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: AdS 2 × S 1 1) Localization on J.David, E.Gava, R.Gupta, K.Narain : JHEP 1703 (2017) 050 2) Boundary Conditions and Localization on :Part 1 AdS J.David, E.Gava, R.Gupta, K.Narain : arXiv:1802.00427 3) Boundary Conditions and Localization on :Part 2 AdS J.David, E.Gava, R.Gupta, K.Narain : to appear soon

Introductions and Motivations Supersymmetric localization • is a powerful technique to evaluate supersymmetric observables 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. ( Dabholkar, Drukker,Gomes, Grassi, Marino, Putrov, Sen.) • has been focussed mostly on susy QFT defined on curved but compact spaces without boundary.

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 AdS n × S m evaluating black hole entropy as well as entanglement entropy in conformal field theories. 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. These boundary conditions tell us which fluctuations to integrate over in the path integral to compute the observable. Typically the natural boundary conditions one imposes on the quantum fluctuations are normalizable boundary conditions w.r.t Z d d x √ g | Φ | 2 < ∞ .

Black Hole Entropy Quantum entropy of an extremal black hole is given as Z ∂ AdS 2 A ( i ) e S = H [ dg µ ν ][ d Φ ] e − S Eucld. − iq i (Sen) AdS 2 , finite One loop computations were performed with normalizable (Banerjee,Gupta, boundary conditions on all the fields. Mandal,Sen) Example: S = A H + C 1 ln A H + C 2 + C 3 H e − A H + .. + ... + D 1 A n A H 4 4 2 N 8 computation uses χ = 2( n v − n H + 1) 2 − χ normalizable bdy. cond. C 1 − 4 0 24 Entropy of black hole in and twisted partition AdS 4 (Cabo-Bizet,Giraldo-Rivera, function on . AdS 2 × S 1 Pando Zayas )

Supersymmetric Localization Basic idea : If there exist a fermionic symmetry Q such that Q 2 = L v is satisfied off shell, combinations of various and QS = 0 symmetry like gauge transformation, R-symmetry etc. Z e − S Q ( ... ) = 0 path integral measure < Q ( ... ) > = Then is also invariant Partition function does not change under S → S + t QV ⇒ Q 2 V = 0 for any fermionic functional such that . V Z Z e − S − t QV = e − S Z = lim M loc = { QV = 0 } t →∞ M loc

Supersymmetry on AdS 2 × S 1 We want to put susy QFT with on a background U (1) R N = 2 which admits one or more rigid supercharges. (Closset,Dumitrescu, Festuccia,Komargodski) The background metric is ds 2 = d τ 2 + L 2 ( dr 2 + sinh 2 r d θ 2 ) We want to solve ( r µ � iA µ ) ✏ = � 1 2 H � µ ✏ � iV µ ✏ � 1 2 ✏ µ νρ V ν � ρ ✏ , ✏ = � 1 ✏ + 1 ( r µ + iA µ ) ˜ 2 H � µ ˜ ✏ + iV µ ˜ 2 ✏ µ νρ V ν � ρ ˜ ✏ We require that the killing vector is ✏� µ ✏ @ µ = K µ = @ @ @⌧ + 1 e L @✓

Killing Spinor We need to turn on background value of some supergravity fields A τ = V τ = 1 A r, θ = V r, θ = H = 0 L The killing spinors are given by ✓ sinh( r ✓ i cosh( r ◆ ◆ 2 ) 2 ) i θ ✏ = e − i θ ✏ = e ˜ , 2 2 sinh( r 2 ) i cosh( r 2 ) The supersymmetry algebra on the background Q 2 = L K + δ gauge transf + δ R − symm . Λ 1 2 L x θ τ · Here Q = δ ✏ + δ e ✏ AdS 2 × S 1

Chern-Simons+Matter We consider Chern-Simons theory coupled to a fundamental matter. (David,Gava,Gupta,Narain;16) The theory is described by S = S C.S. + S matter where ✓ ◆ S C.S. = k Z A µ ∂ ν A ρ − 2 i λλ + i h i d 3 x √ g Tr − ˜ i ε µ νρ 3 A µ A ν A ρ 2 G σ , 4 π equivalent to boson theory ✓ ◆ Z 4 qG − ∆ − 1 4 R + 1 ∆ − 1 h ⇣ V 2 − q 2 σ 2 ⌘ i d 3 x √ g D µ ¯ ¯ φφ − F ¯ S matter = φ D µ φ + F + ... 2 2 ∆ =R-charge of chiral multiplet fermionic terms

SUSY of Action The vector multiplet+chiral multiplet is supersymmetric upto boundary terms. In particular, the action of the chiral multiplet is Q-exact. S matter = Q-exact + boundary terms where Z h 1 d 3 x p g r µ )( ✏� µ ) � i i 2 V µ ¯ ✏ )¯ ✏ e boundary terms = cosh r ( e cosh r " µ ρν ( ✏� ν e � �� � � D ρ � i ✏ ) iq � ¯ +( ✏� µ e �� . cosh r The boundary terms go to zero with normalizable and smoothness conditions on all the fields.

