Half BPS solutions in type IIB and M-theory

Half BPS solutions in type IIB and M-theory Based on work done in collaboration with Eric D Hoker, John Estes, Darya Krym (UCLA) and Paul Sorba (Annecy) E.D'Hoker, J.Estes and M.G., Exact half-bps type IIB interface solutions I&II: Flux solutions and multi-janus'', [arxiv:0705.0024],[arxiv:0705.0022] E.D'Hoker, J.Estes,M.G. and D. Krym, Exact half-bps solutions in M-theory I, [arxiv:0806.0605] E.D'Hoker, J.Estes,M.G., D. Krym and Paul Sorba, Half-BPS Supergravity Solutions and Superalgebras, to appear See also: E.D'Hoker, J.Estes and M.G., Gravity duals of half-bps Wilson loops, [arxiv: 0705.1004] E.D'Hoker, J.Estes and M.G., Interface Yang-Mills, supersymmetry, and Janus, [arxiv: hep-th/0603013]

Plan of talk Supersymmetric defects Janus solution and interface CFT Half BPS Janus solution in type IIB Half BPS solutions in M-theory Superalgebras and half BPS solutions

Conformal defects in conformal field theories Conformal symmetry in d-dimensions: Poincare, scaling, special conformal symmetries: SO(2,d) Conformal defects: Objects which preserve a subgroup of the conformal symmetry which is itself a conformal symmetry group (of lower dimension) Example from 2dim CFT (euclidean): SL(2,C): z (az + b)/(cz + d), a, b, c, d C, ad bc =1 z Defect Im(z)=0 preserved by SL(2,R): a, b, c, d R

Half-BPS defects dim conformal symmetry supergroup R-symmetry N=4 SYM SO(2, 4) P SU(2, 2 4) SO(6) 1/2 BPS Wilson Loops 0+1 1/2 BPS surface 1+1 SO(2, 1) SO(3) OSp(4 4) SO(5) SO(2, 2) U(1) P SU(1, 1 2) P SU(1, 1 2) U(1) SO(4) 1/2 BPS domain wall/interface 2+1 SO(2, 3) OSp(4 4,R) SO(3) SO(3) Half-BPS defects realizations in string theory: Intersecting branes in flat space (before near horizon limit) Probe branes in AdS 5 S 5 Exact supergravity solutions which are locally asymptotic to AdS 5 S 5

Half-BPS defects Intersecting branes (before near horizon limit) D3 D1 D5 D3 D5 0 1 2 3 4 5 6 7 8 9 Wilson loop surface domain wall Probe branes in AdS 5 S 5 D1 probe brane with worldvolume D5 probe brane with worldvolume D3 probe brane with worldvolume D5 probe brane with worldvolume AdS 2 AdS 2 S 4 AdS 3 S 1 AdS 4 S 2

Janus solution in AdS/CFT The Janus solution is a dilatonic deformation of the AdS 5 S 5 solution of type IIB supergravity D. Bak, S. Hirano and M.G. hep-th/0304129 Named after the two faced roman god of string dualities Metric ansatz: ds 2 = f(µ) ( dµ 2 + ds 2 AdS 4 ) + ds 2 S 5 5-form and dilaton: F 5 = 2f(µ) 5 2 dµ ωads4 + 2ω S 5, φ = φ(µ) The undeformed solution AdS 5 S 5 is given by a constant dilaton and f(µ) = 1/cos 2 (µ). The coordinate μ ranges from µ [ π/2, π/2]

Janus solution in AdS/CFT equations of motion can be reduced to first order differential equation f f = 4f 3 4f 2 + c2 0 6 1 f, φ (µ) = c 0 f 3 2 (µ) coordinate range: µ = [ µ 0, µ 0 ], µ 0 > π/2 dilaton behavior: lim φ(µ) = φ (0) ± + φ (1) ± (µ µ 0 ) 4 + µ ±µ 0 The Janus solution is a fat dilatonic domain wall with preserved symmetry SO(6)xSO(2,3) All supersymmetries broken since 0 δλ = i ν φ Γ ν B 1 ɛ i AdS 4 24 Γµνρ G µνρ ɛ 0 worldvolume

