Superconformal Algebras and Mock Theta Functions

It is known that characters of BPS representations of extended superconformal algebras do not have good modular properties due to extra singular vectors coming from the BPS condition. In order to improve their modular properties we apply the method of Zwegers which has recently been developed to analyze modular properties of mock theta functions. We consider the case of N=4 superconformal algebra at general levels and obtain the decomposition of characters of BPS representations into a sum of simple Jacobi forms and an infinite series of non-BPS representations. We apply our method to study elliptic genera of hyper-Kahler manifolds in higher dimensions. In particular we determine the elliptic genera in the case of complex 4 dimensions of the Hilbert scheme of points on K3 surfaces K^{[2]} and complex tori A^{[[3]]}.Comment: 28 page

Surface Shubnikov-de Hass oscillations and non-zero Berry phases of the topological hole conduction in Tl1−x_{1-x}Bi1+x_{1+x}Se2_2

We report the observation of two-dimensional Shubnikov-de Hass (SdH) oscillations in the topological insulator Tl$_{1-x}$Bi$_{1+x}$Se$_2$. Hall effect measurements exhibited electron-hole inversion in samples with bulk insulating properties. The SdH oscillations accompanying the hole conduction yielded a large surface carrier density of $n_{\rm{s}}=5.1 \times10^{12}$/cm$^2$, with the Landau-level fan diagram exhibiting the $\pi$ Berry phase. These results showed the electron-hole reversibility around the in-gap Dirac point and the hole conduction on the surface Dirac cone without involving the bulk metallic conduction.Comment: 5 pages, 4 figure

N=4 Superconformal Algebra and the Entropy of HyperKahler Manifolds

We study the elliptic genera of hyperKahler manifolds using the representation theory of N=4 superconformal algebra. We consider the decomposition of the elliptic genera in terms of N=4 irreducible characters, and derive the rate of increase of the multiplicities of half-BPS representations making use of Rademacher expansion. Exponential increase of the multiplicity suggests that we can associate the notion of an entropy to the geometry of hyperKahler manifolds. In the case of symmetric products of K3 surfaces our entropy agrees with the black hole entropy of D5-D1 system.Comment: 25 pages, 1 figur

Towards A Topological G_2 String

We define new topological theories related to sigma models whose target space is a 7 dimensional manifold of G_2 holonomy. We show how to define the topological twist and identify the BRST operator and the physical states. Correlation functions at genus zero are computed and related to Hitchin's topological action for three-forms. We conjecture that one can extend this definition to all genus and construct a seven-dimensional topological string theory. In contrast to the four-dimensional case, it does not seem to compute terms in the low-energy effective action in three dimensions.Comment: 15 pages, To appear in the proceedings of Cargese 2004 summer schoo

Comments on geometric and universal open string tachyons near fivebranes

In a recent paper (hep-th/0703157), Sen studied unstable D-branes in NS5-branes backgrounds and argued that in the strong curvature regime the universal open string tachyon (on D-branes of the wrong dimensionality) and the geometric tachyon (on D-branes that are BPS in flat space but not in this background) may become equivalent. We study in this note an example of a non-BPS suspended D-brane vs. a BPS D-brane at equal distance between two fivebranes. We use boundary worldsheet CFT methods to show that these two unstable branes are identical.Comment: 8 pages, 1 figure; ver. 2 to appear in JHEP: one comment, refs and appendices adde

On the Genus Two Free Energies for Semisimple Frobenius Manifolds

We represent the genus two free energy of an arbitrary semisimple Frobenius manifold as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so-called "genus two G-function". Conjecturally the genus two G-function vanishes for a series of important examples of Frobenius manifolds associated with simple singularities as well as for ${\bf P}^1$-orbifolds with positive Euler characteristics. We explain the reasons for such Conjecture and prove it in certain particular cases.Comment: 37 pages, 3 figures, V2: the published versio

On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket

In our recent paper we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of PDE's associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin-Zhang, and the bracket is the first Poisson structure of their hierarchy. Our approach was based on a very involved computation of a deformation formula for the bracket with respect to the Givental-Y.-P. Lee Lie algebra action. In this paper, we discuss the structure of that deformation formula. In particular, we reprove it using a deformation formula for weak quasi-Miura transformation that relates our hierarchy of PDE's with its dispersionless limit.Comment: 21 page

Melting Crystal, Quantum Torus and Toda Hierarchy

Searching for the integrable structures of supersymmetric gauge theories and topological strings, we study melting crystal, which is known as random plane partition, from the viewpoint of integrable systems. We show that a series of partition functions of melting crystals gives rise to a tau function of the one-dimensional Toda hierarchy, where the models are defined by adding suitable potentials, endowed with a series of coupling constants, to the standard statistical weight. These potentials can be converted to a commutative sub-algebra of quantum torus Lie algebra. This perspective reveals a remarkable connection between random plane partition and quantum torus Lie algebra, and substantially enables to prove the statement. Based on the result, we briefly argue the integrable structures of five-dimensional $\mathcal{N}=1$ supersymmetric gauge theories and $A$-model topological strings. The aforementioned potentials correspond to gauge theory observables analogous to the Wilson loops, and thereby the partition functions are translated in the gauge theory to generating functions of their correlators. In topological strings, we particularly comment on a possibility of topology change caused by condensation of these observables, giving a simple example.Comment: Final version to be published in Commun. Math. Phys. . A new section is added and devoted to Conclusion and discussion, where, in particular, a possible relation with the generating function of the absolute Gromov-Witten invariants on CP^1 is commented. Two references are added. Typos are corrected. 32 pages. 4 figure

Non-Critical String Duals of N=1 Quiver Theories

We construct N=1 non-critical strings in four dimensions dual to strongly coupled N=1 quiver gauge theories in the Coulomb phase, generalizing the string duals of Argyres-Douglas points in N=2 gauge theories. They are the first examples of superstrings vacua with an exact worldsheet description dual to chiral N=1 theories. We identify the dual of the non-critical superstring using a brane setup describing the field theory in the classical limit. We analyze the spectrum of chiral operators in the strongly coupled regime and show how worldsheet instanton effects give non-perturbative information about the gauge theory. We also consider aspects of D-branes relevant for the holographic duality.Comment: JHEP style; 40 pages, 3 figures; v2: minor corrections, refs added, version to appear in JHE