Interpolating geometries, fivebranes and the Klebanov-Strassler theory

Transcription

1 Interpolating geometries, fivebranes and the Klebanov-Strassler theory Dario Martelli King s College, London Based on: [Maldacena,DM] JHEP 1001:104,2010, [Gaillard,DM,Núñez,Papadimitriou] to appear Universitá degli Studi di Padova 14 Aprile 2010

2 1 Supersymmetric geometries Plan of the seminar 2 Interpolating supersymmetric geometries 3 The unwarped resolved deformed conifold 4 The baryonic branch from fivebranes 5 Fivebranes from the baryonic branch 6 Adding flavours 7 A G 2 story (if there is time - probably there isn t) 8 Outlook

3 Motivations Systematic studies of supersymmetric geometries of String/M theory interesting mathematically and provide useful tools for addressing problems in string phenomenology (scanning the landscape) and gauge/gravity dualities In the context of the gauge/gravity duality: New perspectives on familiar examples Methods to address more complicated models Can lead to the discovery of new gauge/gravity duals

4 General supersymmetric geometries Supersymmetric geometries of d = 11, 10 and d < 10 supergravities may be analysed systematically in the framework of G-structures/generalised geometry Input: general metric ansatz plus Killing spinors Output: set of equations for RR+NS fluxes and (multi-) forms

5 General supersymmetric geometries Supersymmetric geometries of d = 11, 10 and d < 10 supergravities may be analysed systematically in the framework of G-structures/generalised geometry Input: general metric ansatz plus Killing spinors Output: set of equations for RR+NS fluxes and (multi-) forms In Type IIB supergravity: ds 2 = e 2 [ dx 2 1,3 + ] ds2 6 F 5 = e 4 +Φ ( )vol 4 f F 1 = 0 (for simplicity)

6 Type IIB supersymmetric geometries e 2 +Φ/2 (d H 3 )e 2 Φ/2 Ψ 1 = d ( + Φ ) 4 Ψ 1 + ie +5Φ/4 [f 6 F 3 ] 8 (d H 3 )e 2 Φ/2 Ψ 2 = 0 [Graña,Minasian,Petrini,Tomasiello] Ψ 1, Ψ 2 are pure spinors in the sense of generalised geometry. Alternatively: multi-forms We restrict to the case when these take the form Ψ 1 = e iζ e (1 +Φ/4 ie 2 +Φ/2 J 1 ) 2 e4 +Φ J J Ψ 2 = e 4 +Φ Ω J, Ω define a more familiar SU(3) structure. Non-constant phase ζ allows interpolation between different classes

7 Geometry Interpolating SU(3) structures ( ) d e 6 +Φ/2 Ω = 0 d ( e 8 J J ) = 0 ( ) d e 2 Φ/2 cos ζ = 0

8 Interpolating SU(3) structures Geometry ( ) d e 6 +Φ/2 Ω = 0 d ( e 8 J J ) = 0 ( ) d e 2 Φ/2 cos ζ = 0 Fluxes 6 F 3 = e 2 3Φ/2 sec ζd ( e 4 +Φ J ) H 3 = sin ζe Φ 6 F 3 f = e 4 Φ d ( e 4 sin ζ )

9 Interpolating SU(3) structures Geometry ( ) d e 6 +Φ/2 Ω = 0 d ( e 8 J J ) = 0 ( ) d e 2 Φ/2 cos ζ = 0 Fluxes 6 F 3 = e 2 3Φ/2 sec ζd ( e 4 +Φ J ) H 3 = sin ζe Φ 6 F 3 f = e 4 Φ d ( e 4 sin ζ ) sin ζ 1: warped Calabi-Yau with ISD 3-form dj = dω = 0, G 3 = F 3 + ih 3 = i G 3, e Φ = g s [Giddings,Kachru,Polchinski] E.g.: Klebanov-Strassler cos ζ 1: superstrings with torsion [Strominger] E.g.: Maldacena-Núñez

