Holomorphic-topological (HT) field theories form a fascinating class of quantum field theories. These theories combine features of topological quantum field theories (TQFT) and conformal field theories (CFT).

 

Due to the mixed holomorphic-topological nature of such theories, they create interactions between TQFT data (e.g., algbras, monoidal categories, etc) and CFT data (e.g., chiral algebras and chiral categories). This leads to exciting new mathematical structures, and connections to integrable systems, quantum topology and many other areas of mathematics. Recently. much progress has been made on the representation-theoretic aspects of HT theories. Examples include:

 

1. (Shifted) Poisson vertex algebras and their quantizations are constructed from local operators in HT theories.

 

2. Dimensional reduction of 4d HT theories lead to integrable systems and solutions of quantum Yang-Baxter equations.

 

3. 4d N=2 theories are linked to representation theory of K-theoretic Coulomb branches, cluster algebra categorifications, wall crossings and elliptic stable envelops.

 

4. New examples of chiral algebras and their dualities are derived from boundary conditions and dualities of 3d HT theories.

 

Moreover, many interesting TQFTs are given by deformations of holomorphic-topological theories. Examples include topological twists of 3d N=4 and 4d N=2 theories. These theories have attracted considerable attention in recent years for their connections to 3d mirror symmetry and the Langlands program. Some of these TQFTs only admit Lagrangian descriptions as HT QFTs, and therefore studying HT theories offers a possible approach for understanding these non-Lagrangian TQFTs.

 

This conference will focus on the representation-theoretic aspects of HT theories, particularly:

 

1. Chiral algebras arising from observables of HT QFT.

 

2. Quantum algebras, including Yangians and quantum affine algebras, and their relation to HT theories.

 

3. Chiral categories and OPE of line operators in HT theories.

 

4. Deformation of HT theories and their relation to chiral algebra deformations.

 

5. Relation between various HT theories under dimensional-reduction.

 

We aim to bring together leading mathematicians and physicists, to inform each other about the recent progress made in this area.