Generalised T-duality and Integrable Deformations

Transcription

1 Generalised T-duality and Integrable Deformations Daniel C. Thompson Sevrin, DT [1XXX.XXXXX] ; Dorigoni, DT [160X.XXXXX] Sfetsos, Siampos, DT [ ] Demulder, Sfetsos, DT [ ] Sfetsos, DT [ ] Duality and Novel Geometry in M-theory APCTP February 1 st 2016

2 Motivation and Outline Explore the landscape of dualities; DFT/EFT beyond the section; Applications!!! e.g. Holography TODAY: 1. Recap/Fresh Angle on world-sheet Doubled Formalism 2. Generalised T-dualities: Non-Abelian, Poisson-Lie 3. New integrable 2d (C)QFTs & Generalised Duality

3 1. World-sheet Doubled Formalism

4 The Tseytlin Model Revisited Amazing progress in D/EFT mostly from a space-time perspective, but lets not ignore the world sheet. Important if we want to get to grips with the non-leea nature Consider a non-linear σ-model L[X ] = + X I E IJ X J + X I (G IJ + B IJ ) X J Adapted coordinates for U(1) d isometry X I = {x i, Y α }. Buscher part I: gauge flat connection with Lag. multipliers x i. L g = + xg x + J + A + J A + + A ga + J + = E T + x x, J = E + x + x Buscher part II: gauge fix on x and integrate out connection Dual model

5 Instead keep both x and x, partially fix on A and partial integrate out A Double model Axial Gauge Choice [Rocek Tsyetlin 98] A + = A = a Integrate out a and get a theory for X = {x, x} L Tseytlin = H IJ σ X I σ X J + η IJ σ X I τ X J + Ω IJ σ X I τ X J +... ( ) ( ) ( ) H = g bg 1 b bg 1 g 1 b g 1, η = , Ω = Note 1: Topological term inevitable! Note 2: Residual gauge invariance δx = ξ(τ) 1 st order eqm for chiral Bosons dx I = (ηh) I JdX J

6 PST Model via Gauging Cuter fixing condition with some auxiliary function f (σ +, σ ) A + f = A + f A ± = a ± f, Covariant Doubled Action à la PST [ Pasti Sorokin Tonin] L PST = H + X X +f f (P + X) 2 f + f (P + X) 2 Ω + X X +... Projectors 2P ± = 1 ± ηh Residual gauge invariance δx = ξ(f ) 1 st order eqm for chiral Bosons Duality symmetric dilaton emerges from gaussian integral over single component a a i g ij a j Φ = φ 1 log det g 4

7 Given f this defines a theory Z [f ] = [dx][db][dc]e i L PST +L gh L gh = + f b c f b + c For full covariance we should allow f to be a field i.e. average over gauge fixing choices Z = 1 [df ]Z [f ] vol Physics shouldn t (hopefully) depend on f PST symmetry for all fields: ( P+ X δf = ɛ, δx = ɛ + P ) + X f + f Fixing PST symmetry f (σ +, σ ) = τ (no more ghosts!) gives back Tseytlin action Cautions: measure on of f ; fixing need be only locally defined Work in progress: extend gauging approach to superspace, open sector etc.

8 2. Generalised T-dualities

9 Non-Abelian T-duality Abelian dualities have a natural analogue when space time Killing vectors forming a group: Non-Abelian T-duality [ de la Ossa & Quevedo 93 ] Simplest example: Principal Chiral Model Lagrangian L = L a +E ab L b A theory of maps g : Σ G with one-forms L = g 1 dg = L a i dxi T a, dl a = 1 2 f ab c L b L c Buscher the G L global symmetry g hg. Gauge with minimal coupling g Dg = g Ag, A = ia a T a Lagrange multiplier term x a F a + = x a ( + A A + [A +, A ]) a

10 Integration out of the gauge fields and fixing g = 1 T-dual theory L T dual = + x T (E + f ) 1 x, f = f abc x c Unlike Abelian T-duality not expected to be exact CFT duality Isometries (and e.g. super symmetries) of the target space that don t commute with duality are destroyed. Realisation as a canonical transformation of phase space variables Principal Chiral Models on groups or cosets are important AdS n S n Non-abelian T-duality can lift to the RR-sector and given life in SUGRA [Sfetsos & Thompson 10] Remarkable utility in context of holography as a solution generating tool [Sfetsos, Thompson, Lozano, O Colgain, Nunez, Itsios, Macpherson ]

11 A Double model for non-abelian T-duality Instead, choose Axial gauge to get a theory for X M = {x, x} ( ) L A L a =, L L a = (ad g ) b a d x b a L Tseytlin = H AB L A σ L B σ + η AB L A σ L B τ + Ω AB L A σ L B τ +... In coordinate basis H MN (X) = L A M H ABL B N depends in general on all the x (but not the x) Topological term contributes a potential for a three form flux H = dω = f ab c L a L b L c Suggests chiral-wzw [ Klimcik & Severa, Sfetsos, Hull & Reid-Edwards ] S = Σ d 2 σ H AB L A σ L B σ + η AB L A σ L B τ + M 3 f AB D η DC L A L B L C

