Quantum Entropy of Black Holes

Transcription

1 Quantum Entropy of Black Holes A"sh Dabholkar ICTP Sorbonne Universités & CNRS Eurostrings 2015 Cambridge NONPERTURBATIVE 1

2 References A Dabholkar, João Gomes, Sameer Murthy , , A Dabholkar, João Gomes, Sameer Murthy, ValenBn Reys, 1504.nnnn A Dabholkar, Sameer Murthy, Don Zagier

3 New Results Some of these results have been reported elsewhere, for example, at Strings 2014 and earlier. I will give a review and then will focus more on the nonperturba@ve aspects of this computa@on. There has also been some recent progress in addressing aspects of perturba@ve computa@on that were poorly understood earlier. In par@cular the computa@on of one- loop determinants. Extensions to less supersymmetric examples. 3

4 Hurdles for String Theory We don t have a super- LHC to probe the theory directly at Planck scale. We don t even know which phase of the theory may correspond to the real world. How can we be sure that string theory is the right approach to quantum gravity in the absence of direct experiments? A useful strategy is to focus on universal features that must hold in all phases of the theory. 4

5 Quantum Black Holes Any black hole in any phase of the theory should be interpretable as an ensemble of quantum states including finite size effects. Universal and extremely stringent constraint An IR window into the UV Connects to a broader problem of Quantum Holography at finite N. 5

6 Near horizon of a BPS black hole has factor. More generally, near horizon physics of black p- branes leads to AdS p+2 /CF T p+1 AdS p+2 /CF T p+1 AdS 2 holography. A bulk of the work in holography is in infinite N limit, using classical gravity to study quantum CFT. Our interest will be in quantum gravity in the bulk. ANer all, a primary mobvabon for string theory is unificabon of General RelaBvity with QM. 6

7 Quantum Holography Does holography really hold at finite N? The black hole story seems to suggest Yes! More generally: AdS 2 : NonperturbaBve quantum entropy of black holes including all finite size correc@ons. AdS 3 : An unexpected connecbon to the mathemabcs of mock modular forms. AdS 4 : New localizing instantons in bulk supergravity for finite N Chern- Simons- MaXer. (See talk by JOÃO GOMES) 7

8 in Supergravity has enabled many powerful in QFT over the past two decades. One could access aspects of strongly coupled QFT which were otherwise inaccessible. Our results can be viewed as a beginning of a similar program for quantum gravity. Rules are less clear. Comparison with boundary results is a useful guide. One hopes these methods will develop further in the coming years. 8

9 One of the most important clues about quantum gravity is the entropy of a black hole: What is the exact quantum generaliza"on of the celebrated Bekenstein- Hawking formula? S = A 4 + c 1 1 log(a)+c 2 A...+ e A +... How to define it? How to compute it? The exponen@al of the quantum entropy must yield an integer. This is extremely stringent. Subleading correc@ons depend sensi@vely on the phase & provide a window into the UV structure. 9

10 Defining Quantum Entropy The near horizon of a BPS black hole of charge vector Q is AdS 2 so one can use holography. Quantum entropy can then be defined as a path integral W(Q) in AdS 2 over all string fields with appropriate boundary condibons, operator inserbon, and a renormalizabon procedure. Sen (09) For large charges, logarithm of W(Q) reduces to Bekenstein- Hawking- Wald entropy. 10

11 Quantum Entropy. Integrate out massive string modes to get a Wilsonian effec@ve ac@on for massless fields. S@ll need to make sense of the formal path integral of supergravity fields. Using it do explicit computa@ons is fraught with danger. It helps to have microscopic degeneracies from brane coun@ng to compare with: W (Q) =d(q) d(q) 11

12 One- eighth BPS states in N=8 Type- II on Dyonic states with charge vector (Q, P) U- duality invariant = Q 2 P 2 (Q P ) 2 Degeneracy given by Fourier coefficients C( ) of #(,z) 2 ( ) 6 T 6 Maldacena Moore Strominger (99) d( )=( 1) +1 C( ) 12

