Black Hole Microstate Counting and Their Macroscopic Counterpart - - PowerPoint PPT Presentation

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

Black Hole Microstate Counting and Their Macroscopic Counterpart

Ashoke Sen

Harish-Chandra Research Institute, Allahabad, India

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

Plan

• 1. Introduction and motivation
• 2. Microstate counting
• 3. Macroscopic analysis

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

By now there are many examples in string theory where the correspondence between black hole entropy and statistical entropy has been tested for extremal BPS black holes. SBH(Q) = Sstat(Q) SBH = A/4GN, Sstat = ln dmicro A: Area of the event horizon Q: charges carried by the black hole dmicro: microscopic degeneracy of the system of branes which carry the same charges as the black hole. Initial tests were carried out for large charges for which the computation simplifies on both sides.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

This suggests that in the large charge limit a black hole represents an ensemble of microstates whose total number is given by exp[SBH]. What happens beyond the large charge limit? On the microscopic side we can, in principle, count states to arbitrary accuracy. Is the microscopic description more fundamental, and black holes only capture some average properties in the limit of large size? Or, does a black hole contain complete information about the ensemble?

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

For example:

• 1. Do black holes encode systematically corrections to the

entropy due to finite size effect?

• 2. Are black holes capable of computing the distribution of

global quantum numbers among the microstates?

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

In order to answer the first question we need to understand how to compute corrections to the Bekenstein-Hawking formula for finite size black holes. In order to address the second question we need to go beyond black hole thermodynamics and compute finer properties e.g. Tr(g) for a global Z ZN symmetry generator g. We shall see that AdS2/CFT1 correspondence helps us address both these questions. At the same time we shall develop the microscopic counting techniques so that we can compute the finite size corrections, distribution of global quantum numbers etc.

• n the microscopic side.

We can then compare the two sides.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

The role of index

The counting of microstates is always done in a region where gravity is weak and hence the states do not form a black hole. In order to be able to compare it with the black hole entropy we must focus on quantities which do not change as we change the coupling from small to large value. – needs appropriate supersymmetric index. The appropriate index in D=4 is the helicity trace index.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

Suppose we have a BPS state that breaks 2n supersymmetries. → there will be 2n fermion zero modes (goldstino) on the world-line of the state. Quantization of these zero modes will produce Bose-Fermi degenerate states. Thus Tr(−1)F vanishes. Define: Bn = (−1)n/2 Tr(−1)F(2h)n = (−1)n/2 Tr(−1)2h(2h)n

Bachas, Kiritsis

h: third component of angular momentum in rest frame. For every pair of fermion zero modes Tr(−1)F(2h) gives a non-vanishing result, leading to a non-zero Bn.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

Example: For 1/4 BPS black holes in N = 4 supersymmetric string theories we have 2n = 12. Thus the relevant index is B6. If g is a Z ZN symmetry generator that commutes with supersymmetry generators, then we can also consider Bg

6 = −Tr

• (−1)2h(2h)6g
• Note: Since on the microscopic side we compute an index,

we must ensure that on the black hole side also we compute an index. Otherwise we cannot compare the two results.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

Progress in microscopic counting

In a wide class of N = 4 string theories in (3+1) dimensions one now has a complete understanding of the microscopic index of supersymmetric black holes. Typically such theories have multiple Maxwell fields. ⇒ the black hole is characterized by multiple electric and magnetic charges, collectively denoted by (Q, P). The index B6 is expressed as a function of the charges.

Dijkgraaf, Verlinde, Verlinde; Shih, Strominger, Yin; David, Jatkar, A.S.; Dabholkar, Gaiotto, Nampuri; Banerjee, Srivastava, A.S.; Dabholkar, Gomes, Murthy; · · ·

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

In these theories B6 is given by the triple Fourier transform

• f the inverse of a modular form of a subgroup of Sp(2, Z

Z): B6 = (−1)Q.P+1

• dρ
• dσ
• dv e−πi(ρQ2+σP2+2vQ·P)

1 Φ(ρ, σ, v) Q2, P2, Q.P: three T-duality invariant bilinears in charges Φ(ρ, σ, v): explicitly known in each of the examples, and transform as modular forms of certain weights under subgroups of Sp(2, Z Z).

