Exact Half-BPS Solutions in Type IIB and M-theory, John Estes, Michael Gutperle Amsterdam 2008 Exact half-bps Type IIB interface solutions I, Local solutions and supersymmetric Janus, arxiv:0705.0022 Exact half-bps Type IIB interface solutions II, Flux solutions and multi-janus, arxiv:0705.0024 Gravity duals of half-bps Wilson loops, arxiv:0705.1004 Exact Half-BPS Flux Solutions in M-theory I, Local Solutions, & Darya Krym, arxiv:0806.0605 Half-BPS supergravity solutions and supergroups, & Darya Krym, & Paul Sorba, arxiv:080*.**** 1
Geometry of the solutions Type IIB sugra and M-theory families of solutions with 16 susy, IIB on AdS 4 S 2 S 2 Σ symmetry OSp(4 4,R) IIB on AdS 2 S 4 S 2 Σ symmetry OSp(4 4) M on AdS 3 S 3 S 3 Σ symmetry OSp(4 2) OSp(4 2) products warped over 2-dim parameter space Σ (to be specified) General local solution: exactly in terms of harmonic functions on Σ General global solution (for Type IIB): all non-singular solutions are obtained whose boundary is locally asymptotically AdS 5 S 5 Solutions generally have varying fluxes, non-trivial charges, and moduli. 2
Reduction to new integrable systems Solved by reducing the supergravity BPS eqs to integrable systems; Σ is a Riemann surface with boundary, and local coordinate w; Integrable system of the sine-gordon type for a single real field ϑ on Σ ) ( ) w ( w ϑ 2e +iϑ w lnf(h) + w w ϑ 2e iϑ w lnf(h) = 0 h is a real harmonic function on Σ f is a real function which depends on the supergravity system, Type IIB M theory f(h) = tg(h) f(h) = h Not just integrable, but complete explicit solution may be derived exactly. BPS eqs supergravity Bianchi identities and field eqs. 3
Motivation Construction of AdS solutions dual to half-bps states Lin, Lunin, Maldacena dual to local half-bps operators in N = 4 AdS 4 S 2 S 2 Σ dual to N = 4 SYM with a planar interface Janus and interface CFTs possible applications to 2+1-dim CM problems AdS 2 S 4 S 2 Σ dual to N = 4 SYM with a Wilson line AdS 3 S 3 S 3 Σ dual to 32 susy CFT 3 with line interface dual to (0, 2) susy CFT 6 with surface operator Exact solutions: starting point for spectrum and correlation functions 4
Relation to Branes Near horizon limits of brane intersections with 8 susys, Type IIB on AdS 4 S 2 S 2 Σ D3/D5/NS5 with 4 intersecting dims Type IIB on AdS 2 S 2 S 4 Σ D3/D1/D5 with 2 int. dims M-theory on AdS 3 S 3 S 3 Σ M2/M5 or M5/M5 with 3 int. dims see e.g. (Tseytlin Boonstra, Peeters, Skenderis) Probe branes in maximally symmetric geometries, Type IIB on AdS 4 S 2 S 2 Σ D5 in AdS 5 S 5, Type IIB on AdS 2 S 2 S 4 Σ D1 and D5 in AdS 5 S 5 M-theory on AdS 3 S 3 S 3 Σ M2 or M5 in AdS 7 S 4 or M5 in AdS 4 S 7 Here, we shall obtain the fully back-reacted solutions. 5
Type IIB case : Interface AdS/CFT SO(2, 3) = conformal group of planar interface Coupling may vary across interface Allowed susy in CFT driven by interface operators (D Hoker, Estes, Gutperle) - 0 susys original Janus solution (Bak, Gutperle, Hirano) - 4 susys - 8 susys - 16 susys global SO(3) SO(3) R-symmetry; total global bosonic symmetry SO(2, 3) SO(3) SO(3) suitable AdS space AdS 4 S 2 S 2 Σ This talk! Also varying Axion across interface with 16 susys (Gaiotto and Witten) 6
