We describe in more detail our approach to the conformal bootstrap which uses the Mellin representation of CFT
 d
 four point functions and expands them in terms of crossing symmetric combinations of AdS
 d+1 Witten exchange functions. We consider arbitrary external scalar operators and set up the conditions for consistency with the operator product expansion. Namely, we demand cancellation of spurious powers (of the cross ratios, in position space) which translate into spurious poles in Mellin space. We discuss two contexts in which we can immediately apply this method by imposing the simplest set of constraint equations. The first is the epsilon expansion. We mostly focus on the Wilson-Fisher fixed point as studied in an epsilon expansion about d = 4. We reproduce Feynman diagram results for operator dimensions to O(ϵ
 3) rather straightforwardly. This approach also yields new analytic predictions for OPE coefficients to the same order which fit nicely with recent numerical estimates for the Ising model (at ϵ = 1). We will also mention some leading order results for scalar theories near three and six dimensions. The second context is a large spin expansion, in any dimension, where we are able to reproduce and go a bit beyond some of the results recently obtained using the (double) light cone expansion. We also have a preliminary discussion about numerical implementation of the above bootstrap scheme in the absence of a small parameter.

https://link.springer.com/content/pdf/10.1007/JHEP05(2017)027.pdf

A Mellin space approach to the conformal bootstrap

We analyze in detail the global symmetries of various (2 + 1)d quantum field theories and couple them to classical background gauge fields. A proper identification of the global symmetries allows us to consider all non-trivial bundles of those background fields, thus finding more subtle observables. The global symmetries exhibit interesting ’t Hooft anomalies. These allow us to constrain the IR behavior of the theories and provide powerful constraints on conjectured dualities.

/pdf/comments-on-global-symmetries-anomalies-and-duality-in-2-1-d-2y5k4l8kyr.pdf

Comments on global symmetries, anomalies, and duality in (2 + 1)d

The method of large spin perturbation theory allows to analyse conformal field theories (CFT) by turning the crossing equations into an algebraic problem. We apply this method to a generic CFT with weakly broken higher spin (HS) symmetry, to the first non-trivial order in the breaking parameter. We show that the spectrum of broken currents, for any value of the spin, follows from crossing symmetry. After discussing a generic model of a single scalar field, we focus on vector models with O(N ) global symmetry. We rediscover the spectrum of several models, including the O(N ) Wilson-Fisher model around four dimensions, the large O(N ) model in 2 < d < 4 and cubic models around six dimensions, not necessarily unitary. We also discuss models where the fundamental field is not part of the spectrum. Examples of this are weakly coupled gauge theories and our method gives an on-shell gauge invariant way to study them. At first order in the coupling constant we show that again the spectrum follows from crossing symmetry, to all values of the spin. Our method provides an alternative to usual perturbation theory without any reference to a Lagrangian.

/pdf/solving-cfts-with-weakly-broken-higher-spin-symmetry-1o6dwg4v4y.pdf

Solving CFTs with weakly broken higher spin symmetry

Three dimensional bosonization is a conjectured duality between non-super-symmetric Chern-Simons theories coupled to matter fields in the fundamental representation of the gauge group. There is a well-established supersymmetric version of this duality, which involves Chern-Simons theories with $$ \mathcal{N}=2 $$
 supersymmetry coupled to fundamental chiral multiplets. Assuming that the supersymmetric duality is valid, we prove that non-supersymmetric bosonization holds for all planar correlators of single-trace operators. The main tool we employ is a double-trace flow from the supersymmetric theory to an IR fixed point, in which the scalars and fermions are effectively decoupled in the planar limit. A generalization of this technique can be used to derive the duality mapping of all renormalizable couplings, in non-supersymmetric theories with both a scalar and a fermion. Our results do not rely on an explicit computation of planar diagrams.

/pdf/three-dimensional-bosonization-from-supersymmetry-54s8swmm5j.pdf

Three dimensional bosonization from supersymmetry

We apply analytic conformal bootstrap ideas in Mellin space to conformal field theories with O(N) symmetry and cubic anisotropy. We write down the conditions arising from the consistency between the operator product expansion and crossing symmetry in Mellin space. We solve the constraint equations to compute the anomalous dimension and the OPE coefficients of all operators quadratic in the fields in the epsilon expansion. We reproduce known results and derive new results up to O(ϵ
 3). For the O(N) case, we also study the large N limit in general dimensions and reproduce known results at the leading order in 1/N.

/pdf/mellin-space-bootstrap-for-global-symmetry-1mecvei8a3.pdf

Mellin space bootstrap for global symmetry

Chern-Simons gauge theories coupled to massless fundamental scalars or fermions define interesting non-supersymmetric 3d CFTs that possess approximate higher-spin symmetries at large N  In this paper, we compute the scaling dimensions of the higher-spin operators in these models, to leading order in the 1/N expansion and exactly in the ’t Hooft coupling We obtain these results in two independent ways: by using conformal symmetry and the classical equations of motion to fix the structure of the current non-conservation, and by a direct Feynman diagram calculation The full dependence on the ’t Hooft coupling can be restored by using results that follow from the weakly broken higher-spin symmetry This analysis also allows us to obtain some explicit results for the non-conserved, parity-breaking structures that appear in planar three-point functions of the higher-spin operators At large spin, we find that the anomalous dimensions grow logarithmically with the spin, in agreement with general expectations This logarithmic behavior disappears in the strong coupling limit, where the anomalous dimensions turn into those of the critical O(N ) or Gross-Neveu models, in agreement with the conjectured 3d bosonization duality

