Integrability, Poisson-Lie Symmetry and Double Field Theory Falk Hassler University of North Carolina at Chapel Hill University of Pennsylvania based on work in progress, 1707.08624, 1611.07978 and 1502.02428 with Pascal du Bosque, Dieter L¨ ust and Ralph Blumenhagen April 23rd, 2018
Holography, Strings and Exceptional Field Theory Canonical motivation for Exception/Double Field Theory S SO ( 32 ) heterotic Type I T M-theory S S Type IIA E 8 × E 8 heterotic T = T-duality T S = S-duality Type IIB S
Holography, Strings and Exceptional Field Theory But there is also another interesting story. . . AdS/CFT correspondence ∞ classical string theory perturbative gauge theory perturbative string theory classical gauge theory quantum gravity N ∼ g s 1 1 / N expansion planar limit classical SUGRA 0 ∞ 0 1 λ ∼ α ′ 2
Outline 1. Motivation 2. Integrability and AdS/CFT 3. Poisson-Lie Symmetry 4. Double Field Theory on Drinfeld doubles 5. Summary
Integrability or how to “solve” 4D maximal SYM completely Motivation Integrability Poisson-Lie Symmetry Double Field Theory Summary
Anomalous dimension in 4D N = 4 SYM ◮ CFT two point function of primaries � � δ ij O i ( x ) O j ( y ) = | x − y | 2 ∆ ◮ scaling dimension gets renormalized ∆ = ∆ 0 + λ ∆ 1 + . . . ◮ example single trace operator Tr Z L 1 Z = 2 ( φ 1 + i φ 2 ) √ � � � 4 F µν F µν − 1 2 D µ φ i D µ φ i − g 2 d 4 x Tr − 1 4 [ φ i , φ j ][ φ i , φ j ] + fermions S = ◮ ∆ 0 = L what about ∆ 1 , . . . ◮ more general single trace operators with 1 ( L − M ) × Z and M × W = 2 ( φ 3 + i φ 4 ) √ Motivation Integrability Poisson-Lie Symmetry Double Field Theory Summary
SU(2) spin chain and the Bethe ansatz ◮ ∆ 1 ↔ eigenvalues of the Heisenberg spin chain � � � L 4 − � S l � 1 S l = 1 H = 2 2 � σ l S l + 1 l = 1 Z = ↑ , W = ↓ , and for L = 3 Tr ZZW = | ↑↑↓� ◮ Bethe ansatz gives rise to eigenvalues and vectors ◮ just possible because spin chain is integrable ◮ integrability is so powerful that it also to find all corrections ∆ 1 , ∆ 2 , ∆ 3 . . . Motivation Integrability Poisson-Lie Symmetry Double Field Theory Summary
Where is the integrability in string theory? Ingredients for classical/quantum integrability: 1. Hamiltonian/Hamilton operator 2. Poisson-bracket/commutator 3. Lax pair ◮ example Principal Chiral Model (PCM) � S = 1 d 2 σ Tr( g − 1 ∂ + g g − 1 ∂ − g ) 2 � H = 1 d σ Tr( j 2 0 + j 2 j 0 = g − 1 ∂ τ g j 1 = g − 1 ∂ σ g 1 ) 2 L ± ( λ ) = j 0 ± j 1 { j 0 a ( σ ) , j 0 b ( σ ′ ) } = f ab c j 0 c 1 ± λ { j 0 a ( σ ) , j 1 b ( σ ′ ) } = f ab c j 1 c + δ ab { j 1 a ( σ ) , j 1 b ( σ ′ ) } = 0 Motivation Integrability Poisson-Lie Symmetry Double Field Theory Summary