AdS dual to Interface in Type IIB SO(2, 3) SO(3) SO(3) invariant Ansatz on AdS 4 S 2 1 S 2 2 Σ ds 2 = f 2 1ds 2 S 2 1 + f 2 2ds 2 S 2 2 + f 2 4ds 2 AdS 2 4 + 4ρ 2 dw 2 Σ G 3 = g a e 45a + ih a e 67a, ε = χ ζ F 5 = ( e 0123a + ε a b e 4567b )k a a,b = 8, 9 (Σ) Reduce BPS eqs for Type IIB sugra (see also Gomis and Rommelsberger) φ, f 1,f 2, f 4,ρ, k a are real, and g a,h a, ζ are complex functions on Σ χ is a basis of Killing spinors on AdS 4 S 2 1 S2 2 8 complex algebraic reduced eqs (2 from dilatino eq) 4 complex differential reduced gravitino eqs. Every solution may be mapped by a global SL(2,R) transformation onto a solution with vanishing axion, and real g a,h a. 7
Further Reducing the BPS equations Solve algebraic equations for f 1, f 2, f 4, g a, h a, k a in terms of φ, ρ, ζ = (α β) f 1 = α β + ᾱβ f 2 = iᾱβ iα β f 4 = αᾱ + β β g z + ih z = 4α(ρβ) 1 w φ g z ih z = +4β(ρα) 1 w φ k z = iν 2αβ α4 β 4 4ρα 2 β 2 wφ Then, eliminate these functions from the differential BPS equations, Two of the differential BPS eqs are equivalent to Cauchy-Riemann eqs, ρ(α 2 β 2 ) = κ e ±(φ+ λ) κ, λ arbitrary holomorphic functions on Σ 8
Mapping to a new integrable system Eliminating α,β from the remaining 2 differential eqs leaves 2 complex differential eqs for φ,ρ in terms of κ, λ. Drastic simplification by changing variables to µ Im(λ), and sh(2φ + 2λ) sh(2φ + 2 λ) = e2iϑ ρ 8 = ˆρ 8 (sin 2µ) 2 κ 4 κ 4 16 Two differential eqs for ϑ, ˆρ in terms of κ, µ become, sin ϑ + sin µ (sin ϑ sin µ) 3 w ϑ (e iϑ + isinµ)(cosµ) 1 w µ = iκˆρ 2 e iϑ/2 w ϑ 2e iϑ (cos µ) 1 w µ = 2i w ln ˆρ 2 = System of Bäcklund transfs for the partial differential eq, w ( w ϑ 2(cosµ) 1 ( w µ)e iϑ) + c.c. = 0 This system is automatically integrable. 9
General Exact Solution A final change of variables, ψ ˆρ 2 (cosµ)e iϑ/2 maps to a linear system, w ψ w ψ = κ cosµ = ( w µ)(iψ ψ sin µ)/cosµ All fields are given in terms of harmonic functions h 1, h 2 on Σ, e.g. e 4φ = 2h 1h 2 w h 2 2 h 2 2W 2h 1 h 2 w h 1 2 h 2 1 W f 2 1f 2 4 = 4e +2φ h 2 1 W w h 1 w h 2 + w h 2 w h 1 f 2 2f 2 4 = 4e 2φ h 2 2 10
AdS 5 S 5 and Janus with 16 susys We readily obtain a 2-parameter family of non-singular solutions, h 1 = Im ( e w φ + e w φ ) h 2 = Re ( e w+φ + + e w+φ ) For φ + = φ gives AdS 5 S 5 For φ + φ, dilaton varies = Janus solution with 16 susys 11
General Regularity Conditions Regularity inside Σ requires (e.g. e 4φ > 0) : h 1 h 2 > 0 and W 0 Regularity on the boundary Σ Assume solution is locally asymptotic to AdS 5 S 5 ; Asymptotic AdS 5 S 5 regions originate from isolated points on Σ; In between, either S 2 1 or S 2 2 must shrink to zero on Σ f 1 = 0 : h 1 = 0, w h 2 = 0 f 2 = 0 : h 2 = 0, w h 1 = 0 Equivalent to two coupled electro-statics problems with alternating Neumann and vanishing Dirichlet conditions on Σ 12
Solution by hyperelliptic curves Map Σ onto upper half-plane with boundary Σ = R Points e i on Σ where Dirichlet Neumann, i = 1,2,,2g + 2. hyperelliptic curve of genus g, s(u) 2 = (u e 1 )(u e 2 ) (u e 2g+1 ) The meromorphic differentials h 1, h 2 are given by, h 1 = i P 1(u)du s(u) 3 h 2 = P 2(u)du s(u) 3 for two real polynomials P 1,P 2 of degree 3g + 1, Neumann and Dirichlet conditions automatically satisfied, behavior at branch points du/(u e i ) 3/2 guarantees asymptotic AdS 5 S 5 W < 0 requires definite ordering of e i and the real zeros of P 1,P 2, Vanishing Dirichlet requires the vanishing of 2g period integrals. 13
