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Durham Research Online Durham Research Online Deposited in DRO: 21 February 2017 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: Beem, Christopher and Lemos, Madalena and Rastelli, Leonardo and van Rees, Balt C. (2016) 'The (2, 0) superconformal bootstrap.', Physical review D., 93 (2). 025016. Further information on publisher's website: https://doi.org/10.1103/PhysRevD.93.025016 Publisher's copyright statement: This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Additional information: Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk PHYSICAL REVIEW D 93, 025016 (2016) The (2, 0) superconformal bootstrap Christopher Beem,1 Madalena Lemos,2 Leonardo Rastelli,2 and Balt C. van Rees3 1Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540, USA 2C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, New York 11794-3840, USA 3Theory Group, Physics Department, CERN, CH-1211 Geneva 23, Switzerland (Received 5 October 2015; published 21 January 2016) We develop the conformal bootstrap program for six-dimensional conformal field theories with (2, 0) supersymmetry, focusing on the universal four-point function of stress tensor multiplets. We review the solution of the superconformal Ward identities and describe the superconformal block decomposition of this correlator. We apply numerical bootstrap techniques to derive bounds on operator product expansion (OPE) coefficients and scaling dimensions from the constraints of crossing symmetry and unitarity. We also derive analytic results for the large spin spectrum using the light cone expansion of the crossing equation. Our principal result is strong evidence that the A1 theory realizes the minimal allowed central charge (c ¼ 25) for any interacting (2, 0) theory. This implies that the full stress tensor four-point function of the A1 theory is the unique unitary solution to the crossing symmetry equation at c ¼ 25. For this theory, we estimate the scaling dimensions of the lightest unprotected operators appearing in the stress tensor operator product expansion. We also find rigorous upper bounds for dimensions and OPE coefficients for a general interacting (2, 0) theory of central charge c. For large c, our bounds appear to be saturated by the holographic predictions obtained from eleven-dimensional supergravity. DOI: 10.1103/PhysRevD.93.025016 I. INTRODUCTION AND SUMMARY algebra for which there can exist a stress tensor multiplet is the (2, 0) algebra ospð8Ãj4Þ [7]. Thus the six-dimensional In this work we introduce and develop the modern (2, 0) theories are singled out as the maximally supersym- conformal bootstrap program for (2, 0) superconformal metric local conformal field theories (CFTs) in the maximum theories in six dimensions. These theories provide a number of dimensions. powerful organizing principle for lower-dimensional super- While it is easy to identify a free-field theory that realizes symmetric dynamics. From their existence one can infer a (2, 0) superconformal symmetry—namely the Abelian vast landscape of supersymmetric field theories in various tensor model—the existence of interacting (2, 0) theories dimensions and rationalize many deep, nonperturbative was only inferred in the mid 1990s on the basis of string- dualities that act within this landscape (see, e.g., [1–5]). theory constructions [8,9]. A decoupling limit of type IIB Despite their increasingly central role, the (2, 0) theories string theory on asymptotically locally Euclidean spaces have proved stubbornly resistant to study by traditional predicts the existence of a list of interacting (2, 0) theories quantum field theory techniques. This situation, coupled labeled by the simple and simply laced Lie algebras (A ≥1, with the high degree of symmetry and conjectured unique- n D ≥4, E6, E7, E8) [8]. The A −1 model can also be realized ness of these theories, suggests that the (2, 0) theories may n N as the low-energy limit of the world volume theory of N be a prime target for the conformal bootstrap approach. coincident M5 branes in M theory [9]. Two important properties of these theories can be deduced almost immedi- A. (2, 0) theories ately from their string/M-theory constructions: It has been known since the work of Nahm [6] that (i) On R6, the (2, 0) theory of type g has a moduli space superconformal algebras can only be defined for spacetime of vacua given by the orbifold dimension less than or equal to six. In six dimensions, the 5 M ≔ R rg possible superconformal algebras are the ðN ; 0Þ algebras g ð Þ =Wg; ð1:1Þ ospð8⋆j2N Þ. The maximum six-dimensional superconformal where rg