Frobenius Algebras & Geometric Invariants

Frobenius algebras are finite-dimensional, associative, unital algebras with an invariant non-degenerate bilinear form. They occur in many areas of mathematics, e.g. algebra and geometry, and are known to be categorically equivalent to two-dimensional topological quantum field theories. Prominent examples include the quantum cohomology or quantum K-theory rings of certain varieties, such as Grassmannians, or Verlinde algebras in conformal field theory, all of which are active areas of research in enumerative geometry and mathematical physics. 

In the context of quantum integrable systems Frobenius algebras occur as solutions of the Bethe ansatz or quantum inverse scattering method. The latter is used when computing the partition function of a statistical lattice model or when diagonalising the Hamiltonians of a many-body system. Integrable systems can be used to compute the structure constants of these Frobenius algebras which in the mentioned examples are called Gromov-Witten invariants. There is also a link to soliton theory via Witten’s conjecture (proved by Kontsevich) linking solutions (tau-functions) of the Korteweg-de Vries hierarchy to these geometric invariants. 

Selected publications:

• Gorbounov, V.G., Korff, C.  and Stroppel, C. (2020) Yang-Baxter algebras, convolution algebras, and Grassmannians. Russian Mathematical Surveys, 75(5), 791. (doi: 10.1070/RM9959) 

• Gorbounov, V. and Korff, C.  (2017) Quantum integrability and generalised quantum Schubert calculus. Advances in Mathematics, 313, pp. 282-356. (doi: 10.1016/j.aim.2017.03.030) 

• Korff, C.  (2016) From quantum Bäcklund transforms to topological quantum field theory. Journal of Physics A: Mathematical and Theoretical, 49(10), 104001. (doi: 10.1088/1751-8113/49/10/104001) 

Grants & Workshops 

• Royal Society Theo-Murphy Meeting (2018) competitive international scientific meeting scheme funded by the Royal Society of London and held at the Society’s International Research Centre (joint with V. Gorbounov, Aberdeen now HSE Moscow).