Abstract

We consider the long-standing problem of obtaining the 3-point functions of Toda CFT. Our main tools are topological strings and the AGT-W relation between gauge theories and 2D CFTs. In [1] we computed the partition function of 5D T N theories on S 4 × S 1 and suggested that they should be interpreted as the three-point structure constants of q-deformed Toda. In this paper, we provide the exact AGT-W dictionary for this relation and rewrite the 5D T N partition function in a form that makes taking the 4D limit possible. Thus, we obtain a prescription for the computation of the partition function of the 4D T N theories on S 4, or equivalently the undeformed 3-point Toda structure constants. Our formula, has the correct symmetry properties, the zeros that it should and, for N = 2, gives the known answer for Liouville CFT.

Highlights

  • In [1] we computed the partition function of 5D TN theories on S4×S1 and suggested that they should be interpreted as the three-point structure constants of q-deformed Toda

  • We provide the exact AGT-W dictionary for this relation and rewrite the 5D TN partition function in a form that makes taking the 4D limit possible

  • The conformal blocks of the 2D CFTs are given by the appropriate instanton partition functions, while the three point structure constants should be obtained by the S4 partition functions of the TN superconformal theories

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Summary

The AGT dictionary

The main goal of this section is to provide the dictionary needed to relate the topological string amplitudes of section 4 to the Toda CFT correlation functions of section 3. When we need to relate the topological string partition functions to the Toda CFT correlators, the Ω background parameters need to be specialized as ǫ1 = b, ǫ2 = b−1,. We claim that the exact AGT-W dictionary relates the Weyl-invariant structure constants C to the 4D TN partition function on S4 (ZNS4) as. Where again the constant parts can only depend on N and of the Omega deformation parameters but cannot be functions of the parameters that define the theory, i.e. the masses. The above equation gives the complete relationship between the Toda 3-point structure constants and the partition functions of the TN theories

Toda 3-point functions
Review The Lagrangian of the Toda CFT theory is given by
Enhanced symmetry of the Weyl invariant part
Pole structure of the Weyl invariant part
The q-deformed Toda field theory
The TN partition function from topological strings
The 5-brane webs
The topological vertex computation
The 4D limit
Liouville from topological strings
W3 from topological strings
Conclusions and outlook
A Parametrization of the TN junction
The Υ function
The q-deformed Υ function
The finite product functions
D Computation of the TN partition function

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