Boundary Conditions and Localization on AdS Rajesh Gupta Kings - - PowerPoint PPT Presentation

Boundary Conditions and Localization on AdS

Rajesh Gupta King’s College London

Workshop on Supersymmetric Localization and Holography: Black Hole Entropy and Wilson Loops ICTP, Trieste

References: 1) Localization on

AdS2 × S1

JHEP 1703 (2017) 050

2) Boundary Conditions and Localization on :Part 1 AdS 3) Boundary Conditions and Localization on :Part 2 AdS

J.David, E.Gava, R.Gupta, K.Narain : J.David, E.Gava, R.Gupta, K.Narain : J.David, E.Gava, R.Gupta, K.Narain : arXiv:1802.00427 to appear soon

Introductions and Motivations

Supersymmetric localization

• is a powerful technique to evaluate supersymmetric
• bservables exactly.
• has provided non trivial checks of several conjectural

dualities in susy QFT in various dim.

• has also provided highly non trivial checks for AdS/CFT.
• has been focussed mostly on susy QFT defined on curved

but compact spaces without boundary.

( Dabholkar, Drukker,Gomes, Grassi, Marino, Putrov, Sen.)

Introductions and Motivations

Rigid susy QFT can also be defined on curved but non compact spaces. Non compact spaces of the form are relevant in evaluating black hole entropy as well as entanglement entropy in conformal field theories. AdSn × Sm Localization on non compact spaces is a harder problem for two reasons:

• 1. needs to worry about boundary terms in susy variations,
• 2. needs to include boundary conditions.

On Boundary Conditions

On non compacts spaces the boundary conditions defines the problem. Typically the natural boundary conditions one imposes on the quantum fluctuations are normalizable boundary conditions w.r.t These boundary conditions tell us which fluctuations to integrate

• ver in the path integral to compute the observable.

Z ddx√g |Φ|2 < ∞ .

eS = Z

AdS2,finite

[dgµν][dΦ] e−SEucld.−iqi

H

∂AdS2 A(i)

Black Hole Entropy

Quantum entropy of an extremal black hole is given as One loop computations were performed with normalizable boundary conditions on all the fields.

S = AH 4 + C1 ln AH + C2 + C3 AH + ... + D1 An

H e−AH + ..

Example:

(Banerjee,Gupta, Mandal,Sen)

χ = 2(nv − nH + 1)

(Sen)

N

C1

8

−4

4 2

2 − χ 24

(Cabo-Bizet,Giraldo-Rivera, Pando Zayas )

Entropy of black hole in and twisted partition function on .

AdS4

AdS2 × S1

computation uses normalizable bdy. cond.

Basic idea : If there exist a fermionic symmetry Q such that

Q2 = Lv

combinations of various symmetry like gauge transformation, R-symmetry etc.

QS = 0

< Q(...) >= Z e−SQ(...) = 0

Partition function does not change under for any fermionic functional such that .

S → S + t QV V

Q2V = 0

Z = lim

t→∞

Z e−S−t QV = Z

Mloc

e−S

Mloc = {QV = 0}

and is satisfied off shell, Then

path integral measure is also invariant

Supersymmetric Localization

⇒

We want to put susy QFT with on a background which admits one or more rigid supercharges.

N = 2

U(1)R

The background metric is

ds2 = dτ 2 + L2(dr2 + sinh2 r dθ2)

We want to solve We require that the killing vector is

Supersymmetry on AdS2 × S1

(Closset,Dumitrescu, Festuccia,Komargodski)

(rµ iAµ) ✏ = 1 2Hµ✏ iVµ✏ 1 2✏µνρV νρ✏ ,

(rµ + iAµ) ˜ ✏ = 1 2Hµ˜ ✏ + iVµ˜ ✏ + 1 2✏µνρV νρ˜ ✏

e ✏µ✏ @µ = Kµ = @ @⌧ + 1 L @ @✓

We need to turn on background value of some supergravity fields

Aτ = Vτ = 1 L

Ar,θ = Vr,θ = H = 0

The killing spinors are given by The supersymmetry algebra on the background

Q2 = LK + δgauge transf

Λ

+ δR−symm

1 2L

.

