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.

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

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.

The classification and construction of symmetry-protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that (generalized) group cohomology theory or cobordism theory gives rise to a complete classification of SPT phases in interacting boson or spin systems. The construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in three dimensions. In this work, we revisit this problem based on an equivalence class of fermionic symmetric local unitary transformations. We construct very general fixed-point SPT wave functions for interacting fermion systems. We naturally reproduce the partial classifications given by special group supercohomology theory, and we show that with an additional \tilde{B}H^2(G_b,\mathbb{Z}_2) structure [the so-called obstruction-free subgroup of H^2(G_b,\mathbb{Z}_2)], a complete classification of SPT phases for three-dimensional interacting fermion systems with a total symmetry group G_f = G_b × \mathbb{Z}^f_2 can be obtained for unitary symmetry group G_b. We also discuss the procedure for deriving a general group supercohomology theory in arbitrary dimensions.

In their proposed compactification of superstrings [4], Candelas, Horowitz, Strominger and Witten took the matric product of a maximal symmetric four dimensional spacetime M with a six dimensional CalabiYau vacua X as the ten dimensional spacetime; they identified the YangMills connection with the SU (3) connection of the CalabiYau metric and set the dilaton to be a constant. To make this theory compatible with the standard grand unified field theory, Witten [28] and HoravaWitten [20] proposed to use higher rank bundles for strong coupled heterotic string theory so that the gauge groups can be SU (4) or SU (5). Mathematically, this approach relies on UhlenbeckYaus theorem on constructing HermitianYangMills connections on stable bundles [27]. Many authors, including Friedman, Morgan and Witten [18]; Donagi, Ovrat, Pantev and Reinbacher [12]; Andreas [1], Kachru [21] and others, have worked