Conformal Bootstrap Method

Bagga, Aakash Singh and Roy, Shubho Ranjan (2017) Conformal Bootstrap Method. Masters thesis, Indian Institute of Technology.

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Abstract

The aim of this project is to study Conformal Field Theories in d ≥ 3 dimensions and derive the Conformal Bootstrap Equation which may be used to solve theories with conformal invariance. The identification conformal derives from the property that the transformation does not affect the angle between two arbitrary curves cross- ing each other at some point, conformal transformations preserve angles. Mathe- matically, a conformal transformation comprises of two steps - doing a conformal isometry, followed by a Weyl rescaling of the metric. The bootstrap approach is an algebraic recursive procedure, which heavily relies on determining the consequences of the symmetries and imposing consistency conditions. We started by considering infinitesimal form of conformal diffeomorphisms for d ≥ 3 spacetime dimensions. We identified translations, dilatations, rotations, boosts and special conformal transfor- mations as members of the conformal group and that the algebra is isomorphic to a pseudo-orthogonal SO ( d + 1 , 1). A number of fundamental systems have been shown to obey conformal invariance and we explicitly worked out the case for the massless scalar field and in a heuristic fashion, for the case of the source-free Maxwell field. Conformally invariant systems, in particular field theories are described by fields which transform homogeneously under conformal transformations i.e. which consti- tute irreducible representations (irreps) of conformal group. Seeking the irreducible representations of the group, we used the little group method and derived the com- plete set of transformation rules for Φ( x ). Next, we saw that the two and three point functions were determined upto some multiplicative constants by imposing confor- mal invariance alone, but this success stopped here and it was seen that for four(or more) points, the functions have an arbitrary dependence on what are known as anharmonic ratios or cross ratios (which are invariants unchanged by all conformal transformations). Next, we establish state operator correspondence by using radial quantization and show that any operator at a point away from the origin is a linear combination of the primary and its descendants which are together known as a conformal family. Then, we will show that in a CFT, radial quantization leads us to a useful tool known as the Operator Product Expansion (OPE). And that by using this tool, we can recursively reduce any n-point function to 2 point functions. And we show that they are completely determined by conformal invariance upto some constants known as OPE coefficients. Following that, we identify a consistency condition for filtering junk CFT data, this is known as crossing symmetry or OPE associativity . Using this condition we derived the very powerful conformal bootstrap equation . For unitary theories, we can have even more constraints on the spectrum, for which we derived the unitarity bounds before finishing with an algorithm which tells us how we can use the bootstrap equation to rule out inconsistent families of spectra.

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