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Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary

Supersymmetry, Localization and Quantum Entropy Function

Ipsita Mandal 10th June, 2010 Based on arXiv:0905.2686 [hep-th] Collaborators : N. Banerjee, S. Banerjee, R. Gupta and A. Sen

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary

Outline

1

Prologue

2

Symmetries of Euclidean AdS2 × S2

3

Localization of Path Integral

4

Some Applications

5

Summary

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Quantum Entropy Function The Theory on AdS2 Classical Limit of QEF Motivation The Logic Results

Quantum Entropy Function

Supersymmetric extremal black holes typically have a near horizon geometry of the form AdS2 × K, where K contains compact directions and angular coordinates, fibred over AdS2. Quantum Entropy Function is a proposal, based on AdS2/CFT1 correspondence, which relates the degeneracy of a single-centred extremal black hole horizon with the partition function ZAdS2 of string theory on AdS2 × K : dhor = fi exp » −iqi I dθA(i)

θ

–flfinite

AdS2

, where <>AdS2denotes the unnormalized path integral weighted by e−A.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Quantum Entropy Function The Theory on AdS2 Classical Limit of QEF Motivation The Logic Results

A is the Euclidean action, with the b.c. that asymptotically the field configuration approaches the near horizon geometry of the black hole containing an AdS2 factor. {A(i)} → set of all U(1) gauge fields living on the AdS2 component of the near horizon geometry; qi → i-th electric charge carried by the black hole; H dθ A(i)

θ

→ integral of the i-th gauge field along the boundary of AdS2. The superscript ‘finite’ → If we represent AdS2 as the Poincare disk, regularize the infinite volume of AdS2 by putting an infrared cut-off and denote by L the length of the boundary, then for large L, the amplitude is eC L+O(L−1) × ∆, where C and ∆ are L-independent constants. The finite part is ∆, and has been named QEF.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Quantum Entropy Function The Theory on AdS2 Classical Limit of QEF Motivation The Logic Results

In order to compare with microscopic degeneracy formula, we also need to take into account the contribution from multicentred BHs. By considering appropriate products of single centred BH degeneracies, the complete expression for a given charge q is dmacro( q) = X

n

X

{

• qi },

qhair Pn i=1 qi +

• qhair =
• q

( n Y

i=1

dhor( qi) ) dhair( qhair; { qi}) . We expect : dmicro( q) = dmacro( q) . In comparing the microscopic and the macroscopic entropies, in the microscopic theory we typically compute the helicity trace index, while the Bekenstein-Hawking entropy or Wald entropy is supposed to compute the logarithm of the absolute degeneracy. So how to compare the two? QEF provides a natural resolution of this puzzle.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Quantum Entropy Function The Theory on AdS2 Classical Limit of QEF Motivation The Logic Results

We focus on 4d BHs, but similar analysis can be carried out in other dimensions. One can show that, in 4d, the helicity trace index on the macroscopic side is : B2k;macro( q) = (−1)k Tr h (−1)2J(2J)2ki /(2k)! = X

n

X

{

• qi },
• qhair

Pn i=1 qi + qhair = q

( n Y

i=1

dhor( qi, Ji = 0) ) B2k;hair( qhair; { qi}) . Relation to verify : B2k;micro( q) = B2k;macro( q) . Since dhor( qi, Ji = 0) is computed by QEF, this provides a way to compare the helicity trace index in the microscopic description to the QEF in the macroscopic description.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Quantum Entropy Function The Theory on AdS2 Classical Limit of QEF Motivation The Logic Results

The Theory on AdS2

We consider a two dimensional theory obtained as a result of compactifying the fundamental theory on K. We can then describe the dynamics in the near horizon geometry by a theory of gravity coupled to a set of U(1) gauge fields {Ai

µ} and

neutral scalar fields {φs}, integrating out all other fields. The most general field configuration consistent with the isometry of AdS2 is : ds2 = v(−r2dt2 + dr2 r2 ), φs = us, F i

rt = ei ,

where v, us, ei are constants.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Quantum Entropy Function The Theory on AdS2 Classical Limit of QEF Motivation The Logic Results

