The superconformal equation
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Springer
Received: July 15, 2020 Accepted: September 6, 2020 Published: October 23, 2020
Ilija Buri´ c,a Volker Schomerusa and Evgeny Sobkob a
DESY, Notkestraße 85, D-22607 Hamburg, Germany b University of Southampton, Highfield, Southampton, SO17 1BJ, U.K.
E-mail: [email protected], [email protected], [email protected] Abstract: Crossing symmetry provides a powerful tool to access the non-perturbative dynamics of conformal and superconformal field theories. Here we develop the mathematical formalism that allows to construct the crossing equations for arbitrary four-point functions in theories with superconformal symmetry of type I, including all superconformal field theories in d = 4 dimensions. Our advance relies on a supergroup theoretic construction of tensor structures that generalizes an approach which was put forward in [1] for bosonic theories. When combined with our recent construction of the relevant superblocks, we are able to derive the crossing symmetry constraint in particular for four-point functions of arbitrary long multiplets in all 4-dimensional superconformal field theories. Keywords: Conformal and W Symmetry, Conformal Field Theory, Field Theories in Higher Dimensions, Superspaces ArXiv ePrint: 2005.13547
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)147
JHEP10(2020)147
The superconformal %ing equation
Contents 1 Introduction
1 4 4 8 11
3 Lifting correlators to the supergroup 3.1 Statement of the result 3.2 Illustration for 1-dimensional superconformal algebra 3.3 Proof of the lifting formula
14 14 17 19
4 Tensor structures and crossing symmetry equations 4.1 Cartan coordinates, tensor and crossing factors 4.2 Blocks and crossing symmetry equation 4.3 Illustration for 1-dimensional superconformal algebra
21 22 27 30
5 Conclusions and outlook
33
A Proof of covariance laws
35
B Supermanifolds and Lie supergroups B.1 Lie supergroups B.2 Supergroup actions
36 37 38
1
Introduction
Conformal field theories describe very special points in the space of quantum field theories that seem to provide unique views into non-perturbative dynamics through a variety of rather complementary techniques, such as holography, integrability, localization and the conformal bootstrap. One of the principal analytical tools for conformal field theory are conformal partial wave (or block) expansions that were proposed early on in [2]. The role they play in the study of models with conformal symmetry is very similar to the role of Fourier analysis in systems with translational symmetry. While conformal blocks are entirely determined by kinematics, they allow to separate very neatly the dynamical meat of a theory from its kinematical bones. For example, an N -point function of local operators in a conformal field theory can be a very complicated object. If expanded in conformal blocks, however, the coefficients factorize into a set of three-point couplings, i.e. most of
–1–
JHEP10(2020)147
2 Superspace and superconformal symmetry 2.1 Some ba
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