Graded Lagrangian Submanifolds

Floer theory assigns, in favourable circumstances, an abelian group HF (L0, L1) to a pair (L0, L1) of Lagrangian submanifolds of a symplectic manifold (M,ω). This group is a qualitative invariant, which remains unchanged under suitable deformations of L0 or L1. Following Floer [7] one can equip HF (L0, L1) with a canonical relative Z/N -grading, where 1 ≤ N ≤ ∞ is a number which depends on (M,ω), L0 and L1 (for N = ∞ we set Z/N = Z). Relative mostly means that the grading is unique up to an overall shift, although there are also cases with more complicated behaviour. In this paper we take a different approach to the grading: we consider Lagrangian submanifolds equipped with certain extra structure (these are what we call graded Lagrangian submanifolds). This extra structure removes the ambiguity and defines an absolute Z/N -grading on Floer cohomology. There is also a parallel notion of graded symplectic automorphism, which bears the same relation to the corresponding version of Floer theory. Both concepts were first discovered by Kontsevich, at least for N = ∞; see [13, p. 134]. Somewhat later, the present author came upon them independently.