Abstract:
We investigate a class of (2,2) supersymmetric string vacua which may be represented as Landau--Ginzburg theories with a quasihomogeneous potential which has an isolated singularity at the origin. There are at least three thousand distinct models in this class. All vacua of this type lead to Euler numbers which lie in the range $-960 \leq \chi \leq 960$. The Euler characteristics do not pair up completely hence the space of Landau--Ginzburg ground states is not mirror symmetric even though it exhibits a high degree of symmetry. We discuss in some detail the relation between Landau--Ginzburg models and Calabi--Yau manifolds and describe a subtlety regarding Landau--Ginzburg potentials with an arbitrary number of fields. We also show that the use of topological identities makes it possible to relate Landau-Ginzburg theories to types of Calabi-Yau manifolds for which the usual Landau-Ginzburg framework does not apply.

Abstract:
Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 4-folds. The main technique is to find exact solutions to moving multiple cover integrals. The resulting invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds. We conjecture the 4-fold invariants to be integers and expect a sheaf theoretic explanation. Several local Calabi-Yau 4-folds are solved exactly. Compact cases, including the sextic Calabi-Yau in CP5, are also studied. A complete solution of the Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic anomaly equation.

Abstract:
We calculate the critical current density $J^J_c$ for c-axis Josephson tunneling between identical high temperature superconductors twisted an angle $\phi_0$ about the c-axis. We model the tunneling matrix element squared as a Gaussian in the change of wavevector q parallel to the junction, $<|t({\bf q})|^2>\propto\exp(-{\bf q}^2a^2/2\pi^2\sigma^2)$. The $J^J_c(\phi_0)/J^J_c(0)$ obtained for the s- and extended-s-wave order parameters (OP's) are consistent with the Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ data of Li {\it et al.}, but only for strongly incoherent tunneling, $\sigma^2\ge0.25$. A $d_{x^2-y^2}$-wave OP is always inconsistent with the data. In addition, we show that the apparent conventional sum rule violation observed by Basov et al. might be understandable in terms of incoherent c-axis tunneling, provided that the OP is not $d_{x^2-y^2}$-wave.

Abstract:
We calculate exactly the Josephson current for $c$-axis coherent tunneling between two layered superconductors, each with internal coherent tight-binding intra- and interlayer quasiparticle dispersions. Our results also apply when one or both of the superconductors is a bulk material, and include the usually neglected effects of surface states. For weak tunneling, our results reduce to our previous results derived using the tunneling Hamiltonian. Our results are also correct for strong tunneling. However, the $c$-axis tunneling results of Tanaka and Kashiwaya are shown to be incorrect in any limit. In addition, we consider the $c$-axis coherent critical current between two identical layered superconductors twisted an angle $\phi_0$ about the $c$-axis with respect to each other. Regardless of the order parameter symmetry, our coherent tunneling results using a tight-binding intralayer quasiparticle dispersion are inconsistent with the recent $c$-axis twist bicrystal Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ twist junction experiments of Li {\it et al.}

Abstract:
For a K3 surface S, we study motivic invariants of stable pairs moduli spaces associated to 3-fold thickenings of S. We conjecture suitable deformation and divisibility invariances for the Betti realization. Our conjectures, together with earlier calculations of Kawai-Yoshioka, imply a full determination of the theory in terms of the Hodge numbers of the Hilbert schemes of points of S. The work may be viewed as the third in a sequence of formulas starting with Yau-Zaslow and Katz-Klemm-Vafa (each recovering the former). Numerical data suggest the motivic invariants are linked to the Mathieu M_24 moonshine phenomena. The KKV formula and the Pairs/Noether-Lefschetz correspondence together determine the BPS counts of K3-fibered Calabi-Yau 3-folds in fiber classes in terms of modular forms. We propose a framework for a refined P/NL correspondence for the motivic invariants of K3-fibered CY 3-folds. For the STU model, a complete conjecture is presented.

Abstract:
The symmetry operations of the crystal groups relevant for the high temperature superconductors HgBa2CuO4+x (Hg1201), YBa2Cu3O7-x (YBCO), and Bi2Sr2CaCu2O8+x (BSCCO) are elucidated. The allowable combinations of the superconducting order parameter (OP) components are presented for both the angular momentum and lattice representations. For tetragonal Hg1201, the spin singlet OP components are composed from four sets of compatible basis functions, which combine to give the generalized s-, dx2-y2-, dxy-, and gxy(x2-y2)- wave OPs. In YBCO, elements of s- and dx2-y2- wave sets are compatible, but in BSCCO, elements of s- and dxy- wave sets are compatible. The Josephson critical current density JcJ across c-axis twist junctions in the vicinity of Tc is then evaluated as a function of the twist angle phi0, for each allowable OP combination, for both coherent and incoherent tunneling. Recent experiments of Li et al. demonstrated the independence of JcJ(phi0)/JcS upon phi0 at and below Tc, where JcS is the critical current density of a constituent single crystal. These experiments are shown to be consistent with an OP containing an s-wave component, but inconsistent with an OP containing the purported dx2-y2-wave component. In addition, they demonstrate that the interlayer tunneling across untwisted layers in single crystal BSCCO is entirely incoherent. We propose a new type of tricrystal experiment using single crystal c-axis twist junctions, that does not employ substrate grain boundaries.

Abstract:
Li et al. found that the critical current density JcJ across atomically clean c-axis twist junctions of Bi2Sr2CaCu2O(8+x) is the same as that of the constituent single crystal, JcS, independent of the twist angle phi0, even at Tc. We investigated theoretically if a dx2-y2-wave order parameter might twist by mixing in dxy components, but find that such twisting cannot possibly explain the data near to Tc. Hence, the order parameter contains an s-wave component, but not any dx2-y2-wave component. In addition, the c-axis Josephson tunneling is completely incoherent. We also propose a c-axis junction tricrystal experiment which does not rely upon expensive substrates.

Abstract:
We calculate Ic(T) and Rn for both coherent and incoherent electron tunneling across a c-axis break junction between nu=s,d_x^2-y^2-wave superconducting half spaces, each with c-axis bandwidth 2J. Coherent quasiparticle tunneling only occurs for voltages V<2J/e, leading to difficulties in measuring Rn for underdoped samples. The coherent part of Ic(0) is independent of Delta_nu(0) for J/Delta_nu(0)<<1, and can be large. Our results are discussed with regard to recent experiments.

Abstract:
The Yau-Zaslow conjecture determines the reduced genus 0 Gromov-Witten invariants of K3 surfaces in terms of the Dedekind eta function. Classical intersections of curves in the moduli of K3 surfaces with Noether-Lefschetz divisors are related to 3-fold Gromov-Witten theory via the K3 invariants. Results by Borcherds and Kudla-Millson determine the classical intersections in terms of vector-valued modular forms. Proven mirror transformations can often be used to calculate the 3-fold invariants which arise. Via a detailed study of the STU model (determining special curves in the moduli of K3 surfaces), we prove the Yau-Zaslow conjecture for all curve classes on K3 surfaces. Two modular form identities are required. The first, the Klemm-Lerche-Mayr identity relating hypergeometric series to modular forms after mirror transformation, is proven here. The second, the Harvey-Moore identity, is proven by D. Zagier and presented in the paper.

Abstract:
Thin V2O3 films were deposited on a piezoelectric substrate by electron-beam evaporation. Surface acoustic waves (SAW) were generated by interdigital-transducers (IDTs). The attenuation and sound velocity was investigated from 260K to 10K, providing an insight into the temperature dependent electrical, dielectrical and elastic properties of V2O3 at the metal to insulator transition.