10 Non-Kähler geometries (Type I) Geometry (ζ = 0) d ( e 2Φ Ω ) = 0 d ( e 2Φ J J ) = 0 = Φ/4

11 Non-Kähler geometries (Type I) Geometry (ζ = 0) d ( e 2Φ Ω ) = 0 d ( e 2Φ J J ) = 0 = Φ/4 Fluxes (ζ = 0) 6 F 3 = e 2Φ d ( e 2Φ J ) H 3 = 0 F 5 = 0 [Gauntlett,DM,Waldram] S-dual version involves only: metric, dilaton, NS 3-form H 3 Type I/Heterotic solutions M 6 is complex but non-kähler. Killing spinors preserved by connection ˆ = spin + H 3 with torsion

12 Generating new solutions [Minasian,Petrini,Zaffaroni], [Gaillard,DM,Núñez,Papadimitriou] Simple solution generating method In: solution to non-kähler equations out: general solution

13 Generating new solutions [Minasian,Petrini,Zaffaroni], [Gaillard,DM,Núñez,Papadimitriou] Simple solution generating method In: solution to non-kähler equations out: general solution Φ new = Φ old sin ζ = κ 2 e Φold e 2 = κ 1 old cos ζ eφ /2 non trivial F 5, H 3 F new 3 = 1 F old κ generated

14 Generating new solutions [Minasian,Petrini,Zaffaroni], [Gaillard,DM,Núñez,Papadimitriou] Simple solution generating method In: solution to non-kähler equations out: general solution Φ new = Φ old sin ζ = κ 2 e Φold e 2 = κ 1 old cos ζ eφ /2 non trivial F 5, H 3 F new 3 = 1 F old κ generated Key point: Bianchi identity of simpler system Bianchi of more general system equations of motion (integrability results) Applies also with supersymmetric sources [Koerber,Tsimpis] Application to gauge/gravity duality: connection between wrapped fivebranes and Klebanov-Strassler theory

15 The conifold and the conifold transition The (Calabi-Yau) conifold singularity: z z2 2 + z2 3 + z2 4 = 0 Two desingularisations of the tip preserving the CY condition 1 Deformation : i z 2 i = ɛ 2 T S 3 S 3 R 3 2 Resolution : O( 1) O( 1) CP 1 S 2 R 4 [Vafa]: for large N, N D5 branes wrapped on S 2 in the resolved conifold deformed conifold with N units of RR F 3 through S 3 Is it possible to see the geometric transition purely in the context of (Type IIB) supergravity?

16 The unwarped resolved deformed conifold M fivebranes wrapped on the S 2 of the resolved conifold. Back-reaction of branes (M large) modifies the geometry work with non-kähler equations

17 The unwarped resolved deformed conifold M fivebranes wrapped on the S 2 of the resolved conifold. Back-reaction of branes (M large) modifies the geometry work with non-kähler equations [Papadopoulos,Tseytlin] ansatz back-reacted solution ds 2 str = dx M 4 ds2 6 H 3 = M 4 w 3 f(t)c(t) e 2Φ(t) = e 2Φ 0 sinh 2 t ds 2 6 depends (simply) on a function c(t). w 3 is a 3-form Solution explicit, up to 1st order ODEs: f = 4 sinh 2 t c c = 1 f [c2 sinh 2 t (t cosh t sinh t) 2 ] Parameters: M, Φ 0, 0 < U <. U is defined at large t and matched (numerically) to a parameter γ 2 1 near t 0

18 Realising the geometric transition t 0: r 2 S Mγ 2 radius of S 3 3 t : r 2 S M log U radius of S 2 2 Parameter U interpolates between deformation and resolution U 0: deformed conifold with large S 3 + H 3 = M flux U : resolved conifold + M NS5 branes (far from branes)

19 Field theory (decoupling) limit: large U Generalised GVW superpotential define a gauge coupling : W = e 2Φ Ω (H 3 + idj) β 8π 2 = 3M M 6 Decoupling limit (near brane): U field theory λ t Hooft = g 2 YM M 1 log U 1 [Maldacena,Núñez] (CV-MN) solution: SU(M) N = 1 SYM g 2 YM

