Abstract
I will show how the flow triggered by deforming two-dimensional conformal field theories on a torus by the TT¯ operator is identical to the evolution generated by the (radial) quantum Shape Hamiltonian in 2 + 1 dimensions. I will discuss how the gauge invariances of the Shape Dynamics, i.e., volume-preserving conformal invariance and diffeomorphism invariance along slices of constant radius are realized as Ward identities of the deformed quantum field theory. I will also comment about the relationship between the reduction to shape space on the gravity side and the solvability of the irrelevant operator deformation of the conformal field theory
Highlights
Symmetry 2021, 13, 2242 in shape dynamics must satisfy
I will relate this flow equation to the global constraint equation that the physical wavefunction of quantum shape dynamics has to satisfy. This is a refinement to the original holographic correspondance between 3D general relativity in a space of negative cosmological constant and the TTdeformation of the dual conformal field theory that inhabits a finite radial cutoff surface put forward in [5], which was a refinement of the observation made earlier in [6]
The key point of this article is the following: when we identify shape dynamics in 2 + 1 dimensions as dual to the TTdeformation of a CFT, we can explain the latter’s solvability
Summary
The key point of this article is the following: when we identify shape dynamics in 2 + 1 dimensions as dual to the TTdeformation of a CFT, we can explain the latter’s solvability. Our ability to write down simple differential equations for the energy levels and the torus partition function and solve them exactly can be seen as arising from the imposition of the volume preserving conformal constraint of the dual quantum 2 + 1 shape dynamics theory This constraint freezes the inhomogenous modes of the metric in the partition function and renders it a function only of the zero modes of the metric components. I will continue to use TTas a stand in for the right-hand side of (9) throughout this article Note that this operator is irrelevant in the renormalization group sense: the energy momentum tensor has scaling dimension 2. The reason why the above factorization formula is a simplification is that the expectation value of the TToperator can be computed from just knowing the diagonal matrix elements of the stress tensor Tμν in energy eigenstates
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