Elliptic genera and real Jacobi forms

We construct real Jacobi forms with matrix index using path integrals. The path integral expressions represent elliptic genera of two-dimensional N=(2,2) supersymmetric theories. They arise in a family labeled by two integers N and k which determine the central charge of the infrared fixed point through the formula c=3N(1+ 2N/k). We decompose the real Jacobi form into a mock modular form and a term arising from the continuous spectrum of the conformal field theory. We argue that the Jacobi form represents the elliptic genus of a theory defined on a 2N dimensional background with U(N) isometry, containing a complex projective space section, a circle fiber and a linear dilaton direction. We also present formulas for the elliptic genera of orbifolds of these models.Comment: 32 page

Localization and real Jacobi forms

We calculate the elliptic genus of two dimensional abelian gauged linear sigma models with (2,2) supersymmetry using supersymmetric localization. The matter sector contains charged chiral multiplets as well as Stueckelberg fields coupled to the vector multiplets. These models include theories that flow in the infrared to non-linear sigma models with target spaces that are non-compact Kahler manifolds with U(N) isometry and with an asymptotically linear dilaton direction. The elliptic genera are the modular completions of mock Jacobi forms that have been proposed recently using complementary arguments. We also compute the elliptic genera of models that contain multiple Stueckelberg fields from first principles.Comment: 19+1 pages, LaTeX. Minor correctio

Counting Strings, Wound and Bound

We analyze zero mode counting problems for Dirac operators that find their origin in string theory backgrounds. A first class of quantum mechanical models for which we compute the number of ground states arises from a string winding an isometric direction in a geometry, taking into account its energy due to tension. Alternatively, the models arise from deforming marginal bound states of a string winding a circle, and moving in an orthogonal geometry. After deformation, the number of bound states is again counted by the zero modes of a Dirac operator. We count these bound states in even dimensional asymptotically linear dilaton backgrounds as well as in Euclidean Taub-NUT. We show multiple pole behavior in the fugacities keeping track of a U(1) charge. We also discuss a second class of counting problems that arises when these backgrounds are deformed via the application of a heterotic duality transformation. We discuss applications of our results to Appell-Lerch sums and the counting of domain wall bound states.Comment: 38 page

Elliptic Genera of Non-compact Gepner Models and Mirror Symmetry

We consider tensor products of N=2 minimal models and non-compact conformal field theories with N=2 superconformal symmetry, and their orbifolds. The elliptic genera of these models give rise to a large and interesting class of real Jacobi forms. The tensor product of conformal field theories leads to a natural product on the space of completed mock modular forms. We exhibit families of non-compact mirror pairs of orbifold models with c=9 and show explicitly the equality of elliptic genera, including contributions from the long multiplet sector. The Liouville and cigar deformed elliptic genera transform into each other under the mirror transformation.Comment: 29 page

Long Strings and Quasinormal Winding Modes

We compute the path integral for a particle on the covering group of SL(2,R) using a decomposition of the Lie algebra into adjoint orbits. We thus intuitively derive the Hilbert space of the particle on the group including discrete and continuous representations. Next, we perform a Lorentzian hyperbolic orbifold of the partition function and relate it to the Euclidean BTZ partition function. We use the particle model to inform further discussion of the spectral content of the one loop vacuum amplitude for strings on BTZ black hole backgrounds. We argue that the poles in the loop integrand code contributions of long string modes that wind the black hole. We moreover identify saddle point contributions of quasinormal winding modes.Comment: 31 page

Batalin-Vilkovisky gauge-fixing of a chiral two-form in six dimensions

We perform the gauge-fixing of the theory of a chiral two-form boson in six dimensions starting from the action given by Pasti, Sorokin and Tonin. We use the Batalin-Vilkovisky formalism, introducing antifields and writing down an extended action satisfying the classical master equation. Then we gauge-fix the three local symmetries of the extended action in two different ways.Comment: 15 pages, latex, no figures, version accepted by Class. Quant. Gra