20 The baryonic branch from fivebranes Using the generating technique, we can add D3 branes and B-field to the unwarped resolved deformed conifold Warp factor generated: h = 1 + cosh 2 β(e 2(Φ Φ ) 1) Transformed solution has all fluxes (except F 1 ) and depends on one new parameter (β): M, Φ, U, β

21 The baryonic branch from fivebranes Using the generating technique, we can add D3 branes and B-field to the unwarped resolved deformed conifold Warp factor generated: h = 1 + cosh 2 β(e 2(Φ Φ ) 1) Transformed solution has all fluxes (except F 1 ) and depends on one new parameter (β): M, Φ, U, β Solution incorporates various limits. In a decoupling limit β we recover the baryonic branch of the KS theory [Butti,Graña,Minasian,Petrini,Zaffaroni]

22 The baryonic branch from fivebranes Using the generating technique, we can add D3 branes and B-field to the unwarped resolved deformed conifold Warp factor generated: h = 1 + cosh 2 β(e 2(Φ Φ ) 1) Transformed solution has all fluxes (except F 1 ) and depends on one new parameter (β): M, Φ, U, β Solution incorporates various limits. In a decoupling limit β we recover the baryonic branch of the KS theory [Butti,Graña,Minasian,Petrini,Zaffaroni] New perspective on the baryonic branch of Klebanov-Strassler Fivebranes on S 2 of resolved conifold + B-field: for large U we expect fivebranes on a fuzzy S 2

23 Fuzzy two-sphere from the baryonic branch of KS Klebanov-Strassler theory: SU(Mk) SU(M(k + 1)) quiver

24 Fuzzy two-sphere from the baryonic branch of KS Klebanov-Strassler theory: SU(Mk) SU(M(k + 1)) quiver Classical baryonic branch vacuum [Dymarsky,Klebanov,Seiberg] A i = C Φ i 1 M M, B i = 0 0 Φ 1 = D terms : 1 k p k p 0 0 k C A { Φ 2 = B A k A 1 A 1 + A 2A 2 B 1 B 1 B 2 B 2 = (k + 1) C 2 1 k A 1 A 1 + A 2 A 2 B 1 B 1 B 2B 2 = k C 2 1 k+1

25 Fuzzy two-sphere from the baryonic branch of KS Klebanov-Strassler theory: SU(Mk) SU(M(k + 1)) quiver Classical baryonic branch vacuum [Dymarsky,Klebanov,Seiberg] A i = C Φ i 1 M M, B i = 0 0 Φ 1 = D terms : 1 k p k p 0 0 k C A { Φ 2 = B A k A 1 A 1 + A 2A 2 B 1 B 1 B 2 B 2 = (k + 1) C 2 1 k A 1 A 1 + A 2 A 2 B 1 B 1 B 2B 2 = k C 2 1 k+1 From the k (k + 1) matrices Φ i we construct matrices spanning two irreducible representations of SU(2)

26 Fuzzy two-sphere from the baryonic branch of KS L 1 = 1 2 (Φ 1Φ 2 + Φ 2Φ 1 ) L 2 = i 2 (Φ 1Φ 2 Φ 2Φ 1 ) L 3 = 1 2 (Φ 1Φ 1 Φ 2Φ 2 ) Define a spin j = k 1 2 irreducible representation of SU(2) R 1 = 1 2 (Φ 1 Φ 2 + Φ 2 Φ 1) R 2 = i 2 (Φ 2 Φ 1 Φ 1 Φ 2) R 3 = 1 2 (Φ 1 Φ 1 Φ 2 Φ 2) Define a spin j = k 2 irreducible representation of SU(2) Looks like these define SU(2) SU(2) SO(4), but in fact they define a fuzzy super two-sphere

