As a continuation of [1], we study modular properties of the periods, the mirror maps and Yukawa couplings for multi-moduli Calabi-Yau varieties. In Part A of this paper, motivated by the recent work of Kachru-Vafa, we degenerate a three-moduli family of Calabi-Yau toric varieties along a codimension one subfamily which can be described by the vanishing of certain Mori coordinate, corresponding to going to the large volume limit in a certain direction. Then we see that the deformation space of the subfamily is the same as a certain family of K3 toric surfaces. This family can in turn be studied by further degeneration along a subfamily which in the end is described by a family of elliptic curves. The periods of the K3 family (and hence the original Calabi-Yau family) can be described by the squares of the periods of the elliptic curves. The consequences include: (1) proofs of various conjectural formulas of physicists [2

We consider a kinetic model of self-propelled particles with alignment interaction and with precession about the alignment direction. We derive a hydrodynamic system for the local density and velocity orientation of the particles. The system consists of the conservative equation for the local density and a non-conservative equation for the orientation. First, we assume that the alignment interaction is purely local and derive a first order system. However, we show that this system may lose its hyperbolicity. Under the assumption of weakly non-local interaction, we derive diffusive corrections to the first order system which lead to the combination of a heat flow of the harmonic map and Landau-Lifschitz-Gilbert dynamics. In the particular case of zero self-propelling speed, the resulting model reduces to the phenomenological Landau-Lifschitz-Gilbert equations. Therefore the present theory provides a kinetic formulation of classical micromagnetization models and spin dynamics.

We review first Azumaya geometry and D-branes in the realm of algebraic geometry along the line of Polchinski-Grothendieck Ansatz from our earlier work and then use it as background to introduce Azumaya C^{\infty} -manifolds with a fundamental module and morphisms therefrom to a projective complex manifold. This gives us a description of D-branes of A-type. Donaldson's picture of Lagrangian and special Lagrangian submanifolds as selected from the zero-locus of a moment map on a related space of maps can be merged into the setting. As a pedagogical toy model, we study D-branes of A-type in a Calabi-Yau torus. Simple as it is, it reveals several features of D-branes, including their assembling/disassembling. The 4th theme of Sec. 2.4, the 2nd theme of Sec. 4.2, and Sec. 4.3 are to be read respectively with Gmez-Sharpe (arXiv: hep-th/0008150), Donagi-Katz-Sharpe (arXiv: hep-th/0309270), and Denef (arXiv: hep-th/0107152). Some string-theoretical remarks are given at the end of each section.

Warped conformal field theory (WCFT) is a two dimensional quantum field theory whose local symmetry algebra consists of a Virasoro algebra and a U(1) Kac-Moody algebra. In this paper, we study correlation functions for primary operators in WCFT. Similar to conformal symmetry, warped conformal symmetry is very constraining. The form of the two and three point functions are determined by the global warped conformal symmetry while the four point functions can be determined up to an arbitrary function of the cross ratio. The warped conformal bootstrap equation are constructed by formulating the notion of crossing symmetry. In the large central charge limit, four point functions can be decomposed into global warped conformal blocks, which can be solved exactly. Furthermore, we revisit the scattering problem in warped AdS spacetime (WAdS), and give a prescription on how to match the bulk result to a WCFT retarded Green’s function. Our result is consistent with the conjectured holographic dualities between WCFT and WAdS.

In this Part II of D (11), we introduce new objects: super- C^ k -schemes and Azumaya super- C^ k -manifolds with a fundamental module (or, synonymously, matrix super- C^ k -manifolds with a fundamental module), and extend the study in D (11.1)([L-Y3], arXiv: 1406.0929 [math. DG]) to define the notion ofdifferentiable maps from an Azumaya/matrix supermanifold with a fundamental module to a real manifold or supermanifold'. This allows us to introduce the notion offermionic D-branes' in two different styles, one parallels Ramond-Neveu-Schwarz fermionic string and the other Green-Schwarz fermionic string. A more detailed discussion on the Higgs mechanism on dynamical D-branes in our setting, taking maps from the D-brane world-volume to the space-time in question and/or sections of the Chan-Paton bundle on the D-brane world-volume as Higgs fields, is also given for the first time in the D-project. Finally note that mathematically string theory begins with the notion of a differentiable map from a string world-sheet (a C^ k -manifold) to a target space-time (a real manifold). In comparison to this, D (11.1) and the current D (11.2) together bring us to the same starting point for studying D-branes in string theory as dynamical objects.