Defects in Classical and Quantum WZW models

Transcription

1 Defects in Classical and Quantum WZW models Ingo Runkel (King s College London) joint work with Rafał Suszek (King s College London) [hep-th]

2 Outline Defects in classical sigma models Jump defects in WZW models Defects in conformal quantum field theory & defect junctions Comparison of classical and quantum WZW model with jump defects Orbifolds

3 Applications of defects in 2d CFT (continuum limit of) order-disorder duality in lattice models perturbative symmetries of string theory, in particular T-duality functors between D-brane categories...

4 Classical σ-models world sheet Σ (metric γ) target space M (metric G, closed 3-form H) X : Σ M once differentiable S[X] = Skin[X] + Stop[X]

5 Topological term If H = db : If [H] H 3 (M, 2πZ) : Alvarez 85 Gawedzki 88 Brylinski 93 Ui : good open cover of M G S top [X] = i exp( S top ) ( = exp i = ( Bi : 2-form on Ui, dbi = H Σ X B Σ gerbe ) X G Aij : 1-form on Ui Uj gijk : function Ui Uj Uk U(1) )

6 Boundaries X M Target space : D-brane ι : D M 2-form ω on D s.t. ι H = dω If H=dB : twisted line bundle with connection on D, ω = B + F (bulk action) + log Hol ( X( Σ) ) General: gerbe module on D Gawedzki 99, Kapustin 99 Carey, Johnson, Murray 02 = stable isomorphism Φ : ι G I ω on D

7 Defects Λ X1 M Fuchs, Schweigert, Waldorf 07 X2 M Target space : bi-brane 2-form ω on Q s.t. (ι 1, ι 2 ) : Q M M ι 1H ι 2H = dω If H=dB : twisted line bundle with connection on Q, ω = ι1 * B - ι2 * B + F (bulk action) + log Hol ( (X1,X2)(Λ) ) General: gerbe bimodule on Q = stable isomorphism on Q Φ : ι 1G ι 2G I ω

8 Defect condition for fields X1 M y x p X2 M 1) (X 1 (p), X 2 (p)) Q M M 2) for all v1 v2 T(X1,X2)Q G X1 (p)(v 1, y X 1 ) G X2 (p)(v 2, y X 2 ) Suszek, IR 08 = i 2 ω (X 1,X 2 )(p)(v 1 v 2, x (X 1, X 2 ))

9 ... defect condition for fields G X1 (p)(v 1, y X 1 ) G X2 (p)(v 2, y X 2 ) = i 2 ω (X 1,X 2 )(p)(v 1 v 2, x (X 1, X 2 )) boundary term in variation of action world sheet boundary : usual mixed D/N condition with T = G( X, X), T = G( X, X) : T 1 (p) T 1 (p) = T 2 (p) T 2 (p) on defect defect is conformal

10 Moving defects

11 Moving defects Choose X : U Q M M such that 1) ι 1 X Ū1 = X 1, ι 2 X Ū2 = X 2 2)for all v T ˆX(q) Q, q U, u1,u2 oriented ON-basis at q G bx(q) (v, X u 2 ) = i 2 ω ˆX(q) (v, X u 1 ) U1 U2 M X1 X2 M U where G = ι 1G ι 2G

12 ... moving defects G bx(q) (v, X u 2 ) = i 2 ω ˆX(q) (v, X u 1 ) U1 U2 M X1 X2 M U if X exists it is unique T 1 = T 2 and T1 = T 2 on defect line defect is topological S[ ] = S[ ]

13 Jump defects in WZW model M = G : compact, simple, connected and simply connected Lie group H G k Z >0 : Cartan 3-form : basic gerbe on G : level Gawedzki, Reis 02 Meinrenken 02 use gerbe G k, curvature k H

14 ... jump defects in WZW model Z(G) : centre of G jump defects: for z Z(G) Q z = { (zg, g) g G } z ω = 0 stable isomorphism ι 1G k ι 2G k from Gawedzki, Reis 03

15 Defects in 2d conformal quantum field theory Afflek, Oshikawa 96 Conformal defects T 1 (p) T 1 (p) = T 2 (p) T 2 (p) Topological defects T 1 = T 2 T1 = T 2 Petkova, Zuber vs. 1 2

16 Defects in quantum WZW model g ĝ k : (complexified) Lie algebra of Lie group G : affine Lie algebra at level k V λ : irreducible highest weight representations, λ P + k Space of states on the circle H = V λ C V λ λ P + k ( Fusion product V λ ˆ V µ = ( V ν ) N ν λµ ) ν P + k