/pdf/on-the-higher-spin-spectrum-in-large-n-chern-simons-vector-xpmvrxptg7.pdf

On the higher-spin spectrum in large N Chern-Simons vector models

Chern-Simons gauge theories coupled to massless fundamental scalars or fermions define interesting non-supersymmetric 3d CFTs that possess approximate higher-spin symmetries at large N. In this paper, we compute the scaling dimensions of the higher-spin operators in these models, to leading order in the 1/N expansion and exactly in the 't Hooft coupling. We obtain these results in two independent ways: by using conformal symmetry and the classical equations of motion to fix the structure of the current non-conservation, and by a direct Feynman diagram calculation. The full dependence on the 't Hooft coupling can be restored by using results that follow from the weakly broken higher-spin symmetry. This analysis also allows us to obtain some explicit results for the non-conserved, parity-breaking structures that appear in planar three-point functions of the higher-spin operators. At large spin, we find that the anomalous dimensions grow logarithmically with the spin, in agreement with general expectations. This logarithmic behavior disappears in the strong coupling limit, where the anomalous dimensions turn into those of the critical O(N) or Gross-Neveu models, in agreement with the conjectured 3d bosonization duality.

/pdf/on-the-higher-spin-spectrum-in-large-n-chern-simons-vector-s411vdpjl2.pdf

On the Higher-Spin Spectrum in Large N Chern-Simons Vector Models

Non-abelian Chern-Simons theories coupled to fermions are known to provide an interesting class of non-supersymmetric conformal fixed points [1]. These theories, particularly those based on bifundamental matter, are important because they may provide simple non-supersymmetric examples of the AdS/CFT correspondence. For instance, it seems natural to conjecture that O(N)−k
 × O(N)
 k
 Chern-Simons theory coupled to Majorana fermions transforming in a bi-vector representation may be dual to pure Einstein gravity with a small negative cosmological constant in the “M-theory” limit where k = 1 and N is large. While it is extremely difficult to directly study such bifundamental theories when k = 1 or even at strong ’t Hooft coupling $$ \lambda =\frac{N}{k} $$
 , it is possible to calculate physical quantities to all orders in λ in a $$ \mathrm{U}{(M)}_{k_M}\times \mathrm{U}{(N)}_{k_N} $$
 theory, in the limit M ≪ N, in an M/N expansion. To illustrate this, we calculate the anomalous dimension of the primary operator tr $$ \overline{\psi}\psi $$
 , to first order in M/N, to all orders in $$ {\lambda}_M=\frac{N}{k_M} $$
 , but with $$ {\lambda}_N=\frac{N}{k_N} = 0 $$
 . We also comment on possible bosonization dualities for bifundamental Chern-Simons theories.

/pdf/anomalous-dimensions-in-non-supersymmetric-bifundamental-4wj3gy9ezw.pdf

Anomalous dimensions in non-supersymmetric bifundamental Chern-Simons theories

Non-abelian Chern-Simons theories coupled to fermions are known to provide an interesting class of non-supersymmetric conformal fixed points \cite{Giombi:2011kc}. These theories, particularly those based on bifundamental matter, are important because they may provide simple non-supersymmetric examples of the AdS/CFT correspondence. For instance, it seems natural to conjecture that $O(N)_{-k}\times O(N)_k$ Chern-Simons theory coupled to Majorana fermions transforming in a bi-vector representation may be dual to pure Einstein gravity with a small negative cosmological constant in the "M-theory" limit where $k=1$ and $N$ is large. While it is extremely difficult to directly study such bifundamental theories when $k=1$ or even at strong 't Hooft coupling $\lambda=\frac{N}{k}$, it is possible to calculate physical quantities to all orders in $\lambda$ in a $U(M)_{k_M} \times U(N)_{k_N}$ theory, in the limit $M \ll N$, in an $M/N$ expansion. To illustrate this, we calculate the anomalous dimension of the primary operator $\bar{\psi}{\psi}$, to first order in $M/N$, to all orders in $\lambda_M=\frac{N}{k_M}$, but with $\lambda_N=\frac{N}{k_N}=0$. We also comment on possible bosonization dualities for bifundamental Chern-Simons theories.

/pdf/anomalous-dimensions-in-non-supersymmetric-bifundamental-4jlpt7kqys.pdf

Anomalous Dimensions in Non-Supersymmetric Bifundamental Chern-Simons Theories

/pdf/erratum-to-anomalous-dimensions-in-non-supersymmetric-41mva4q0qh.pdf

V. Gurucharan

Papers

On the higher-spin spectrum in large N Chern-Simons vector models

On the Higher-Spin Spectrum in Large N Chern-Simons Vector Models

Anomalous dimensions in non-supersymmetric bifundamental Chern-Simons theories

Anomalous Dimensions in Non-Supersymmetric Bifundamental Chern-Simons Theories

Erratum to: Anomalous dimensions in non-supersymmetric bifundamental Chern-Simons theories