### 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 language | English (US) |
---|---|

Article number | 070 |

Journal | Journal of High Energy Physics |

Volume | 2006 |

Issue number | 12 |

DOIs | |

State | Published - Dec 1 2006 |

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### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics

### Cite this

*Journal of High Energy Physics*,

*2006*(12), [070]. https://doi.org/10.1088/1126-6708/2006/12/070

}

*Journal of High Energy Physics*, vol. 2006, no. 12, 070. https://doi.org/10.1088/1126-6708/2006/12/070

**Topological wave functions and heat equations.** / Günaydin, Murat; Neitzke, Andrew; Pioline, Boris.

Research output: Contribution to journal › Review article

TY - JOUR

T1 - Topological wave functions and heat equations

AU - Günaydin, Murat

AU - Neitzke, Andrew

AU - Pioline, Boris

PY - 2006/12/1

Y1 - 2006/12/1

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.

UR - http://www.scopus.com/inward/record.url?scp=33846177579&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846177579&partnerID=8YFLogxK

U2 - 10.1088/1126-6708/2006/12/070

DO - 10.1088/1126-6708/2006/12/070

M3 - Review article

AN - SCOPUS:33846177579

VL - 2006

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

SN - 1126-6708

IS - 12

M1 - 070

ER -