In these lecture notes we first assemble the basic ingredients of supersymmetric gauge theories (particularly N=4 super-Yang-Mills theory), supergravity, and superstring theory. Brane solutions are surveyed. The geometry and symmetries of anti-de Sitter space are discussed. The AdS/CFT correspondence of Maldacena and its application to correlation functions in the the conformal phase of N=4 SYM are explained in considerable detail. A pedagogical treatment of holographic RG flows is given including a review of the conformal anomaly in four-dimensional quantum field theory and its calculation from five-dimensional gravity. Problem sets and exercises await the reader.

The graviton exchange diagram for the correlation function of arbitrary scalar operators is evaluated in anti-de Sitter space, AdS d+1 . This enables us to complete the computation of the 4-point amplitudes of dilaton and axion fields in IIB supergravity on AdS 5 × S 5 . By the AdS/CFT correspondence, we obtain the 4-point functions of the marginal operators Tr(F 2 + . . .) and Tr(FF + . . .) in N = 4, d = 4 SU (N ) SYM at large N , large g 2 Y M N . The short distance asymptotics of the amplitudes are studied. We find that in the direct channel the leading power singularity agrees with the expected contribution of the stressenergy tensor in a double OPE expansion. Logarithmic singularities occur in the complete 4-point functions at subleading orders. containing the contribution of various primary operators O p and their descendents ▽ k O p in the intermediate state. For simplicity we have assumed that these are scalars, but vector and tensor operators contribute in a similar way, each with a characteristic tensor structure. (For primaryRecognizing in the supergravity 4-point results a structure of the form (1.1) should allow to determine the operator content of the theory and its OPE structure in the large N, large λ limit. Preliminary computations [11,12] have indicated that the supergravity diagrams contain the expected contributions to (1.1) of chiral primary operators and their superconformal descendents.It is however clear that these contributions alone do not reproduce the supergravity result [9]. A natural expectation is that appropriately defined normal-ordered products of chiral primaries and descendents also contribute to the OPE and form the full operator content of the theory in this limit. This set of operators has a dual interpretation in terms of multi-particle Kaluza-Klein states in supergravity. Massive string states are expected to decouple in this limit 1 . The computation of a complete realistic 4-point correlator presented here should allow to put these ideas to test.An interesting issue raised in [9] is the presence in the 4-point supergravity amplitudes of logarithms of the coordinate separation between two points in the limit when the points come close. Logarithmic singularities appear to be a generic feature of all the AdS processes studied so far [10,12,13], and we find the same situation for the graviton exchange. The question then is whether the logarithms cancel when the various contributions to a realistic correlator are assembled. If not, we should ask whether the logarithms can still be incorporated in the OPE framework. Here we find that logarithmic singularities do indeed occur in the complete 4-point functions.As pointed out by Witten [23], logarithms can generically arise in the perturbative expansion of a CFT 4-point correlator as renormalization effects like mixings and corrections to the 1 Group-theoretic aspects of multi-particle and string states have been considered in [28].

The complete Type IIB supergravity solutions with 16 supersymmetries are obtained on the manifold AdS 4 × S 2 × S 2 × Σ with SO(2, 3) × SO(3) × SO(3) symmetry in terms of two holomorphic functions on a Riemann surface Σ, which generally has a boundary. This is achieved by reducing the BPS equations using the above symmetry requirements, proving that all solutions of the BPS equations solve the full Type IIB supergravity field equations, mapping the BPS equations onto a new integrable system akin to the Liouville and Sine-Gordon theories, and mapping this integrable system to a linear equation which can be solved exactly. Amongst the infinite class of solutions, a non-singular Janus solution is identified which provides the AdS/CFT dual of the maximally supersymmetric Yang-Mills interface theory discovered recently. The construction of general classes of globally non-singular solutions, including fully back-reacted AdS 5 × S 5 and supersymmetric Janus doped with D5 and/or NS5 branes, is deferred to a companion paper [1]. 2 The bulk Lagrangian may be put in a more standard from by scaling the scalar fields asφ i → g −2φi at the cost of introducing interface operators of the form (∂ π g) 2φiφj .6

This paper is devoted to recent progress made towards the understanding of closed bosonic and fermionic string perturbation theory, formulated in a Lorentz-covariant way on Euclidean space-time. Special emphasis is put on the fundamental role of Riemann surfaces and supersurfaces. The differential and complex geometry of their moduli space is developed as needed. New results for the superstring presented here include the supergeometric construction of amplitudes, their chiral and superholomorphic splitting and a global formulation of supermoduli space and amplitudes.

Abstract:The analytic contribution to the low energy expansion of Type II string amplitudes at genus-one is a power series in space-time derivatives with coefficients that are determined by integrals of modular functions over the complex structure modulus of the world-sheet torus. These modular functions are associated with world-sheet vacuum Feynman diagrams and given by multiple sums over the discrete momenta on the torus. In this paper we exhibit exact differential and algebraic relations for a certain infinite class of such modular functions by showing that they satisfy Laplace eigenvalue equations with inhomogeneous terms that are polynomial in non-holomorphic Eisenstein series. Furthermore, we argue that the set of modular functions that contribute to the coefficients of interactions up to order D 10 R 4 are linear sums of functions in this class and quadratic polynomials in Eisenstein series and odd Riemann zeta values. Integration over the complex structure results in coefficients of the low energy expansion that are rational numbers multiplying monomials in odd Riemann zeta values.