Multi-centered higher spin solutions from $$ {\mathcal{W}}_N $$ conformal blocks

Abstract

Motivated by the question of bulk localization in holography, we study the problem of constructing multi-centered solutions in higher spin gravity which describe point particles in the interior of AdS 3. In the Chern-Simons formulation these take into account the backreaction after adding Wilson line sources. We focus on chiral solutions where only the left-moving sector is excited. In that case it is possible to choose a gauge where the dynamical variables are a set of Toda fields living in the bulk. The problem then reduces to solving the A N −1 Toda equations with delta function sources, which in turn requires solving an associated monodromy problem. We show that this monodromy problem is equivalent to the monodromy problem for a particular W N vacuum conformal block at large central charge. Therefore, knowledge of the W N vacuum block determines the multi-centered solution. Our calculations go beyond the heavy-light approximation by including the backreaction of all higher spin particles.

Information recovery from pure state geometries in 3D

“…From the first equality in (4.18) and from (4.16) we see that inserting a chiral vertex operator in the CFT corresponds to adding a chiral particle source in the bulk. The above comments on the backreaction of spinning particles provide a direct derivation in the metric formalism of properties which were derived in [13] using Chern-Simons variables, where the coupling of spinning particles becomes particularly simple [8].…”

“…Thus, in contrast to studies of the Vaidya metric [9][10][11] they do not collapse on each other. Using the methods of [12,13] we construct the metric explicitly in the rotationally symmetric limit where the centers are distributed continuously on a circular shell. This geometry contains a throat region which is capped off inside the shell of matter.…”

“…In this subsection we will see that the left-thermal geometries contain a submanifold with constant negative curvature which is conveniently described by a solution of Liouville theory [12,13]. For simplicity we will limit the discussion in this subsection to the regime of interest (3.6) where the right-movers are in the ground state, J = M + 1 or T −− = −1, though it could easily be generalized to the regime where −1 ≤ T −− ≤ 0.…”

More on Wilson toroidal networks and torus blocks

“…Recently, CFT d conformal blocks were interpreted in the AdS d+1 /CFT d correspondence as geodesic (Witten) networks stretched in the asymptotically AdS d+1 spaces [3][4][5][6][7][8][9][10][11][12][13][14][15]. The alternative description of conformal blocks in terms of Wilson lines was extensively studied in [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. 1 On the other hand, there is an intriguing relation between the space of quantum states in the three-dimensional Chern-Simons theory in the presence of the Wilson lines and the space of conformal blocks in two-dimensional conformal field theory noticed a long ago [58][59][60].…”

A note on the $$ \mathcal{N} $$ = 2 super-$$ {\mathcal{W}}_3 $$ holographic dictionary

“…The analysis here is one very modest example in the context of supersymmetric dualities. In recent years this asset has been applied to Wilson lines in Chern-Simons theory as one efficient approach to evaluate W-conformal blocks in the CFT [44][45][46]. In our case it would be interesting to evaluate a supersymmetric Wilson line and include quantum corrections following the approach of [47][48][49][50][51][52].…”