Classical Limit of QEF

Let L0 be the classical lagrangian density and A be the action : A[gµν, {AI

µ}, {φs}] = −

Z d2x√−gL0 . We define : f ( u, v, e) = √−gL0 = vL0 . Then the Wald entropy is given by SWald( q) = 2π(eI qI − f ( u, v, q)) , at ∂f ∂us = 0 , ∂f ∂v = 0 , qi = ∂f ∂ei . In the classical limit, the QEF reduces to SWald( q) : ln dhor( q) → SWald( q) .

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Quantum Entropy Function The Theory on AdS2 Classical Limit of QEF Motivation The Logic Results

Why Study Localization?

For 1/8 BPS BHs in N = 8 susic theories, 1/4 BPS BHs in N = 4 susic theories and 1/2 BPS BHs in N = 2 susic theories, the SL(2, R) × SO(3) isometry of near horizon geometry gets enhanced to the SU(1, 1|2) supergroup. Hence in 4d, SUSY requires BHs to be spherically symmetric with near horizon geometry having AdS2 × S2 factor. Goal → To simplify the path integral over string fields by making use of these

• isometries. We shall use localization techniques to show that the path integral

receives non-zero contribution only from field configurations which preserve a particular subgroup of SU(1, 1|2).

Duistermaat, Heckman, Witten, Schwarz, Zaboronsky, Nekrasov, Pestun Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Quantum Entropy Function The Theory on AdS2 Classical Limit of QEF Motivation The Logic Results

The Logic

Consider an arbitrary quantum field theory with function space M over which one wishes to integrate. Let F be a supergroup of symmetries generated by Q and a compact U(1) generator X such that Q2 = X. Suppose F acts freely on M. In that case one can form the quotient space M/F. A point in the space M/F corresponds to an orbit of the elements of F. This orbit contains an element of M and its images under the action of the supergroup F. Thus by integrating first over the orbit, one can reduce the integral to an integral

• ver M/F. The integral over the orbit simply gives a factor of svol(F) :

Z

M

e−LO = svol(F) Z

M/F

e−LO . Since the integration over the bosonic parameter gives a finite result, the volume

• f the supergroup F is zero :

svol(F) = Z dxbos dθfer = 0 .

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Quantum Entropy Function The Theory on AdS2 Classical Limit of QEF Motivation The Logic Results

In general, the group F does not act freely and has fixed point locus M0. Let C be an arbitrary neighbourhood of M0 and let M′ be its complement. Then the path integral restricted to M′ vanishes and the entire contribution comes from the integration over C. Since the neighbourhood C is arbitrary, the integral in this sense is said to be localised on M0.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Quantum Entropy Function The Theory on AdS2 Classical Limit of QEF Motivation The Logic Results

Results

1

The global symmetry group of AdS2 × S2 is SU(1, 1|2). It is possible to construct supergroup H1 (which is an analogue of F) generated by supercharge Q1 and compact bosonic generator Q2

1.

2

Using the arguments of localization, we will show that the path integral receives non-vanishing contribution only from integration around H1-invariant field configurations.

3

We will show the explicit construction of a class of H1-invariant saddle points.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary The Algebra Coordinate Frame

The Algebra

The global part of the N = 4 superconformal algebra in 1 + 1 dimensions generates the supergroup SU(1, 1|2), with the generators obeying the following algebra : [Lm, Ln] = i(m − n)Lm+n , [J3, J±] = ±J± , [J+, J−] = 2J3 , [Ln, G α±

r

] = i( n 2 − r)G α±

r+n ,

[J3, G α±

r

] = ± 1 2 G α±

r

, [J±, G α∓

r

] = G α±

r

, {G +α

r

, G −β

s

} = 2ǫαβLr+s − 2i(r − s)(ǫσi)βαJi , where ǫ+− = −ǫ−+ = 1 , ǫ++ = ǫ−− = 0, m, n = 0, ±1 , r, s = ± 1 2 , α, β = ± .

