Searching for gauge theories with the conformal bootstrap
Zhijin Li, David Poland
Abstract
A bstract Infrared fixed points of gauge theories provide intriguing targets for the modern conformal bootstrap program. In this work we provide some preliminary evidence that a family of gauged fermionic CFTs saturate bootstrap bounds and can potentially be solved with the conformal bootstrap. We start by considering the bootstrap for SO( N ) vector 4-point functions in general dimension D . In the large N limit, upper bounds on the scaling dimensions of the lowest SO( N ) singlet and traceless symmetric scalars interpolate between two solutions at ∆ = D/ 2 − 1 and ∆ = D − 1 via generalized free field theory. In 3D the critical O ( N ) vector models are known to saturate the bootstrap bounds and correspond to the kinks approaching ∆ = 1 / 2 at large N . We show that the bootstrap bounds also admit another infinite family of kinks $$ {\mathcal{T}}_D $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>D</mml:mi> </mml:msub> </mml:math> , which at large N approach solutions containing free fermion bilinears at ∆ = D − 1 from below. The kinks $$ {\mathcal{T}}_D $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>D</mml:mi> </mml:msub> </mml:math> appear in general dimensions with a D -dependent critical N * below which the kink disappears. We also study relations between the bounds obtained from the bootstrap with SO( N ) vectors, SU( N ) fundamentals, and SU( N ) × SU( N ) bi-fundamentals. We provide a proof for the coincidence between bootstrap bounds with different global symmetries. We show evidence that the proper symmetries of the underlying theories of $$ {\mathcal{T}}_D $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>D</mml:mi> </mml:msub> </mml:math> are subgroups of SO( N ), and we speculate that the kinks $$ {\mathcal{T}}_D $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>D</mml:mi> </mml:msub> </mml:math> relate to the fixed points of gauge theories coupled to fermions.