The Homotopy Moment Map for Chern–Simons Field Theory

The homotopy moment map for Chern–Simons field theory Homotopy moment maps 1 / 16 Homotopy moment maps Let X be a manifold, with de Rham complex Ω ∗ ( X ). Let n be a natural number. Definition An n-multisymplectic form on X is a closed differential form 𝜔 ∈ Ω n+1 ( X ) such that the morphism 𝜉 ∈ Vec( X ) ↦→ 𝜄 (𝜉)𝜔 ∈ Ω n ( X ) is injective. The case n = 1 of symplectic forms is special, since if X is finite-dimensional, the morphism 𝜉 ↦→ 𝜄 (𝜉)𝜔 is not just injective, but bijective. Homotopy moment maps 2 / 16 Let 𝔥 be a Lie algebra with differential action on X : 𝜌 : 𝔥 → Vec( X ). Let be the Chevalley–Eilenberg complex of 𝔥, with differential 𝛿 0 . Denote the tensor product of C∗ (𝔥) and Ω ∗ ( X ) by C∗ (𝔥) C∗ (𝔥) ⊗ Ω ∗ ( X ). Choose a basis {ci } of 𝔥, and dual basis {c i } of 𝔥∗ ; denote the structure coefficients of 𝔥 by Akij . Form the operator ∑︁ i 𝜄= c ⊗ 𝜄 (𝜌 i ) : Cp (𝔥) ⊗ Ω q ( X ) → Cp+1 (𝔥) ⊗ Ω q − 1 ( X ), i of total degree 0. Definition (Callies, Frégier, Rogers, Zambon) A homotopy moment map for the action 𝜌 of 𝔥 on an n-multisymplectic manifold ( X , 𝜔) is an element 𝜇 ∈ C∗ (𝔥) ⊗ Ω ∗ ( X ) of total degree n − 1 such that (𝛿 0 + d )𝜇 = e𝜄 𝜔 − 𝜔. Homotopy moment maps 3 / 16 The case of symplectic manifolds: n = 1 P If n = 1, we have 𝜄𝜔 = i c i 𝜄 (𝜌 i )𝜔 ∈ 𝔥∗ ⊗ Ω 1 ( X ). Thus, a homotopy moment P i map is an element 𝜇 = i c 𝜇i ∈ 𝔥∗ ⊗ Ω 0 ( X ) such that d 𝜇 i = −𝜄 (𝜌 i )𝜔 and ∑︁ Akij 𝜇 k = 𝜄 (𝜌 i )𝜄 (𝜌 j )𝜔 = −L (𝜌 i )𝜇 j = {𝜇 i , 𝜇 j }. k In other words, 𝜇 is a moment map for the action 𝜌 of 𝔥 on the symplectic manifold ( X , 𝜔). Homotopy moment maps 4 / 16 Reformulation for exact n-multisymplectic manifolds Definition An n-multisymplectic manifold ( X , 𝜔) is exact if 𝜔 = d 𝛼, for 𝛼 ∈ Ωn ( X ). The Lie derivative ci ↦→ L (𝜌 i ) makes the de Rham complex Ω ∗ ( X ) into a differential graded 𝔥-module. This motivates replacing the complex C∗ (𝔥) ⊗ Ω ∗ ( X ) by the complex C∗ (𝔥, Ω ∗ ( X )), with differential 𝛿 + d , where ∑︁ i 𝛿 = 𝛿0 + c L (𝜌 i ). i Lemma 𝛿 + d = e𝜄 ◦ (𝛿 0 + d ) ◦ e−𝜄 Homotopy moment maps 5 / 16 The equation for a homotopy moment map becomes (𝛿 0 + d ) (𝛼 + 𝜇) = e𝜄 𝜔. Define 𝜈 = e𝜄 (𝛼 + 𝜇) ∈ C∗ (𝔥, Ω ∗ ( X )), or equivalently, 𝜇 = e−𝜄 𝜈 − 𝛼 . For exact n-multisymplectic manifolds, a homotopy moment map is an equivariant extension of 𝛼: (𝛿 + d )𝜈 = 𝜔. Homotopy moment maps 6 / 16 