Let’s generalize this construction! ◮ Hamiltonian (Poisson-Lie σ -model) : � H = 1 d σ j A ( σ ) H AB j B ( σ ) 2 ◮ Poisson-bracket: { j A ( σ ) , j B ( σ ′ ) } = F ABC j C ( σ ) δ ( σ − σ ′ ) + η AB δ ′ ( σ − σ ′ ) ◮ Lax pair: L ± ( λ ) = J ± R 1 ± λ Examples: ◮ η -deformation All know integrable 2D non-linear ◮ with/without WZW term σ -models can be brought in this ◮ on group mainfolds form. They are fixed completely by ◮ and coset spaces specifying the constants H AB and ◮ λ -deformation F ABC . Motivation Integrability Poisson-Lie Symmetry Double Field Theory Summary
Poisson-Lie symmetry Poisson as in Poisson-bracket: required for the Hamiton formalism Lie as in Lie-algbra: e.g. required for Lax’s equation dL dt = [ P , L ] Motivation Integrability Poisson-Lie Symmetry Double Field Theory Summary
Drinfeld double [Drinfeld, 1988] Definition: A Drinfeld double is a 2 D -dimensional Lie group D , whose Lie-algebra d 1. has an ad -invariant bilinear for � · , ·� with signature ( D , D ) 2. admits the decomposition into two maximal isotropic subalgebras g and ˜ g ◮ � � t a ∈ ˜ t a t a = t A ∈ d , t a ∈ g and g � 0 � δ a ◮ � t A , t B � = η AB = b δ b 0 a ◮ [ t A , t B ] = F ABC t C with non-vanishing commutators [ t a , t b ] = ˜ [ t a , t b ] = f abc t c f bca t c − f acb t c [ t a , t b ] = ˜ f abc t c ◮ ad -invariance of � · , ·� implies F ABC = F [ ABC ] Motivation Integrability Poisson-Lie Symmetry Double Field Theory Summary
Poisson-Lie Symmetry [Klimcik and Severa, 1995] ◮ 2D σ -model on target space M with action � z E ij ∂ x i ¯ ∂ x j dzd ¯ S ( E , M ) = ◮ E ij = g ij + B ij captures metric and two-from field on M ◮ inverse of E ij is denoted as E ij ◮ left invariant vector field v ai on G is the inverse transposed of right invariant Maurer-Cartan form t a v ai dx i = dg g − 1 ◮ adjoint action of g ∈ G on t A ∈ d : Ad g t A = gt A g − 1 = M AB t B ◮ analog for ˜ G Definition: S ( E , D / ˜ G ) has Poisson-Lie Symmetry if E ij = v ci M ac ( M ae M be + E ab d v d j 0 ) M b holds, where E ab is constant and invertible with the inverse E 0 ab . 0 Motivation Integrability Poisson-Lie Symmetry Double Field Theory Summary
Immediate consequence: Poisson-Lie T-duality ◮ exchanging G and ˜ G results in dual σ -model with E ij = ˜ v ci ˜ M be + E 0 ab ) ˜ ˜ M ac ( ˜ M ae ˜ M bd ˜ v dj ˜ abelian T-d. G abelian and G abelian ˜ ◮ captures non-abelian T-d. G non-abelian and G abelian [Ossa and Quevedo, 1993;Giveon and Rocek, 1994; Alvarez, Alvarez-Gaume, and Lozano, 1994;. . . ] ◮ dual σ -models related by canonical transformation [Klimcik and Severa, 1995;Klimcik and Severa, 1996;Sfetsos, 1998] → equivalent at the classical level ◮ preserves conformal invariance at one-loop [Alekseev, Klimcik, and Tseytlin, 1996;Sfetsos, 1998;. . . ;Jurco and Vysoky, 2017] ◮ dilaton transformation [Jurco and Vysoky, 2017] � �� � � � 0 (˜ φ = − 1 g − 1 1 + ˜ 2 log � det B 0 + Π) � � � �� � � φ = − 1 ˜ 1 + g − 1 0 ( B 0 + ˜ 2 log � det Π) � Motivation Integrability Poisson-Lie Symmetry Double Field Theory Summary
