A Geometric Interpretation of the χy Genus on Hyper-Kähler Manifolds
نویسنده
چکیده
The group SL(2) acts on the space of cohomology groups of any hyper-Kähler manifold X . The χy genus of a hyper-Kähler X is shown to have a geometric interpretation as the super trace of an element of SL(2). As a by product one learns that the generalized Casson invariant for a mapping torus is essentially the χy genus. email: [email protected]
منابع مشابه
A path integral derivation of χy-genus
The formula for the Hirzebruch χy-genus of complex manifolds is a consequence of the Hirzebruch–Riemann–Roch formula. The classical index formulae for Todd genus, Euler number and signature correspond to the case when the complex variable y = 0,−1 and 1 respectively. Here we give a direct derivation of this nice formula based on supersymmetric quantum mechanics. PACS numbers: 12.60.Jv, 02.40.Ma...
متن کاملA Survey on Hyper-kähler with Torsion Geometry
Manifolds with special geometric structures play a prominent role in some branches of theoretical physics, such as string theory and supergravity. For instance, it is well known that supersymmetry requires target spaces to have certain special geometric properties. In many cases these requirements can be interpreted as restrictions on the holonomy group of the target space Riemannian metric. Ho...
متن کاملHIRZEBRUCH χy GENERA OF THE HILBERT SCHEMES OF SURFACES BY LOCALIZATION FORMULA
We use the Atiyah-Bott-Berline-Vergne localization formula to calculate the Hirzebruch χy genus χy(S), where S[n] is the Hilbert schemes of points of length n of a surface S. Combinatorial interpretation of the weights of the fixed points of the natural torus action on (C2)[n] is used.
متن کاملElliptic Genus of Calabi–yau Manifolds and Jacobi and Siegel Modular Forms
In the paper we study two types of relations: a one is between the elliptic genus of Calabi–Yau manifolds and Jacobi modular forms, another one is between the second quantized elliptic genus, Siegel modular forms and Lorentzian Kac–Moody Lie algebras. We also determine the structure of the graded ring of the weak Jacobi forms with integral Fourier coefficients. It gives us a number of applicati...
متن کاملQuaternionic Kähler Manifolds with Hermitian and Norden Metrics
Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kähler manifolds are considered. Some necessary and sufficient conditions the investigated manifolds be isotropic hyper-Kählerian and flat are found. It is proved that the quaternionic Kähler manifolds with the considered metric structure are Einstein for dimension at least 8. The c...
متن کامل