Conformal Partial Waves and the Operator Product Expansion

By solving the two variable differential equations which arise from finding the eigenfunctions for the Casimir operator for $O(d,2)$ succinct expressions are found for the functions, conformal partial waves, representing the contribution of an operator of arbitrary scale dimension $\Delta$ and spin $\ell$ together with its descendants to conformal four point functions for $d=4$, recovering old results, and also for $d=6$. The results are expressed in terms of ordinary hypergeometric functions of variables $x,z$ which are simply related to the usual conformal invariants. An expression for the conformal partial wave amplitude valid for any dimension is also found in terms of a sum over two variable symmetric Jack polynomials which is used to derive relations for the conformal partial waves.Comment: 17 pages, uses harvmac, v2 correction to eq. 2.2

Conjectures for Large N Superconformal N=4 Chiral Primary Four Point Functions

An expression for the four point function for half-BPS operators belonging to the [0,p,0] SU(4) representation in N=4 superconformal theories at strong coupling in the large N limit is suggested for any p. It is expressed in terms of the four point integrals defined by integration over AdS_5 and agrees with, and was motivated by, results for p=2,3,4 obtained via the AdS/CFT correspondence. Using crossing symmetry and unitarity, the detailed form is dictated by the requirement that at large N the contribution of long multiplets with twist less than 2p, which do not have anomalous dimensions, should cancel corresponding free field contributions.Comment: 50 pages, 1 figure, uses harvmac, version 2 extra reference, minor change

Conformal Partial Wave Expansions for N=4 Chiral Four Point Functions

The conformal partial wave analysis of four point functions of half BPS operators belonging to the SU(4) [0,p,0] representation is undertaken for p=2,3,4. Using the results of N=4 superconformal Ward identities the contributions from protected short and semi-short multiplets are identified in terms of the free field theory. In the large N limit contributions corresponding to long multiplets with twist up to 2p-2 are absent. The anomalous dimensions for twist two singlet multiplets are found to order g^4 and agree with other perturbative calculations. Results for twist four and six are also found.Comment: 53 pages, uses harvmac, includes 1 figure, version 2 some corrections and minor extensions, version 3 some further corrections, version 4 as to be publishe

Noncommutative vector bundles over fuzzy CP^N and their covariant derivatives

We generalise the construction of fuzzy CP^N in a manner that allows us to access all noncommutative equivariant complex vector bundles over this space. We give a simplified construction of polarization tensors on S^2 that generalizes to complex projective space, identify Laplacians and natural noncommutative covariant derivative operators that map between the modules that describe noncommuative sections. In the process we find a natural generalization of the Schwinger-Jordan construction to su(n) and identify composite oscillators that obey a Heisenberg algebra on an appropriate Fock space.Comment: 34 pages, v2 contains minor corrections to the published versio

Superconformal Symmetry, Correlation Functions and the Operator Product Expansion

Superconformal transformations are derived for the $\N=2,4 supermultiplets corresponding to the simplest chiral primary operators. These are applied to two, three and four point correlation functions. When$\N=4$, results are obtained for the three point function of various descendant operators, including the energy momentum tensor and SU(4) current. For both$\N=2$ or 4 superconformal identities are derived for the functions of the two conformal invariants appearing in the four point function for the chiral primary operator. These are solved in terms of a single arbitrary function of the two conformal invariants and one or three single variable functions. The results are applied to the operator product expansion using the exact formula for the contribution of an operator in the operator product expansion in four dimensions to a scalar four point function. Explicit expressions representing exactly the contribution of both long and possible short supermultiplets to the chiral primary four point function are obtained. These are applied to give the leading perturbative and large N corrections to the scale dimensions of long supermultiplets.Comment: 75 pages, plain TeX file using harvmac; revised version, minor corrections and extra referenc

Superconformal Indices for Orbifold Chern-Simons Theories

We calculate the superconformal indices of recently discovered three-dimensional N=4,5 Chern-Simons-matter theories and compare them with the corresponding indices of supergravity on AdS4 times orbifolds of S7. We find perfect agreement in the large N and large k limit, provided that the twisted sector contributions at the fixed loci of the orbifolds are properly taken into account. We also discuss the index for the so-called "dual ABJM" proposal.Comment: 27 pages, 1 figure; v2. reference added, minor correction

Conformal Four Point Functions and the Operator Product Expansion

Various aspects of the four point function for scalar fields in conformally invariant theories are analysed. This depends on an arbitrary function of two conformal invariants u,v. A recurrence relation for the function corresponding to the contribution of an arbitrary spin field in the operator product expansion to the four point function is derived. This is solved explicitly in two and four dimensions in terms of ordinary hypergeometric functions of variables z,x which are simply related to u,v. The operator product expansion analysis is applied to the explicit expressions for the four point function found for free scalar, fermion and vector field theories in four dimensions. The results for four point functions obtained by using the AdS/CFT correspondence are also analysed in terms of functions related to those appearing in the operator product discussion.Comment: 38 pages, plain TeX file using harvmac; revised version minor changes and an extra reference, revised version2, further correction

New methods in conformal partial wave analysis

We report on progress concerning the partial wave analysis of higher correlation functions in conformal quantum field theory.Comment: 16 page

Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories

The results of Romelsberger for a N=1 superconformal index counting protected operators, satisfying a BPS condition and which cannot be combined to form long multiplets, are analysed further. The index is expressible in terms of single particle superconformal characters for N=1 scalar and vector multiplets. For SQCD, involving SU(N_c) gauge groups and appropriate numbers of flavours N_f, the formula used to construct the index may be proved to give identical results for theories linked by Seiberg duality using recently proved theorems for q-series elliptic hypergeometric integrals. The discussion is also extended to Kutasov-Schwimmer dual theories in the large N_c, N_f limit and to dual theories with Sp(N) and SO(N) gauge groups. For the former, a transformation identity for elliptic hypergeometric integrals directly verifies that the index is the same for the electric and magnetic theories. For SO(N) theories the corresponding result may also be obtained from the same basic identity. An expansion of the index to several orders is also obtained in a form where the detailed protected operator content may be read off. Relevant mathematical results are reviewed.Comment: 47 pages, uses harvmac, v2. minor corrections, SO(N) cases proved, ref. added, v3. minor additions and correction