Thomas Invariants Research Articles

Deformation quantisation for $(-1)$-shifted symplectic structures and vanishing cycles

We formulate a notion of $E_0$ quantisation of $(-1)$-Poisson structures on derived Artin $N$-stacks, and construct a map from $E_0$ quantisations of $(-1)$-shifted symplectic structures to power series in de Rham cohomology. For a square root of the dualising line bundle, this gives an equivalence between even power series and self-dual quantisations. In particular, there is a canonical quantisation of any such square root, which localises to recover the perverse sheaf of vanishing cycles on derived DM stacks, thus giving a form of derived categorification of Donaldson--Thomas invariants.

The Critical CoHA of a Quiver With Potential

Pursuing the similarity between the Kontsevich--Soibelman construction of the cohomological Hall algebra of BPS states and Lusztig's construction of canonical bases for quantum enveloping algebras, and the similarity between the inetgrality conjecture for motivic Donaldson--Thomas invariants and the PBW theorem for quantum enveloping algebras, we build a coproduct on the cohomological Hall algebra associated to a quiver with potential. We also prove a cohomological dimensional reduction theorem, further linking a special class of cohomological Hall algebras with Yangians, and explaining how to connect the study of character varieties with the study of cohomological Hall algebras.