Elliptic boundary value problem on non-compactG-manifolds

Abstract

In this paper, an equality between the Hochs-Mathai type index and the Atiyah-Patodi-Singer type index is established when the manifold and the group action are both non-compact, which generalizes a result of Ma and Zhang for compact group actions. As a technical preparation, a problem concerning the Fredholm property of the global elliptic boundary value problems of the Atiyah-Patodi-Singer type on a non-compact manifold is studied.Elliptic boundary value problem on non-compact G-manifolds 2 on which a locally compact group G acts symplectically. Suppose the group action is both proper and cocompact. Then we prove that an indices equality of type (1.2) still holds on M. 2 The main difficulty for such a generalization is to find proper definitions for both sides of (1.2) in the non-compact setting. On the left hand side, an index introduced by Hochs and Mathai [17] is a natural candidate. Therefore, the main effort of this paper centers around the right hand side of (1.2), i.e., establishing the well-definedness of the APS type index for non-compact manifolds. Admittedly, this is more or less of technical flavor.Indeed, let (M, g TM ) be an even dimensional non-compact Spin c Riemannian manifold with boundary. To carry out the usual argument about L 2 sections, one relies on a cut-off function, say f , as in [24]. Let S(TM) = S + (TM) ⊕ S − (TM) be the spinor bundle and E be the coefficient Hermitian vector bundle respectively. Suppose that Γ(M, S(TM) ⊗ E) G is the space of invariant smooth sections. An orthogonal projection operator P f is defined by f on the space of L 2 sections of S(TM) ⊗ E, whose range, denoted by H 0 f (M, S(TM) ⊗ E) G , is the closure of f Γ(M, S(TM) ⊗ E) G in the space of L 2 sections. In the same fashion, higher Sobolev spaces H i f (M, S(TM) ⊗ E) G associated to f are also defined.Roughly, two aspects should be checked when one tries to adapt the definition of the APS type index in order to take f into consideration. Firstly, it's well-known that the APS type index involves a kind of global boundary condition, the so-called APS boundary condition, whose definition relies on the spectral projection of the boundary operator. In this paper, with the help of f , certain spectral projection, denoted by P ≥0, f , plays a similar role on a non-compact manifold. Secondly, the Fredholm property of the Dirac operator with the APS boundary condition should be reestablished. For this purpose, we will apply the elementary functional analysis method as in [1] and [4]. More concretely, our strategy is as follows. A partial result, the Fredholmness of the boundary value problem with the product structure assumption, is proved in Subsection 4.1. Then the result is extended to the general case in Subsection 4.2, using the Rellich perturbation theorem about self-adjoint operators.Put it briefly, the major technical problem is the solution of the following boundary value problem on a non-compact manifold.

Twisted Holography and Celestial Holography from Boundary Chiral Algebra

“…(2.12) the field A A A 2d of the original 2d theory has no pole at r = 0 and is well defined on the whole complex plane C. For general 1d BF theory, this boundary condition is analyzed in [27,28], and is called the A boundary condition. The corresponding space of boundary local operators is given by the Chevalley-Eilenberg algebra of g[w]:…”