Janus solution and interface theories Asymptotic behavior near the boundary ds 2 In Poincare coordinates the spatial section of the boundary consists of two three dimensional half planes joined by a two dimensional interface. µ = ±µ 0 in Poincare patch 1 (µ µ 0 ) 2 z 2 (z2 dµ 2 dt 2 + dx 2 1 + dx 2 2) The standard field operator mapping of AdS/CFT relates the constant part of the dilaton with the insertion of the operator dimension 4: L = trf 2 +

Janus solution and interface theories The constant part of the dilaton is therefore related to the YM coupling S = d 4 x 1 g 2 Y M The dual of the Janus solution is given by a YM coupling which jumps across a 2+1 dim interface. x π = 0 L x 1,2,3 g Y M (x π ) = g (0) Y M (1 + ɛθ(xπ )) g Y M = g + x π g Y M = g Interface theory: No localized degrees of freedom on the 2+1 space Supersymmetry broken by terms localized at x π = 0 (2+1 dim conformal and R-symmetry) symmetry SO(2,3) x SO(6) Restoration of supersymmetry: add terms localized at interface

Supersymmetric Interface theories Dimension3, gauge invariant, no new degrees of freedom (N=4 SYM) L = 1 g(x π ) 2 L {N=4} + δ(x π )L int Supersymmetry variation δ = δ {N=4} + δ int Susy preserved if δl = δ 0 L N=4 + δ N=4 L int + δ int L N=4 + δ int L int For example: L ψ = δ(x π )tr(y 1 ψγ π ψ + i 4 yij 2 ψγ π ρ ij ψ i 2 yijk 3 ψt ρ ijk ψ + cc) Tensors y break SU(4) R-symmetry

Supersymmetric Interface theories Classification of supersymmetric interface theories given in hep-th/0603013 Number of superconformal charges unbroken R-symmetry 4 N=1 interface susy SU(3) 8 N=2 interface susy SU(2) x U(1) 16 N=4 interface susy SU(2) x SU(2) Generalization by Gaiotto and Witten: Varying Θ angle, lozalized matter, S- duality of half BPS boundary conditions : arxiv:0804.2907,0804.2902 Bosonic symmetry of N=4 interface susy: SO(2,3) x SU(2) x SU(2) AdS 4 S 2 S 2

Half BPS Janus solution Ansatz: Fibration over two dimension surface Σ ds 2 = f 2 4 ds 2 AdS 4 + f 2 1 ds 2 S 1 + f 2 2 ds 2 S 2 + ρ 2 dz dz P = p a e a, Q = q a e a G = g a ω S1 e a + ih a ω S2 e a F 5 = f a ( ω AdS4 e a + ɛ a b ω S1 ω S2 e b ) Find solution to the BPS equations δλ = ip Γ B 1 ɛ i 4 (G (3) Γ)ɛ δψ M = D M ɛ + i 4 (F (5) Γ) Γ M ɛ 1 16 (Γ M (G (3) Γ) + 2(G (3) Γ)Γ M )B 1 ɛ

Half BPS Janus solution Dilatino variation projection on 16 supercharges ε Gravitino variation does not impose additional projection: ε can be expanded in terms of Killing spinors on AdS 4 S 2 S 2 Reduce BPS equations in terms of 2 dim spinor ξ 1. Discrete symmetries 2. SU(1,1) and reality condition Q=0 3. functions in ansatz expressed in terms of spinor bilinears Reduced BPS equations can be mapped into a classically integrable system via various maps Local solutions completely determined by two functions h 1 (z, z), h 2 (z, z) z z h 1,2 = 0 details in arxiv:0705.0024

Local half-bps Janus solution Useful quantity: W = z h 1 z h 2 + z h 2 z h 1 dilaton: Σ-metric: ρ 8 = W 2 e 4φ = 2h 1h 2 z h 2 2 h 2 2W 2h 1 h 2 z h 1 2 h 2 1 W h 3 1 h3 2 metric factors: ρ 4 f 2 1 f 2 2 = 4W 2 (2h 1 z h 2 2 h 2 W )(2h 2 z h 1 2 h 1 W ) f 2 1 f 2 4 = 4e 2φ h 2 1 f 2 2 f 2 4 = 4e 2φ h 2 2 regularity: real finite dilaton, no singularity in the metric

