Index Theorems on Torsional Geometries

Abstract

We study various topological invariants on a torsional geometry in the presence of a totally anti-symmetric torsion H under the closed condition dH = 0, which appears in string theory compactification scenarios. By using the identification between the Clifford algebra on the geometry and the canonical quantization condition of fermions in quantum mechanics, we construct N=1 quantum mechanical sigma model in the Hamiltonian formalism. We extend this model to N=2 system, equipped with the totally anti-symmetric tensor associated with the torsion on the target space geometry. Next we construct transition elements in the Lagrangian path integral formalism and apply them to the analyses of the Witten indices in supersymmetric systems. We explicitly show the formulation of the Dirac index on the torsional manifold which has already been studied. We also formulate the Euler characteristic and the Hirzebruch signature on the torsional manifold.

45 pages, the version to appear in JHEP

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• Chiral Generations on Intersecting 5-branes in Heterotic String Theory
• The Holst Action by the Spectral Action Principle
• Chiral Zero Modes on Intersecting Heterotic 5-branes
• Landau Levels in graphene in the presence of emergent gravity
• Comments on HKT supersymmetric sigma models and their Hamiltonian reduction
• AdS Vacua, Attractor Mechanism and Generalized Geometries
• On Nieh-Yan Transport
• N=4 mechanics with diverse (4,4,0) multiplets: explicit examples of HKT, CKT, and OKT geometries
• Bi-HKT and bi-Kaehler supersymmetric sigma models
• Solitons and Yukawa Couplings in Nearly Kahler Flux Compactifications
• Moduli in General $SU(3)$-Structure Heterotic Compactifications