Topological wave functions and heat equations

Murat Günaydin, Andrew Neitzke, Boris Pioline

Research output: Contribution to journalReview article

30 Citations (Scopus)

Abstract

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 G/K, we show that the general solution of the anomaly equations is a matrix element 〈;Ψ|g|Ω〉 of the Schrödinger-Weil representation of a Heisenberg extension of G, between an arbitrary state 〈Ψ| and a particular vacuum state |Ω〉. 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 G′ in three dimensions, and on its relations to hypermultiplet couplings, nonabelian Donaldson-Thomas theory and black hole degeneracies.

Original languageEnglish (US)
Article number070
JournalJournal of High Energy Physics
Volume2006
Issue number12
DOIs
StatePublished - Dec 1 2006

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wave functions
anomalies
thermodynamics
string theory
partitions
strings
tubes
vacuum
matrices

All Science Journal Classification (ASJC) codes

  • Nuclear and High Energy Physics

Cite this

Günaydin, Murat ; Neitzke, Andrew ; Pioline, Boris. / Topological wave functions and heat equations. In: Journal of High Energy Physics. 2006 ; Vol. 2006, No. 12.
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Topological wave functions and heat equations. / Günaydin, Murat; Neitzke, Andrew; Pioline, Boris.

In: Journal of High Energy Physics, Vol. 2006, No. 12, 070, 01.12.2006.

Research output: Contribution to journalReview article

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AU - Neitzke, Andrew

AU - Pioline, Boris

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N2 - 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 G/K, we show that the general solution of the anomaly equations is a matrix element 〈;Ψ|g|Ω〉 of the Schrödinger-Weil representation of a Heisenberg extension of G, between an arbitrary state 〈Ψ| and a particular vacuum state |Ω〉. 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 G′ in three dimensions, and on its relations to hypermultiplet couplings, nonabelian Donaldson-Thomas theory and black hole degeneracies.

AB - 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 G/K, we show that the general solution of the anomaly equations is a matrix element 〈;Ψ|g|Ω〉 of the Schrödinger-Weil representation of a Heisenberg extension of G, between an arbitrary state 〈Ψ| and a particular vacuum state |Ω〉. 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 G′ in three dimensions, and on its relations to hypermultiplet couplings, nonabelian Donaldson-Thomas theory and black hole degeneracies.

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