1. Localization and Supersymmetric Entanglement Renyi entropy
Questions:
(i) Is it possible to stop the flow of time, or say, go backwards in time? (ii) Is the topology of space dynamical? (ii) From the classical point of view, Can complete knowledge of the initial Bing Bang singularity
the beginning of time) be enough to predict the future?
(iv) Are there other Universes inside black holes?
Eric Howard
Macquarie University
SUSY MELBOURNE 2016

2. Outline
-consider supersymmetric Renyi entropy defined for 3d N = 2 CFTs
(using localized partition functions)
-employ a simplication used for CFT allowing to rewrite Renyi entropy as a partition function over a curved manifold: the branched 3-sphere.
-restrict to superconformal theories with N =2 supersymmetry in 3d spacetime
- Renyi entropy of a CFT is given by the partition function on this space, but conical singularities break the supersymmetry preserved in the bulk.  use a compensating R-symmetry gauge field, calc. partition function using localization.
-theory can be coupled to a fixed gravitational background while preserving
supersymmetry
-under conformal transformation, the branched covering is mapped to S 1 x H2
with gravity dual a charged AdS4 TBH.
-BH embedded into 4d N = 2 gauged supergravity (mass and charge are related so that it preserves half of the supersymmetries).
-compute supersymmetric Renyi entropy (using Wilson loop op.) in gravity theory
 agreement with those of the dual field theories in large-N limit.

3. Introduction
consider gravity duals of supersymmetric Renyi entropies for 3d N = 2 SCFT, defined:
Zn = supersymmetric partition function on a branched n-covering of 3-sphere
Supersymmetric Renyi entropies ≠ usual Renyi entropies of a disk entangling region in |R2,1 due to nontrivial R-symmetry background gauge field for preserving half of supersymmetries. results of holographic Renyi entropies can not be applied here
find gravity duals  look for half-BPS solution in 4d N = 2 gauged supergravities. Instead of branched n-covering 3-sphere, conformally map it by cotθ = sinh u to spacetime S 1 x H2
Coord. advantage: conical singularity, previous at θ = 0 in (1.2) is send to spatial infinity u ∞
AdS/CFT  dual gravity theory should have U(1) gauge symmetry corresp. to R-symmetry background gauge field of the field theory.

4. Introduction
Consider charged AdS4 topological BH with half supersymmetries  construct holographic duals of supersymmetric Renyi entropies
conformal map from the branched covering of a 3-sphere to S 1 x H2. Red circle at θ = π/2 (left) mapped to circle along direction u = 0 of hyperbolic space H2 (right).
1. supersymmetric Renyi entropies of 3d N=2 SCFT. Partition function Zn given by matrix model on squashed 3-sphere S3 b. Calc Zn for class of theories with gravity duals described by M-theory in large-N limit. Add Wilson loop, estimate shift of supersymmetric Renyi entropies for the ABJM theory (holographic dual to M-theory) in large-N limit. Partition function given by Euclidean on-shell action I(n) in dual gravity theory, Zn = e-I(n) , of charged topological AdS4 BH at finite temp. T = 1/(2πn). As a gravity dual of Wilson loop, consider fundamental string, calc. supersymmetric Renyi entropies.  holographic calc. agree with field theory results in large-N limit.

5. Supersymmetric Renyi entropy
partition function Zn was calculated by localization method; partition function on squashed 3-sphere S3b (squashing parameter b = p√n)
σi = eigenvalues of the matrix σ
free energy
hyperbolic gamma function
|W| = order of Weyl group W of gauge group G; k = Chern-Simons level. I = types of chiral multiplets; ρ = a weight in a representation RI of gauge group G. Δ I = R-charge of scalar field in chiral multiplet

6. Large N-limit
class of Chern-Simons gauge theories dual to M-theory in large-N limit: non-chiral gauge theories with gauge group G = U(N)k1 x U(N)k2 x U(N)kp and bifundamental fields.
large-N analysis on a round 3-sphere shows the theories must satisfy
for every gauge group a = 1…p and bifund. fields of representation I of R-charge Δ I .
and
assume these conditions for taking the large-N limit of partition function
-for each unitary group U(N)ka; a = 1,…,p, there are eigenvalues σa;i with i = 1,…,N with respect to which Fn is extremized
-solve saddle-point eq.take large-N limit  eigenvalue density
assume eigenvalues are
 consistent solution to saddle-point equations. point =separation of long/short range forces between eigenvalues. Employ approx. of hyperbolic gamma function with
ω1= ω2-1