is the rank of the simply laced Lie algebra g and Wg its Weyl group. At a generic point in the moduli space, the long distance physics is described Published by the American Physical Society under the terms of by a collection of rg decoupled free tensor multiplets. the Creative Commons Attribution 3.0 License. Further distri- 5 1 bution of this work must maintain attribution to the author(s) and (ii) Upon supersymmetric compactification on R × S , the published article’s title, journal citation, and DOI. the (2, 0) theory of type g reduces at low energies to a 2470-0010=2016=93(2)=025016(37) 025016-1 Published by the American Physical Society CHRISTOPHER BEEM et al. PHYSICAL REVIEW D 93, 025016 (2016) maximally supersymmetric five-dimensional Yang- one is instructed to send k → ∞. This moduli space is Mills theory with gauge algebra g, whose dimensionful singular due to small instanton singularities. A specific 1 2 ∼ gauge coupling is controlled by the S radius, gYM R. regularization procedure has been advocated in [28], but its The string/M-theory constructions give physically compel- conceptual status seems somewhat unclear, and one may be ling evidence for the existence of the interacting (2, 0) concerned that important features of the theory might be theories, but the evidence is indirect and requires an hidden in the details of the UV regulator. The DLCQ extensive conceptual superstructure. Moreover, at present proposal remains largely unexplored due to the inherent these constructions do not provide tools for computing difficulty of performing calculations in a strongly coupled anything in the conformal phase of the theory beyond a quantum mechanical model and the need to take the large k very limited set of robust observables, such as anomalies.1 limit. To the best of our knowledge, the only explicit result At large n, the An theories can be described holograph- obtained in this framework is the calculation of the half BPS ically in terms of eleven-dimensional supergravity on AdS7× spectrum of the An theory, carried out in [28]. According to 4 2 4 S [18]. The AdS/CFT correspondence then renders these another little-explored proposal, the An (2, 0) theory on R × large n theories extremely tractable. However, the extension T2 (with a finite-size T2) can be “deconstructed” in terms of to finite n is presently only possible at leading order. Higher- four-dimensional N ¼ 2 quiver gauge theories [29]. Finally order corrections require a method for computing quantum there is the suggestion [30,31] that five-dimensional max- corrections in M theory. An intrinsic field-theoretic con- imally supersymmetric Yang-Mills theory is a consistent struction of the (2, 0) theories would therefore be indispen- quantum field theory at the nonperturbative level, without sable. Turning this logic around, such an independent additional UV degrees of freedom, and that it gives a definition would offer a window into quantum M theory. complete definition of the (2, 0) theory on S1. The con- However, most standard field theory methods are inad- ceptual similarities shared by these three proposals have equate for describing the (2, 0) theories. The mere existence been emphasized in [32]. It would be of great interest to of unitary, interacting quantum field theories in d>4 develop them further, ideally to the point where quantitative appears surprising from effective field theory reasoning— information for the nonprotected operator spectrum could be conventional local Lagrangians are ruled out by power derived and compared to the bootstrap results obtained here. counting. The (2, 0) theories are isolated, intrinsically quantum mechanical conformal field theories, which can- B. Bootstrap approach not be reached as infrared fixed points of local renormal- ization group flows starting from a Gaussian fixed point. In the present work, we will eschew the problem of “ ” This is in sharp contrast to more familiar examples of identifying the fundamental degrees of freedom of the isolated CFTs in lower dimensions, such as the critical (2, 0) theories. Indeed, we are not certain that this question Ising model in three dimensions. It is unclear whether an makes sense. Instead, we will do our best to rely exclusively unconventional (nonlocal?) Lagrangian for the interacting on symmetry. We regard the (2, 0) theories as abstract — (2, 0) theories could be written down (see, e.g., [21–26] for conformal field theories (CFTs), to be constrained ideally, — a partial list of attempts), but in any event it would be completely determined by bootstrap methods.
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