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary The Algebra Coordinate Frame

Choice of Coordinates

We will work in coordinates w, z which represent AdS2 as a disk and S2 in the stereographic coordinates : ds2 = 4u dw d ¯ w (1 − w ¯ w)2 + 4v dz d¯ z (1 + z¯ z)2 , where u and v are constants. Defining ˆ L0 = 1 2 (L1 + L−1) , ˆ L± = L0 ± i 2 (L1 − L−1) , ˆ G αβ

±

= G αβ

1/2 ∓ iG αβ −1/2 ,

we get : ˆ L0 = (w∂w − ¯ w∂ ¯

w) ,

ˆ L+ = −i(w2∂w − ∂ ¯

w) ,

ˆ L− = i(∂w − ¯ w2∂ ¯

w) .

Thus ˆ L0 and J3 are the generators of rotation about the origins in the w and z-planes respectively.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary The Algebra Coordinate Frame

Redifinition of the Fermionic Generators

We define Q1 = b G ++

+

+ b G −−

−

, Q2 = −i “ b G ++

+

− b G −−

−

” , Q3 = −i “ b G −+

+

+ b G +−

−

” , Q4 = b G −+

+

− b G +−

−

, e Q1 = b G ++

−

+ b G −−

+

, e Q2 = −i “ b G ++

−

− b G −−

+

” , e Q3 = −i “ b G −+

−

+ b G +−

+

” , e Q4 = b G −+

−

− b G +−

+

, so that {Qi, Qj} = 8 δij (b L0 − J3) , {e Qi, e Qj} = 8 δij (b L0 + J3) , [b L0 − J3, Qi] = 0 , [b L0 + J3, e Qi] = 0 .

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary The Subgroup H1 Choice of Integration Variables Proof

The Subgroup H1

The subgroup H1 of the supergroup SU(1, 1|2) is generated by Q1 and Q2

1 = 4(ˆ

L0 − J3) . Using the supergroup H1 and localization arguments, we can show that the path integral receives nonvanishing contribution only from integration around a field configuration Φ which is at least invariant under H1, i.e., (ˆ L0 − J3)Φ = 0 , Q1Φ = 0 .

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary The Subgroup H1 Choice of Integration Variables Proof

Labelling the Fluctuations

We choose the coordinates of the configuration space measuring the fluctuations about Φ as follows : We decompose the fluctuations in terms of the eigenfunctions of (ˆ L0 − J3) with eigenvalues m ∈ Z. We label all the bosonic fluctuations with m by zs

m and −m by zs∗ m , where m > 0.

Since Q2

1 = 4(ˆ

L0 − J3), the action of Q1 on zs

m for m = 0 can not vanish.

For m = 0 , let Q1zs

m = ζs m . Then Q1ζs m = 4mzs m .

For the complex conjugate deformations : Q1zs∗

m = ζs∗ m ,

Q1ζs∗

m = −4mzs∗ m .

We shall call the m = 0 bosonic and fermionic modes collectively as y .

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary The Subgroup H1 Choice of Integration Variables Proof

The Path Integral

The path integral over the various fields can be regarded as an integral over the parameters zs

m, zs∗ m , ζs m, ζs∗ m (for m > 0) and

y : I = Z d y Y

m>0,s

dzs

m dzs∗ m dζs m dζs∗ m I e−A ,

where I represents any measure factor which might arise from changing the integration variables to ( y, z, z∗, ζ, ζ∗) .

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary The Subgroup H1 Choice of Integration Variables Proof

Proof

Now we deform the above integral to another integral : I(t) = Z d y Y

m>0,s

dzs

m dzs∗ m dζs m dζs∗ m I e−A−t Q1F .

t is a positive real parameter and A is the euclidean action satisfying Q1A = 0 . F = P

m>0

P

s zs∗ m ζs m . This gives

Q1F = X

m>0

X

s

[4mzs∗

m zs m + ζs mζs∗ m ] , and

Q2

1F = 0 ,

noting that F has m = 0 by construction. By construction, I(t = 0) = I .