7. Large N-limit
the long range force vanishes and the free energy is proportional to N3/2
proportional to ω
proportional to ω3
yI(x) = ya(x) - yb(x) for the indices of the gauge groups a, b the bifund. field I belongs.
If we rescale Chern-Simons level by ka = ω2 ka, the term =ω3 times the free energy for the theory with the levels ka. Knowing free energy on S3 (n = 1) of the gauge theories dual to M-theory is prop. to square root of the levels 
with explicit dependence of free energy on the parameters.
Supersymmetric Renyi entropy in the large-N limit is
eigenvalue density of ABJM theory with gauge group G = U(N)k x U(N)-k =ct. function on a compact support

8. Wilson loop
consider a Wilson loop at θ = π/2 wrapping around τ direction of the branched n-covering 3.
-conformal mapping equivalent to temporal Wilson loop along τ at u = 0 in CFT on S1 x H2
-addition of the Wilson loop W shifts the entanglement entropy by
expectation value on the
branched n-covering 3-sphere.
shift of Renyi entropy due to the loop:
for N = 2 Chern-Simons gauge theories  supersymmetric Wilson loop in represent. R:
location of the Wilson loop
expectation value of the Wilson loop on the n-covering three-sphere is obtained by localization and reduces to the matrix model:

9. Wilson loop
-in large-N limit, expectation value of Wilson loop in fund. representation using eigenvalue density ρ(x) as
 apply to 1=6-BPS Wilson loop in ABJM theory. Eigenvalue
density is already calc. but need to scale Chern-Simons level by ω2 on n-covering 3-sphere
λ='t Hooft coupling λ = N/k  supersymmetric Renyi entropy of the Wilson loop of the ABJM theory in large-N limit is
does not depend on parameter n.
 consider gravity duals of the supersymmetric Renyi entropies with(out) a Wilson loop do they agree with these results?

10. Charged topological AdS black hole
-construct gravity duals of supersymmetric Renyi entropies. Instead of finding a solution whose boundary is a branched covering of 3-sphere search solution which asymptotes to S1 X H2 near the boundary.
-turn on U(1) R-symmetry background gauge field for boundary SCFT (to preserve half of supersymmetries)  there exists an U(1) gauge field in the bulk theory.
consider Einstein-Maxwell theory in (Euclidean) 4-dimensions
 there exists a charged topological AdS BH solution whose boundary is
with
and U(1) gauge field is
let rH =largest root of f(r). Chemical potential µ can be det. by requiring that A vanishes on the horizon at r = rH:

11. Charged topological AdS black hole
Embed theory into N = 2 gauged supergravity (Killing spinor equation =variation of gravitino
Γa = gamma matrices for local Lorentz coord. satisfying
=the pullback by the vielbein
Solution preserves 1/2 supersymmetries when the condition M = Q holds
only consider supersymmetric solution Q is
temperature of BH
thermal entropy
Calc. free energy  evaluate action of the solution with the boundary & counter terms
γαβ =induced metric on boundary B, K= extrinsic curvature, RB=Ricci scalar of induced metric. (to regularize UV divergence from ∞ vol. of boundary r = r∞  ∞ free energy of supersymmetric charged topological AdS4 BH

12. Holographic supersymmetric
Renyi entropy
compare free energy with dual field theory. Temp. T should be identified with Tn = 1/(2n), that det. radius of the horizon
Calc. holographic free energy from
satisfies same relation to large-N limit of free energy of dual field theory.  holographic supersymmetric Renyi entropy agrees with previous results here (in large N-limit)
usual holographic Renyi entropy can be written as integration of thermal entropy: 
discrepancy between large N-limit supersymmetric Renyi entropy and this formula arises from the fact that there is the electric charge Q and chemical potential depending on T. Here, we used the thermodynamical relation

13. Holographic Wilson loop
Wilson loop in a fund. repres. = holographically dual to the fund. string in space-time The string world sheet action with a target space
The gravity dual of a Wilson loop in a fund. repres. is given by the fund. string in AdS4 spacetime. Here, it has two ends at UV boundary r = r∞ and the horizon r = rH.

14. Holographic Wilson loop
with the expectation value
find the world sheet coordinates with the target space as
the rest of the target space coordinates don’t depend on because of the rotational symmetry of Wilson loop.
 on-shell action of the fundamental string on the background is
UV divergence near the boundary is regulated by the UV cutoff at r = r∞. For 1/6 BPS Wilson loop in ABJM theory with U(N)k x U(N)-k gauge group,  rewrite the result in terms of 't Hooft coupling 
agrees with the field theory computation in large-N limit.
the holographic supersymmetric Renyi entropy of the Wilson loop agrees with
supersymmetric Renyi entropy of the Wilson loop of the ABJM theory in large-N limit

15. Conclusions
considered the partition function in a specific supergravity background
super-Renyi entropy can be calculated using the method of localization
this quantity differs from Renyi entropy of SCFT (using the R-symmetry of the theory)
the usual Renyi entropy can be used to recover the entanglement entropy
Renyi entropy has nice analytic properties in the large-N limit

16. Questions?