13 Hardy- Ramanjuan- Rademacher Expansion An exact convergent expansion (using modularity) 1X C( )=N c 9/2 p Ĩ 7/2 K c ( ) c Ĩ 7/2 (z) = 1 2 i c=1 Z +i1 ds z2 exp[s + i1 s9/2 4s ] exp hz 4 log z + c i z +... z = A/4 The c=1 Bessel func@on sums all perturbabve (in 1/z) correc@ons to entropy. The c>1 are non- perturba@ve 13

14 Generalized Kloosterman Sum K c ( ) X cappled<0; (d,c)=1 e 2 i d c ( /4) M 1 ( c,d ) 1 e 2 i a c ( 1/4) = mod 2 Relevant only in exponen@ally subleading nonperturba@ve correc@ons. Even though highly subleading, conceptually very important for integrality. New results concerning these nonperturba@ve phases 14

15 System M 1 ( ) is a par@cular two- dimensional representa@on of the SL(2, Z) matrix M 1 (T )= i M 1 (S) = e i p Use the con@nued frac@on expansion: = a c b d = T m t S...T m 1 S 15

16 W ( ) The structure of the microscopic answer suggests that W ( ) should have an expansion 1X W ( )= W c ( ) c=1 Each W c ( ) corresponds to a Z c orbifold of the Euclidean near horizon black hole geometry. The higher c are exponen@ally subleading. Unless one can evaluate each of them exactly it is not par@cularly meaningful to add them. LocalizaBon enables us to do this. 16

17 Path Integral on AdS 2 Including the M- theory circle, there is a family of geometries that are asympto@cally AdS 2 S 1 : ds 2 =(r 2 1 c 2 )d 2 + Z c dr2 r 2 1 c 2 + R 2 dy These are orbifolds of BTZ black hole. These geometries M c,d all have the same asympto@cs and contribute to the path integral. Related to the SL(2, Z) family in AdS 3 i R (r 1 c )d + d c d Maldacena Strominger (98), Sen (09), Pioline Murthy (09) 2 17

18 If a supersymmetric integral is invariant under a localizing supersymmetry Q which squares to a compact generator H, then the path integral localizes onto fixed manifold of the symmetry Q. We consider localiza@on in N=2 supergravity coupled to n v vector mul@plets whose chiral ac@on is governed by a prepoten@al F. We find the localizing submanifold len invariant by Q and evaluate the renormalized acbon. 18

19 Off- shell Localizing We found off- shell localizing instantons in AdS 2 for supergravity coupled to n v vector mul@plets with scalars X I and auxiliary fields Y I X I = X I + CI r, Y I = CI r 2, C I 2 R I =0,...,n v These solu@ons are universal in that they are independent of the physical acbon and follow en@rely from the off- shell susy transforma@ons. 19

20 Renormalized The renormalized for F is 1 S ren (,q,p)= q I I + F(,p) F(,p)= 2 i apple F I + ip I is the off- shell value of X at the origin. 2 ( I + ip I ) I For each orbifold one obtains a Laplace integral of Z top 2 reminiscent of OSV conjecture. Ooguri Strominger Vafa (04) Cardoso de Wit Mohaupt (00) 2 F I 2 ip I 20

21 Final integral The for the truncated theory is F (X) = 1 2 X 1 X 0 7X a,b=2 C ab X a X b (n v = 7) (dropping the extra gravi@ni mul@plets) The path integral reduces to the Bessel integral Z ds W 1 ( )=N s exp [s + 2 9/2 4s ] W 1 ( )=Ĩ7/2( p ) 21

22 Weyl In conformal supergravity in our gauge, conformal factor of the metric has been traded for a scalar field. The fiducial metric is held fixed but the physical metric has a nontrivial profile for the off- shell solu@on. Rather like Liouville in 2d. There are no addi@onal solu@ons from the Weyl mul@plet that contains the metric Gupta Murthy (12) Subtle@es with localiza@on (because metric is fluctua@ng) that need to be understood in beqer. 22

