Xiaonan Ma

Toeplitz Operators on Symplectic Manifolds

We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for non-compact manifolds and orbifolds are also established.

BerezinToeplitz quantization on Khler manifolds

Abstract We study BerezinToeplitz quantization on Khler manifolds. We explain first how to compute various associated asymptotic expansions, then we compute explicitly the first terms of the expansion of the kernel of the BerezinToeplitz operators, and of the composition of two BerezinToeplitz operators. As an application, we estimate the norm of Donaldsons Q-operator.

The spin

Abstract. We study the asymptotic of the spectrum of the

^{\bf c}

{\rm spin^c}

Dirac operator on high tensor powers of a line bundle

Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin–Uribe [13, Theorem 2], which was originally proved by using the analysis of Toeplitz operators of Boutet de Monvel and Guillemin [10].

ON FAMILY RIGIDITY THEOREMS, I

Note that Ind(P ) is a virtual G-representation. Let chg(Ind(P )) with g ∈G be the equivariant Chern character of Ind(P ) evaluated at g. In this paper, we first prove a family fixed-point formula that expresses chg(Ind(P )) in terms of the geometric data on the fixed points X of the fiber of π . Then by applying this formula, we generalize the Witten rigidity theorems and several vanishing theorems proved in [Liu3] for elliptic genera to the family case. Let G = S1. A family elliptic operator P is called rigid on the equivariant Chern character level with respect to this S1-action, if chg(Ind(P )) ∈H ∗(B) is independent of g ∈ S1. When the base B is a point, we recover the classical rigidity and vanishing theorems. When B is a manifold, we get many nontrivial higher-order rigidity and vanishing theorems by taking the coefficients of certain expansion of chg . For the history of the Witten rigidity theorems, we refer the reader to [T], [BT], [K], [L2], [H], [Liu1], and [Liu4]. The family vanishing theorems that generalize those vanishing theorems in [Liu3], which in turn give us many higher-order vanishing theorems in the family case. In a forthcoming paper, we extend our results to general loop group representations and prove much more general family vanishing theorems that generalize the results in [Liu3]. We believe there should be some applications of our results to topology and geometry, which we hope to report on a later occasion. This paper is organized as follows. In Section 1, we prove the equivariant family index theorem. In Section 2, we prove the family rigidity theorem. In the last part of Section 2, motivated by the family rigidity theorem, we state a conjecture. In Section 3, we generalize the family rigidity theorem to the nonzero anomaly case. As corollaries, we derive several vanishing theorems.

Functoriality of real analytic torsion forms

In this paper, we prove the functoriality of the analytic torsion forms of Bismut and Lott [BLo] with respect to the composition of two submersions.

Geometric Quantization on Kähler and Symplectic Manifolds

We explain various results on the asymptotic expansion of the Bergman kernel on Kahler manifolds and also on symplectic manifolds. We also review the “quantization commutes with reduction” phenomenon for a compact Lie group action, and its relation to the Bergman kernel. Mathematics Subject Classification (2010). Primary 53D; Secondary 58J, 32A.

Quantum Hall Effect and Quillen Metric

We study the generating functional, the adiabatic curvature and the adiabatic phase for the integer quantum Hall effect (QHE) on a compact Riemann surface. For the generating functional we derive its asymptotic expansion for the large flux of the magnetic field, i.e., for the large degree k of the positive Hermitian line bundle Lk. The expansion consists of the anomalous and exact terms. The anomalous terms are the leading terms of the expansion. This part is responsible for the quantization of the adiabatic transport coefficients in QHE. We then identify the non-local (anomalous) part of the expansion with the Quillen metric on the determinant line bundle, and the subleading exact part with the asymptotics of the regularized spectral determinant of the Laplacian for the line bundle Lk, at large k. Finally, we show how the generating functional of the integer QHE is related to the gauge and gravitational (2+1)d Chern–Simons functionals. We observe the relation between the Bismut-Gillet-Soulé curvature formula for the Quillen metric and the adiabatic curvature for the electromagnetic and geometric adiabatic transport of the integer Quantum Hall state. We then obtain the geometric part of the adiabatic phase in QHE, given by the Chern–Simons functional.

Exponential Estimate for the asymptotics of Bergman kernels

We prove an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. Further, we give Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line bundles.