Σ is infinite strip Simple solutions: AdS_5 x S_5 and Janus five sphere: S 2 S 2 fibration over interval V ol(s 2) 0 V ol(s 1 ) 0 h 2 = 0 h 1 = 0 AdS 5 : AdS 4 fibration over real line y Σ = {z = x + iy, 0 < y < π/2} Janus deformation: π/2 h 1 = ie z + ie z + ie z ie z = 4 sinh x sin y h 2 = e z + e z + e z + e z = 4 cosh x cos y 0 h 2 = 0 h 1 = 0 h 1 = ie φ + (e z e z ) + ie φ (e z e z ) h 2 = e φ + (e z + e z ) + e φ + (e z + e z ) x

General solution this solution is the supergravity dual to the half BPS interface theory, much more general solutions are possible Map strip to upper half plane: y π/2 h 2 = 0 z w 0 h 1 = 0 x h 2 = 0 0 h 1 = 0

Local half-bps Janus solution Useful quantity: W = z h 1 z h 2 + z h 2 z h 1 dilaton: Σ-metric: ρ 8 = W 2 e 4φ = 2h 1h 2 z h 2 2 h 2 2W 2h 1 h 2 z h 1 2 h 2 1 W h 3 1 h3 2 metric factors: ρ 4 f 2 1 f 2 2 = 4W 2 (2h 1 z h 2 2 h 2 W )(2h 2 z h 1 2 h 1 W ) f 2 1 f 2 4 = 4e 2φ h 2 1 f 2 2 f 2 4 = 4e 2φ h 2 2

General solution this solution is the supergravity dual to the half BPS interface theory, much more general solutions are possible Map strip to upper half plane: y π/2 h 2 = 0 z w 0 h 1 = 0 x h 2 = 0 0 h 1 = 0 Volume of either two sphere vanishes W = 0 V ol(s 1 ) = 0, h 1 = 0, n h 2 = 0 V ol(s 2 ) = 0, h 2 = 0, n h 1 = 0

General solution boundary conditions on harmonic functions, regularity conditions on dilaton, non-sigular metric factors and local Janus limit can be solved genus g hyperelliptic surface with boundary h 1 = 0, n h 2 = 0 h 2 = 0, n h 1 = 0 e 2g+1 e 2g e 2g 1 e 2g 2... e 2 e 1 explicit solution for harmonic functions in terms of zeros of z h 1, z h 2 4g+6 real moduli 2g+2 Janus boundaries g 3-spheres with NS-NS three form flux g 3-spheres with R-R three form flux

Genus one solution

Half-BPS Wilson loops Wilson loop in N=4 SYM: W (R, C) = tr R {P exp i C dτ(a µ dx µ dτ + Φi dyi dτ )} C: curve in R 1,3 R 6 parameterized by {x µ (τ),y i (τ)} R: representation of Wilson loop 1/2 BPS Wilson loop: C is a straight line x 0 = τ,x 1 = x 2 = x 3 =0, y i = n i τ Symmetry SO(2,1)x SO(5) xso(3) Supergravity ansatz: ds 2 = f 2 1 ds 2 AdS 2 + f 2 2 ds 2 S 2 + f 2 3 ds 2 S 4 + ds 2 Σ G = g a ω AdS2 e a + ih a ω S2 e a Solution: Very similar to BPS defect, boundary conditions and regularity conditions are slightly different

Half-BPS solution in M-theory Maximally super symmetric solutions of 11dim supergravity AdS 4 S 7 AdS 7 S 4 near horizon limit of M2-branes near horizon limit of M5-branes 2+1 dim CFT 5+1 dim CFT Intersecting branes 1/4 BPS before near horizon limit M2 M5 0 1 2 3 4 5 6 7 8 9 10 M5 probe brane in AdS 4 S 7 M2 probe brane in AdS 7 S 4 worldvolume worldvolume AdS 3 S 3 AdS 3 Symmetry: SO(2, 2) SO(4) SO(4)

