Abstract
We extend the holographic duality between 3d pure gravity and 2d Ising CFT proposed in ref. [1] to CFTs with boundaries. Besides the usual asymptotic boundary, the dual bulk spacetime now has a real cutoff, on which live branes with finite tension, giving Neumann boundary condition on the metric tensor. The strongly coupled bulk theory requires that we dress the well-known semiclassical AdS/BCFT answer with boundary gravitons, turning the partition function into the form of Virasoro characters. Using this duality, we relate the brane tensions to the modular S-matrix elements of the dual BCFT and derive the transformation between gravitational solutions with different brane tensions under modular S action.
Highlights
Charge c = 3 AdS/2GN on the asymptotic AdS3 boundary. This indicates that, under certain assumptions, the Virasoro characters of the corresponding CFT should be the result of a determination of the gravity partition function, which turns out to be given by the summation of modular images of a “vacuum seed”, i.e., the vacuum conformal block of the dual CFT, over a finite-index mapping class group representing the enhanced “gauge” symmetry at strong coupling
We use the results for the CFT partition function on the disk, more precisely the boundary contribution to this partition function which is given by the boundary entropy, to fix the free parameter in the bulk: the brane tension
We summarize the semiclassical duality between AdS and boundary CFT (BCFT) and the partition function of BCFT on an annulus
Summary
We provide a short review of the Ising/gravity duality on closed manifolds, a brief introduction to boundary conformal field theory (BCFT) as well as the AdS/BCFT duality in the semi-classical regime. Since the theory is strongly coupled, there is no sense in which the latter sum can be done perturbatively It is one of the key assumptions of the proposal of [1] that this procedure accounts for the full bulk partition function. The modular sum can be organized into a summation of the modular images of the gravitational partition function of the thermal AdS3 saddle Zvac and the boundary graviton fluctuations around it. The result of the summation yields the gravitational partition functions Zgrav = 8ZIsing At this level, a similar duality can be proposed for the tricritical Ising model with c = 7/10, where Zgrav = 48Ztri-Ising. It is shown using topological quantum field theory techniques that the gravitational partition function for
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