One of the main purposes of this paper is to understand the geometry of the moduli and the Teichmller spaces of Riemann surfaces. The most interesting results we have in this paper are the detailed understanding of two new complete Khler metrics with nice properties and the KhlerEinstein metric on the Teichmller and the moduli spaces of Riemann surfaces. The two new metrics, the Ricci metric and the perturbed Ricci metric, are naturally defined as the negative Ricci curvature of the WeilPetersson metric and a combination of it with the WeilPetersson metric. We prove that these new metrics and the KhlerEinstein metric on the Teichmller and moduli spaces all have Poincar type boundary behavior, and further, in [7], we prove that they all have bounded geometry. Note that the KhlerEinstein metric is the key link between the differential geometric and algebraic geometric aspects of these spaces

Mirror principle is a general method developed in [LLY1]-[LLY4] to compute characteristic classes and characteristic numbers on moduli spaces of stable maps in terms of hypergeometric type series. The counting of the numbers of curves in Calabi-Yau manifolds from mirror symmetry corresponds to the computation of Euler numbers. This principle computes quite general Hirzebruch multiplicative classes such as the total Chern classes.

The BMS (Bondi-van der Burg-Metzner-Sachs) symmetry arises as the asymptotic symmetry of flat spacetime at null infinity. In particular, the BMS algebra for three dimensional flat spacetime (BMS3) is generated by the super-rotation generators which form a Virasoro sub-algebra with central charge cL, together with mutually-commuting super-translation generators. The super-rotation and super-translation generators have non-trivial commutation relations with another central charge cM. In this paper, we study a free scalar theory in two dimensions exhibiting BMS3 symmetry, which can also be understood as the ultra-relativistic limit of a free scalar CFT2. Upon canonical quantization on the highest weight vacuum, the central charges are found to be cL=2 and cM=0. Because of the vanishing central charge cM=0, the theory features novel properties: there exist primary states which form a multiplet, and the Hilbert space can be organized by an enlarged version of BMS modules dubbed the staggered modules. We further calculate correlation functions and the torus partition function, the later of which is also shown explicitly to be modular invariant.

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We compute the partition function on the hemisphere of a class of two-dimensional (2,2) supersymmetric field theories including gauged linear sigma models. The result provides a general exact formula for the central charge of the D-brane placed at the boundary. It takes the form of Mellin-Barnes integral and the question of its convergence leads to the grade restriction rule concerning branes near the phase boundaries. We find expressions in various phases including the large volume formula in which a characteristic class called the Gamma class shows up. The two sphere partition function factorizes into two hemispheres glued by inverse to the annulus. The result can also be written in a form familiar in mirror symmetry, and suggests a way to find explicit mirror correspondence between branes.