### Abstract

We examine the problem of counting bound states of BPS black holes on local Calabi-Yau threefolds which are fibrations over a Riemann surface by computing the partition function of q-deformed Yang-Mills theory on the Riemann surface. We study in detail the genus zero case and obtain, at finite N, the instanton expansion of the gauge theory. It can be written exactly as the partition function for U(N) Chern-Simons gauge theory on a Lens space, summed over all non-trivial vacua, plus a tower of non-perturbative instanton contributions. The correspondence between two and three dimensional gauge theories is elucidated by an explicit mapping between two-dimensional Yang-Mills instantons and flat connections on the Lens space. In the large N limit we find a peculiar phase structure in the model. At weak string coupling the theory reduces exactly to the trivial flat connection sector with instanton contributions exponentially suppressed, and the topological string partition function on the resolved conifold is reproduced in this regime. At a certain critical point all non-trivial vacua contribute, instantons are enhanced and the theory appears to undergo a phase transition into a strong coupling regime. We rederive these results by performing a saddle-point approximation to the exact partition function. We obtain a q-deformed version of the Douglas-Kazakov equation for two-dimensional Yang-Mills theory on the sphere, whose one-cut solution below the transition point reproduces the resolved conifold geometry. Above the critical point we propose a two-cut solution that should reproduce the chiral-antichiral dynamics found for black holes on the Calabi-Yau threefold and the Gross-Taylor string in the undeformed limit. The transition from the strong coupling phase to the weak coupling phase appears to be of third order. © SISSA 2006.

Original language | English |
---|---|

Pages (from-to) | 797-863 |

Number of pages | 67 |

Journal | Journal of High Energy Physics |

Issue number | 1 |

DOIs | |

Publication status | Published - 2006 |

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### Keywords

- Brane Dynamics in Gauge Theories
- Field Theories in Lower Dimensions
- Nonperturbative Effects
- Topological Strings

### Cite this

*Journal of High Energy Physics*, (1), 797-863. https://doi.org/10.1088/1126-6708/2006/01/035

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*Journal of High Energy Physics*, no. 1, pp. 797-863. https://doi.org/10.1088/1126-6708/2006/01/035

**Topological strings and large N phase transitions I : Nonchiral expansion of q-deformed Yang-Mills theory.** / Caporaso, Nicola; Cirafici, Michele; Griguolo, Luca; Pasquetti, Sara; Seminara, Domenico; Szabo, Richard J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Topological strings and large N phase transitions I

T2 - Nonchiral expansion of q-deformed Yang-Mills theory

AU - Caporaso, Nicola

AU - Cirafici, Michele

AU - Griguolo, Luca

AU - Pasquetti, Sara

AU - Seminara, Domenico

AU - Szabo, Richard J.

PY - 2006

Y1 - 2006

N2 - We examine the problem of counting bound states of BPS black holes on local Calabi-Yau threefolds which are fibrations over a Riemann surface by computing the partition function of q-deformed Yang-Mills theory on the Riemann surface. We study in detail the genus zero case and obtain, at finite N, the instanton expansion of the gauge theory. It can be written exactly as the partition function for U(N) Chern-Simons gauge theory on a Lens space, summed over all non-trivial vacua, plus a tower of non-perturbative instanton contributions. The correspondence between two and three dimensional gauge theories is elucidated by an explicit mapping between two-dimensional Yang-Mills instantons and flat connections on the Lens space. In the large N limit we find a peculiar phase structure in the model. At weak string coupling the theory reduces exactly to the trivial flat connection sector with instanton contributions exponentially suppressed, and the topological string partition function on the resolved conifold is reproduced in this regime. At a certain critical point all non-trivial vacua contribute, instantons are enhanced and the theory appears to undergo a phase transition into a strong coupling regime. We rederive these results by performing a saddle-point approximation to the exact partition function. We obtain a q-deformed version of the Douglas-Kazakov equation for two-dimensional Yang-Mills theory on the sphere, whose one-cut solution below the transition point reproduces the resolved conifold geometry. Above the critical point we propose a two-cut solution that should reproduce the chiral-antichiral dynamics found for black holes on the Calabi-Yau threefold and the Gross-Taylor string in the undeformed limit. The transition from the strong coupling phase to the weak coupling phase appears to be of third order. © SISSA 2006.

AB - We examine the problem of counting bound states of BPS black holes on local Calabi-Yau threefolds which are fibrations over a Riemann surface by computing the partition function of q-deformed Yang-Mills theory on the Riemann surface. We study in detail the genus zero case and obtain, at finite N, the instanton expansion of the gauge theory. It can be written exactly as the partition function for U(N) Chern-Simons gauge theory on a Lens space, summed over all non-trivial vacua, plus a tower of non-perturbative instanton contributions. The correspondence between two and three dimensional gauge theories is elucidated by an explicit mapping between two-dimensional Yang-Mills instantons and flat connections on the Lens space. In the large N limit we find a peculiar phase structure in the model. At weak string coupling the theory reduces exactly to the trivial flat connection sector with instanton contributions exponentially suppressed, and the topological string partition function on the resolved conifold is reproduced in this regime. At a certain critical point all non-trivial vacua contribute, instantons are enhanced and the theory appears to undergo a phase transition into a strong coupling regime. We rederive these results by performing a saddle-point approximation to the exact partition function. We obtain a q-deformed version of the Douglas-Kazakov equation for two-dimensional Yang-Mills theory on the sphere, whose one-cut solution below the transition point reproduces the resolved conifold geometry. Above the critical point we propose a two-cut solution that should reproduce the chiral-antichiral dynamics found for black holes on the Calabi-Yau threefold and the Gross-Taylor string in the undeformed limit. The transition from the strong coupling phase to the weak coupling phase appears to be of third order. © SISSA 2006.

KW - Brane Dynamics in Gauge Theories

KW - Field Theories in Lower Dimensions

KW - Nonperturbative Effects

KW - Topological Strings

UR - http://www.scopus.com/inward/record.url?scp=33645517443&partnerID=8YFLogxK

U2 - 10.1088/1126-6708/2006/01/035

DO - 10.1088/1126-6708/2006/01/035

M3 - Article

SP - 797

EP - 863

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

SN - 1029-8479

IS - 1

ER -