Cold holographic matter and a color group-breaking instability - PowerPoint PPT Presentation

Cold holographic matter and a color group-breaking instability Javier Tarr´ ıo Universit´ e Libre de Bruxelles in collaboration with A. Faedo, A. Kundu, D. Mateos, C. Pantelidou arXiv:1410.4466 arXiv:1505.00210 arXiv:1511.05484 work in progress Oxford, May 3 rd 2016

Take home message ⇒ Take (super)Yang–Mills theories at strong coupling and large N ⇒ Add fundamental matter (quarks) at finite charge density ⇒ Then... ...the simplest ( ∗ ) ground states are described in the IR by a non-relativistic field theory with specific scaling properties...

Take home message ⇒ Take (super)Yang–Mills theories at strong coupling and large N ⇒ Add fundamental matter (quarks) at finite charge density ⇒ Then... ...the simplest ( ∗ ) ground states are described in the IR by a non-relativistic field theory with specific scaling properties... ...and they are probably unstable towards color-superconducting phases

Table of Contents Motivation A quick review of strongly coupled SYM theories Adding fundamental matter and baryon density Looking for an instability Conclusions

A cartoon of the QCD phase diagram T QGP Hadronic matter CS CFL µ NM

A cartoon of the QCD phase diagram T QGP We don’t Hadronic matter know much CS about this CFL region µ NM

Interest and use of holography ⇒ Why care? Color-superconductivity phases and transitions; neutron stars... ⇒ Region of strong coupling suggests a holographic approach, but no QCD dual, we take N = 4SYM as ballpark ⇒ Try to extract qualitative lessons of the effects of the chemical potential in strongly coupled systems ◮ What is the equation of state? ◮ How to observe color superconductivity?

Strings ho! I describe results from top-down models, where we extremize type II SUGRA, DBI+WZ and NG actions.

Take home message ⇒ Take (super)Yang–Mills theories at strong coupling and large N ⇒ Add fundamental matter (quarks) at finite charge density ⇒ Then... ...the simplest ( ∗ ) ground states are described in the IR by a non-relativistic field theory with specific scaling properties... ...and they are probably unstable towards color-superconducting phases

Take home message ⇒ Take the geometry sourced by a set of Dp-branes on (flat) spacetime ⇒ Add fundamental matter (quarks) at finite charge density ⇒ Then... ...the simplest ( ∗ ) ground states are described in the IR by a non-relativistic field theory with specific scaling properties... ...and they are probably unstable towards color-superconducting phases

Take home message ⇒ Take the geometry sourced by a set of Dp-branes on (flat) spacetime ⇒ Backreact Dq-branes with gauge field on the w.v. turned on ⇒ Then... ...the simplest ( ∗ ) ground states are described in the IR by a non-relativistic field theory with specific scaling properties... ...and they are probably unstable towards color-superconducting phases

Take home message ⇒ Take the geometry sourced by a set of Dp-branes on (flat) spacetime ⇒ Backreact Dq-branes with gauge field on the w.v. turned on ⇒ Then... ...the simplest ( ∗ ) supergravity solutions at T = 0 are described in the IR by a Hyperscaling-Violating Lifshitz spacetime... ...and they seem to be unstable towards Higgs-branch phases

In this talk I will have the gauge/gravity duality in mind (non-)AdS spacetime ← → (non-)conformal theory Domain wall solution ← → Renormalization group flow Boundary ← → UV (high energies) Origin ← → IR (low energies)

System action � � � 1 R ∗ 1 − 1 2 H ∧ ∗ H − 1 e − 2 φ 2 d φ ∧ ∗ d φ S = 2 κ 2 � 1 1 2 F 1 ∧ ∗ F 1 + 1 2 F 3 ∧ ∗ F 3 + 1 − 4 F 5 ∧ ∗ F 5 2 κ 2 � 1 1 − 2 C 4 ∧ H ∧ F 3 2 κ 2 � � � e d A + ˆ B ˆ d 8 x e − φ −| ˆ G + d A + ˆ − N f T 7 B | + N f T 7 C q

Table of Contents Motivation A quick review of strongly coupled SYM theories Adding fundamental matter and baryon density Looking for an instability Conclusions

SYM theories from type II SUGRA SU ( N ) SYM with 16 super- r charges S 8 − p d s 2 = h − 1 / 2 d x 2 1 , p + h 1 / 2 � d r 2 + r 2 d Ω 2 � 8 − p � S 8 − p ∗ F 8 − p ∼ N

SYM theories from type II SUGRA [Itzhaki et al ’98] ⇒ For D3-branes one has N = 4 SYM, conformal ⇒ For D2-branes λ is dimensionful and there is a running λ UV IR D2-brane perturbative description description ⇒ The holographic radius is related to the energy scale r = E ℓ 2 S .

