The TT‾T\overline T deformation of quantum field theory as random geometry

We revisit the results of Zamolodchikov and others on the deformation of two-dimensional quantum field theory by the determinant $\det T$ of the stress tensor, commonly referred to as $T\overline T$. Infinitesimally this is equivalent to a random coordinate transformation, with a local action which is, however, a total derivative and therefore gives a contribution only from boundaries or nontrivial topology. We discuss in detail the examples of a torus, a finite cylinder, a disk and a more general simply connected domain. In all cases the partition function evolves according to a linear diffusion-type equation, and the deformation may be viewed as a kind of random walk in moduli space. We also discuss possible generalizations to higher dimensions.Comment: 32 pages. Final published version! Solution for t>0 clarifie

Bulk Renormalization Group Flows and Boundary States in Conformal Field Theories

We propose using smeared boundary states $e^{-\tau H}|\cal B\rangle$ as variational approximations to the ground state of a conformal field theory deformed by relevant bulk operators. This is motivated by recent studies of quantum quenches in CFTs and of the entanglement spectrum in massive theories. It gives a simple criterion for choosing which boundary state should correspond to which combination of bulk operators, and leads to a rudimentary phase diagram of the theory in the vicinity of the RG fixed point corresponding to the CFT, as well as rigorous upper bounds on the universal amplitude of the free energy. In the case of the 2d minimal models explicit formulae are available. As a side result we show that the matrix elements of bulk operators between smeared Ishibashi states are simply given by the fusion rules of the CFT.Comment: 17 pages, 3 figures. v3: Reference to related work added; analysis of minimal models clarified; reformatted to conform with SciPost submission guidelines. v4: discussion of tricritical Ising expanded; minor improvements and added references suggested by referee

The Stress Tensor in Quenched Random Systems

The talk describes recent progress in understanding the behaviour of the stress tensor and its correlation functions at a critical point of a generic quenched random system. The topics covered include:(i) the stress tensor in random systems considered as deformed pure systems; (ii) correlators of the stress tensor at a random fixed point: expectations from the replica approach and c-theorem sum rules; (iii) partition function on a torus; (iv) how the stress tensor enters into correlation functions: subtleties with Kac operators.Comment: 5 pages; talk presented at Workshop on Statistical Field Theory, Como, June, 200