From Strings to AdS4-Black holes

by Marco Rabbiosi 13 October 2015

Why quantum theory of Gravity? The four foundamental forces are described by Electromagnetic, weak and strong QFT (Standard Model) Gravity Dierential Geometry (General Relativity) These are two very dierent mathematical frameworks.....but a common belief is the existance of a foundamental microscopic theory able to describe all together. Canonical quatization of GR fails because of problems with unitarity and renormalizability......however for a complete understanding of Black holes and Big Bang we need Quantum Gravity: they exibit a singular behaviour in GR, curvature scalars diverge. Characteristic lenght: l ph = hg c 3 10 35 m direct epxerimental data inaccessible...but on the theoretical side...

Supersymmetry Enlargment of ISO(3, 1) Coleman-Mandula No-go theorem. 4d SuperPoincarè algebra s N = {P µ, J µν,q i α = ( Q i A, QȦi ) t,t r } {Q i A, QḂj } = 2i (σ µ ) AḂ δ i j P µ, {Q i A,Q j B } = ε AB ( U ij + iv ij), [Q i A,T r] = (U r ) i j Q j A [Q i A,J µν] = 1 2 ( σµν ) B plus complex conjugate relations and the usual Poincaré algebra. A Q i B The representations are particular multiplets of elds with Fermionic d.o.f. = Bosonic d.o.f on shell. Very strict conditions on a supersymmetric eld theory (dimensions, interactions...).

M-theory Gauging supersymmetry requires dieomorphism invariance,...the symmetry of GR!...we obtain Supergravity! Moreover, the quantum spectrum of a string always has a massless spin 2 particle...again Gravity! These ingredients lead to M-theory d = 11 supersymmetric theory of strings and branes. Beyond an ordinary QFT: a net of duality between superstring theories that describe it in dierent limits.

Compactication Interesting to study the low energy limit (α 0) of these theories, the massless part of the spectrum. The eective eld theory results to be a Supergravity theory in d = 10,11! Compactications means they admit solutions like C s M d s. They link the dierent Supergravities and their deformations in all the dimensions! In particular N = 2 d = 4 gauged supergravity comes from compactications on a ux background. We search black hole solutions in the e.o.m. of this theory...main motivations:

Motivations AdS 4 S 7 /Z k d = 3N = 6 ABJM U(N) k U(N) k Asintotically Ads 4 black hole solutions could be dual to some deformations of ABJM. Lifting to vacuas of M-theory...M2-branes wrapping a Riemann surface. Study of microstates of black holes. Integrability properties of classical GR in d = 4.

N = 2 d = 4 abelian gauged supergravity The bosonic part of the action (fermionic conguration to zero is a consistent truncation) reads L = g( R 2 h uv µ q u µ q v g i j µ z i µ z j + 1 4 I ΛΣH Λµν H Σ µν + R ΛΣH Λ µν ε µνρσ H Σ ρσ 8 g ( 4h uv k u Λ k v Σ X Λ X Σ +(f Λ i g i j f Σ j 3X Λ X Σ )PΛ x P ) Σ x 1 ε µνρσ 1 ) Θ Λa B µνa ρ A σλ + 4 g 32 g ΘΛa Θ b Λ ε µνρσ B µνa B ρσ b, Where the covariant derivative of the hyperscalars is µ q u = µ q u + A Λ µ k u Λ A Λµk Λu, and H Λ µν = F Λ µν + 1 2 ΘΛa B µνa are modied eld strenghts (necessary for having also magnetically charged hyperscalars).

Ansatz We choose the more simple possible ansatz for the metric where ds 2 = e 2U(r) dt 2 + e 2U(r) dr 2 + e 2ψ(r) 2U(r) dω 2 l, f l (θ) = 1 sin( { sinθ, l = 1, lθ) = l sinhθ, l = 1, and for the gauge elds A Λ = A Λ t dt lpλ f l (θ)dφ, A Λ = A Λt dt lq Λ f l (θ)dφ, B Λ = 2lp Λ f (θ)dr dφ, B l Λ = 2lq Λ f l (θ)dr dφ H Λ tr = e 2U 2ψ I ΛΣ (R ΛΓ p Γ q Σ ), H Λ θφ = p Λ f l (θ). Now all the elds dening this ansatz depend only on the radial coordinate r!

Eective eld theory The substitution of this ansatz in the e.o.m. shows the possibility to derive these equations from a nite dimensional dynamical eective system S e = dr[e 2ψ (U 2 ψ 2 + h uv q u q v + g i jz i z j + 1 4 e4u 4ψ Q H 1 Q ) V e ], with V e ( ) = e 2U 2ψ V BH + e 2ψ 2U V g l. ensuring the constaints H e = 0 and p Λ k u Λ q Λk uλ = 0 The e.o.m. now are switched from PDE to ODE......however highly coupled and of the second order...we can do something better!

Hamilton-Jacobi For a n-dimensional dynamical system like ( ) 1 S = dr 2 g abφ a Φ b V (Φ a ), with the condition H = 0, Hamilton-Jacobi equation for the principal function W = W (Φ a ;α 1,...,α n ) reads 1 2 g ab a W b W = V (Φ a ). In general, too dicult to nd the complete integral...easier to have a particular solution! In this case we have not solved completly the system, however we can rewrite the action as sum of square S = dr 1 2 g ( ab Φ a ± g ac c W )( ) Φ b ± g bd d W.

First order equations Applaing this technique to S e showed before, we nd: U = ε(e U 2ψ ReZ + le U ImL ), ψ = 2l εe U ( ImL, ) z i = εe iα g i j e U 2ψ Z ile D j U L, D j q u = 2lεh uv e U Im(e iα v L ), Q = ε4e 2ψ 3U H ΩReV, 2e U Re V,K u = ε A,K u K u,p x = 0, Where the principal Hamilton-Jacobi function reads W = e U Z + ile 2ψ 2U Q x W x, We have generalized the result of BPS analysis of Halmagyi, Petrini, Zaffaroni [arxiv : 1305.0730v 3[hep th]] and Dall Agata, Gnecchi [arxiv : 1012.37565v 1[hep th]].

Summary and Outlook Summary 1 Very quickly, we have given an idea of what superstring theories/m-theory are. 2 We have seen a formal development of the Einstein equations aims to nd out new solutions....are both part of our work! Outlook The next step is to solve the equations in some particular models and further extend the result to a large class of black holes: non-extremal, rotating and NUT-charged.....however many things can be again done for a better understanding of the nature of Gravity, both at classical and quantum level!

Thank you! Thank you for the attention!