Some questions about the index of quantized contact transformations

On the geometric quantization of contact manifolds

Suppose that $(M,E)$ is a compact contact manifold, and that a compact Lie group $G$ acts on $M$ transverse to the contact distribution $E$. In an earlier paper, we defined a $G$-transversally elliptic Dirac operator $\dirac$, constructed using a Hermitian metric $h$ and connection $\nabla$ on the symplectic vector bundle $E\rightarrow M$, whose equivariant index is well-defined as a generalized function on $G$, and gave a formula for its index. By analogy with the geometric quantization of symplectic manifolds, the $\mathbb{Z}_2$-graded Hilbert space $Q(M)=\ker \dirac \oplus \ker \dirac^{*}$ …

The Atiyah-Singer index formula for subelliptic operators on contact manifolds, Part II

We present a new solution to the index problem for hypoelliptic operators in the Heisenberg calculus on contact manifolds, by constructing the appropriate topological K-theory cocycle for such operators. Its Chern character gives a cohomology class to which the Atiyah-Singer index formula can be applied. Such a K-cocycle has already been constructed by Boutet de Monvel for Toeplitz operators, and, more recently, by Melrose and Epstein for the class of Hermite operators. Our construction applies to general hypoelliptic pseudodifferential operators in the Heisenberg calculus. As in the Hermite I…

Moduli Spaces of Contact Instantons

We construct the moduli space of contact instantons, an analogue of Yang-Mills instantons defined for contact metric $5$-manifolds and initiate the study of their structure. In the $K$-contact case we give sufficient conditions for smoothness of the moduli space away from reducible connections and show the dimension is given by the index of an operator elliptic transverse to the Reeb foliation. The moduli spaces are shown to be Kähler when the $5$-manifold $M$ is Sasakian and hyperKähler when $M$ is transverse Calabi-Yau. We show how the transverse index can be computed in various cases, in pa…

The topology of a subspace of the Legendrian curves in a closed contact 3-manifold

In this paper we study a subspace of the space of Legendrian loops and we show that the injection of this space into the full loop space is an S1-equivariant homotopy equivalence. This space can be also seen as the space of zero Maslov index Legendrian loops and it shows up as a suitable space of variations in contact form geometry.

接触海藻代数的分类

Classification of contact seaweeds

本研究针对海藻代数的接触性分类开放问题，证明指数为1的海藻代数是接触的当且仅当它是拟可约的，结合已有拟可约海藻代数分类结果完成接触海藻代数的完整分类，且A型、C型中指数为1即可保证存在接触形式。

An index formula for the extended Heisenberg algebra of Epstein, Melrose and Mendoza

The extended Heisenberg algebra for a contact manifold has a symbolic calculus that accommodates both Heisenberg pseudodifferential operators as well as classical pseudodifferential operators. We derive here a formula for the index of Fredholm operators in this extended calculus. This formula incorporates in a single expression the Atiyah-Singer formula for elliptic operators, as well as Boutet de Monvel's Toeplitz index formula.