Here

Killing Spinor x AdS2 × S1

θ

τ

·

✏ = e

iθ 2

✓i cosh( r

2)

sinh( r

2)

◆ , ˜ ✏ = e− iθ

2

✓ sinh( r

2)

i cosh( r

2)

◆

Q = δ✏ + δe

✏

Chern-Simons+Matter

Smatter = Z d3x√g h Dµ ¯ φDµφ + ⇣ − 1 4qG − ∆ 4 R + 1 2 ✓ ∆ − 1 2 ◆ V 2 − q2σ2⌘ ¯ φφ − F ¯ F + ... i

We consider Chern-Simons theory coupled to a fundamental matter. The theory is described by

S = SC.S. + Smatter

where

SC.S. = k 4π Z d3x√g Tr h iεµνρ ✓ Aµ∂νAρ − 2i 3 AµAνAρ ◆ − ˜ λλ + i 2Gσ i ,

∆ =R-charge of chiral multiplet

fermionic terms

equivalent to boson theory

(David,Gava,Gupta,Narain;16)

boundary terms = Z d3xpg rµ h

• 1

cosh r(e ✏ e )(✏µ ) i 2V µ ¯ i cosh r"µρν(✏νe ✏)¯ Dρ +(✏µe ✏) iq cosh r ¯

• i

.

Smatter = Q-exact + boundary terms

SUSY of Action

The vector multiplet+chiral multiplet is supersymmetric upto boundary terms. where The boundary terms go to zero with normalizable and smoothness conditions on all the fields. In particular, the action of the chiral multiplet is Q-exact.

Boundary Condition on Chiral Multiplet

The normalizable boundary conditions on are

AdS2 × S1

φ ∼ e− r

2 ,

ψ ∼ e− r

2

These conditions are not consistent with susy transformations

Q = ✏

Q = Γµ˜ ✏ Dµ + ...

This is due to the fact that susy parameters grow exponentially for large , r

✏ ∼ e

r 2 ,

e ✏ ∼ e

r 2 .

This seems generic fact for any .

AdSd

Localization Manifold

Vloc = Z d3x√g 1 (˜ ✏✏)2 Tr h Ψµ(QΨµ)† + Ψ(QΨ)† + (Q )† + ˜ (Q ˜ )†i

Ψ = i 2(˜ ✏ + ✏˜ ) , Ψµ = 1 2(✏µ˜ + ˜ ✏µ)

Here We add following positive definite Q-exact terms Solutions:

φ = ¯ φ = F = ¯ F = 0 .

aµ = 0 , σ = iα cosh r , G = 4iα L cosh2 r ,

real parameter

Z = Z

R

[dα] exp(−πikLTrα2)Zvec.

1−loop(α)Zchiral 1−loop(α, ∆)

δ δα ln Z1−loop(α) = Tr[GF δ δαDF (α)] − 1 2Tr[GB δ δαDB(α)] .

DF (α): Fermionic kinetic operator DB(α): Bosonic kinetic operator

Partition Function

Thus the partition function is given as The one loop determinant can be evaluated using Green’s function of the kinetic operator.

fermionic Green’s function bosonic Green’s function

D(x)S(x) = 0

Green’s Function : Methodology 1

We first find the solutions to the differential equation

This does not have a global solution.

Let Then the Green’s function is given as

G(x, y) = c h Θ(y − x) S1(x)S2(y) + Θ(x − y)S1(y)S2(x) i .

S1(x) : is a valid solution near .

x → 0

S2(x)

x → ∞ : is a valid solution near .

}

Smoothness and

• bdy. conditions.

Green’s function is a solution to the differential equation

D(x)G(x, y) = δ(x, y) D(x): is a differential operator without zero modes.

Green’s Function : Methodology 2

G(x, y) = c h Θ(y − x) S1(x)S2(y) + Θ(x − y)S1(y)S2(x) i .

The Green’s function is

• The Green’s function is continuous at .

x = y Properties :

• The first derivative of the Green’s function is discontinuous

at . The discontinuity fixes the constant .

x = y

c

For fermionic case, the Green’s function is discontinuous.

E.O.M

then its super partner also satisfies the same equation

,

D X0 = 0

D QX0 = 0 .

Consequence of the supersymmetry: if satisfies

X0

A solution for is also sol. for provided it is consistent with the respective bdy. and smoothness conditions.

