Nonperturba ve Quantum Entropy of Black Holes
A sh Dabholkar ICTP Sorbonne Universités & CNRS
Eurostrings 2015 Cambridge
NONPERTURBATIVE QUANTUM ENTROPY OF BLACK HOLES ATISH DABHOLKAR 1 References
• A Dabholkar, João Gomes, Sameer Murthy 1404.0033, 1111.1161, 1012.0265
• A Dabholkar, João Gomes, Sameer Murthy, Valen n Reys, 1504.nnnn
• A Dabholkar, Sameer Murthy, Don Zagier 1208.4074
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 2 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.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 3 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.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 4 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.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 5 AdSp+2/CF Tp+1
• Near horizon of a BPS black hole has factor. AdS2 More generally, near horizon physics of black p-
branes leads to holography. AdSp+2/CF Tp+1 • 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. A er all, a primary mo va on for string theory is unifica on of General Rela vity with QM.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 6 Quantum Holography
Does holography really hold at finite N? The black hole story seems to suggest Yes! More generally:
• AdS : 2 Nonperturba ve quantum entropy of black holes including all finite size correc ons.
• AdS 3 : An unexpected connec on to the mathema cs of mock modular forms.
• AdS : 4 New localizing instantons in bulk supergravity for finite N Chern-Simons-Ma er. (See talk by JOÃO GOMES)
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 7 Localiza on in Supergravity
• Localiza on has enabled many powerful computa ons 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.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 8 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?
A 1 A S = + c log(A)+c ...+ e + ... 4 1 2 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.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 9 Defining Quantum Entropy • The near horizon of a BPS black hole of charge vector Q is AdS2 so one can use holography. • Quantum entropy can then be defined as a path integral W(Q) in AdS2 over all string fields with appropriate boundary condi ons, operator inser on, and a renormaliza on procedure. Sen (09) • For large charges, logarithm of W(Q) reduces to Bekenstein-Hawking-Wald entropy.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 10 Compu ng 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 d(Q) from brane coun ng to compare with:
W (Q)=d(Q)
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 11 One-eighth BPS states in N=8
• Type-II compac fied on T 6 • Dyonic states with charge vector (Q, P) • U-duality invariant = Q2P 2 (Q P )2 · • Degeneracy given by Fourier coefficients of C( ) #(⌧,z)2 ⌘(⌧)6 Maldacena Moore Strominger (99)
d( )=( 1) +1C( )
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 12 Hardy-Ramanjuan-Rademacher Expansion An exact convergent expansion (using modularity)
1 9/2 ⇡p C( )=N c I˜ K ( ) 7/2 c c c=1 X ✏+i 2 ˜ 1 1 ds z I7/2(z)= 9/2 exp[s + ] 2⇡i ✏ i s 4s Z 1 c exp z 4 log z + + ... z = A/4 ⇠ z The c=1 Bessel func on sums h all perturba vei (in 1/z) correc ons to entropy. The c>1 are non-perturba ve
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 13 Generalized Kloosterman Sum Kc( )
2⇡i d ( /4) 1 2⇡i a ( 1/4) e c M ( c,d)⌫1 e c c d<0; (d,cX)=1 ⌫ = 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
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 14 Mul plier System
1 is a par cular two-dimensional M ( ) representa on of the matrix SL(2, Z) ab = cd ✓ ◆ ⇡i 1 10 1 e 4 11 M (T )= M (S)= 0 i p2 1 1 ✓ ◆ ✓ ◆ Use the con nued frac on expansion: = T mt S...Tm1 S
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 15 Compu ng W ( ) • The structure of the microscopic answer suggests that should have an expansion W ( ) 1 W ( )= Wc( ) c=1 X • Each corresponds to a orbifold of the Wc( ) Zc 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. Localiza on enables us to do this.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 16 Path Integral on AdS2
• Including the M-theory circle, there is a family of geometries that are asympto cally : AdS S1 2 ⇥ 1 dr2 i 1 d 2 ds2 =(r2 )d✓2 + + R2 dy (r )d✓ + d✓ c2 r2 1 R c c c2 ✓ ◆ • These are orbifolds of BTZ black hole. These Zc geometries all have the same asympto cs Mc,d and contribute to the path integral.
• Related to the family in SL(2, Z) AdS3 Maldacena Strominger (98), Sen (09), Pioline Murthy (09)
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 17 Localiza on
• 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 vector mul plets whose chiral nv ac on is governed by a prepoten al F. • We find the localizing submanifold le invariant by Q and evaluate the renormalized ac on.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 18 Off-shell Localizing Solu ons
• We found off-shell localizing instantons in AdS2 for supergravity coupled to vector mul plets nv with scalars XI and auxiliary fields YI
I I I I I C I C C R X = X + ,Y= 2 , 2 ⇤ r r I =0,...,nv
• These solu ons are universal in that they are independent of the physical ac on and follow en rely from the off-shell susy transforma ons.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 19 Renormalized Ac on
• The renormalized ac on for prepoten al F is ( ,q,p)= ⇡q I + ( ,p) Sren I F I + ipI I ipI ( ,p)= 2⇡i F F¯ F 2 2 ⇣ ⌘ ⇣ ⌘ 1 ( I + ipI ) XI is the off-shell value of at the origin. 2 • For each orbifold one obtains a Laplace integral of reminiscent of OSV conjecture. Z 2 | top| Ooguri Strominger Vafa (04) Cardoso de Wit Mohaupt (00)
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 20 Final integral
The prepoten al for the truncated theory is 1 X1 7 F (X)= C XaXb (n = 7) 2 X0 ab v a,bX=2 (dropping the extra gravi ni mul plets) The path integral reduces to the Bessel integral ds ⇡2 W ( )=N exp [s + ] 1 s9/2 4s Z ˜ W1( )=I7/2(⇡p )
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 21 Weyl Mul plet
• 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 be er.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 22 Degeneracy, Quantum Entropy, Wald Entropy
C( ) W1( ) exp(⇡p ) 3 8 7.972 230.765 4 -12 12.201 535.492 7 39 38.986 4071.93 8 -56 55.721 7228.35 11 152 152.041 22506. 12 -208 208.455 53252. 15 513 512.958 192401
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 23 • Note that are alterna ng in sign so that C( ) is strictly posi ve. d( )=( 1) +1C( ) 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.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 24 Kloosterman from Supergravity
• Our analysis thus far is local and insensi ve to global topology. The Chern-Simons terms in the bulk and the boundary terms are sensi ve to the global proper es of c,d M • 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!
