Twisted elliptic genus for K3 and Borcherds product

We further discuss the relation between the elliptic genus of K3 surface and the Mathieu group M24. We find that some of the twisted elliptic genera for K3 surface, defined for conjugacy classes of the Mathieu group M24, can be represented in a very simple manner in terms of the eta-product of the corresponding conjugacy classes. It is shown that our formula is a consequence of the identity between the Borcherds product and additive lift of some Siegel modular forms.Comment: 17 page

Generalization of Calabi-Yau/Landau-Ginzburg correspondence

We discuss a possible generalization of the Calabi-Yau/Landau-Ginzburg correspondence to a more general class of manifolds. Specifically we consider the Fermat type hypersurfaces $M_N^k$: $\sum_{i=1}^N X_i^k =0$ in ${\bf CP}^{N-1}$ for various values of k and N. When k<N, the 1-loop beta function of the sigma model on $M_N^k$ is negative and we expect the theory to have a mass gap. However, the quantum cohomology relation $\sigma^{N-1}={const.}\sigma^{k-1}$ suggests that in addition to the massive vacua there exists a remaining massless sector in the theory if k>2. We assume that this massless sector is described by a Landau-Ginzburg (LG) theory of central charge $c=3N(1-2/k)$ with N chiral fields with U(1) charge $1/k$. We compute the topological invariants (elliptic genera) using LG theory and massive vacua and compare them with the geometrical data. We find that the results agree if and only if k=even and N=even. These are the cases when the hypersurfaces have a spin structure. Thus we find an evidence for the geometry/LG correspondence in the case of spin manifolds.Comment: 19 pages, Late

Seiberg-Witten Curve for E-String Theory Revisited

We discuss various properties of the Seiberg-Witten curve for the E-string theory which we have obtained recently in hep-th/0203025. Seiberg-Witten curve for the E-string describes the low-energy dynamics of a six-dimensional (1,0) SUSY theory when compactified on R^4 x T^2. It has a manifest affine E_8 global symmetry with modulus \tau and E_8 Wilson line parameters {m_i},i=1,2,...,8 which are associated with the geometry of the rational elliptic surface. When the radii R_5,R_6 of the torus T^2 degenerate R_5,R_6 --> 0, E-string curve is reduced to the known Seiberg-Witten curves of four- and five-dimensional gauge theories. In this paper we first study the geometry of rational elliptic surface and identify the geometrical significance of the Wilson line parameters. By fine tuning these parameters we also study degenerations of our curve corresponding to various unbroken symmetry groups. We also find a new way of reduction to four-dimensional theories without taking a degenerate limit of T^2 so that the SL(2,Z) symmetry is left intact. By setting some of the Wilson line parameters to special values we obtain the four-dimensional SU(2) Seiberg-Witten theory with 4 flavors and also a curve by Donagi and Witten describing the dynamics of a perturbed N=4 theory.Comment: 35 pages, 2 figures, LaTeX2

Prepotentials of N=2N=2 Supersymmetric Gauge Theories and Soliton Equations

Using recently proposed soliton equations we derive a basic identity for the scaling violation of $N=2$ supersymmetric gauge theories $\sum_i a_i\partial F/\partial a_i-2F=8 \pi i b_1 u$. Here $F$ is the prepotential, $a_i$'s are the expectation values of the scalar fields in the vector multiplet, $u=1/2\, {\rm Tr}\, \langle\phi^2\rangle$ and $b_1$ is the coefficient of the one-loop $\beta$-function. This equation holds in the Coulomb branch of all $N=2$ supersymmetric gauge theories coupled with massless matter.Comment: 10 pages, latex, no figures, a reference adde