QX0

X0

{X1, QX0} for chiral, the fermions are

If bdy. conditions are consistent with susy, the Green’s function for bosonic fields and fermionic fields are related. for

Vchiral =

• Q ¯

X0 ¯ X1 ✓ D00 D01 D10 D11 ◆ ✓ X0 QX1 ◆ for chiral X0 = {φ}

X1 = M QX0

E.O.M 2

DµDµφ − ⇣ − 1 4qG − ∆ 4 R + 1 2 ✓ ∆ − 1 2 ◆ V 2 − q2σ2⌘ φ = 0

and the remaining equation is Explicitly the equation of motion for is

φ

QX0 = ✏

X1 = e ✏

In terms of chiral multiplet fermion the twisted variables are

, . and

X1 = iL (∆ − 2) − 2Lq(e ✏✏) + 2iQ2 (e ✏µe ✏)DµQX0

Solutions to E.O.M

Explicitly obtaining the solutions for fields in chiral multiplet and analysing their asymptotic behaviour, we find that

{fn,p, (bn,p, cn,p)}|normalizable =      {f +, (b+

n,p, c+ n,p)} for n > ∆ 2L

{f +, (b−

n,p, c− n,p)} for ∆−1 2L < n < ∆ 2L

{f −, (b−

n,p, c− n,p)} for n < ∆−1 2L .

X0 ∼ fn,p einτ+ipθ, QX0 ∼ cn,p einτ+ipθ

QX1 ∼ bn,p einτ+i(p−1)θ

Here the modes are coefficients of Fourier expansion:

x AdS2 × S1

θ

τ

·

r

e

r 2 fn,p → 0,

bn,p(z) → 0, cn,p(z) → 0

r → ∞

for

Solutions to E.O.M:2

{fn,p, (bn,p, cn,p)}|susy = ( {f +, (b+

n,p, c+ n,p)} for n > ∆−1 2L

{f −, (b−

n,p, c− n,p)} for n < ∆−1 2L .

For susy bdy. conditions, we have

e

r 2 fn,p → 0,

e

r 2 cn,p → 0,

e− r

2 bn,p → 0

r → ∞

for The susy boy. conditions are

δ δαDf(α) = L2q 2√1 − z σ3, δ δαDb(α) = Lq(−i + 2Lqα) 2√1 − z .

z = tanh2 r

When normalizable bdy. conditions are consistent with susy, after integration by parts, we find

B.T. : bdy. terms, functions of solns. and .

S1(z) S2(z)

δ δα ln Z1−loop(α) = Tr[GF δ δαDF (α)] − 1 2Tr[GB δ δαDB(α)] .

One Loop Determinant 1

We want to evaluate For chiral multiplet

δ δα ln e Zchiral

1−loop(α) =

Z 1 dz h ∂ ∂z B.T. + E.O.M i

This is the case for and

n > ∆ 2L n < ∆ − 1 2L

.

When normalizable bdy. conditions are not consistent with susy, after integration by parts, we find

One Loop Determinant 2

Bulk Terms : bulk terms, functions of solns. and .

S1(z) S2(z)

ln Zchiral

1−loop(α) = ln e

Zchiral

1−loop(α) + ln ˆ

Zchiral

1−loop(α)

The complete partition function is

Bulk term = − Z 1 dz Lq(−i + 2Lqα)S1−(z)S2−(z) 4c1−− √1 − z + Z 1 dz Lq(−i + 2Lqα)S1−(z)S2+(z) 4c1−+ √1 − z .

δ δα ln ˆ Zchiral

1−loop(α) =

Z 1 dz h Bulk Terms + ∂ ∂z B.T. + E.O.M i

ˆ x = 2|p| + ∆ − 2Ln + 2iLqα , ˆ y = 2|p| − ∆ + 2Ln − 2iLqα .

Here

Chiral Multiplet Result

The one loop determinant is

Boundary terms from susy region Bulk terms

ln Zchiral

1loop(α, ∆) =

X

p>0,nd ∆

2L e

ln ⇣ p + L(n + iqα) − ∆ 2 ⌘ − X

p0,n< ∆−1

2L

ln ⇣ − p − L(n + iqα) + ∆ 2 ⌘

− X

p≤0

d ∆−1

2L en< ∆ 2L

ln ⇣ − p − L(n + iqα) + ∆ 2 ⌘ − X

∆−1 2L <n< ∆ 2L

X

p2Z

ln Γ( 1

2 + 1 4 ˆ

x)Γ( 1

4 ˆ

x⇤) Γ( 1

2 + 1 4 ˆ

y)Γ(1 + 1

4 ˆ

y⇤)

