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Conformal field theories with infinitely many conservation laws

### Abstract

Globally conformal invariant quantum field theories in a D-dimensional space-time (D even) have rational correlation functions and admit an infinite number of conserved (symmetric traceless) tensor currents. In a theory of a scalar field of dimension D-2 they were demonstrated to be generated by bilocal normal products of free massless scalar fields with an O(N), U(N), or Sp(2N) (global) gauge symmetry [B. Bakalov, N. M. Nikolov, K.-H. Rehren, and I. Todorov, “Unitary positive energy representations of scalar bilocal fields,” Commun. Math. Phys.271, 223–246 (Year: 2007)10.1007/s00220-006-0182-2; e-print arXiv:math-ph/0604069v3; B. Bakalov, N. M. Nikolov, K.-H. Rehren, and I. Todorov, “Infinite dimensional Lie algebras in 4D conformal quantum field theory,” J. Phys. A Math Theor.41, 194002 (Year: 2008)10.1088/1751-8113/41/19/194002; e-print arXiv:0711.0627v2 [hep-th]]. Recently, conformal field theories “with higher spin symmetry” were considered for D = 3 by Maldacena and Zhiboedov [“Constraining conformal field theories with higher spin symmetry,” e-print arXiv:1112.1016 [hep-th]] where a similar result was obtained (exploiting earlier study of CFT correlators). We suggest that the proper generalization of the notion of a 2D chiral algebra to arbitrary (even or odd) dimension is precisely a conformal field theory (CFT) with an infinite series of conserved currents. We recast and complement (part of) the argument of Maldacena and Zhiboedov into the framework of our earlier work. We extend to *D* = 4 the auxiliary Weyl-spinor formalism developed by Giombi et al. [“A note on CFT correlators in three dimensions,” e-print arXiv:1104.4317v3 [hep-th]] for *D* = 3. The free field construction only follows for *D* > 3 under additional assumptions about the operator product algebra. The problem of whether a rational CFT in 4D Minkowski space is necessarily trivial remains open.

© 2013 American Institute of Physics

Received 20 September 2012
Accepted 11 January 2013
Published online 19 February 2013

Acknowledgments:
I thank Yassen Stanev for collaboration, Karl-Henning Rehren and Raymond Stora for their interest and critical remarks, and Ali Tavanfar for inviting me to present a talk on this topic to the TH Journal Club (in February 2012). The hospitality and support of the Theory Division of CERN in Geneva and of IHES in Bures-sur-Yvette is gratefully acknowledged. The author's work has been supported in part by Grant No. DO 02-257 of the Bulgarian National Science Foundation.

Article outline:

I. INTRODUCTION
II. HUYGENS LOCALITY: TWIST *D* − 2 BILOCAL FIELDS
III. TWO AND THREE POINT SPIN-TENSOR INVARIANTS
IV. REDUCTION TO *D* = 3: INFINITE LIE ALGEBRAS
V. DISCUSSION

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