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary The Subgroup H1 Choice of Integration Variables Proof

We find that ∂tI(t) = Z d y Y

m>0

dzs

m dzs∗ m dζs m dζs∗ m I (−Q1F) e−A−t Q1 F

= Z d y Y

m>0

dzs

m dzs∗ m dζs m dζs∗ m (Q1I ) F e−A−t Q1 F

= 0 , using the Q1 invariance of the measure. Thus I(t) has the same value in the limits t → 0 and t → ∞ . In the t → ∞, the zs

m and ζs m dependent terms in A are subleading. Thus the

• riginal integral reduces to

I = Z d y I ′ ( y) e−A′(

y) ,

where (ˆ L0 − J3) y = 0 .

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary The Subgroup H1 Choice of Integration Variables Proof

We can further localize the y integral onto Q1-invariant subspace. Intuitively this can be understood by noting that unless y is invariant under Q1, the orbit of Q1 through a point y will give a vanishing contribution to the integral. This establishes that, after taking into account appropriate measure factors, we can express the path integral as integration over an H1 invariant slice passing through Φ.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Application 1 Application 2

Integrating Over the Orbits of the Superconformal Current Algebra

String theory on AdS2 × S2, describing the near horizon geometry of a BPS BH, has an infinite group of asymptotic symmetries besides the global SU(1, 1|2). These group of symmetries does not leave AdS2 × S2 invariant, but preserve the asymptotic boundary conditions. Since the action does not change under these transformation, these represent zero modes. In a non-supersymmetric theory, integration over the bosonic zero modes will generate an infinite factor. Hence integration over these directions must be restricted by declaring the corresponding transformations as gauge transformations. In a supersymmetric theory, there is a possibility of cancellation between the bosonic and fermionic zero mode integrals, yielding a finite result. We now demonstrate that this is indeed what happens.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Application 1 Application 2

The Generators of The Superconformal Current Algebra

The generators of the extended superconformal algebra may be labelled as e Ln, e Ji

n

and e G αβ

r

with n ∈ Z, r ∈ Z + 1

2 , 1 ≤ i ≤ 3 and α, β = ± .

The generators of su(1, 1|2) discussed earlier are special cases of these generators with the identifications b L0 = e L0, b L± = e L∓1, Ji = e Ji

0,

b G αβ

±

= e G αβ

∓ 1

2

. The commutators relevant for our analysis are : [b L0 − J3, e Ln] = −n e Ln , [b L0 − J3, e J3

n] = −n e

J3

n ,

[b L0 − J3 , e J±

n ] = (−n ∓ 1) e

J±

n ,

[b L0 − J3, e G αβ

r

] = „ − 1 2 β − r « e G αβ

r

.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Application 1 Application 2

Integration gives a Finite Result

We note that most of the modes generated by the superconformal current algebra carry non-zero eigenvalues under (b L0 − J3). They are part of the deformations labelled by zs

m and ζs m, and are eliminated by the localization procedure.

Thus we only need to worry about deformations generated by (b L0 − J3)-invariant

• generators. Of these several are part of the global symmetry group SU(1, 1|2)

and have already been taken into account in the analysis before. Such generators, which are not part of SU(1, 1|2), are e J+

−1 and e

J−

1 . Together

with e J3

0 = J3, they generate an SU(2) group. Thus integration over these zero

modes gives a finite factor proportional to the volume of SU(2). This shows that, around an H1-invariant saddle point, integration over the fermionic and bosonic zero modes, generated by the full superconformal current algebra, gives a finite result.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Application 1 Application 2

