2 research outputs found
Bergman Kernel from Path Integral
We rederive the expansion of the Bergman kernel on Kahler manifolds developed
by Tian, Yau, Zelditch, Lu and Catlin, using path integral and perturbation
theory, and generalize it to supersymmetric quantum mechanics. One physics
interpretation of this result is as an expansion of the projector of wave
functions on the lowest Landau level, in the special case that the magnetic
field is proportional to the Kahler form. This is relevant for the quantum Hall
effect in curved space, and for its higher dimensional generalizations. Other
applications include the theory of coherent states, the study of balanced
metrics, noncommutative field theory, and a conjecture on metrics in black hole
backgrounds. We give a short overview of these various topics. From a
conceptual point of view, this expansion is noteworthy as it is a geometric
expansion, somewhat similar to the DeWitt-Seeley-Gilkey et al short time
expansion for the heat kernel, but in this case describing the long time limit,
without depending on supersymmetry.Comment: 27 page
Topological wave functions and heat equations
It is generally known that the holomorphic anomaly equations in topological
string theory reflect the quantum mechanical nature of the topological string
partition function. We present two new results which make this assertion more
precise: (i) we give a new, purely holomorphic version of the holomorphic
anomaly equations, clarifying their relation to the heat equation satisfied by
the Jacobi theta series; (ii) in cases where the moduli space is a Hermitian
symmetric tube domain , we show that the general solution of the anomaly
equations is a matrix element \IP{\Psi | g | \Omega} of the
Schr\"odinger-Weil representation of a Heisenberg extension of , between an
arbitrary state and a particular vacuum state .
Based on these results, we speculate on the existence of a one-parameter
generalization of the usual topological amplitude, which in symmetric cases
transforms in the smallest unitary representation of the duality group in
three dimensions, and on its relations to hypermultiplet couplings, nonabelian
Donaldson-Thomas theory and black hole degeneracies.Comment: 50 pages; v2: small typos fixed, references added; v3: cosmetic
changes, published version; v4: typos fixed, small clarification adde