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

It is also possible to find the systematic expansion of B6 for large charges. ln |B6| = π

• Q2P2 − (Q.P)2 + f
• Q.P

P2 ,

• Q2P2 − (Q.P)2

P2

• +O(charge−2)

f: a known function.

Cardoso, de Wit, Kappeli, Mohaupt; David, Jatkar, A.S.

For example, for heterotic string theory compactified on a six dimensional torus, f(τ1, τ2) = 12 ln τ2 + 24 ln η(τ1 + iτ2) + 24 ln η(−τ1 + iτ2) η: Dedekind function

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

On special subspaces of the parameter space of the N = 4 supersymmetric string theories in (3+1) dimensions, the theory develops Z ZN discrete symmetries which commute with supersymmetry. Each theory has a certain set of allowed values of N. In each case we can calculate the twisted index Bg

6, and

find that the result is again given by Fourier integrals of inverses of modular forms of subgroups of Sp(2, Z Z). Bg

6 = (−1)Q.P+1

• dρ
• dσ
• dv e−πi(ρQ2+σP2+2vQ·P)

1 Φg(ρ, σ, v)

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

Furthermore for large charges we find Bg

6 = exp[π

• Q2P2 − (Q · P)2/N + · · · ]

All these results provide us with the ‘experimental data’ to be explained by a ‘theory of black holes’.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

Progress in black hole entropy computation

We shall first describe how to compute B6 ≡ − 1 6!Tr[(−1)2h(2h)6]

• f a black hole.

First step: Relate index to degeneracy

A.S., arXiv:0903.1477 Dabholkar, Gomes, Murthy, A.S., to appear

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

A quarter BPS black hole in N = 4 supersymmetric string theory breaks 12 supersymmetries. This leads to 12 fermion zero modes with support outside the horizon (called the hair modes). The trace in B6 receives contribution from these hair modes and also the horizon. After tracing over the hair modes we get B6 = Trhor(−1)2hhor Supersymmetry = ⇒ hhor = 0 = ⇒ B6 = Trhor(1) = dhor → degenercay associated with the horizon.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

Computation of dhor

In the leading order dhor = exp[SBekenstein−Hawking] In string theory this receives two types of corrections.

1

Higher derivative (α′) corrections in classical string theory.

2

Quantum (gs) corrections. Of these the α′ corrections are captured by Wald’s modification of the Bekenstein-Hawking formula. Thus in classical string theory dhor = exp[Swald]

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

Near horizon geometry of an extremal, BPS black hole in four dimensions always has the form of AdS2 × S2 × K. Using this Wald’s formula takes a simple form:

A.S.

Swald = 2π

• qiei −
• det gAdS2 LAdS2
• ei: near horizon electric fields

qi: electric charges conjugate to ei gAdS2: metric on AdS2 det gAdS2LAdS2: Classical Lagrangian density, evaluated

• n the near horizon geometry and integrated over S2 × K.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

What are the quantum corrections to dhor? We can apply the rules of AdS/CFT correspondence due to the AdS2 factor.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

Steps for computing dhor

• 1. Consider the euclidean AdS2 metric:

ds2 = v

• (r2 − 1)dθ2 +

dr2 r2 − 1

• ,

1 ≤ r < ∞, θ ≡ θ + 2π = v(sinh2 η dθ2 + dη2), r ≡ cosh η, 0 ≤ η < ∞ Regularize the infinite volume of AdS2 by putting a cut-off r ≤ r0f(θ) for some smooth periodic function f(θ). This makes the AdS2 boundary have a finite length L.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

• 2. Define ZAdS2: Path integral over string fields in the

euclidean near horizon background geometry weighted by exp[−Action − iqk

• ∂(AdS2)

dθ A(k)

θ ]

{qk}: electric charges carried by the black hole under the U(1) gauge field A(k).