Boundary terms from non susy region

The one loop determinant from another Q-exact deformations is

ln Z(α, ∆) = X

p>0,nd ∆

2L e

ln ⇣ p + L(n + iqα) − ∆ 2 ⌘ − X

p0,n< ∆−1

2L

ln ⇣ − p − L(n + iqα) + ∆ 2 ⌘ − X

p0,d ∆−1

2L en< ∆ 2L

ln ⇣ − p − L(n + iqα) + ∆ 2 ⌘ − X

∆−1 2L <n< ∆ 2L

X

p>0

ln Γ(e a2)Γ(b2) Γ(e b2)Γ(a2) − X

∆−1 2L <n< ∆ 2L

X

p0

ln Γ(e a2 − p)Γ(b2 − p) Γ(e b2 − p)Γ(a2 − p)

Chiral Multiplet Result

Some Comments

• Bulk terms depend on the Q-exact action.
• The result is consistent with explicit one loop calculation

for free chiral multiplet using eigen function method.

• If the bdy. conditions consistent with susy, the variation is given

in terms of bdy. terms. Boundary terms are independent of Q- exact action.

Index Result

ln Z = −1 2index D10 ln Q2

We also computed the one loop result using the index of

D10

In the case of chiral multiplet, we get

ln Zindex = X

p=1,n> ∆−1

2L

ln(p + L(n + iqα) − ∆ 2 ) − X

p=0,n> 1−∆

2L

ln(p + L(n − iqα) + ∆ 2 ) .

indexD10 = dim. of kernel − dim. of cokernel

The one loop determinant of the vector multiplet using the index computation results

ρ : roots of the Lie algebra

Zvec

1-loop,index =

Y

ρ>0

(ρ.α)2 Y

ρ

sY

n6=0

(n − iρ · α) Y

p6=0

( p L − iρ · α) = Y

ρ>0

sinh(πρ · α) sinh(πLρ · α)

L2 > 3 4

This is consistent with normalisable bdy. cond. if .

This computation is relevant for the black hole entropy. The metric background admits killing spinors with or without graviphoton background.

AdS2 × S2

We have computed partition function of a hypermultiplet on background with equal radius. The computations are performed using normalizable boundary

• conditions. Naively it is not consistent with susy.

However, we find that in this case there are no bulk terms.

Hypermultiplet on AdS2 × S2

Result on AdS2 × S2

When T = ¯ T = 0 , we find

T = ¯ T 6= 0

On the other hand for the black hole background i.e.

ln Zhyper =

∞

X

`=0 ∞

X

p>0

ln(` + p − 2iq↵) +

∞

X

`=0

X

p<0

ln(−` + p − 2iq↵)

= X

k∈Z

|k| ln(k − 2iqα) ln Zhyper = −

∞

X

`=0 ∞

X

p>0

ln(` + p − 2iq↵) −

∞

X

`=0

X

p<0

ln(−` + p − 2iq↵) = − X

k∈Z

|k| ln(k − 2iqα)

(Murthy, Reys)

same as S4

• We have computed one loop partition function of Chern

Simons theory coupled to a chiral multiplet on .

AdS2 × S1

• One loop result of vector multiplet based on normalizable

boundary conditions for gauge field is same as that on with some conditions on . Also consistent with explicit index calculations.

S3

• One loop result for the chiral multiplet based on normalizable

boundary conditions depends on the Q-exact actions.

Summary of Results : AdS2 × S1

• Disagreement if there exist an integer

n ∈ D : (∆ − 1 2L , ∆ 2L)

L : ratio of the radius of to that of . AdS2

S1

∆ : R-charge of the chiral multiplet

L

AdS2 × S2 Summary of Results :

• We have computed the one loop partition function of a

hypermultiplet on using normalizable boundary conditions.

AdS2 × S2

• In this case, we find that the normalizable boundary conditions

are consistent with susy.

• The result of hypermultiplet is consistent with the index answer.

(Murthy, Reys)

Conclusion

• We developed Green’s function method to compute one loop

determinant and incorporate bdy. conditions on AdS-spaces.

• When bdy. conditions are consistent with susy, the variation
• f the one loop determinant is a total derivative.

A general proof is still missing.

• The Green’s function is harder to compute for higher spins
• fields. The difficulty increases with space time dimension.
• One of the advantage of the Green’s function method compared

to index calculation is that one just needs asymptotic behaviour

• f the solutions.

Future Directions

• Next we want to apply the Green’s function method to

higher spin fields.

• It will be interesting to compute the one loop determinant in

the case for black holes in AdS spaces.

• It will be interesting to generalize to higher dimensional

AdS spaces.

Thank You.