arXiv:math/0603498v1 [math.SG] 21 Mar 2006 SEMI-GLOBAL INVARIANTS OF PIECEWISE SMOOTH LAGRANGIAN FIBRATIONS R. CASTA ˜ NO-BERNARD AND D. MATESSI Abstract. We study certain types of piecewise smooth Lagrangian fibrations of smooth symplectic manifolds, which we call stitched Lagrangian fibrations. We extend the classical theory of action-angle coordinates to these fibrations by encoding the information on the non-smoothness into certain invariants consisting, roughly, of a sequence of closed 1-forms on a torus. The main motivation for this work is given by the piecewise smooth Lagrangian fibrations previously constructed by the authors [3], which topologically coincide with the local models used by Gross in Topological Mirror Symmetry [5]. 1. Introduction Lagrangian fibrations arise naturally from integrable systems. It is a standard fact of Hamiltonian mechanics that such fibrations are locally given by maps of the type: f =(f 1 ,...,f n ), where the function components of f are Poisson commuting functions on a symplectic man- ifold and such that the differentials df 1 ,...,df n are pointwise linearly independent almost everywhere. It is customary to assume f to be C ∞ differentiable (smooth). Under this regularity assumption, a classical theorem of Arnold-Liouville says that a smooth proper La- grangian submersion with connected fibres has locally the structure of a trivial Lagrangian T n -bundle. In particular, all proper Lagrangian submersions are locally modelled on U × T n , where U ⊆ R n is a contractible open set and U × T n has the standard symplectic form induced from R 2n . Standard coordinates with values in U × T n are known as action-angle coordinates. Since these are defined on a fibred neighbourhood, action-angle coordinates are semi-global canonical coordinates. Thus proper Lagrangian submersions have no semi-global symplectic invariants. In this article we investigate the semi-global symplectic topology of proper Lagrangian fibrations given by piecewise smooth maps. In [3]§6 we introduced the notion of stitched Lagrangian fibration. These are continuous proper S 1 invariant fibrations of smooth sym- plectic manifolds X which fail to be smooth only along the zero level set Z = µ −1 (0) of the moment map of the S 1 action and whose fibres are all smooth Lagrangian n-tori. Essentially, these fibrations consist of two honest smooth pieces X + = {µ ≥ 0} and X − = {µ ≤ 0}, stitched 1 together along Z , which we call the seam. These fibrations, roughly speaking, can be expressed locally as: f =(µ, f ± 2 ,...,f ± n ), where f + j and f − j are smooth functions defined on X + and X − , respectively, whose differen- tials do not necessarily coincide along Z . Fibrations of this type are implicit in the examples proposed earlier by the authors [3]§5 and may also be implicit in those in [14]. In this paper, we develop a theory of action-angle coordinates for this class of piecewise smooth fibrations. Contrary to what happens in the smooth case, we found that these fibrations do give rise to semi-global symplectic invariants. To the authors’ knowledge, the kind of non-smoothness we investigate here does not seem to be of relevance to Hamiltonian mechanics. Nevertheless it is an important issue in symplectic topology and mirror symmetry. Over the past ten years, Lagrangian torus fibrations, in particular those which are special Lagrangian, have been discovered to play a fundamental 1 We have chosen to use ‘stitching’ rather than ‘gluing’ since the resulting map is in general non smooth; the term ‘gluing’ usually has a smoothness meaning attached to it. 1