Examples of H1-invariant Saddle Points

In the computation of dhor, we must include contributions from all saddle points preserving the asymptotic boundary conditions. We consider type IIB string theory on K3 and consider 6d geometries which are asymptotic to S1 × ˜ S1 × AdS2 × S2 with background 3-form fluxes. The simplest H1-invariant saddle point is S1 × e S1 × AdS2 × S2 with background fluxes, invariant under the full SU(1, 1|2) : ds2 = v(dη2 + sinh2 ηdθ2) + u(dψ2 + sin2 ψdφ2) + R τ2 |dx4 + τdx5|2 , G I = 1 8π2 [QI sin ψdx5 ∧ dψ ∧ dφ + PI sin ψdx4 ∧ dψ ∧ dφ + dual] , V i

I = constant,

V r

I = constant ,

where V i

I , V r I

(1 ≤ i ≤ 5, 6 ≤ r ≤ 26) are matrix-valued scalar fields and BI

MN

(1 ≤ I ≤ 26) are 2-form fields with field strengths G I = dBI .

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Application 1 Application 2

Contribution to Entropy

The classical contribution to the QEF from this saddle point is given by exp(SWald). We shall now construct other H1-invariant saddle points by taking orbifold of the above background by some discrete Z Zs group. The classical contribution to the QEF from this kind of saddle points is given by exp(SWald/s). This matches with the microscopic result.

Sen Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Application 1 Application 2

Orbifolding by Z Zs

Since H1 contains both Q1 and Q2

1, the generators of Z

Zs must commute with Q1. Typically the generators of Z Zs will involve an element of SU(1, 1|2), together with an internal symmetry transformation that commutes with SU(1, 1|2). The only bosonic generator of su(1, 1|2) which commutes with Q1 is (ˆ L0 − J3). The orbifold of the background is given by : (θ, φ, x5) → (θ + 2π s , φ − 2π s , x5 + 2πk s ) , k, s ∈ Z , gcd(s, k) = 1 . This is a freely acting orbifold as the circle parametrized by x5 is non-contractible. Since (ˆ L0 − J3) shifts θ and φ in opposite directions, the above transformation is generated by (ˆ L0 − J3) together with a shift along x5.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary Application 1 Application 2

Asymptotic Behaviour of the Orbifolded Space

After taking the orbifold, the new configuration is : ds2 = v n dη2 + sinh2 η „ 1 + (1 − s−2)e−η 2 sinh η «2 dθ2o + u n dψ2 + sin2 ψ(dφ + dθ − s−1dθ)2o + R2 τ2 ˛ ˛dx4 + τ(dx5 + ks−1dθ) ˛ ˛2 , G I = 1 8π2 ˆ QI sin ψ (dx5 + ks−1dθ) ∧ dψ ∧ (dφ + dθ − s−1dθ) + PI sin ψ dx4 ∧ dψ ∧ (dφ + dθ − s−1dθ) + dual ˜ , (θ, φ, x5) ≡ ` θ + 2π, φ, x5´ ≡ ` θ, φ + 2π, x5´ ≡ ` θ, φ, x5 + 2π ´ . Since the asymptotic region lies at large η , this has the same asymptotic behaviour as the S1 × e S1 × AdS2 × S2 background.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary

Summary

QEF gives a prescription for computing the degeneracy of extremal BHs in terms

• f path integral over string fields on the near horizon geometry.

Using the global symmetry group SU(1, 1|2) , we have shown that there exists a subgroup H1, generated by the fermionic generator Q1 and the compact bosonic generator (ˆ L0 − J3) , such that Q2

1 = 4(ˆ

L0 − J3) . Using the subgroup H1, we have shown that the path integral receives nonvanishing contribution only from integration around field configurations Φ which are invariant at least under H1. We have shown that the bosonic and fermionic zero modes generated by the action of the superconformal current algebra cancel with each other to give a finite result. We have also shown the construction of a class of H1-invariant saddle points from freely acting orbifolds of the near horizon geometry of the BH.

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function

Outline Prologue Symmetries of Euclidean AdS2 × S2 Localization of Path Integral Some Applications Summary

THANK YOU !

Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function