We find some integrability conditions for low-dimensional manifolds to admit metrics with nonnegative scalar curvature. In particular, we solve the positive action conjecture in general relativity in the affirmative.

In this paper, we construct the bilinear identities for the wave functions of an extended B-type Kadomtsev-Petviashvili (BKP) hierarchy, which contains two types of (2+1)-dimensional Sawada-Kotera equation with a self-consistent source (2d-SKwS-I and 2d-SKwS-II). By introducing an auxiliary variable corresponding to the extended flow for the BKP hierarchy, we find the tau-function and the bilinear identities for this extended BKP hierarchy. The bilinear identities can generate all the Hirota's bilinear equations for the zero-curvature forms of this extended BKP hierarchy. As examples, the Hirota's bilinear equations for the two types of 2d-SKwS (both 2d-SKwS-I and 2d-SKwS-II) will be given explicitly.

Classical two-dimensional Liouville gravity is often considered in conformal gauge which has a residual left and right Virasoro symmetry algebra. We consider an alternate, chiral, gauge which has a residual right Virasoro Kac-Moody algebra, and no left Virasoro algebra. The Kac-Moody zero mode is the left-moving energy. Dirac brackets of the constrained Hamiltonian theory are derived, and the residual symmetries are shown to be generated by integrals of the conserved chiral currents. The central charge and Kac-Moody level are computed. The possible existence of a corresponding quantum theory is discussed.

In this continuation of [L-Y1],[LLSY],[L-Y2], and [L-Y3](arXiv: 0709.1515 [math. AG], arXiv: 0809.2121 [math. AG], arXiv: 0901.0342 [math. AG], arXiv: 0907.0268 [math. AG]), we study D-branes in a target-space with a fixed B -field background B along the line of the Polchinski-Grothendieck Ansatz, explained in [L-Y1] and further extended in the current work. We focus first on the gauge-field-twist effect of B -field to the Chan-Paton module on D-branes. Basic properties of the moduli space of D-branes, as morphisms from Azumaya schemes with a twisted fundamental module to B , are given. For holomorphic D-strings, we prove a valuation-criterion property of this moduli space. The setting is then extended to take into account also the deformation-quantization-type noncommutative geometry effect of B -field to both the D-brane world-volume and the superstring target-space (-time) B . This brings the notion of twisted B -modules that are realizable as twisted locally-free coherent modules with a flat connection into the study. We use this to realize the notion of both the classical and the quantum spectral covers as morphisms from Azumaya schemes with a fundamental module (with a flat connection in the latter case) in a very special situation. The 3rd theme (subtitled" Sharp vs. Polchinski-Grothendieck") of Sec. 2.2 is to be read with the work [Sh3](arXiv: hep-th/0102197) of Sharp while Sec. 5.2 (subtitled less appropriately" Dijkgraaf-Holland-Sukowski-Vafa vs. Polchinski-Grothendieck") is to be read with the related sections in [DHSV](arXiv: 0709.4446 [hep-th]) and [DHS](arXiv: 0810.4157 [hep-th]) of Dijkgraaf, Hollands, Sukowski, and Vafa.

Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.