Conformal Bootstrap : Old and New

Abstract

Conformal Field Theories (CFT) are Quantum Field Theories characterized by enhanced (conformal) symmetries. They are interesting to Theoretical Physicists because they occur at critical points in phase transitions of various systems and also in the world sheet formulation of String Theory. CFTs allow Operator Product Expansion (OPE) in their correlators. The idea of Conformal Bootstrap is to solely use the conformal symmetries and crossing symmetry in the OPE to solve a conformal led theory and not explicitly use a lagrangian. Solving a CFT is equivalent to obtaining the anomalous dimensions and OPE coe client’s of the operators. The work presented in this thesis shows how ideas of bootstrap can be used to get analytic results for dimensions and OPE coe client’s of various operators in CFTs. In the conventional bootstrap program, the OPE in the direct (s-) channel is compared with the OPE of a crossed (t-) channel. This requirement of crossing symmetry is called the bootstrap equation. The flow of logic is somewhat reversed in the \new" idea that is formulated in this thesis. The trick is to expand a CFT correlator in terms of Witten diagrams, in all channels. This is a manifestly crossing symmetric description, and is in contrast to the usual expansion in terms of conformal blocks, which is in only one channel. For convenience we work with the Mellin transforms of Witten diagrams. For consistency of the Witten diagrams expansion with the conformal block expansion in a certain channel, we require the satisfaction of some equations, which we call the bootstrap equations in Mellin space. This scheme was rest chalked out by Polyakov in 1973, where he proposed the use of \unitary amplitudes" to expand a correlator. The unitary amplitudes had similar symmetry and analytic properties as the Witten diagrams. Even though he did not take his idea forward, replacing unitary amplitudes with Witten diagrams seems to work very well for obtaining analytic results. The working of bootstrap equations in Mellin space is demonstrated for the 4 Wilson-Fisher fixed point in d = 4 , O(N) theory at Wilson-Fisher point (in d = 4 ), as well as with large N (in general d), and large spin operators in strongly coupled and weakly coupled theories. For the case of global symmetry we have also analysed the somewhat unexplored case of cubic anisotropy. The results are obtained as perturbative series in , 1=N, or 1=` as applicable, and they are consistent with known results in literature. We also obtain various new results, for instance the OPE coe client’s of general higher spin operators. These results are otherwise very di cult to end from Feynman diagrams, but in this approach they come out very simply, essentially by solving some algebraic equations. We also show the use of the conventional bootstrap strategy, for analytically obtaining anomalous dimensions of large spin operators having higher twists, in a O(N) theory, by working in the light cone limit. One can question the validity of the proposal of using Witten diagrams to expand a correlator. One such issue is convergence of the sum over Witten diagrams. Convergence can be shown to hold for the operator spectrum we have worked with. Also there are operators that might upset convergence under some conditions. Resolutions of such cases, and ways to improve convergence have also been discussed. The conventional bootstrap method has been very successful in giving numerical results in nonpertur-bative CFTs, like the 3 dimensional Ising model. Numerical analysis can also be made possible with the new bootstrap in Mellin space approach. Having a convergent basis of expansion improves the prospect of numeric. The goal is to formulate a bootstrap scheme that, under a single framework, can make most of all the CFT properties. It should be systematic, so that one can obtain anomalous dimensions and OPE coe client’s of all operators up to any desired order, and works for all strongly/weakly coupled and perturbative/nonpertur-bative CFTS, both analytically and numerically. Finally, the use of Witten diagrams also indicates the possibility of Ising CFT or weakly coupled CFTs having connections with AdS/CFT, and hence String Theory. It does seem we have a right direction towards achieving our goal.