27 The fuzzy sphere spectrum (weak coupling) Fluctuations: A i = Φ i + δa i, B i = δb i, a L,R µ = δa L,R µ fields on S 2 SU(2) spin N = 1 multiplet eigenvalues a L µ, ar µ scalar j = l 1 vector λ l,, λ l,+ δa i vector j = l 1 vector λ l,, λ l,+ B i spinor j = l chiral λ l,b Eigenvalues for l k λ l, g2 + C 2 2k + 1 l(l + 1), λ l,b = C 4 h 2 (l + 1) 2 Agrees with spectrum of Maldacena-Núñez compactification of D5 branes wrapped on S 2 [Andrews,Dorey] Parameters: θ Fuzzy 1 k C 2 R 2 Fuzzy k g 2 +

28 Comparison with gravity (strong coupling) Compare with the parameters computed in the gravity solution (in the intermediate fivebrane region) B g s M log U k # cascade steps S 2 U Tr[A i A i B i B i] C 2 MUΛ 2 0 i=1,2 [Dymarsky,Klebanov,Seiberg] Large B-field use open string metric: r open [Seiberg,Witten] B r closed Parameters: θ NC 1 B m 2 KK C g2 + 2 k

29 Adding flavours [Gaillard,DM,Núñez,Papadimitriou] Branes wrapped on an infinitely extended surface effective 4d coupling constant vanishes flavours [Karch,Katz] Back-reacted solution with N f N c smeared D5 flavour branes constructed in [Casero,Nuñez,Paredes] SU(3) structure transformation flavoured warped resolved deformed conifold, includes: N c colour D5, N f flavour D5, plus bulk and source D3 branes Different from previous flavoured solutions, obtained with D7 branes. Possible because D5 probes (with D3 charge) are supersymmetric on the resolved deformed conifold

30 The fivebranes set up S 2 base S 2 fiber ψ r N c D5 x x x x x x N f D5 x x x x x x Directions span a resolved conifold: R 4 S 2 base. S 2 fiber R4 fiber To include back-reaction use non-kähler equations: F 3, Φ, g µν ( ) S = S type I e 6 +Φ/2 vol (4) J C (6) Ξ (4) N f D5 branes smeared on 4567 smearing form Ξ (4) Susy (non-kähler) equations unchanged. Bianchi identity df 3 = Ξ (4)

31 The flavoured fivebrane solution Ansatz unchanged. BPS equations slightly modified: f = 4 sinh 2 t c c = 1 f [c2 sinh 2 t (t cosh t sinh t) 2 ] N f 2N c Parameters of the solution: N f, N c, e Φ = g s, U UV asymptotics unchanged. E.g. Φ constant Singularity in the IR. We impose boundary conditions such that the smooth solution is recovered for N f 0

32 The flavoured fivebrane field theory In the decoupling limit U there is a field theory dual SU(N c ) SQCD + N f quarks q i, q i [Casero,Nuñez,Paredes] Quartic superpotential W (q q) 2 Beta function β 8π 2 = 3 (N c N f /2) g 2 YM Smearing on S 2 45 S2 67 extra SU(2) SU(2) symmetry SU(N c ) [SU(N f /2) SU(2) SU(2)] flavour U(1) R (U(1) R broken to Z 2 )

33 The flavoured, deformed, resolved, conifold Using the transformation we add D3 branes, B-field and generate a warp factor h = 1 + cosh 2 β(e 2(Φ Φ ) 1) In decoupling limit β we obtain new solution generalising that of [Butti et al] by the addition of flavour D5 branes

34 The flavoured, deformed, resolved, conifold Using the transformation we add D3 branes, B-field and generate a warp factor h = 1 + cosh 2 β(e 2(Φ Φ ) 1) In decoupling limit β we obtain new solution generalising that of [Butti et al] by the addition of flavour D5 branes A distribution of source D3 branes is generated (as required by κ-symmetry) h (N f/u)r 2 + (N c N f /2) 2 log r r 4 + O(1/r 4 ) Define two types of running number of D3 branes n f N f U r2 νr 2 n (N c N f /2) 2 log r for r

