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
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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 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 -