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 25 Kloosterman and Chern-Simons
2 I(A)= Tr A dA + A3 ^ 3 ZMc,d ✓ ◆ In our problem we have three relevant groups
nv +1 U(1) SU(2)L SU(2)R
2⇡i d ( /4) 1 2⇡i a ( 1/4) e c M ( c,d)⌫1 e c c d<0; (d,cX)=1
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 26 Dehn Twis ng
The geometries are topologically a solid Mc,d 2-torus and are related to by Dehn-filling. M1,0 Relabeling of cycles of the boundary 2-torus:
Cn ab C1 ab = for SL(2, Z) Cc cd C2 cd 2 ✓ ◆ ✓ ◆✓ ◆ ✓ ◆
C is contrac ble and is noncontrac ble in C 1,0 1 2 M Cc C c,d is contrac ble and n is noncontrac ble in M
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 27 Boundary Condi ons and Holonomies
• The cycle is the M-circle and is the boundary C2 C1 of for the reference geometry AdS2 1,0 M • This implies the boundary condi on
AI =fixed, AI = not fixed IC2 IC1 and a boundary term 2 Ib(A)= TrA1A2d x 2 Elitzur Moore Schwimmer Seiberg (89)ZT
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 28 Contribu on from SU(2)R 3 3 A =2⇡i A =2⇡i 2 2 IC1 IC2 3 3 A =2⇡i↵ A =2⇡i 2 2 ICc ICn 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.
2 2 Ib[AR]=2⇡ I[AR]=2⇡ ↵ Kirk Klassen (90)
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 29 2⇡i 3 A = , A =0 R c 2 R IC1 IC2
Supersymmetric orbifold Zc JR =0 = 1/c , =0, ↵ = 1 , = a/c (using ) ↵ = c + d , = a + b The total contribu on to renormalized ac on is 2⇡ik a S = R k =1 ren 4 c R in perfect agreement with a term in Kloosterman.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 30 Mul plier System from SU(2)L
There is an explicit representa on of the Mul plier matrices that is suitable for our purposes.
c 1 1 i⇡ [d(⌫+1)2 2(⌫+1)(2rn+✏(µ+1))+a(2rn+✏(µ+1))2] M ( )⌫µ = C ✏ e 2rc ✏= n=0 X± X
Unlike the holonomies of are not SU(2)R SU(2)L constrained by supersymmetry and have to be summed over which gives precisely this matrix. (Assuming usual shi of k going to k +2 )
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 31 Knot Theory and Kloosterman
• This computa on is closely related to knot invariants of Lens space using the surgery Lc,d formula of Wi en. Wi en (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 c,d M • Intriguing rela on between topology and number theory for an appropriate CS theory.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 32 Quantum Entropy: An Assessment
✓ Choice of Ensemble: boundary condi ons AdS2 imply a microcanonical ensemble. Sen (09)
✓ Supersymmetry and boundary condi ons AdS2 imply that index = degeneracy and JR =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)
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 33 ✓Contribu ons from orbifolded localizing instantons can completely account for all nonperturba ve correc ons 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 solu ons de Wit Katamadas Zalk (10) Murthy Reys (13) Path integral of quantum gravity (a complex analy c con nuous object) can yield a precise integer (a number theore c discrete object). S[ ] W (Q)= d e = integer Z ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 34 Open Problems
? Computa on of the measure including one-loop determinants. Subtle es with gauge fixing. ? We used an N=2 trunca on 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 trunca on. ? We dropped the hypermul plets. They are known not to contribute to Wald entropy but could contribute to off-shell one-loop determinants.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 35 One-loop determinants
• Computa on of one-loop determinants is subtle because of gauge fixing. The localiza on 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.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 36 Off-shell supergravity
? It would be useful to have off-shell realiza on of the two localizing supercharges on all fields of N=8 supermul plet. Hard but doable technical problem. ? We have treated gravity mul plet as any other mul plet in a fixed background. This is not fully jus fied for very off-shell configura ons. Kloosterman sum arising from topological terms is essen ally independent of these subtle es.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 37 An IR Window into the UV
• The degeneracies d(Q) count brane bound states. These are nonpertuba ve states whose masses are much higher than the string scale. • Our supergravity computa on of W(Q) can apparently access this informa on 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.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 38 Platonic Elephant of M-theory
• Quantum gravity seems more like an equivalent dual descrip on 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.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 39 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.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 40 AdS 3/CF T2 and Mock Modular Forms
• Euclidean AdS 3 has a 2-torus boundary and we expect modular symmetry for the partition function. • O en 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!
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 41 • 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.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 42 Decomposi on 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)
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 43 Instantons and Polar Terms
• Expanding the instanton contributions appear to give infinite number of Bessel integrals. • But the microscopic results indicate a charge- dependent 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.
ATISH DABHOLKAR QUANTUM ENTROPY OF BLACK HOLES 44