12 Mathematica Detour: The Drinfeld Double The Drinfeld Double is a Lie Algebra D which can be decomposed as the sum D = G + G for two maximally isotropic sub-algebras. If T A = {T a, T a } then T a T b = T a T b = 0 T A T B = η AB Mixed Jacobi identity restrict choices of G and G. Some examples: Abelian Double D = u(1) d + u(1) d Semi-Abelian Double D = G + u(1) d Non-Abelian Double D = G + G (e.g. so(3, 1) = su(2) + e 3 ) The Doubled σ-model lives on the Drinfeld Double [Klimcik & Severa] (also [Sfetsos, Hull & Reid-Edwards ])! Structure constants & WZW structure torsion = ± spin connection

13 Drinfeld Double & The Section Condition Regular T-dual pairs of σ-models extracted by parameterising h D as h = g g and integrating out g or as h = gg and integrating out g D = u(1) d + u(1) d Abelian T-duality H MN constant D = G + u(1) d non-abelian T-duality H MN (x) on-section D = G + G Poisson-Lie T-duality H MN (x, x) beyond-section β-function of H AB implies scalar potential of gauged supergravity!! [Dall agata,prezas; Sfetsos-Siampos-DT] dh AB = 1 ( ) dt 4 (H AC H BF η AC η BF ) H KD H HE η KD η HE fkh C f DE F Compare with DFT Scherk-Schwarz!

14 PLT details PL T-duality which is an equivalence between two σ-models S[g] = 1 d 2 σl T 2πt +(E Π) 1 L, g G, S[ g] = 1 d 2 σ L T 2πt +(E 1 Π) 1 L, g G. The group theoretic matrix Π a a b = g 1 T a g, T b, b ab = g 1 T a g, T b, Π = b T a Typically these backgrounds have no isometries so don t expect conserved currents however non-commutative conservation with respect to dual group d J a = f bc a J b J b (1) Thus J should be pure gauge in a dual algebra (Field Equations Bianchi identity)

15 3. New Integrable Models

16 Integrability in AdS-CFT Integrability has been a game changer for N = 4 SYM Source: Main image from Beisert et al. Others from Wikipedia/DCT

17 Integrability in AdS-CFT Is integrability just a feature of maximal supersymmetry? QCD in high energy limits [Lipatov; Faddeev, Korchemsky] Marginal (real)-β deformations that break SUSY to N = 1 ( ) d 4 x d 4 θ Tr e i β XYZ e i β ZXY String holographic spacetime given by doing T-dualities (TsT) [Lunin,Maldacena] is integrable [Frolov] Other generalisations e.g. null TsT transformations and Schrodinger deformations [Bobev, Kundu], non-comm YM etc. recently classified in terms of Yang-Baxter equation [Matsumoto,Yoshida; Van Tongeren] New η-deformations [Delduc,Magro, Vicedo] based on Yang-Baxter σ-models [Klimcik] preserve no SUSY and only some U(1)

18 Integrability: a game-changer for AdS/CFT Can we deform AdS/CFT and keep integrability? Two new ideas: η and λ integrable deformations Closely connected via generalised T-dualities

19 η-deformations Recap: the PCM Toy model: Principal Chiral Model [PCM] on S 3 S = κ2 ( ) d 2 σtr g 1 + gg 1 g 4π Σ Non-conformal proto-qcd model SU(2) L SU(2) R symmetry Integrable: Lax formulation and conserved charges, g : Σ SU(2) A(z) = 1 1 z 2 L + z 1 z 2 L, da A A = 0 T (z) = P exp dσa σ

20 η-deformations A toy model for η Deform [Cherednik 81]: S = κ2 ( ) d 2 σtr g 1 + gg 1 g + CJ 4π +J 3 3 Σ Integrable but SU(2) L SU(2) R SU(2) L U(1) R Non-local charges recover semi-classical version of U q (sl 2 ) [Kawaguchi, Matsumoto, Yoshida 11, 12] ( ) {Q + R, Q R } P.B. = qq3 R q QR 3 C q q 1, q = exp 1 + C

21 η-deformations Yang-Baxter and η Arbitrary groups [Klimcik 02] based on modified Yang-Baxter eq: [RA, RB] R([RA, B] + [A, RB]) = [A, B], ( g 1 + g, S η = 1 d 2 σtr 2πt Σ 1 1 ηr g 1 g A, B g ) Poisson algebra of G R gives quantum group [Delduc, Magro, Vicedo 1308] Cosets and super-cosets i.e. AdS 5 S 5 superstring [Delduc, Magro, Vicedo 1309] Perturbative matching to q-deformed S-matrix [Artyunov, Borsato, Frolov 1312] [Beisert, Krooteev 08; Hoare, Hollowood, Miramontes 11; de Leeuw, Regelskis, Matsumoto 11 ]