35 Klebanov-Strassler + source D3 branes Instructive to consider a special limit: U 0 In the unflavoured [Butti et al] solution this gives the ordinary Klebanov-Strassler solution With N f 0 this limit is not possible. Reason: probe D5 branes are not susy on the deformed conifold Consider a new limit: U 0, N f 0 at fixed N f /U = ν. The internal metric is the deformed conifold, warped by: h = 2ν ρ dρ ( sinh(4ρ ) 4ρ ) 1/3 + hks df 5 H 3 F 3 = νb Ξ (4) 0 source D3 branes

36 Comments on the field theory UV asymptotics would-be Goldstone boson mode for spontaneously broken U(1) B is not normalisable Indeed presence of n f extra D3 branes suggests changing the theory from (1) SU(N c (k + 1)) SU(kN c ) to (2) SU(N c (k + 1) + n f ) SU(kN c + n f ) In (1) there is a baryonic branch, whereas in (2) there isn t theory on the mesonic branch! [Dymarsky,Klebanov,Seiberg] Two different types of varying F 5 flux interpreted as: n N 2 c log r cascade of Seiberg dualities n f νr 2 Higgsing [Aharony] Adding back N f complicates the picture in an interesting + puzzling way..

37 A G 2 story Flash out a similar construction in Type IIA supergravity ds 2 = h 1/2 dx 2 1,2 + h1/2 ds 2 7 F 2 = 0, F 4, H 3 unconstrained [in progress]

38 A G 2 story Flash out a similar construction in Type IIA supergravity ds 2 = h 1/2 dx 2 1,2 + h1/2 ds 2 7 F 2 = 0, F 4, H 3 unconstrained [in progress] Supersymmetry associative three-form φ and function ζ, defining an interpolating G 2 structure. BPS equations follow from [DM,Sparks 2003]

39 A G 2 story Flash out a similar construction in Type IIA supergravity ds 2 = h 1/2 dx 2 1,2 + h1/2 ds 2 7 F 2 = 0, F 4, H 3 unconstrained [in progress] Supersymmetry associative three-form φ and function ζ, defining an interpolating G 2 structure. BPS equations follow from [DM,Sparks 2003] Geometry d(e 2Φ 7 φ) = 0 φ dφ = 0 cos 2 ζ = h

40 A G 2 story Flash out a similar construction in Type IIA supergravity ds 2 = h 1/2 dx 2 1,2 + h1/2 ds 2 7 F 2 = 0, F 4, H 3 unconstrained [in progress] Supersymmetry associative three-form φ and function ζ, defining an interpolating G 2 structure. BPS equations follow from [DM,Sparks 2003] Geometry d(e 2Φ 7 φ) = 0 φ dφ = 0 cos 2 ζ = h Fluxes F 4 = vol 3 dh 1 + c 2 d(e 2Φ φ) 7 H 3 = c 1 e 2Φ d(e 2Φ φ) ζ 0: 7d counterpart of non-kähler geometries [GMPW] ζ π/2: warped G 2 holonomy manifold [Cvetic,Lu,Pope]

41 A G 2 story Solution generating method: start with M fivebranes wrapped on the S 3 inside the G 2 -manifold X = S 3 R 4. After backreaction the geometry is G 2 with torsion : there is an interpolating parameter U U 0: G 2 -manifold X with large S 3 + H 3 flux U : G 2 -manifold X + NS5 branes on S 3 Realises G 2 geometric transition in Type IIA supergravity Decoupling limit (near brane): U field theory T-dual [Maldacena,Nastase]: SU(M) M 2 N = 1 Chern-Simons N = 1, 3d field theory dual to warped G 2 manifold not known presumably it is a Chern-Simons theory

42 Outlook Solution generating transformation for classes of supersymmetric geometries of Type IIA/IIB Solutions realising geometric transitions in Type IIB (torsional SU(3)) and Type IIA (torsional G 2 ) Perhaps there exist other classes with similar features, besides SU(3) and G 2. Eleven dimensions? Relation between baryonic branch of KS and fuzzy two-sphere may be explored for more general quiver theories Constructed new flavoured resolved deformed conifold solution. Field theory interpretation seems different from baryonic branch (and any other solutions so far) The G 2 story: the geometry works as in the SU(3) case. It would be nice to have a field theory picture