[Paper Review] Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems

Robbert Dijkgraaf, Cumrun Vafa|ArXiv.org|Sep 13, 2009

Black Holes and Theoretical Physics45 references212 citations

TL;DR

This paper establishes a string-theoretic derivation of the AGT correspondence for $\mathcal{N}=2$ $SU(n)$ gauge theories by showing that the topological string partition function on $A_{n-1}$ singularities over Riemann surfaces is captured by chiral blocks of $A_{n-1}$ Toda conformal field theory. Using large $N$ dualities and matrix model holography, it identifies the Seiberg-Witten curve as the spectral curve of Penner-like matrix models, with the Nekrasov deformation mapping to a $\beta$-ensemble and general background charge in Toda theory.

ABSTRACT

We consider the topological string partition function, including the Nekrasov deformation, for type IIB geometries with an A_{n-1} singularity over a Riemann surface. These models realize the N=2 SU(n) superconformal gauge systems recently studied by Gaiotto and collaborators. Employing large N dualities we show why the partition function of topological strings in these backgrounds is captured by the chiral blocks of A_{n-1} Toda systems and derive the dictionary recently proposed by Alday, Gaiotto and Tachikawa. For the case of genus zero Riemann surfaces, we show how these systems can also be realized by Penner-like matrix models with logarithmic potentials. The Seiberg-Witten curve can be understood as the spectral curve of these matrix models which arises holographically at large N. In this context the Nekrasov deformation maps to the beta-ensemble of generalized matrix models, that in turn maps to the Toda system with general background charge. We also point out the notion of a double holography for this system, when both n and N are large.

Motivation & Objective

• To explain the AGT correspondence for $\mathcal{N}=2$ $SU(n)$ gauge theories using topological string theory and large $N$ dualities.
• To show that the topological string partition function on $A_{n-1}$-singular geometries computes Toda chiral blocks.
• To realize the Seiberg-Witten geometry holographically via the spectral curve of Penner-like matrix models at large $N$.
• To map the Nekrasov deformation to a $\beta$-ensemble and relate it to general background charge in Toda theory.
• To explore double holography in the limit of large $n$ and large $N$.

Proposed method

• Employ large $N$ dualities between topological strings and gauge theories on branes to relate closed string amplitudes to open string invariants.
• Use the B-model topological string on mirror Calabi-Yau geometries to compute the prepotential and higher-genus amplitudes.
• Map the $\mathcal{N}=2$ gauge theory to a system of $N$ M5 branes wrapping a Riemann surface, with the geometry encoding the Seiberg-Witten curve.
• Realize the topological string partition function via Penner-like matrix models with logarithmic potentials for genus-zero Riemann surfaces.
• Identify the spectral curve of the matrix model as the Seiberg-Witten curve in the large $N$ limit.
• Map the Nekrasov deformation $\epsilon_1, \epsilon_2$ to the $\beta$-ensemble parameter and the background charge $Q$ in Toda CFT.

Experimental results

Research questions

• RQ1 How does the topological string partition function on $A_{n-1}$-singular Calabi-Yau geometries relate to chiral blocks of $A_{n-1}$ Toda conformal field theory?
• RQ2 What is the role of the Seiberg-Witten curve in the large $N$ limit of matrix models dual to topological strings?
• RQ3 How does the Nekrasov deformation emerge from the $\beta$-ensemble formulation of matrix models?
• RQ4 Can the AGT correspondence be derived from a geometric transition in topological string theory?
• RQ5 What is the nature of double holography when both $n$ (rank of gauge group) and $N$ (number of branes) are large?

Key findings

• The topological string partition function on $A_{n-1}$-singular geometries with a Riemann surface base is equivalent to the chiral block of $A_{n-1}$ Toda CFT.
• For genus-zero Riemann surfaces, the partition function is captured by a Penner-like matrix model with logarithmic potential, whose spectral curve is the Seiberg-Witten curve.
• The Nekrasov deformation $\epsilon_1, \epsilon_2$ maps to the $\beta$-ensemble parameter in the matrix model, which in turn corresponds to the background charge $Q$ in Toda theory.
• The genus-zero free energy $\mathcal{F}_0$ is determined by the spectral curve, while higher-genus amplitudes $\mathcal{F}_g$ receive odd-integer powers of $g_s$ due to the non-invariance of the Toda stress tensor under $\phi \to -\phi$.
• The system exhibits double holography: one from the large $N$ limit of the matrix model, and another from the large $n$ limit of the $A_{n-1}$ singularity, corresponding to $A_\infty$ quiver matrix models.
• The connection extends to correlation functions of vertex operators, which are expressed as $\langle \prod_i V_m(q_i) \rangle_N$ and computed in the large $N$ limit via the matrix model.