The gauged WZ term with boundary

Transcription

1 The gauged WZ term with boundary José Figueroa-O Farrill Edinburgh Mathematical Physics Group LMS Durham Symposium, 29 July 2005

2 A quaternion of sigma models WZ bwz gwz gbwz

3 ... and one of cohomology theories de Rham relative de Rham equivariant relative equivariant

4 References hep-th/ Phys. Lett. B341 (1994) , hep-th/ JHEP 01 (2001) 006, hep-th/ with Sonia Stanciu hep-th/ with Nouri Mohammedi

5 Sigma models Two oriented pseudo-riemannian manifolds: Σ d, X n Σ may or may not be empty ϕ : Σ X, dϕ Ω 1 (Σ, ϕ T X) sigma model action I σ = Σ 1 2 dϕ 2 defines variational problem for harmonic maps d Σ dϕ = 0

6 The Wess Zumino term H Ω d+1 closed (X) assume Σ = and ϕ(σ) X bounds ϕ(σ) = M M X Wess Zumino term I WZ = M H

7 dh = 0 = field equations for ϕ δi WZ = L δϕ H = M M dı δϕ H = ϕ(σ) ı δϕ H classical theory is independent of choice of M quantum theory depends on I WZ modulo 2πZ quantum theory is independent of M if [ 1 2πH] is an integral class in H d+1 dr (X)

8 Example: the WZW model Σ two-dimensional X a compact simple Lie group H 2 (X) = 0 so ϕ(σ) always bounds [Cartan] [ 1 2π H] is k times the generator of H3 (X; Z) = Z, where k is the level of the WZW model

9 The presence of a boundary suppose Σ = need to specify boundary conditions let i : Y X, i H = db for some B Ω d (Y ) BCs: ϕ( Σ) Y = a theory of relative maps ϕ : (Σ, Σ) (X, Y )

10 The boundary Wess Zumino term ϕ(σ) X is not a cycle, but it is a cycle modulo Y : ϕ(σ) = ϕ( Σ) Y so suppose that it bounds modulo Y : M X, D Y s.t. M = ϕ(σ) + D whence D = ϕ( Σ)

11 boundary Wess Zumino term I bwz = M H D B B only enters in the boundary conditions: δi bwz = M dı δϕ H D L δϕ B = ϕ(σ) ı δϕ H + ϕ( Σ) ı δϕ B and field equations are not otherwise changed

12 = classical theory is again independent on choice of M and D, but quantum theory is not unless [ 1 2πH] is an integral class in the relative de Rham cohomology H d+1 dr (X, Y )

13 Example: the boundary WZW model Y X a (twisted) conjugacy class integrality of [ 1 2π H] H3 dr (X, Y ) selects a discrete set of such conjugacy classes corresponding to unitary highest weight representations of the (twisted) affine Lie algebra at level k

14 Symmetries of sigma model with WZ term G a connected Lie group, with Lie algebra g with basis Z a G acts on X isometrically preserving H Z a v a a Killing vector, [v a, v b ] = f ab c v c let ı a := ı va and L a := L va, then L a = dı a + ı a d L a H = 0, equivalently dı a H = 0

15 Gauging a sigma model means coupling it to a gauge field A Ω 1 (Σ, g) so that the action is invariant under (infinitesimal) gauge transformations: δ λ A = dλ + [A, λ] δ λ ω = dλ a ı a ω + λ a L a ω where ω Ω(X) and λ C (Σ, g) can write δ λ A = dλ a ı a A + λ a L a A by defining ı a A b = δ b a ı a F b = 0 and L a = dı a + ı a d where F = da [A, A] is the field-strength

16 Minimal coupling I σ = Σ 1 2 dϕ 2 can be gauged by minimal coupling: dϕ d A ϕ := dϕ A a ı a dϕ but I WZ is a different matter: the minimally coupled H need not be closed

17 Gauging the WZ term means extending H to a closed gauge-invariant form H : H = H + terms involving A and F such that dh = 0 and δ λ H = dλ a ı a H + λ a L a H = 0 equivalently ı a H = 0 (and L a H = 0)

18 Differential graded algebras a g-dga A: A = i 0 Ai (associative, supercommutative) product : A i A j A i+j differential d : A i A i+1, d 2 = 0 derivation ı a : A i A i 1 derivation L a = dı a + ı a d defines g-action

19 Ω(X) is a g-dga so is the Weyl algebra W = Λg Sg with generators A a Λ 1 g and F a S 1 g and where ı a A b = δ b a and ı a F b = 0 and da a = F a 1 2 f bc a A b A c Weyl homomorphism w : W Ω(Σ, g) defined by A a A a and F a F a Ω(X) W is a g-dga

20 Equivariant forms A a g-dga, then φ A is horizontal, if ı a φ = 0 invariant, if L a φ = 0 equivariant, if both Ω g (X): subcomplex of equivariant elements of Ω(X) W {w(φ) φ Ω g (X)} are gauge-invariant gauging the WZ term is finding an equivariant closed extension H Ω d+1 g (X) of H Ω d+1 (X)

