Higher Spin Operators in Conformal Field Theories

Abstract

In this thesis we study higher spin operators in conformal field theories with weakly broken higher spin symmetry. Exact higher spin symmetry is known to be very constraining, containing the usual conformal group as a subgroup, and actually enforcing the theory to be free. Moreover, even in the presence of some weak, perturbative breaking of this symmetry, it still constrains the correlation functions.

In particular, it simplifies the calculation of anomalous dimensions of those currents to the lowest order in perturbation theory, reducing it to a two-point function of the non-conservation operator corresponding to the current. We apply this method to a variety of vector models, both bosonic and fermionic. Specifically, we reproduce some known results for Wilson-Fisher model in 4 − ε expansion, as well as the 1/N expansion of the model and use those to interpolate with good accuracy to d = 3 Ising model. We also get some new results for non-linear sigma model in 2 + ε expansion and cubic models in 6 − ε. We also apply this technique to fermionic models, the Gross-Neveu-Yukawa model in 4 − ε expansion and 1/N expansion. Further, we use a combination of direct Feynman diagram calculation and analytic bootstrap methods to calculate the anomalous dimension of some composite operators.

In the last chapter, we study vector models in d = 3 with a Chern-Simons interaction, which were an object of close study recently as a testing ground of a whole family of boson-fermion dualities. In particular, these dualities imply a matching of anomalous dimensions and three-point functions of higher spin currents between bosonic and fermionic theories under a certain mapping of the Chern- Simons coupling and the rank of the gauge group. We confirm those predictions using both the non-conservation operator formalism and a direct Feynman diagram calculation.