Topological strings and large<i>N</i>phase transitions II: chiral expansion of<i>q</i>-deformed Yang-Mills theory

Caporaso, Nicola; Cirafici, Michele; Pasquetti, Sara; Seminara, Domenico; Szabo, Richard J.

doi:10.1088/1126-6708/2006/01/036

Cited by 32 publications

(44 citation statements)

References 51 publications

(138 reference statements)

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“…This expression is easily integrated to give the free energy 28) which as expected coincides with the genus 0 free-energy of closed topological string theory on the resolved conifold. The a-independent function c(t) can be easily determined by looking at the asymptotic expansion in t. In the limit t → 0, p → ∞ the free-energy of ordinary Yang-Mills theory in the weak-coupling phase is easily recovered.…”

Section: Resolved Conifold From the One-cut Solutionmentioning

confidence: 67%

“…These string theoretic issues will be discussed in full detail in a forthcoming paper [28]. Nevertheless, we can present some evidences for how the topological string expansion emerges which also resolves an apparent puzzle.…”

Section: Evidence For the String Picturementioning

confidence: 77%

“…Our strong-coupling solution which is based on a symmetric ansatz probably misses the non-trivial fluxes, because it only computes the Q = 0 sector of the U (1) component of the gauge group, but it should capture the correct Calabi-Yau geometry as well as the D-brane contributions. These aspects will be the subjects of a subsequent paper devoted to the comparison of the strong-coupling regime with the topological string amplitudes and the black hole partition function [28]. Here we concentrated our attention on the construction of the strong-coupling solution based on a two-cut ansatz.…”

Section: Discussionmentioning

confidence: 99%

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Topological strings and largeNphase transitions I: Nonchiral expansion ofq-deformed Yang-Mills theory

Caporaso

Cirafici

Pasquetti

et al. 2006

J. High Energy Phys.

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121

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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 YangMills 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.

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Section: Resolved Conifold From the One-cut Solutionmentioning

confidence: 67%

Section: Evidence For the String Picturementioning

confidence: 77%

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Topological strings and largeNphase transitions I: Nonchiral expansion ofq-deformed Yang-Mills theory

Caporaso

Cirafici

Pasquetti

et al. 2006

J. High Energy Phys.

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121

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“…. , N r and 12) where m i ∈ Z and i m i = m is the total Chern class. For fixed Chern class m, the absolute minimum value of the Yang-Mills functional …”

Section: The Nonabelian Localization Formulamentioning

confidence: 99%

“…3.2. As we discuss in detail below, the calculation is somewhat subtle and must be handled with care because, unlike the case of local P 1 studied in [12], the local elliptic threefold is not a toric manifold and so the standard topological vertex methods do not immediately apply. We will also describe, for completeness, an alternative derivation that follows the techniques of [26] where the topological string theory is interpreted as a two-dimensional topological quantum field theory.…”

Section: Topological String Theory On the Local Elliptic Threefoldmentioning

confidence: 99%

Topological strings, two-dimensional Yang-Mills theory and Chern-Simons theory on torus bundles

Caporaso¹,

Cirafici²,

Pasquetti³

et al. 2008

Advances in Theoretical and Mathematical Physics

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Abstract:We study the relations between two-dimensional Yang-Mills theory on the torus, topological string theory on a Calabi-Yau threefold whose local geometry is the sum of two line bundles over the torus, and Chern-Simons theory on torus bundles. The chiral partition function of the Yang-Mills gauge theory in the large N limit is shown to coincide with the topological string amplitude computed by topological vertex techniques. We use Yang-Mills theory as an efficient tool for the computation of Gromov-Witten invariants and derive explicitly their relation with Hurwitz numbers of the torus. We calculate the Gopakumar-Vafa invariants, whose integrality gives a non-trivial confirmation of the conjectured nonperturbative relation between two-dimensional Yang-Mills theory and topological string theory. We also demonstrate how the gauge theory leads to a simple combinatorial solution for the Donaldson-Thomas theory of the Calabi-Yau background. We match the instanton representation of Yang-Mills theory on the torus with the nonabelian localization of Chern-Simons gauge theory on torus bundles over the circle. We also comment on how these results can be applied to the computation of exact degeneracies of BPS black holes in the local Calabi-Yau background.

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Chiral expansion and Macdonald deformation of two-dimensional Yang-Mills theory

2016

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We derive the analog of the large N Gross-Taylor holomorphic string expansion for the refinement of q-deformed U(N) Yang-Mills theory on a compact oriented Riemann surface. The derivation combines Schur-Weyl duality for quantum groups with the Etingof-Kirillov theory of generalized quantum characters which are related to Macdonald polynomials. In the unrefined limit we reproduce the chiral expansion of q-deformed Yang-Mills theory derived by de Haro, Ramgoolam and Torrielli. In the classical limit q = 1, the expansion defines a new β-deformation of Hurwitz theory wherein the refined partition function is a generating function for certain parameterized Euler characters, which reduce in the unrefined limit β = 1 to the orbifold Euler characteristics of Hurwitz spaces of holomorphic maps. We discuss the geometrical meaning of our expansions in relation to quantum spectral curves and β-ensembles of matrix models arising in refined topological string theory.

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Topological strings and largeNphase transitions II: chiral expansion ofq-deformed Yang-Mills theory

Cited by 32 publications

References 51 publications

Topological strings and largeNphase transitions I: Nonchiral expansion ofq-deformed Yang-Mills theory

Topological strings and largeNphase transitions I: Nonchiral expansion ofq-deformed Yang-Mills theory

Topological strings, two-dimensional Yang-Mills theory and Chern-Simons theory on torus bundles

Chiral expansion and Macdonald deformation of two-dimensional Yang-Mills theory

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