Topology of regular solutions 5 AdS 5x S φ=φ 1 S5 f 2=0 S 5 AdS 5 5x S φ=φ 4 f 1=0 2 S x S 3 3 2 S x S f =0 1 5 5 S S f =0 2 AdS 5 5x S AdS 5 5x S φ=φ 2 φ=φ 3 2g + 2 asymptotic AdS 5 S 5, each with independent dilaton limit φ i g NSNS 3-form cycles S 3 and g RR 3-form cycles S 3 4g + 6 free (real) moduli Limiting behavior of moduli, topology change g g 1 the D5 or NS5 probe limits 14
CFT dual to AdS 4 solutions H 3,1 g = e φ 1 YM H 3,1 g YM = eφ 2 I 2,1 H 3,1 H 3,1 g = e φ 4 YM g = e φ YM 3 g + 1 different species of N =4, interact only through interface are coupled via extra massless fields on the interface recover massless string excitations from probe D5 branes De Wolfe, Freedman, Ooguri Skenderis, Taylor Gaiotto, Witten 15
Exact Half-BPS solutions in M-theory SO(2, 2) SO(4) SO(4) invariant Ansatz on AdS 3 S 3 2 S 3 3 ds 2 = f 2 1ds 2 AdS 3 + f 2 2ds 2 S 3 2 + f 2 3ds 2 S 3 3 + ρ 2 dw 2 Σ F 4 = g 1a e 012a + g 2a e 345a + g 3a e 678a a = 9, 10 (Σ) Reducing BPS eqs, (see also Yamaguchi; Lunin) using ε = P χ ξ with χ a basis of Killing spinors on AdS 3 S 3 2 S3 3 solving for the metric factors f i introduces real constants c i, c i f i = ξ σ i ξ c 1 + c 2 + c 3 = 0 Special cases: AdS 4 S 7 has c 2 = c 3 and AdS 7 S 4 has c 1 = c 2,c 3 Any solution asymptotic to AdS 4 S 7 has c 2 = c 3 AdS 7 S 4 has c 1 = c 2,c 3 No solutions with mixed AdS 4 S 7 and AdS 7 S 4 asymptotics 16
Deriving the general solution Eliminating f i, the remaining differential BPS eqs are D z ξ = 1 4 σ3 G z σ 3 ξ D z ξ = 1 12 G zξ where G z = g 1 z σ 1 ig 2 z σ 2 + ig 3 z σ 3 is a generalized connection A c i -dependent linear combination in ρξ (ξ ) 3 κ = c 3 ρ(αᾱ 3 β β 3 ) (c 1 c 2 )ρᾱ β(α β βᾱ) ξ = (α β) is a holomorphic 1-form on Σ allows one to further reduce the BPS equation We have derived the complete solutions in exact form Asymptotic to either AdS 4 S 7 or AdS 7 S 4 corresponding to either c 2 = c 3 or c 1 = c 2, c 3 17
The exact local solution for AdS 7 S 4 All fields of the solution may be expressed in terms of G and h, e.g. f 1 f 2 f 3 = ±h ρ 6 = wh 6 16h 4 (1 G 2 )W 2 f 6 1,2 = 4h2 W 4(1 G 2 ) ( G Ḡ ± 2 G 2) 3 where h is harmonic on Σ, and G satisfies a linear equation, 2 w Ḡ = (G + Ḡ) w lnh κ = w h 18
Full solution space for M-theory case II (includes AdS7xS4) c1=c3 c3=o c1=o case III (includes AdS7xS4) OSp(2,2 2)xOSp(4 2) decompactified OSp(4 2)xOSp(2,2 2) c1=c2 decompactified OSp(4 2,R)xOSp(4 2,R) decompactified c2=o case I (includes AdS4xS7) c2=c3 Supergroup throughout this diagram is D(2,1;c 2 /c 1 ) D(2,1; c 2 /c 1 ) for Σ Euclidean, de Boer, Pasquinucci, Skenderis, 1999 19
Supergroups with 16 susys in Type IIB Can one construct all solutions with 16 susys which have a CFT dual? View solutions as AdS duals to deformations of N = 4 SYM Expect a subgroup H of SU(2,2 4) with 16 susys to be preserved Semi-simple H first, with maximal bosonic subgroup H B H H B space-time SU(2, 2 4) SO(2, 4) SO(6) AdS 5 S 5 SU(2 2) SU(2 2) SO(4) SO(4) R M 4 S 3 S 3 (LLM) OSp(4 4, R) SO(2, 3) SO(3) SO(3) AdS 4 S 2 S 2 Σ 2 OSp(4 4) SO(2, 1) SO(3) SO(5) AdS 2 S 2 S 4 Σ 2 SU(2, 2 2) *?* SO(2, 4) SO(3) SO(2) AdS 5 S 2 S 1 Σ 2 SU(1, 1 4) *?* SO(2, 1) SO(5) SO(2) AdS 2 S 5 S 1 Σ 2 20