23 Degeneracy, Quantum Entropy, Wald Entropy C( ) W 1 ( ) p exp( )

24 Note that C( ) d( )=( 1) +1 C( ) are alterna@ng in sign so that is strictly posi@ve. This is a predic@on from IR quantum gravity for black holes which is borne out by the UV. This explains the Bessel func@ons for all c with correct argument because for each orbifold the ac@on is reduced by a factor of c What about the Kloosterman sums? It was a long standing puzzle how this intricate number theore@c structure could possibly arise from a supergravity path integral. 24

25 Kloosterman from Supergravity Our analysis thus far is local and to global topology. The Chern- Simons terms in the bulk and the boundary terms are to the global of M c,d Addi@onal contribu@ons to renormalized ac@on and addi@onal saddle points specified by holonomies of flat connec@ons. Various phases from CS terms for different groups assemble neatly into precisely the Kloosterman sum! 25

26 Kloosterman and Chern- Simons Z I(A) = Tr A ^ da + 23 A3 M c,d In our problem we have three relevant groups U(1) n v+1 SU(2) L SU(2) R X cappled<0; (d,c)=1 e 2 i d c ( /4) M 1 ( c,d ) 1 e 2 i a c ( 1/4) 26

27 Dehn M c,d The geometries are topologically a solid 2- torus and are related to by Dehn- filling. M 1,0 Relabeling of cycles of the boundary 2- torus: Cn C c = a c b d C1 C 2 for a c b d 2 SL(2, Z) is contrac@ble and is contrac@ble and is noncontrac@ble in C 1 C 2 M 1,0 is noncontrac@ble in C c C n M c,d 27

28 Boundary and Holonomies The cycle C 2 is the M- circle and C 1 is the boundary of AdS 2 for the reference geometry M 1,0 This implies the boundary condi@on I A I =fixed, A I = not fixed C 2 C 1 and a boundary term I b (A) = Z I T 2 TrA 1 A 2 d 2 x Elitzur Moore Schwimmer Seiberg (89) 28

29 from I I C 1 A =2 i C c A =2 i SU(2) R Chern- Simons contribu@on to the renormalized ac@on is completely determined knowing the holonomies. For abelian the bulk contribu@on is zero for flat connec@ons and only boundary contributes. I I C 2 A =2 i C n A =2 i I b [A R ]=2 2 I[A R ]=2 2 Kirk Klassen (90) 29

30 I C 1 A R = 2 i c 3 2, I C 2 A R =0 Supersymmetric Z c orbifold J R =0 = 1/c, =0, = 1, = a/c (using = c + d, = a + b ) The total contribu@on to renormalized ac@on is S ren = 2 ik R 4 in perfect agreement with a term in Kloosterman. a c k R =1 30

31 System from SU(2) L There is an explicit representa@on of the Mul@plier matrices that is suitable for our purposes. M 1 ( ) µ = C X =± Xc 1 n=0 e i 2rc[d( +1) 2 2( +1)(2rn+ (µ+1))+a(2rn+ (µ+1)) 2 ] Unlike SU(2) R the holonomies of SU(2) L are not constrained by supersymmetry and have to be summed over which gives precisely this matrix. (Assuming usual shir of k going to k +2 ) 31

32 Knot Theory and Kloosterman This is closely related to knot invariants of Lens space L c,d using the surgery formula of Wiqen. WiXen (89) Jeffrey (92) This is not an accident. Lens space is obtained by taking two solid tori and gluing them by Dehn- twis@ng the boundary of one of them. But Dehn- twisted solid torus is our M c,d Intriguing rela@on between topology and number theory for an appropriate CS theory. 32

33 Quantum Entropy: An Assessment Choice of Ensemble: AdS 2 boundary condi@ons imply a microcanonical ensemble. Sen (09) Supersymmetry and AdS 2 boundary condi@ons imply that index = degeneracy and J R =0 Sen (10) Dabholkar Gomes Murthy Sen (12) Path integral localizes and the localizing solu@ons and the renormalized ac@on have simple analy@c expressions making it possible to even evaluate the remaining finite ordinary integrals. DGM (10, 11) 33

34 from orbifolded localizing instantons can completely account for all to the quantum entropy. All intricate details of Kloosterman sum arise from topological terms in the path integral. DGM (14) (Most) D- terms evaluate to zero on the localizing de Wit Katamadas Zalk (10) Murthy Reys (13) Path integral of quantum gravity (a complex analybc conbnuous object) can yield a precise integer (a number theorebc discrete object). W (Q) = Z d e S[ ] = integer 34