• 3. By AdS2/CFT1 correspondence:

ZAdS2 = ZCFT1 ZCFT1 = Tr(e−LH) = d0 e−L E0 H: Hamiltonian of dual CFT1 at the boundary of AdS2. (d0, E0): (degeneracy, energy) of the states of CFT1.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

ZAdS2 = ZCFT1 = d0 e−L E0

• 4. Thus we can define dhor by expressing ZAdS2 as

ZAdS2 = eCL × dhor as L → ∞ C: A constant dhor: ‘finite part’ of ZAdS2. With this definition dhor calculates d0, ı.e. the degeneracy

• f the dual CFT1.

Consistency check: dhor gives us back exp[Swald] in the classical limit.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

One loop correction to dhor from massive string states can be incorporated using Wald’s formula, with classical action replaced by one loop effective action. This reproduces correctly all the order 1 corrections to ln |B6|.

Cardoso, de Wit,Kappeli, Mohaupt; David, Jatkar, A.S.

The full quantum expression for dhor has been used to explain some logarithmic and exponentially suppressed terms in the known microscopic results.

Banerjee, Jatkar, A.S; A.S. Murthy, Pioline Banerjee, Gupta, A.S.

More tests are underway.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

The twisted index

Suppose in an N = 4 SUSY string theory we want to compute the twisted index Bg

6 = − 1

6! Tr(−1)2h (2h)6 g g: some Z ZN symmetry generator that commutes with SUSY. After separating out the contribution from the hair degrees

• f freedom, one finds that the relevant quantity associated

with the horizon is Trhor((−1)2hhorg) = Trhor(g) What macroscopic computation should we carry out?

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

By following the logic of AdS/CFT correspondence we find that Trhor(g) is given by the finite part of a twisted partition function – the path integral is to be carried out over fields satisfying g twisted boundary condition under θ → θ + 2π. Other than this, the asymptotic boundary condition must be identical to that of the original geometry since the charges have not changed

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

Recall euclidean AdS2 metric: ds2 = v

• (r2 − 1)dθ2 +

dr2 r2 − 1

• = v
• dη2 + sinh2 ηdθ2

r = cosh η, 1 ≤ r < ∞, 0 ≤ η < ∞, θ ≡ θ + 2π The boundary circle, parametrized by θ, is contractible at the origin r = 1. Thus a g twist under θ → θ + 2π is not admissible. → the original near horizon geometry is not a valid saddle point of the path integral.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

Question: Are there other saddle points which could contribute to the path integral? Main constraints:

• 1. It must have the same asymptotic geometry as the
• riginal near horizon geometry.
• 2. It must have a g twist under θ → θ + 2π.
• 3. It must preserve sufficient amount of supersymmetries

so that integration over the fermion zero modes do not make the integral vanish.

Beasley, Gaiotto, Guica, Huang, Strominger, Yin; Banerjee, Banerjee, Gupta, Mandal, A.S.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

There are indeed such saddle points in the path integral, constructed as follows.

• 1. Take the original near horizon geometry of the black

hole.

• 2. Take a Z

ZN orbifold of this background with Z ZN generated by simultaneous action of a) 2π/N rotation in AdS2 (θ → θ + 2π N ) b) g c) 2π/N rotation in S2 (φ → φ + 2π N ) (needed for preserving SUSY, but otherwise does not affect

• ur analysis)

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

The classical action associated with this saddle point, after removing the divergent part proportional to the length of the boundary, is Swald/N. Thus the contribution to the twisted index Bg

6 from this

saddle point is |Bg

6| = Zfinite g

= exp [Swald/N] in agreement with the microscopic results.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion

Conclusions

Quantum gravity in the near horizon geometry contains detailed information about not only the total number of

• microstates. but also finer details e.g. the

Z ZN quantum numbers carried by the microstates. Thus at least for extremal black holes there seems to be an exact duality between Gravity description ⇔ Microscopic description The gravity description contains as much information as the microscopic description, but in quite different way.