Eric D'Hoker

Graviton exchange and complete four point functions in the AdS / CFT correspondence

Abstract 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 four-point amplitudes of dilaton and axion fields in IIB supergravity on AdS 5 × S 5 . By the AdS/CFT correspondence, we obtain the four-point functions of the marginal operators Tr( F 2 +…) and Tr (F F +…) in N =4 , d =4 SU ( N ) SYM at large N , large g YM 2 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 stress-energy tensor in a double OPE expansion. Logarithmic singularities occur in the complete four-point functions at subleading orders.

Field theory tests for correlators in the AdS-CFT correspondence

The order g{sup 2} radiative corrections to all two- and three-point correlators of the composite primary operators TrthinspX{sup k} are computed in N=4 supersymmetric Yang-Mills theory with the gauge group SU(N). Corrections are found to vanish for all N. For k=2 this is a consequence of known superconformal nonrenormalization theorems, and for general k the result confirms an N{r_arrow}{infinity}, fixed large g{sup 2}N supergravity calculation and further conjectures by S. Lee {ital et al.}, hep-th/9806074. A three-point correlator involving N=4 descendents of TrthinspX{sup 2} is calculated, and its order g{sup 2} contribution also vanishes, giving evidence for the absence of radiative corrections in correlators of descendent operators. {copyright} {ital 1999} {ital The American Physical Society}

Exact half-BPS Type IIB interface solutions. I. Local solution and supersymmetric Janus

The complete Type IIB supergravity solutions with 16 supersymmetries are obtained on the manifold AdS4 ? S2 ? S2 ? ? with SO(2, 3) ? SO(3) ? SO(3) symmetry in terms of two holomorphic functions on a Riemann surface ?, 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 AdS5 ? S5 and supersymmetric Janus doped with D5 and/or NS5 branes, is deferred to a companion paper [1].

Two-loop superstrings.: I. Main formulas☆

Abstract An unambiguous and slice-independent formula for the two-loop superstring measure on moduli space for even spin structure is constructed from first principles. The construction uses the super-period matrix as moduli invariant under worldsheet supersymmetry. This produces new subtle contributions to the gauge-fixing process, which eliminate all the ambiguities plaguing earlier gauge-fixed formulas. The superstring measure can be computed explicitly and a simple expression in terms of modular forms is obtained. For fixed spin structure, the measure exhibits the expected behavior under degenerations of the surface. The measure allows for a unique modular covariant GSO projection. Under this GSO projection, the cosmological constant, the 1-, 2- and 3-point functions of massless supergravitons all vanish pointwise on moduli space without the appearance of boundary terms. A certain disconnected part of the 4-point function is shown to be given by a convergent, finite integral on moduli space. A general slice-independent formula is given for the two-loop cosmological constant in compactifications with central charge c =15 and N =1 worldsheet supersymmetry in terms of the data of the compactification conformal field theory. In this Letter, a summary of the above results is presented with detailed constructions, derivations and proofs to be provided in a series of subsequent publications.

Two-loop superstrings IV: The cosmological constant and modular forms☆

Abstract The slice-independent gauge-fixed superstring chiral measure in genus 2 derived in the earlier papers of this series for each spin structure is evaluated explicitly in terms of theta-constants. The slice-independence allows an arbitrary choice of superghost insertion points q 1 , q 2 in the explicit evaluation, and the most effective one turns out to be the split gauge defined by S δ ( q 1 , q 2 )=0. This results in expressions involving bilinear theta-constants M . The final formula in terms of only theta-constants follows from new identities between M and theta-constants which may be interesting in their own right. The action of the modular group Sp(4, Z ) is worked out explicitly for the contribution of each spin structure to the superstring chiral measure. It is found that there is a unique choice of relative phases which insures the modular invariance of the full chiral superstring measure, and hence a unique way of implementing the GSO projection for even spin structure. The resulting cosmological constant vanishes, not by a Riemann identity, but rather by the genus 2 identity expressing any modular form of weight 8 as the square of a modular form of weight 4. The degeneration limits for the contribution of each spin structure are determined, and the divergences, before the GSO projection, are found to be the ones expected on physical grounds.

Two-loop superstrings II: The chiral measure on moduli space☆

Abstract A detailed derivation from first principles is given for the unambiguous and slice-independent formula for the two-loop superstring chiral measure which was announced in the first paper of this series. Supergeometries are projected onto their superperiod matrices, and the integration over odd supermoduli is performed by integrating over the fibers of this projection. The subtleties associated with this procedure are identified. They require the inclusion of some new finite-dimensional Jacobian superdeterminants, a deformation of the worldsheet correlation functions using the stress tensor, and perhaps paradoxically, another additional gauge choice, “slice μ choice”, whose independence also has to be established. This is done using an important correspondence between superholomorphic notions with respect to a supergeometry and holomorphic notions with respect to its superperiod matrix. Altogether, the subtleties produce precisely the corrective terms which restore the independence of the resulting gauge-fixed formula under infinitesimal changes of gauge-slice. This independence is a key criterion for any gauge-fixed formula and hence is verified in detail.

