zbMATH — the first resource for mathematics

Conformal partial waves and the operator product expansion. (English) Zbl 1097.81735
Summary: 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.

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
33C90 Applications of hypergeometric functions
Full Text: DOI arXiv
[1] Ferrara, S.; Grillo, A.F.; Gatto, R.; Parisi, G.; Ferrara, S.; Grillo, A.F.; Gatto, R.; Parisi, G.; Ferrara, S.; Gatto, R.; Grillo, A.F., Properties of partial-wave amplitudes in conformal invariant theories, Nucl. phys. B, Nuovo cimento A, Nuovo cimento A, 26, 226, (1975)
[2] Dobrev, V.K.; Petkova, V.B.; Petrova, S.G.; Todorov, I.T., Dynamical derivation of vacuum operator product expansion in conformal field theory, Phys. rev. D, 13, 887, (1976)
[3] Lang, K.; Rühl, W., The critical O(N) sigma model at dimensions 2<D<4: fusion coefficients and anomalous dimensions, Nucl. phys. B, 400, 597, (1993) · Zbl 0941.81585
[4] Dolan, F.A.; Osborn, H., Conformal four point functions and the operator product expansion, Nucl. phys. B, 599, 459, (2001) · Zbl 1097.81734
[5] Dolan, F.A.; Osborn, H., Superconformal symmetry, correlation functions and the operator product expansion, Nucl. phys. B, 629, 3, (2002) · Zbl 1039.81551
[6] Arutyunov, G.; Sokatchev, E., Implications of superconformal symmetry for interacting (2,0) tensor multiplets, Nucl. phys. B, 635, 3, (2002) · Zbl 0996.81089
[7] Dirac, P.A.M., Wave equations in conformal space, (), 37, 823, (1936) · Zbl 0014.08004
[8] Ferrara, S.; Gatto, R.; Grillo, A.F., Conformal algebra in space – time and operator product expansion, Springer tracts in modern physics, Vol. 67, (1973), Springer Heidelberg
[9] Heslop, P.J.; Howe, P.S., Four-point functions in N=4 SYM, Jhep, 0301, 043, (2003) · Zbl 1226.81265
[10] Muirhead, R.J., Systems of partial differential equations for hypergeometric functions of matrix argument, Ann. math. statist., 41, 991, (1970) · Zbl 0225.62078
[11] Vilenkin, N.Ja.; Klimyk, A.U., Representation of Lie groups and special functions: recent advances, (1995), Kluwer Academic Publishers Boston · Zbl 0826.22001
[12] Jack, H., A class of symmetric polynomials with a parameter, Proc. R. soc. Edinburgh, 69, 1, (1970) · Zbl 0198.04606
[13] Gradshteyn, I.S.; Ryzhik, I.M., ()
[14] Koornwinder, T.; Sprinkhuizen-Kuyper, I., Generalized power series expansions for a class of orthogonal polynomials in two variables, SIAM J. math. anal., 9, 457, (1978) · Zbl 0395.33008
[15] M. Nirschl, H. Osborn, in preparation
[16] Vretare, L., Formulas for elementary spherical functions and generalized Jacobi polynomials, SIAM J. math. anal., 15, 805, (1984) · Zbl 0549.43006
[17] Desrosiers, P.; Lapointe, L.; Mathieu, P.; Desrosiers, P.; Lapointe, L.; Mathieu, P., Jack polynomials in superspace, Commun. math. phys., 233, 383, (2003), Commun. Math. Phys., in press · Zbl 1078.81033
[18] Andrews, G.E.; Askey, R.; Roy, R., Special functions, (1999), Cambridge Univ. Press Cambridge
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.