Gauge invariance and Holography

Transcription

1 Gauge invariance and Holography Sang-Jin Sin (Hanyang U.) Based on arxiv:

2 collaborators Keun-Young Kim*, Kyung-Kyu Kim*, Yunseok Seo# *GwangJu Institute of Science and technology, #Hanyang univ. Based on arxiv: , see also 2

3 What is Holography Viewing different scale simultaneously With RG encoded in Einstein equation. Exact duality `proven only for N=4 SYM/AdS5 however.. 3

4 Emergence of holography in other context MERA is a discretization of AdS! (Multiscale Entanglement Renomaliation Ansatz Vidal 2007) [Swingle] Hope: Holography is a generic phenomena in strongly coupled system near criticality 4

5 [Moviation] Challenging Physics goal StCorElectron systems: ex: the Cuprate High Tc Royal Society publishing, D. Galanakis et.al 5

6 Here: Back to the Basic Q How holography works? 1. d vs d+1: huge extra killed by gauge inv. GI is the key and holo Ren. Is the holder. 2. Infalling bc to all fields? No! crisis in matching d.o.f. & its resolution 3. Global vs Local sym Translation sym Diffeomorphism 4. Ward identity in CFT is encoded in eqs of moti on in the bulk (Skendris). Why? 5. We will see that basic reason is the way hologr aphy treat the symmetry as a gauge inv. 6

7 Bulk vs boundary Boundary theory is equivalent to bulk theory Boundary has to be a part of the bulk. These sometimes raises subtleties. 7

8 Start with a specific model RN AdS black holes + scalar Bardoux, Caldarelli, Charmousis (2012), Andrade, Withers(2013) 8

9 EOMs RN-AdS solution + two scalars 9

10 Fluctuation dynamics green functions Fluctuations EOMs in Independent Boundary action 10

11 How Holography works Embedding boundary theory into bulk theory introdu ceshuge tower of extra spectrum. (d d+1) All these extra d.o.f should be constrained by local g auge invariance. That is, Diffeomorphism invariance is thekey to the equivalence. Holographic renormalization plays a role to GI. 11

12 Dependence on the gauge hrx=0 12

13 GI and Holographic renormalizatio n 13

14 Scheme for conductivity calculation : (illustration with RN) Fluctuations EOMs Boundary action Linear response Two issues for generalisation 1. Coupled case 2. identify the sources and currents Hartnoll

15 Infalling Boundary conditions To solve the eqs. We need BC. Two boundaries. At bdy: always Dirichlett. At horizon? For causal green function, we need in-falling bc. Question: Can we impose the infalling bc for all fields involved? No (Surprising!) If a_x decouples, no problem. What if it does not? (here, Lifshitz, HSV, ) See eqs. we developed a formalism [ , ] 15

16 16 Coupled nature of variables

17 BC at horizon Boudnary value problem Initial value problem Dirichlett Infalling Infalling + Dirichlett Determined by horizon regularity by the first three 3 fields 2 fields Due to the specific nature of infalling bc Near bdy, no such constraint. 17

18 Mismatch in degrees of freedo m At the Horizon 2 dgree of freedom with infalling bc At the boundary 3 degree of freedom (3 source fields). Why? 2 3 impossible! 18

19 Resolution? Hartnoll and Kovtun : introduced a constant solutio n: So not a bulk degree of freedom? Then crisis remains But it fullfil the goal to generate 3 boundary d.o.f Possibly one more reason not to accept it? It is on the gauge orbit of a coordinate transformati on. 19

20 Orbit of Zero mode of diffeo. How a configuration on the gauge orbit can be accepted as a new degree of freedom? 漸入佳境 20

21 Way out? Problem is defined by Dirichlett at boundary + infalling at Horizon So any gauge symmetry should fix these two bc. Boundary shifting diffeomorphism generator does n ot satisfy thses. So it is not a gauge symmetry. However, it is a symmetry of eq. of motion. Therefore we should regard the transformtion as sol ution generating symmetry not a gauge symmetry. So its orbit generate solution both in bdy and bulk. 21

22 What is happening? We can not impose independent infallin g b.c. at the horizon to all fields. If we d o, we get sigularity at the horizon even before you evolve along r-direction. If we run back, i.e, start from the bound ary rather than from the horizon, we will generically get a constant piece + infalli ng solution. 22

23 Why this happen? x x+a e^{-iwt} is a translation sym. at bdy theory which generate Such symmetry gives Ward identity of the boundary theory and therefo re should be allowed. Notice the shifted fiedls are bulk solution and constant parts CAN NOT satisfy the infalling bc. To allow such solution at the bdy, bulk solution should not allow 3 infalling bc. This is the way holography encodes the traslational symmetry. REMARK: This argument works in hrx=0 gauge. 23

24 Summary: Anatomy of holography Set of No constraint here R-evolution Set of 24

25 Can non-infalling d.o.f. physical? Yes. Otherwise, we have d.o.f mismatch. Having a physical degree of freedom that does not satisfy the infalling degree of freedom is crutial to h ave translation symmetry. This is the way holography can encodes the translra tional symmetry. If you do not feel good, we can do better: we can e xtend coordinate transformation such that it satisfie s infalling bc. But the idea is the same. 25

26 Ward id and holography. Ward identity: consequence of sym One of Bulk eq. of Motion One can show the equivalence of these two using Q: How bulk e.o.m s include the ward identity at the boundary? 26

27 Ans: 1. Boundary translational symmetry is included as the zero mode of the bulk diffeomorphism. 2. Identification of bulk and boundary on-shell actions Ward identity Gauge invariance constraint Equation of motion 27

28 Summary and Conclusion Holography works by Gauge invariance. Holographic ren. Is a key to GI. Infalling bc is not allowed to all fields to accommodate translation inv of the bdy. Holography treats symmetry and gauge invarian ce in the same way Origin of subtlety. Ward identity is encoded into eq.o.m in bulk sin ce translation inv is embedded into the bulk ga uge invariance. The latter three are different aspect of the same. 28