Half-BPS solution in M-theory Ansatz for exact supergravity solution: ds 2 = f 2 1 ds 2 AdS 3 + f 2 2 ds 2 S 3 2 + f 2 3 ds 2 S 3 3 + ds 2 Σ F (4) = g 1a ω AdS3 e a + g 2a ω S (1) 3 e a + g 3a ω S (1) 3 e a Supersymmetry variation of gravitino M ɛ + 1 ( NP QR ΓM 8δ N M Γ P QR) F NP QR ɛ =0 288 Same strategy for solution as in type IIB case: express ɛ in terms of Killing spinors on AdS and spheres express metric and fieldstrength in terms of spinor bilinears the Σ component of susy variations produces differential equation map differential equation into an integrable system

AdS solutions: Σ is infinite strip Half-BPS solution in M-theory y π/2 Σ = {z = x + iy, 0 < y < π/2} 0 x case I: AdS 4 S 7 f 1 = cosh(2x), f 2 = 2 cos(y) f 3 = 2 cos(y) c 1 = 2, c 2 = c 3 =1 case II: AdS 7 S 4 f 1 = 2 cosh(x), f 2 = 2 sinh(x) f 3 = sin(2y) c 2 = 2, c 1 = c 3 =1 case III: AdS 7 S 4 f 1 = 2 cosh(x), f 2 = sin(2y) f 3 = 2 sinh(x) c 3 = 2, c 1 = c 2 =1 Solution depends on a one parameter family: c 1,c 2,c 3,c 1 + c 2 + c 3 =0 f 1 = ζ ζ c 1 f 2 = ζ σ 3 ζ c 2 f 3 = ζ σ 2 ζ c 3

Half-BPS solution in M-theory Solution is completely determined in terms of a single harmonic function on Riemann surface and the solution of a PDE (for all cases) Σ 2 w G =(G + Ḡ) w ln h exact form of the solution depends on the case (here we show case I) The metric factors are given by f 3 1 = W 2 =4 G 4 +(G Ḡ)2 hw 16( G 2 1) f 6 2 = 2h2 W 4 ( G 2 1) f 6 3 = 2h2 W 4 ( G 2 1) ( G 2 + i2 (G Ḡ) ) 3 ( G 2 i2 (G Ḡ) ) 3 h the following combination is particularly simple f 1 f 2 f 3 = h 4

Half-BPS solution in M-theory Boundary: One of the two spheres shrinks to zero size h 0 V ol(s (3) 1 ) 0 if G +i V ol(s (3) 2 ) 0 if G i However equation for G is not easy to solve Choose coordinates: y = h x = h (dual harmonic function) With G = G y + ig x PDE for G becomes G can be expressed as y G y + x G x = G y y y G x x G y = 0 G y = y (yψ) G x = x (yψ) ( x 2 + y 2 + 1 y y 1 )Ψ(x, y) = 0 y2

Half-BPS solutions and superalgebras Global symmetry: Bosonic isometries and 32 supersymmetries: superalgebra Type IIB: PSU(2,2 4) M-theory: OSp(8 4,R) or OSp(8* 4) Half BPS configuration: sub-superalgebra with 16 fermionic generators. One can produce a classification.

Summary Supersymmetry imposes powerful constraints on lower dimensional defects Holographic description of defects using an ansatz of AdS and sphere factors warped over a 2 dimensional base space BPS-equations can be solved, leading to a new kind of integrable system Explicit solution in terms of harmonic functions Fully backreacted solution instead of probe approximation M-theory BPS defect theories can be treated the same way leading to a very similar integrable system General classification of half BPS solutions using superalgebras: Several case are not known (yet)

Open questions Analyze regularity and moduli space of solutions for defect theories in M-theory Description of holographic defect theories on the CFT side (for M2-brane) Existence of new half-bps solutions as indicated by the subalgebra analysis