holomorphically and topologically twisted field theories directly as worldvolume theories in deformed supergravity backgrounds using their brane engineering approach. Overall, this paper provides new insights into understanding how to reconcile holomorphically twisted field theories with topological twists through careful analysis of boundary conditions and their associated deformations. The authors' approach offers a promising avenue for future research in this area.