17 ... defects in quantum WZW model Conformal defects : difficult, mostly examples Bachas, de Boer, Dijkgraaf, Ooguri 01 Quella, Schomerus 02 Bachas, Gaberdiel 04 Quella, Watts, IR 06 Bachas, Brunner 07 full set only known for ŝu(2) 1 Fuchs, Gaberdiel, Schweigert, IR 07 Topological defects preserving ĝ k ĝ k all known integrable highest weight reps. of Petkova, Zuber 00 Fröhlich, Fuchs, Schweigert, IR 06 ĝ k

18 Defect junctions = φ φ φ r φ r 0 φ Twisted state space H λ1 λ n = ( Vα C Vβ ) Nλ1 λnαβ 0 α,β P + k

19 ... defect junctions Remarks: 1) λ ν Φ0 with left/right conformal weight 0 φ0 μ parametrised by φ 0 Hom( V λ ˆ V µ, V ν ) 2) jump defects simple currents Z(G) P + k z λ z Vλx ˆ V λy = Vλxy Z(G) iso to simple current group (except ) ê(8) 2 Fuchs 91

20 ... defect junctions 3) in the classical theory Suszek, IR 08 ➀ ➁ ➂ M Q M M T3 M(1) M(2) M(3) π12 π23 π13 Q Q Q such that π12ω + π23ω = π13ω on T3 twisted scalar field φ with values in U(1) 2-morphism ϕ : (π23φ id) π12φ = π13φ

21 Classical and quantum jump defects G compact, simple, connected and simply connected Lie group, centre Z(G) xyz xyz Σ L = x xy y z, Σ R = x yz y z Pick XL : ΣL G, fix XR : ΣR G as XL outside shaded region and y XL inside shaded region

22 ... classical and quantum jump defects xyz xyz Σ L = x xy y z, Σ R = x yz y z classical e S[Σ L,X L ] = ψ G k(x, y, z) e S[Σ R,X R ] (geometric calculation using gerbe data from Gawedzki, Reis 03) quantum (representation theory of to simple current sector) Corr ΣL = ψ bgk (x, y, z) Corr ΣR ĝ k, quantum 6j symbols restricted

23 ... classical and quantum jump defects xy ψ G k and ψ bgk x y are defined up to λx,y U(1) obey ψ(x, y, z) ψ(x, y, z) λy,zλ x,yz λ xy,z λ x,y ψ(y, z, w) ψ(x, yz, w) ψ(x, y, z) ψ(xy, z, w) ψ(x, y, zw) contain information independent of choices [ψ] H 3 ( Z(G), U(1) ) = 1

24 ... classical and quantum jump defects find that, for all (compact, simple, connected and simply connected) Lie groups G and levels k Z >0 [ψ G k] = [ψ bgk ]

25 ... classical and quantum jump defects Comments: 1) discrete symmetry group S of CFT implemented by defects [ψ] H 3 (S, U(1)) 2) [ψ] gives obstruction to orbifolding by S (classically : equivariant structure on gerbe quantum : consistent 3-string interactions) Can orbifold by S if and only if [ψ] = 1.

26 ... classical and quantum jump defects 3) Relation [ψ G k] = [ψ bgk ] already partially known from orbifolding obstruction: Fix S Z(G), then the values k for which [ψ G k S ] = 1 are precisely those for which [ψ bgk S ] = 1. Gawedzki, Reis 02 03

27 Orbifolds Take superposition of symmetry generating defects e.g. (classical) Q = (quantum) Q z G G B = z Z(G) z Z(G) V λz if [ψ] = 1 choose twisted scalar φ on T3 ϕ Hom(B ˆ B, B) (classical) (quantum) such that = for e S Corr (classical) (quantum) + non-degeneracy condition

28 ... orbifolds orbifold amplitude: embed fine enough defect network e.g. torus:

29 ... orbifolds In CFT: B = V λz is special case z Z(G) in general: B integrable highest weight rep of with associative ϕ Hom(B ˆ B, B) + non-degeneray condition ĝ k get generalised orbifold e.g. for E7 invariant of ŝu(2) 16 B = V (0) V (8) V (16) take Fuchs, Schweigert, IR 02

30 ... orbifolds In fact: all CFTs well-defined at genus 0 and 1, with - ĝ k ĝ k symmetry - unique vacuum state - non-degenerate two-point function are orbifolds of the charge-conjugate theory in above sense. Kong, IR 08 holds for rational vertex operator algebras

31 Summary defects in classical sigma models classical and quantum defect junctions 3-cocycle from symmetry implemented by defects 3-cocycle agrees for jump defects in classical and quantum WZW models relation to orbifolds