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Via compactification on a circle, the matrix model of M-theory proposed by Banks et al suggests a concrete identification between the large N limit of two-dimensional N = 8 supersymmetric Yang-Mills theory and type IIA string theory. In this paper we collect evidence that supports this identification. We explicitly identify the perturbative string states and their interactions, and describe the appearance of D-particle and D-membrane states. * Here we work in string units α ′ = 1. A derivation of (1) from matrix theory and a discussion of our normalizations is given in the appendix.

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We investigate a dilaton gravity model in AdS 2 proposed by Almheiri and Polchinski [1] and develop a 1d effective description in terms of a dynamical boundary time with a Schwarzian derivative action. We show that the effective model is equivalent to a 1d version of Liouville theory, and investigate its dynamics and symmetries via a standard canonical framework. We include the coupling to arbitrary conformal matter and analyze the effective action in the presence of possible sources. We compute commutators of local operators at large time separation, and match the result with the time shift due to a gravitational shockwave interaction. We study a black hole evaporation process and comment on the role of entropy in this model.

Recently, a duality between Liouville theory and four dimensional N = 2 gauge theory has been uncovered by some of the authors. We consider the role of extended objects in gauge theory, surface operators and line operators, under this correspondence. We map such objects to specific operators in Liouville theory. We employ this connection to compute the expectation value of general supersymmetric 't Hooft-Wilson line operators in a variety of N = 2 gauge theories.

We present a microscopic index formula for the degeneracy of dyons in four-dimensional N = 4 string theory. This counting formula is manifestly symmetric under the duality group, and its asymptotic growth reproduces the macroscopic Bekenstein-Hawking entropy. We give a derivation of this result in terms of the type II five-brane compactified on K3, by assuming that its fluctuations are described by a closed string theory on its world-volume. We find that the degeneracies are given in terms of the denominator of a generalized super Kac-Moody algebra. We also discuss the correspondence of this result with the counting of D-brane states.where the 'contour' integral over σ is from 0 to 1 and η(σ) is the Dedekind η-function.The conjectured electric-magnetic SL(2, Z)-duality predicts that there should also exist a solitonic version of the heterotic string that carries pure magnetic charge q m ∈ Γ 22,6 , and thus that a similar formula counts the pure magnetically charged 1/2 BPS states. The generic dyonic states, however, preserve only 1/4 of the supersymmetries and are more * Here and in the subsequent we omit a factor of 16, and therefore count the number of 1/2 BPS multiplets rather than individual states.

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