Quantum differential equations (qDEs) are a rich object attached to complex smooth projective varieties. They encode information on their enumerative geometry, topology and (conjecturally) on their algebraic geometry. In occasion of the 1998 ICM in Berlin, B. Dubrovin conjectured an intriguing connection between the enumerative geometry of a Fano manifold $X$ with algebro-geometric properties of exceptional collections in the derived category $D_b(X)$. Under the assumption of semisimplicity of the quantum cohomology of $X$, the conjecture prescribes an explicit form for local invariants of $QH^*(X)$, the so-called “monodromy data”, in terms of Gram matrices and characteristic classes of objects of exceptional collections. In this talk I will discuss an equivariant analog of these relations, focusing on the example of projective spaces. The study of the equivariant quantum differential equations for partial flag varieties has been initiated by V. Tarasov and A. Varchenko in 2017. They discovered the existence of a system of compatible qKZ difference equations, which have made the study of the quantum differential equations easier than in the non-equivariant case. I will establish relations between the monodromy data of the joint system of the equivariant qDE and qKZ equations for $\mathbb{P}^n$ and characteristic classes of objects of the derived category of T-equivariant coherent sheaves on $\mathbb{P}^n$.

Based on joint works with B. Dubrovin, D. Guzzetti and A. Varchenko.