35 Open Problems? of the measure including one- loop determinants. with gauge fixing.? We used an N=2 of N=8 supergravity. This should be OK for finding the localizing instantons because the near horizon has N=2 susy. But it s a truncabon.? We dropped the hypermul@plets. They are known not to contribute to Wald entropy but could contribute to off- shell one- loop determinants. 35

36 One- loop determinants of one- loop determinants is subtle because of gauge fixing. The charge is a combina@on of susy and BRST. Similar to computa@on by Pestun. For vector mul@plets and hypermul@plets coupled to gravity this computa@on has recently been done in agreement with on- shell results. Addi@onal subtle@es in supergravity in gauge- fixing conformal supergravity to Poincaré supergravity. Murthy Reys; Gupta Ito Jeon (to appear) In the N=8 case, the net contribu@on of the fields that we have dropped is zero. Jus@fies the trunca@on. 36

37 Off- shell supergravity? It would be useful to have off- shell of the two localizing supercharges on all fields of N=8 Hard but doable technical problem.? We have treated gravity as any other in a fixed background. This is not fully jus@fied for very off- shell configura@ons. Kloosterman sum arising from topological terms is essenbally independent of these subtlebes. 37

38 An IR Window into the UV The degeneracies d(q) count brane bound states. These are nonpertubabve states whose masses are much higher than the string scale. Our supergravity of W(Q) can apparently access this with precision. If we did not know the spectrum of branes a priori in the N=8 theory then we could in principle deduce it. For example, in N=6 models d(q) is not known but the sugra computa@on of W(Q) seems doable. 38

39 Platonic Elephant of M- theory Quantum gravity seems more like an equivalent dual rather than a coarse- graining. It is not only UV- complete (like QCD) but UV- rigid. E. g. Small change in the effec@ve ac@on of an irrelevant operator will destroy integrality. AdS/CFT is just one solitonic sector of the theory. It seems unlikely that we can bootstrap to construct the whole theory from a single CFT which for a black hole is just a finite dimensional vector space. 39

40 One- quarter BPS states in N=4 Now there are three new phenomena. The spectrum jumps upon crossing walls in moduli space. An apparent loss of modularity. There is a nontrivial higher deriva@ve quantum contribu@on to the effec@ve ac@on from brane- instantons. Need to include these effects. The coun@ng func@on is a Siegel modular form and has many more `polar terms. As a result, the Rademacher expansion is more intricate. 40

41 AdS 3 /CF T 2 and Mock Modular Forms AdS 3 Euclidean has a 2-torus boundary and we expect modular symmetry for the partition function. Oren the asymptotic degeneracy includes contributions from not only single-centered black holes but also multi-centered bound states of black holes. Related to Wall-crossing phenomenon. Isolating the single-centered contribution microscopically is subtle. The counting function is then no longer modular as expected. Modular symmetry is apparently lost! 41

42 Loss of modularity is a serious problem. It means loss of general coordinate invariance in the context of AdS3/CFT2. It is far from clear if and how modular symmetry can be restored. We obtained a complete solu@on to this problem for black strings with N=4 supersymmetry. It naturally involves mock modular forms. We obtained a number of new results in the mathema@cs of mock modular forms mo@vated by this physics. Signifies noncompactness of boundary CFT. 42

43 Theorem The counting function of single-centered black hole is a mock Jacobi form. The coun@ng func@on of mul@- centered black holes is an Appel Lerch sum. Neither is modular but both admit a modular comple@on by an addi@ve nonholomorphic correc@on term restoring modular symmetry! Dabholkar Murthy Zagier (12) 43

44 Instantons and Polar Terms Expanding the instanton contributions appear to give infinite number of Bessel integrals. But the microscopic results indicate a chargedependent choice of contour for the Bessel integral. As a result only a finite number give I-Bessel functions. Dabholkar Gomes Murthy Reys (work in progress) The N=4 effective action appears to have the required ingredients. 44