223 research outputs found
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 : in for various values of k and N. When k<N, the 1-loop beta function of
the sigma model on is negative and we expect the theory to have a mass
gap. However, the quantum cohomology relation
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 with N chiral fields with U(1) charge . 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 Supersymmetric Gauge Theories and Soliton Equations
Using recently proposed soliton equations we derive a basic identity for the
scaling violation of supersymmetric gauge theories . Here is the prepotential, 's are the
expectation values of the scalar fields in the vector multiplet, and is the coefficient of the one-loop
-function. This equation holds in the Coulomb branch of all
supersymmetric gauge theories coupled with massless matter.Comment: 10 pages, latex, no figures, a reference adde
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