SYM theories from type II SUGRA SYM with other gauge group r and less supercharges M 8 − p d s 2 = h − 1 / 2 d x 2 1 , p + h 1 / 2 � d r 2 + r 2 d Σ 2 � 8 − p � ∗ F 8 − p ∼ N M 8 − p

SYM theories from type II SUGRA To preserve N = 1 supersymmetry the internal manifold must admit one Killing spinor. This constrains the possible choices. From now on dim base cone D 3-branes 5 Sasaki-Einstein Calabi-Yau D 2-branes 6 nearly K¨ ahler G 2 -cone

Table of Contents Motivation A quick review of strongly coupled SYM theories Adding fundamental matter and baryon density Looking for an instability Conclusions

Fundamental matter [Karch and Katz ’03] x µ θ 1 , 2 , 3 θ 4 , ··· , 8 − p r × − − − Dp × × × − D(p+4) ⇒ The action as sum of two parts: type II SUGRA and � � d p +1 x d 4 y e − φ � ˆ ˆ S flavor = − T p +4 G + T p +4 C 5+ p

Backreaction of the flavor branes ⇒ Consider the RR form the flavor brane sources � � S IIB + D 7 ⊃ 1 d C 8 ∧ ∗ d C 8 + C 8 ∧ ( δ ( f 1 ) δ ( f 2 ) d f 1 ∧ d f 2 ) 2 � �� � Ξ 2 which implies the Bianchi identity for a sourced RR form d F 1 = − Ξ 2 ⇒ The number of flavor branes is given by Gauss law � F 1 ∼ N f

Backreaction with smearing (D3/D7) [Benini et al ’06] ⇒ Two things I have shown in previous slides 1. For D 3-branes the compact manifold is a SE − d 8 x e − φ √ � � � 2. S D 7 = T 7 − G + C 8 ∧ Ξ 2 with Ξ 2 exact ⇒ SE manifolds can be expressed as U(1) fibrations over KE manifolds and are equiped with an SU(2)-structure vol ( SE ) = 1 d η KE = 2 J KE , 2 J KE ∧ J KE ∧ η KE ⇒ Idea: to identify Ξ 2 ∼ J KE and use the SU(2)-structure to write a consistent radial ansatz for the IIB+sources action F 1 ∼ N f η KE ⇒ d F 1 ∼ N f J KE

A taste of the smeared solution (D3/D7) [Benini et al ’06] ⇒ With a simple ansatz d s 2 = g 1 ( r ) d x 2 1 , 3 + g 2 ( r ) d r 2 + g 3 ( r ) d s 2 KE + g 4 ( r ) η 2 KE , with dilaton and RR forms F 5 ∼ N (1 + ∗ ) J KE ∧ J KE ∧ η KE , F 1 ∼ N f η KE , ⇒ A SUSY solution exists 1 φ ′ = N f e φ e φ = ⇒ N f ( r LP − r )

When is backreaction needed? (D3/D7) ⇒ When can we omit backreaction (probe approximation) and when is it necessary? ⇒ Compare energies (effect on metric) | F 1 | | F 5 | ∼ λ N f N and one concludes (wrongly) that for λ N f N ≪ 1 probe approx. is enough at all scales.

Backreaction and smearing (D2/D6) [Faedo et al ’15] For the D2/D6 case a similar situation holds ⇒ Start with a NK (6d) manifold and − d 7 x e − φ √ � � � − G + C 7 ∧ Ξ 3 with Ξ 3 exact S D 6 = T 6 ⇒ There is a SU(3)-structure vol ( NK ) = 1 6 J 3 d J = 3ImΩ , d ReΩ = 2 J ∧ J , ⇒ In this case F 2 ∼ N f J ⇒ F 2 ∼ N f ImΩ ∼ Ξ 3

When is backreaction needed? (D2/D6) ⇒ Backreaction matters in the IR | F 6 | ∼ λ N f | F 2 | ≡ E flavor N E E and there is a change in the dynamics of the theory with a crossover at E ∼ E flavor . ⇒ UV boundary conditions

A taste of the smeared solution (D2/D6) [Faedo et al ’15] The D2/D6 solution e φ 0.100 0.050 0.010 0.005 0.001 5 � 10 � 4 r 0.01 0.1 1 10 100 1000 AdS 4 D2-brane IR UV λ N f N

Including charge in the setup ⇒ My motivation was to add a ‘quark’ density to these setups, but keeping vanishing temperature. ⇒ Dissolved strings in the flavor branes dual U (1) global current ← → U (1) gauge field ⇒ In particular a charge density corresponds to A = A t ( r ) d t � � d 8 x e − φ � S D 7 = − T 7 −| G + F + B | + T 7 C 8 − C 6 ∧ ( F + B )+ · · ·

When does the charge density matter [Chen et al ’09] [Bigazzi et al ’11] ⇒ Take the e.o.m. for the NS form (and set B = 0) d ( e − 2 φ ∗ H ) = 0 = F 3 ∧ F 5 + F 1 ∧ ∗ F 3 + ( DBI ) ∗ d t ∧ d r ⇒ From the second term in RHS we deduce a component F 3 ⊃ C ′ ( r ) d t ∧ d r ∧ η KE + · · · ⇒ From the first term in RHS we deduce a constant (density) term F 3 ⊃ N q d x 1 ∧ d x 2 ∧ d x 3

When does the charge density matter [Faedo et al ’14] ⇒ Same game as before: When is the effect of charge comparable to the effect of color physics? � � 3 � E charge � 3 λ 2 / 3 ( N q / N 2 ) 1 / 3 | F 3 | | F 5 | = ≡ E E so charge becomes important in IR for E < E charge . ⇒ Similarly, we can show from | F 1 | / | F 3 | that the charge dominates always in the IR.

When does the charge density matter [Faedo et al ’14] ⇒ The same argument also works for the D2/D6 system � � 4 � E charge � 4 λ 1 / 2 ( N q / N 2 ) 1 / 4 | F 2 | ≡ | F 6 | = E E so charge becomes important in IR for E < E charge . | / | F charge ⇒ Similarly, we can show from | F flavor | that the charge 2 2 dominates always in the IR.