5d-4d Correspondence in Twisted M-theory on a Conifold

TL;DR

This paper extends Costello’s twisted M-theory framework to a conifold background and shows that twisted M-theory in this geometry is captured by a 5d non-commutative Chern-Simons-matter theory on $\mathbb{R}_+\times\mathbb{C}^2$, which can be reformulated as a CS theory for the supergroup $U(N|K)$. It introduces a boundary sector—a gauged 4d chiral WZW model—whose gauge variation cancels the bulk CS variation, establishing a 5d–4d holographic duality via the BV-BFV formalism, including matter in the adjoint representation. The authors compute both classical and quantum current algebras of the boundary theory, and demonstrate a boundary generating functional that encodes the bulk dynamics, with a non-commutative generalization preserved by star-product structures. The results provide a concrete holographic description of twisted M-theory on conifolds and suggest deep links to affine Yangians, conformal blocks, and higher-dimensional edge states in quantum Hall-type systems.

Abstract

We study twisted M-theory in a general conifold background, and describe it in terms of a 5d non-commutative Chern-Simons-matter theory, which is equivalent to 5d non-commutative Chern-Simons theory for a supergroup. In an equivalent description as twisted type IIA string theory, the matter degrees of freedom arise from topological strings stretched between stacks of D6-branes. In order to study 5d Chern-Simons-matter theories with a boundary, we first construct and investigate the properties of a 4d non-commutative gauged chiral WZW model. We prove the gauge invariant coupling of this 4d theory to the bulk 5d Chern-Simons theory defined on $\mathbb{R}_+ \times \mathbb{C}^2 $, and further generalize our results to the 5d Chern-Simons-matter theory. We also investigate the toroidal current algebra of the 4d chiral WZW model that arises from radial quantization along one of the complex planes. Finally, we show that a gauged non-commutative chiral 4d WZW model arises from the partition function for quantum 5d non-commutative Chern-Simons theory with boundaries in the BV-BFV formalism, and further generalize this 5d-4d correspondence to the 5d non-commutative Chern-Simons-matter theory for the case of adjoint matter.

Figures (2)

• Figure 1: Feynman diagrams
• Figure 2: Feynman diagrams for matter fields