This paper introduces a ‘\(q\)-analogue’ of the classical Picard-Lefschetz formula for the action of a Dehn twist on the middle-dimensional cohomology of a variety, in the presence of certain additional structures and assumptions. This \(q\)-analogue is introduced via a ‘categorification’ of the Picard-Lefschetz formula by Lagrangian Floer cohomology, which we briefly recall. Given two Lagrangian submanifolds \(L_0\), \(L_1\) of a symplectic \(2n\)-manifold \(M\), under suitable assumptions, Lagrangian Floer cohomology [A. Floer, J. Differ. Geom. 28, No. 3, 513–547 (1988; Zbl 0674.57027)] defines a graded vector space \(HF^*(L_0,L_1)\) which categorifies the topological intersection number in the sense that \[ \chi(HF^*(L_0,L_1)) = (-1)^{n+1}L_0 \cdot L_1. \] Seidel’s long exact sequence [P. Seidel, Topology 42, No. 5, 1003–1063 (2003; Zbl 1032.57035)] categorifies the Picard-Lefschetz formula (in the setting that \(M\) is exact and convex at infinity).
One part of the extra structure needed to define the \(q\)-analogue of this story is the notion of a ‘dilation’: a class \(B\) in the symplectic cohomology \(SH^*(M)\) which is mapped to the identity by the BV operator (for a discussion of the BV operator, see [P. Seidel, in: Current developments in mathematics, 2006. Somerville, MA: International Press. 211–253 (2008; Zbl 1165.57020)]). Given a dilation \(B\), the notion of a ‘\(B\)-equivariant Lagrangian submanifold’ is introduced. If \(L_0\) and \(L_1\) are \(B\)-equivariant Lagrangians, then the leading order part of \(\Phi(B)\), where \[ \Phi: SH^*(M) \rightarrow HH^*(\mathcal{F}(M)) \] is the closed-open string map [P. Seidel, in: Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China. Vol. II: Invited lectures. Beijing: Higher Education Press. 351–360 (2002; Zbl 1014.53052)], gives a (graded) endomorphism \[ \Phi^1(B) \in \mathrm{Hom}(HF^*(L_0,L_1), HF^*(L_0,L_1)). \] The generalized eigenspaces of \(\Phi^1(B)\) give a decomposition \[ HF^*(L_0,L_1) \cong \bigoplus_{\lambda \in \mathbb{C}} HF^*(L_0,L_1)^{\lambda}. \] Taking the Euler characteristic yields the ‘\(q\)-intersection number’ \[ L_0 \bullet_q L_1 := \sum_{\lambda} \chi(HF^*(L_0,L_1)^{\lambda}) q^{\lambda}. \] The usual intersection number is recovered by setting \(q=1\).
Various nice properties of the \(q\)-intersection numbers are proven. For example, one has \[ L_1 \bullet_{q^{-1}} L_0 = (-1)^n q^{-1} (L_0 \bullet_q L_1), \] and also the aforementioned \(q\)-analogue of the Picard-Lefschetz formula: if \(\tau_V: M \rightarrow M\) is the generalized Dehn twist along a Lagrangian sphere \(V \subset M\), then \[ \tau_V(L_0) \bullet_q L_1 = L_0 \bullet_q L_1 + (-1)^{n+1} q^{-1} (L_0 \bullet_q V)(V \bullet_q L_1). \] The authors consider the question of the existence or otherwise of dilations for various manifolds, including for cotangent bundles, and prove that if the fibre of a Lefschetz fibration admits a dilation, then so does the total space; this allows them to show, in particular, that \(A_m\) Milnor fibres admit dilations.
We remark that the whole construction is motivated by homological mirror symmetry, as is the nomenclature: the mirror to a dilation is an algebraic vector field which contracts the holomorphic volume form, and equivariant Lagrangians correspond to coherent sheaves which are infinitesimally equivariant under the deformation of the category of coherent sheaves coming from this vector field.