On \(G\)-trace and \(G\)-index in deformation quantization
The author constructs a \(G\)-trace for a compact group of symplectomorphisms acting on a symplectic manifold. This trace is used to define a \(G\)-index for which the author proves a theorem similar to the Atiyah-Segal-Singer equivariant index theorem for elliptic operators. Canonical deformation quantization replaces the algebra of continuous functions on a symplectic manifold by an algebra \(A_h\) of quantum observables whose multiplication * is a deformation of the usual multiplication of functions. The canonical deformation quantization is said to be \(G\)-invariant if for every \(s\in G\) the pull-back map it defines is an automorphism of the algebra \(A_h\). For such a deformation quantization one defines a \(G\)-trace to be a set of linear functionals \(Tr:A_h\to C[[h,h^{-1}]]\) indexed by the elements of \(G\) satisfying: \[ Tr_sa*b= Tr_sb*((s^{-1})^*a) \qquad Tr_{s_1s_2}a= Tr_{s_2s_1} (s_2^{-1})^*a. \] The author constructs locally such a trace using the trace of operators in the Weyl quantization, and for it an explicit formula is proved. The construction for a general manifold is done by first observing that the distributional support of the functional \(Tr_s\) is contained in the fixed point set \(M_0\) of \(s\). The main observation is that the algebra \(A_h\) in the vicinity of \(M_0\) can be treated as an algebra of quantum observables with coefficient bundle the normal bundle of \(M_0\). The map \(s\) acts on coefficients only, and this allows one to define a trace on the coefficient bundle using the previous local trace. Assuming that the fixed point set \(M_0\) is a compact submanifold, the author introduces the \(s\)-index \(\text{ind}_sA_h=Tr1\) and proves an index formula for this quantity in the spirit of the Atiyah-Segal-Singer equivariant index theorem. To this end an equivariant Chern character is defined.
Atiyah-Segal-Singer equivariant index theorem
scientific article; zbMATH DE number 1545425