Boundary Condition on Chiral Multiplet AdS 2 × S 1 The normalizable boundary conditions on are φ ∼ e − r ψ ∼ e − r 2 , 2 These conditions are not consistent with susy transformations Q � = ✏ Q = Γ µ ˜ ✏ D µ � + ... This is due to the fact that susy parameters grow exponentially for large , r 2 . r r e 2 , ✏ ∼ e ✏ ∼ e This seems generic fact for any . AdS d

Localization Manifold We add following positive definite Q-exact terms 1 Z h Ψ µ ( Q Ψ µ ) † + Ψ ( Q Ψ ) † + ( Q ) † + ˜ ) † i ( Q ˜ d 3 x √ g V loc = ✏✏ ) 2 Tr (˜ Ψ µ = 1 Ψ = i ✏� + ✏ ˜ 2( ✏� µ ˜ Here 2(˜ � ) , � + ˜ ✏� µ � ) i α 4 i α Solutions : a µ = 0 , σ = cosh r , G = L cosh 2 r , real parameter φ = ¯ φ = F = ¯ F = 0 .

Partition Function Thus the partition function is given as Z [ d α ] exp( − π ikL Tr α 2 ) Z vec. 1 − loop ( α ) Z chiral Z = 1 − loop ( α , ∆ ) R The one loop determinant can be evaluated using Green’s function of the kinetic operator. δα D F ( α )] − 1 δ δ δ δα ln Z 1 − loop ( α ) = Tr[ G F 2Tr[ G B δα D B ( α )] . fermionic Green’s function bosonic Green’s function D F ( α ) : Fermionic kinetic operator D B ( α ) : Bosonic kinetic operator

Green’s Function : Methodology 1 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. We first find the solutions to the differential equation This does not have D ( x ) S ( x ) = 0 a global solution. Let } S 1 ( x ) : is a valid solution near . x → 0 Smoothness and bdy. conditions. : is a valid solution near . S 2 ( x ) x → ∞ Then the Green’s function is given as h i G ( x, y ) = c Θ ( y − x ) S 1 ( x ) S 2 ( y ) + Θ ( x − y ) S 1 ( y ) S 2 ( x ) .

Green’s Function : Methodology 2 The Green’s function is h i G ( x, y ) = c Θ ( y − x ) S 1 ( x ) S 2 ( y ) + Θ ( x − y ) S 1 ( y ) S 2 ( x ) . Properties : • The Green’s function is continuous at . x = y • The first derivative of the Green’s function is discontinuous at . The discontinuity fixes the constant . c x = y For fermionic case, the Green’s function is discontinuous.

E.O.M If bdy. conditions are consistent with susy, the Green’s function for bosonic fields and fermionic fields are related. Consequence of the supersymmetry: if satisfies X 0 , for chiral X 0 = { φ } D X 0 = 0 then its super partner also satisfies the same equation D QX 0 = 0 . X 1 = M QX 0 for chiral, the fermions are { X 1 , QX 0 } � ✓ ◆ ✓ ◆ D 00 D 01 X 0 for � Q ¯ ¯ V chiral = X 0 X 1 D 10 D 11 QX 1 A solution for is also sol. for provided it is consistent X 0 QX 0 with the respective bdy. and smoothness conditions.

E.O.M 2 φ Explicitly the equation of motion for is ✓ ◆ 4 qG − ∆ − 1 4 R + 1 ∆ − 1 ⇣ V 2 − q 2 σ 2 ⌘ , D µ D µ φ − φ = 0 2 2 and the remaining equation is iL . X 1 = ✏✏ ) � + 2 iQ 2 ( e ✏ ) D µ QX 0 ✏� µ e ( ∆ − 2) − 2 Lq ( e In terms of chiral multiplet fermion the twisted variables are and X 1 = e QX 0 = ✏ ✏

Solutions to E.O.M Explicitly obtaining the solutions for fields in chiral multiplet and analysing their asymptotic behaviour, we find that  n,p ) } for n > ∆ { f + , ( b + n,p , c + 2 L   ∆ − 1 2 L < n < ∆ { f + , ( b − { f n,p , ( b n,p , c n,p ) }| normalizable = n,p ) } for n,p , c − 2 L n,p ) } for n < ∆ − 1  { f − , ( b − n,p , c − 2 L .  Here the modes are coefficients of Fourier expansion: X 0 ∼ f n,p e in τ + ip θ , QX 0 ∼ c n,p e in τ + ip θ r x θ τ · QX 1 ∼ b n,p e in τ + i ( p − 1) θ AdS 2 × S 1 r 2 f n,p → 0 , for b n,p ( z ) → 0 , c n,p ( z ) → 0 e r → ∞

Solutions to E.O.M:2 For susy bdy. conditions, we have ( n,p ) } for n > ∆ − 1 { f + , ( b + n,p , c + 2 L { f n,p , ( b n,p , c n,p ) }| susy = n,p ) } for n < ∆ − 1 { f − , ( b − n,p , c − 2 L . The susy boy. conditions are r r e − r 2 f n,p → 0 , 2 c n,p → 0 , 2 b n,p → 0 for e e r → ∞