SUGRA ◮ DFT makes PL-Symmetry manifest ◮ consistent tructions are central ◮ get the dialton, R/R sector nearly for free Motivation Integrability Poisson-Lie Symmetry Double Field Theory Summary
Additional structure on the Drinfeld double [Blumenhagen, Hassler, and L¨ ust, 2015, Blumenhagen, Bosque, Hassler, and L¨ ust, 2015] ◮ right invariant vector E AI field on D is the inverse transposed of left invariant Maurer-Cartan form t A E AI dX I = g − 1 dg ◮ two η -compatible, covariant derivatives 1 1. flat derivative � � D A V B = E AI ∂ I V B − wF A V B , � det( E BI ) � F A = D A log 2. convenient derivative ∇ A V B = D A V B + 1 3 F ACB V C ◮ generalized metric H AB ( w = 0) H AC η CD H DB = η AB H AB = H ( AB ) , ◮ generalized dilaton d with e − 2 d scalar density of weight w = 1 ◮ triple ( D , H AB , d ) captures the doubled space of DFT 1 definitions here just for quantities with flat indices Motivation Integrability Poisson-Lie Symmetry Double Field Theory Summary
Double Field Theory for ( D , H AB , d ) [Blumenhagen, Bosque, Hassler, and L¨ ust, 2015] see also [Vaisman, 2012; Hull and Reid-Edwards, 2009;Geissbuhler, Marques, Nunez, and Penas, 2013; Cederwall, 2014; . . . ] ◮ action ( ∇ A d = − 1 2 e 2 d ∇ A e − 2 d ) � d 2 D Xe − 2 d � 1 8 H CD ∇ C H AB ∇ D H AB − 1 2 H AB ∇ B H CD ∇ D H AC S NS = D CD H AB � − 2 ∇ A d ∇ B H AB + 4 H AB ∇ A d ∇ B d + 1 6 F ACD F B ◮ generalized diffeomorphisms � ∇ A ξ B − ∇ B ξ A � L ξ V A = ξ B ∇ B V A + V B + w ∇ B ξ B V A ◮ 2 D -diffeomorphisms L ξ V A = ξ B D B V A + wD B ξ B V A ◮ global O( D , D ) transformations V A → T AB V B T AC T BD η CD = η AB with ◮ section condition (SC) η AB D A · D B · = 0 Motivation Integrability Poisson-Lie Symmetry Double Field Theory Summary
Symmetries of the action ◮ S NS invariant for X I → X I + ξ A E AI and 1. H AB → H AB + L ξ H AB e − 2 d → e − 2 d + L ξ e − 2 d and 2. H AB → H AB + L ξ H AB e − 2 d → e − 2 d + L ξ e − 2 d and object gen.-diffeomorphisms 2 D -diffeomorphisms global O( D , D ) H AB tensor scalar tensor ∇ A d not covariant scalar 1-form e − 2 d scalar density ( w =1) scalar density ( w =1) invariant η AB invariant invariant invariant F ABC invariant invariant tensor E AI invariant vector 1-form S NS invariant invariant invariant SC invariant invariant invariant D A not covariant covariant covariant ∇ A not covariant covariant covariant � �� � manifest Motivation Integrability Poisson-Lie Symmetry Double Field Theory Summary
Poisson-Lie T-duality: 1. Solve SC [Hassler, 2016] � � ◮ fix D physical coordinates x i from X I = x ˜ x i i on D � 0 � . . . such that η IJ = E AI η AB E BJ = → SC is solved . . . . . . ◮ fields and gauge parameter depend just on x i ◮ only two SC solutions, relate them by symmetries of DFT � � ˜ d ( X I ) = g ( x i )˜ i ) t a g ( x t A = t a � t a � g ( x ′ i ) g ( x ′ ˜ t A = d ( X ′ I ) = ˜ i ) t a Motivation Integrability Poisson-Lie Symmetry Double Field Theory Summary
Recommend
More recommend
