Title: M2-branes matrix models, topological string and q-difference relations
Abstract:
In this talk we study a class of finite (~N) dimensional integrals (M2 matrix models) which calculates the partition function of N stack of M2-branes in M-theory. These objects play important roles in understanding M-theory since their behavior in the large N limit and the 1/N correction captures the quantum effects in M-theory through the AdS/CFT correspondence.
Through the rewriting called Fermi gas formalism, a large class of M2 matrix models is related to the spectral determinant of the quantum operator associated with some 1d algebraic curves. Interestingly, it turned out that the large N expansion of these M2 matrix models are given by the free energy of refined topological string on the toric Calabi-Yau threefold associated with the algebraic curve. The topological string free energy on toric Calabi-Yau threefold is related to the instanton partition function of 5d super Yang-Mills theory living on the dual toric diagram, which is known to satisfy non-linear relations called Nakajima-Yoshioka blowup relation. From this it is suggested that the generating function of the M2 matrix model with respect to N satisfy bilinear q-difference relations such as q-Painleve equations, which can be confirmed with the exact values of the M2 matrix model at finite N. On one hand, this gives various q-uplifed generalizations of the Fredholm determinant solutions to the Painleve equations studied by McCoy,Tracy,Wu and Tracy,Widom through the M2 matrix models. On the other hand, the q-difference relations may provide a new tool to study the large N expansion of M2 matrix models.