Maslov index and Spectral Flow

Date

2016

Authors

Lim, Johnny

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Abstract

In [1], Arnol'd discussed the Maslov index as an intersection number of Lagrangian loop with the Maslov cycle, whereas in [17], Robbin and Salamon gave a de nition of the Maslov index in term of the signature of crossing form. Furthermore, the Maslov index is characterized by axioms. The index is a homotopy invariant with xed endpoints, and is additive for the concatenation of Lagrangian paths. Application wise, in [8], Floer studied the case where the Hessian (a second order di erential operator) of the symplectic action functional A is taken along a gradient ow line x(t) i.e. a solution satisfying the gradient ow equation x_ = rA (x) connecting two critical endpoints x = limt! 1 x(t): Both the gradient rA and Hessian A = r2A are taken with respect to suitable metric on the underlying manifold. In particular, Floer de ned a relative index at x and showed that the spectral ow of the Hessian of A is equal to the relative Morse index between x which is also equal to the dimension of the space of trajectories of gradient ow between x : In Morse theory (as nite dimensional case of Floer theory), such A is bounded below and the Hessian H has only nitely many negative eigenvalues, when treated as matrix. Then the spectral ow of A(t) is the number of negative eigenvalues (counted with multiplicity) of A at x+ minus the number of negative eigenvalues of A at x?? which is equal to the Fredholm index of linearization operator DA: Moreover, it happens to be the case that the unstable manifold Wu(x??) intersects the stable manifold Ws(x+) transversally if and only if DA is surjective. Then, the moduli space M= Wu(x??) \Ws(x+) is a nite dimensional manifold of dimension equal to the relative Morse index. In this thesis, we will mainly focus on both the notion of Maslov index and spectral ow and their coincidence. Roughly speaking, Maslov index can be seen as the number of times the Lagrangian paths crosses the Maslov cycle and the spectral ow can be seen as the number of eigenvalues of A(t) crossing zero from negative to positive from t = ??1 to t = 1: The organization of this thesis will be distributed as follows: In Chapter 1 we brie y review some backgrounds in Symplectic Geometry, that include the notion of symplectic vector space, symplectic manifold and Darboux Theorem. To prepare for later chapter, we study the relation between Sp(2n) and U(n) and their actions on the Lagrangian Grassmannian (n): In Chapter 2 we study the notion and computational tool for the Maslov index in the sense of crossing operator(adopting the method introduced in [17]). Other related indices such as H ormander index and the well-known Conley-Zehnder index will also be introduced. In Chapter 3, we study the notion of Spectral ow of operator satisfying certain conditions and its coincidence with Maslov index and Fredholm index, together with several examples in which this notion applies to will be given.

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Thesis (Masters)

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DOI

10.25911/5d9efb36b5576

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