New boundary conditions for AdS2

We describe new boundary conditions for AdS2 in Jackiw-Teitelboim gravity. The asymptotic symmetry group is enhanced to Diff(S1) ⋉ C∞(S1) whose breaking to SL(2, ℝ) × U(1) controls the near-AdS2 dynamics. The action reduces to a boundary term which is a generalization of the Schwarzian theory and can be interpreted as the coadjoint action of the warped Virasoro group. This theory reproduces the low-energy effective action of the complex SYK model. We compute the Euclidean path integral and derive its relation to the random matrix ensemble of Saad, Shenker and Stanford. We study the flat space version of this action, and show that the corresponding path integral also gives an ensemble average, but of a much simpler nature. We explore some applications to near-extremal black holes.

Complexity measures from geometric actions onVirasoro and Kac-Moody orbits

We further advance the study of the notion of computational complexity for 2d CFTs based on a gate set built out of conformal symmetry transformations. Previously, it was shown that by choosing a suitable cost function, the resulting complexity functional is equivalent to geometric (group) actions on coadjoint orbits of the Virasoro group, up to a term that originates from the central extension. We show that this term can be recovered by modifying the cost function, making the equivalence exact. Moreover, we generalize our approach to Kac-Moody symmetry groups, finding again an exact equivalence between complexity functionals and geometric actions. We then determine the optimal circuits for these complexity measures and calculate the corresponding costs for several examples of optimal transformations. In the Virasoro case, we find that for all choices of reference state except for the vacuum state, the complexity only measures the cost associated to phase changes, while assigning zero cost to the non-phase changing part of the transformation. For Kac-Moody groups in contrast, there do exist non-trivial optimal transformations beyond phase changes that contribute to the complexity, yielding a finite gauge invariant result. Moreover, we also show that our Virasoro complexity proposal is equivalent to the on-shell value of the Liouville action, which is a complexity functional proposed in the context of path integral optimization. This equivalence provides an interpretation for the path integral optimization proposal in terms of a gate set and reference state. Finally, we further develop a new proposal for a complexity definition for the Virasoro group that measures the cost associated to non-trivial transformations beyond phase changes. This proposal is based on a cost function given by a metric on the Lie group of conformal transformations. The minimization of the corresponding complexity functional is achieved using the Euler-Arnold method yielding the Korteweg-de Vries equation as equation of motion.

EULER–POINCARÉ FLOWS ON THE LOOP BOTT–VIRASORO GROUP AND SPACE OF TENSOR DENSITIES AND (2 + 1)-DIMENSIONAL INTEGRABLE SYSTEMS

Following the work of Ovsienko and Roger ([54]), we study loop Virasoro algebra. Using this algebra, we formulate the Euler–Poincaré flows on the coadjoint orbit of loop Virasoro algebra. We show that the Calogero–Bogoyavlenskii–Schiff equation and various other (2 + 1)-dimensional Korteweg–deVries (KdV) type systems follow from this construction. Using the right invariant H1 inner product on the Lie algebra of loop Bott–Virasoro group, we formulate the Euler–Poincaré framework of the (2 + 1)-dimensional of the Camassa–Holm equation. This equation appears to be the Camassa–Holm analogue of the Calogero–Bogoyavlenskii–Schiff type (2 + 1)-dimensional KdV equation. We also derive the (2 + 1)-dimensional generalization of the Hunter–Saxton equation. Finally, we give an Euler–Poincaré formulation of one-parameter family of (1 + 1)-dimensional partial differential equations, known as the b-field equations. Later, we extend our construction to algebra of loop tensor densities to study the Euler–Poincaré framework of the (2 + 1)-dimensional extension of b-field equations.