22 λ-deformations λ-deformations A simple recipe for integrable λ deformations [Sfetsos 1312] 1. Double the d.o.f.: κ 2 S PCM [ g] + ks WZW [g] 2. Gauge G L in PCM and G diag in WZW 3. Gauge Fix g = 1 4. Integrate out non-propagating gauge fields S λ = ks WZW + k Tr(g g 2π λ 1 gg 1 ) + Ad g λ = k κ 2 + k

23 λ-deformations λ-limits Nice behaviour in limits of small and large deformations: λ 0: current bilinear perturbation S λ λ 0 ks WZW + k λj a π +J a + O(λ 2 ) λ 1: non-abelian T-dual of PCM S λ λ X a (δ π ab + f c ab X c ) 1 X b + O(k 1 )

24 λ-deformations λ-space time Gets pretty messy, so lets explicitly give S 2 : ( ds 2 1 λ = k 1 + λ (dω2 + cot 2 ωdφ 2 ) + 4λ ) (cosφdω + sinφcotωdφ)2 1 λ2 After wick rotation ω iρ, φ it, k k this gives a 2d-geometry that is a deformation of the Witten black hole. In Kruskal coordinates: u = cosh ρ(e t + λe t ), v = cosh ρ(e t + λe t ) ds 2 = k(1 λ 2 ) dudv f (u, v), f (u, v) = (u λv)(v λu) (1 λ2 ) 2. v V u IV II III I VI

25 λ-deformations λ-developments Applies to (super)-cosets [Hollowood, Miramontes, Schmidtt 1408, 1409] Conjectured to be quantum group deformation with q root unity [HMS 1409,1506; Itsios et al. 1409] β-functions [Itsios, Sfetsos, Siampos 1405] vanish for PSU(2, 2 4)/SO(4, 1) SO(5) [Appadu, Hollowood 1507] Type II supergravity embeddings of bosonic λ-deformations [Sfetsos,DT 1410; Demulder, Sfetsos, DT 1504] Multi-parameter integrable λ-deformations [Siampos, Sfetsos, DT 1506]

26 The η-λ connection η, λ and Poisson-Lie η and λ connected by generalised Poisson Lie T-duality [Vicedo 1504; Hoare & Tseytlin 1504; Siampos Sfetsos DT 1506; Klimcik 1508] Modified conservation law for currents of broken G R in η-model: d J a = f bc aj b J c f bc a structure constants for g R [A, B] R = [RA, B] [A, RB] Mathematically g g R g C defines a Drinfel d Double

27 The η-λ connection η, λ and Poisson-Lie Can still T-dualise these currents Drinfel d Double provides the Doubled Formalism! Analytic continue certain Euler angles and deformation parameters Acting on the parameter q we have η i 1 λ π(1 + λ), t 1 + λ (1 λ) q = e ηt q = e i π k

28 Conclusions and Outlook η and λ open a new window onto integrable deformations in the AdS/CFT conjecture and generalised T-dualities What s next? 1. Implications of Poisson-Lie duality for DFT? 2. Consistent CFTs? 3. New scenario s e.g. AdS 4 CP 3? 4. Demonstrate the Mass gap in η-deformed 2 d QFTs 5. Implication on gauge theory side?

29 Appendix

30 Currents for squashed S 3 With J = g 1 dg define U(1) R current and non-local currents j R,3 = 2(1 + C )J 3 j R,± µ = 2e γχ ( η µµ + i C ɛ µν ) Tr(T ± J ν ) with non-local contributions χ(x) dyɛ(x y)jt R,3 (y) Conserved charges Q ± = j R,± t = Q ± 0 ± i CQ ± Q 0 and Q 1 generate Yangian

31 Yangian Schematic picture. Generators Q 0 = J and Q 1 = Q obey [J, J] = FJ, [J, Q] = [Q, J] = FQ Co-product J = J J, Q = Q Q + α/2fj J Serre relations (since acts as structure preserving map) (extra relation needed for SU(2) [Q, [Q, J]] [J, [Q, Q]] F 4 J 3

32 U q (sl 2 ) Classical Lie algebra [H, X ± ] = ±2X ±, [X +, X ] = H Quantum group: algebra generated by 1, X ± and q ±H/2 Co-product: [X +, X ] = qh q H q q 1, Ad q H/2X ± = q ±1 X ± q ±H/2 = q ±H/2 q ±H/2, X ± = X ± q H/2 + q H/2 X ±

33 η-space time For AdS 5 the η deformed space time is given by ds 2 5 = 1 + ρ2 dρ 1 κ 2 ρ 2 dt2 + 2 (1 κ 2 ρ 2 )(1 + ρ 2 ) + ρ 2 cos 2 ζ 1 + κ 2 ρ 4 sin 2 η dψ κ 2 ρ 4 sin 2 η dζ2 + ρ 2 sin 2 ζdψ2 2