Exact half-BPS Type IIB interface solutions II: Flux solutions and multi-Janus

Regularity and topology conditions are imposed on the exact Type IIB solutions on AdS4 × S2 × S2 × Σ with 16 supersymmetries, which were derived in a companion paper [1]. We construct an infinite class of regular solutions with varying dilaton, and non-zero 3-form fluxes. Our solutions may be viewed as the fully back-reacted geometries of AdS5 × S5 (or more generally, Janus) doped with D5 and/or NS5 branes. The solutions are parametrized by the choice of an arbitrary genus g hyper-elliptic Riemann surface Σ with boundary, all of whose branch points are restricted to lie on a line. For genus 0, the Janus solution with 16 supersymmetries and 6 real parameters is recovered; its topology coincides with that of AdS5 × S5. The genus g ≥ 1 solutions are parametrized by a total of 4g + 6 real numbers, 2g−1 of which are the real moduli of Σ. The solutions have 2g + 2 asymptotic AdS5 × S5 regions, g three-spheres with RR 3-form charge, and another g with NSNS 3-form charge. Collapse of consecutive branch points of Σ yields singularities which correspond to D5 and NS5 branes in the probe limit. It is argued that the AdS/CFT dual gauge theory to each of our solutions consists of a 2+1-dimensional planar interface on which terminate 2g + 2 half-Minkowski 3+1-dimensional space-time = 4 super-Yang-Mills theories. Generally, the = 4 theory in each Minkowski half-space-time may have an independent value of the gauge coupling, and the interface may support various operators, whose interface couplings are further free parameters of the dual gauge theory.

The operator product expansion of N=4 SYM and the 4-point functions of supergravity

We give a detailed Operator Product Expansion interpretation of the results for conformal 4-point functions computed from supergravity through the AdS/CFT duality. We show that for an arbitrary scalar exchange in AdSd+1 all the power-singular terms in the direct channel limit exactly match the corresponding contributions to the OPE of the operator dual to the exchanged bulk field and of its conformal descendents. The leading logarithmic singularities in the 4-point functions of protected N=4 super-Yang–Mills operators (computed from IIB supergravity on AdS5×S5) are interpreted as O(1/N2) renormalization effects of the double-trace products appearing in the OPE. Applied to the 4-point functions of the operators Oφ∼(trF2+⋯) and OC∼(trFF+⋯), this analysis leads to the prediction that the double-trace composites [OφOC] and [OφOφ−OCOC] have anomalous dimension −16/N2 in the large N, large gYM2N limit. We describe a geometric picture of the OPE in the dual gravitational theory, for both the power-singular terms and the leading logarithms. We comment on several possible extensions of our results.

AdS / CFT four point functions: How to succeed at z integrals without really trying

Abstract A new method is discussed which vastly simplifies one of the two integrals over AdS d +1 required to compute exchange graphs for four-point functions of scalars in the AdS/CFT correspondence. The explicit form of the bulk-to-bulk propagator is not required. Previous results for scalar, gauge boson and graviton exchange are reproduced, and new results are given for massive vectors. It is found that precisely for the cases that occur in the AdS 5 × S 5 compactification of Type IIB supergravity, the exchange diagrams reduce to a finite sum of graphs with quartic scalar vertices. The analogous integrals in n -point scalar diagrams for n >4 are also evaluated.

Gravity duals of half-BPS Wilson loops

We explicitly construct the fully back-reacted half-BPS solutions in Type IIB supergravity which are dual to Wilson loops with 16 supersymmetries in = 4 super Yang-Mills. In a first part, we use the methods of a companion paper to derive the exact general solution of the half-BPS equations on the space AdS2 × S2 × S4 × Σ, with isometry group SO(2, 1) × SO(3) × SO(5) in terms of two locally harmonic functions on a Riemann surface Σ with boundary. These solutions, generally, have varying dilaton and axion, and non-vanishing 3-form fluxes. In a second part, we impose regularity and topology conditions. These non-singular solutions may be parametrized by a genus g ≥ 0 hyperelliptic surface Σ, all of whose branch points lie on the real line. Each genus g solution has only a single asymptotic AdS5 × S5 region, but exhibits g homology 3-spheres, and an extra g homology 5-spheres, carrying respectively RR 3-form and RR 5-form charges. For genus 0, we recover AdS5 × S5 with 3 free parameters, while for genus g ≥ 1, the solution has 2g+5 free parameters. The genus 1 case is studied in detail. Numerical analysis is used to show that the solutions are regular throughout the g = 1 parameter space. Collapse of a branch cut on Σ subtending either a homology 3-sphere or a homology 5-sphere is non-singular and yields the genus g−1 solution. This behavior is precisely expected of a proper dual to a Wilson loop in gauge theory.