21 The Cartan model in a local gauge-invariant quantity, A only appears in minimally coupled expressions (or through F ) this suggests defining Ω C (X) := (Ω(X) Sg ) g called the Cartan model of Ω g (X) indeed, Ω C (X) = Ω g (X)

22 π : Ω g (X) = Ω C (X) consists in setting A = 0 π 1 : Ω C (X) = Ω g (X) consists of minimal coupling induced differential d C = π d π 1 : Ω p C for ω Ω(X) (X) Ωp+1 C (X) is d C F a = 0 and d C ω = dω ı a ωf a gauging WZ term is equivalent to finding a d C -closed extension H C Ω C (X)

23 The Hull Spence obstructions write the most general H C in the Cartan model H C = H + θ a F a θ abf a F b + where θ a Ω d 1 (X), θ ab Ω d 3 (X),... satisfy L a θ b = f c ab θ c L a θ bc = f d ab θ dc + f d ac θ bd...

24 splitting d C H C = 0 into types: ı a H = dθ a ı a θ b + ı b θ a = dθ ab... which are the first two Hull Spence obstructions overcoming these obstructions yields H C and minimal coupling yields H and the gauged WZ term I gwz = M H

25 The two-dimensional case H C = H + θ a F a d C H C = 0 implies ı a H = dθ a and ı a θ b + ı b θ a = 0 L a H C = 0 implies L a θ b = f ab c θ c

26 the gauged WZ term is I gwz = H + M Σ ( ϕ θ a A a ϕ (ı a θ b )A a A b) [Hull & Spence; Jack, Jones, Mohammedi & Osborn] to this action we can add a Yang Mills term Σ 1 4 F 2 or other topological terms, corresponding to cocycles in Ω 3 g(x)

27 Example: the gauged WZW model we try to gauge g x x defined by homomorphisms l : g x and r : g x the only obstruction is l, = r, with, is the ad-invariant scalar product on x

28 typical anomaly-free g: diagonal: l = r twisted diagonal: l = r τ for some isometry τ Aut(x) chiral: r = 0 and g x an isotropic subalgebra

29 The twisted Courant algebroid on T Λ d 1 T T X Λ d 1 T X has a Λ d 2 T X-valued bilinear v + α, w + β = ı v β + ı w α Λ d 2 T X and also has a Courant bracket: [v + α, w + β] = [v, w] + L v β L w α 1 2 d(ı vβ ı w α)

30 H Ω d+1 closed (X) twists the bracket [v + α, w + β] H = [v + α, w + β] ı v ı w H we say L T X Λ d 1 T X is isotropic if for all v + α C (L), ı v α Ω d 2 exact(x) an isotropic, involutive L defines a Lie algebroid over X

31 The Lie algebroid of the gauged WZ term let L be the image of g C (T X Λ d 1 T X) given by Z a v a + θ a ı a θ b + ı b θ a = dθ ab, whence L is isotropic dθ a = ı a H and L a θ b = f c ab θ c then imply that L is involutive so L defines a Lie algebroid isomorphic to g [cf. Alekseev & Strobl (d = 2), Bonelli & Zabzine]

32 Gauging the boundary WZ term assume G acts preserving (Y, B) gauging I bwz consists in finding equivariant extensions H and B of H and B such that dh = 0 and i H = db on Y or in the Cartan model finding H C and B C with d C H C = 0 and i H C = d C B C the gauged boundary WZ term is then I gbwz = H M D B

33 Two-dimensional gauged boundary WZ term B C = B + h a F a, where h a C (X) obeying L a h b = f ab c h c i H C = d C B C is equivalent to i θ a + i a B = dh a

34 and I gbwz = H B M D ( + ϕ θ a A a ϕ (ı a θ b )A a A b) + Σ Σ ϕ h a A a one can add topological terms, corresponding to relative cocycles in Ω 3 g(x, Y )

35 The boundary Lie algebroid the twisted Courant bracket restricts to T Y Λ d 1 T Y, but since i H = db, it is B-related to the untwisted Courant bracket: [v + α, w + β] db = [e B (v + α), e B (w + β)] the image of the map g C (T Y Λ d 1 T Y ) defined by Z a e B (v a + i θ a ) = v a + ı a B + i θ a = v a + dh a is isotropic and involutive: a Lie algebroid on Y isomorphic to g

36 Example: the gauged boundary WZW model possible boundary conditions are G-orbits: (twisted) diagonal gaugings: (twisted) conjugacy classes chiral gaugings: cosets boundary offers no new obstructions dh a = 0 and h a [g, g] o boundary Lie algebroid is the action Lie algebroid of g on Y

37 Open questions are there CFT constructions for the cosets? relation with boundary conditions on X X? relation with boundary integrable systems? (oidish) interpretation for higher obstructions?

38 Thank you!