The variational bicomplex Let p : E → M be a fiber bundle, with jet-space J∞ ( p). An adapted coordinate system at a point e ∈ E is a coordinate system ( t 1, . . . , t n ; u1, . . . , uN ) such that the coordinates ( t 1, . . . , t n ) are pulled back from a coordinate system around p ( e) ∈ M . The coordinates t 𝜇 are the independent coordinates, and M is the worldsheet; the coordinates ua are the dependent coordinates, identified with the fields of the theory. Denote 𝜕/𝜕 t 𝜇 by 𝜕𝜇 . An adapted coordinate system gives rise to coordinates ( t 1, . . . , t n ; 𝜕 I u1, . . . , 𝜕 I uN ) on the jet-space J∞ ( p), where I ranges over multi-indices ( i1, . . . , in ) ∈ Nn , and 𝜕 I = 𝜕1i1 . . . 𝜕nIn . Denote partial differentiation 𝜕/𝜕(𝜕 I ui ) with respect to a jet coordinate by 𝜕i,I . Let O∞ be sheaf of algebras over M whose sections are smooth functions in the coordinates ( t 1, . . . , t n ; 𝜕 I u1, . . . , 𝜕 I uN ). Homotopy moment maps 7 / 16 The variation de Rham complex ∗ is the de Rham complex associated to The variational de Rham complex Ω∞ O∞ . It is bigraded: the differentials dt 𝜇 have bidegree ( 1, 0), and the differentials ∑︁ 𝜃 Ii = d (𝜕 I ui ) − 𝜕𝜇 𝜕 I ui dt 𝜇 𝜇 have bidegree ( 0, 1). The differential d on Ω∞ breaks into horizontal and vertical parts ∑︁ 𝜇 ∑︁ a p,q p+1,q p,q p,q+1 d 1,0 = dt D𝜇 : Ω∞ → Ω∞ , d 0,1 = 𝜃 I 𝜕i,I : Ω∞ → Ω∞ . i ,I 𝜇 where D𝜇 = 𝜕𝜇 + ∑︁
𝜕 𝜇 𝜕 I u i 𝜕i , I . i ,I Homotopy moment maps 8 / 16 The variational n-multisymplectic form of a classical field theory A first-order Lagrangian density L determines a classical field theory. In an adapated coordinate system, n,0 L = L ( t , ui , 𝜕𝜇 ui ) dt 1 ∧ . . . ∧ dt n ∈ Ω∞ . From L, we construct a Lepage form n,0 n − 1,1 n 𝛼 = L − 𝜕i,𝜇 L 𝜃 i ∧ 𝜄 (𝜕𝜇 ) dt 1 ∧ . . . ∧ dt n ∈ Ω∞ ⊕ Ω∞ ⊂ Ω∞ . n+1 The differential 𝜔 = d 𝛼 ∈ Ω∞ extends Noether’s symplectic form off-shell: 𝛿L n,1 n − 1,2 𝜔= dui ∧ dt 1 ∧ . . . ∧ dt n + · · · ∈ Ω∞ + Ω∞ . 𝛿 ui Modulo mild conditions on L which guarantee nondegeneracy, 𝜔 is a variational analogue of a n-multisymplectic form. Homotopy moment maps 9 / 16 Variational Cartan calculus Derivations 𝜉 ∈ Der(O∞ ) of O∞ are the variational vector fields. There is a variational Cartan calculus: to a variational vector field 𝜉, we associate the contraction ∗ ∗− 1 𝜄 (𝜉) : Ω∞ → Ω∞ and the Lie derivative ∗ ∗ L (𝜉) = d ◦ 𝜄 (𝜉) + 𝜄 (𝜉) ◦ d : Ω∞ → Ω∞ . Homotopy moment maps 10 / 16 n,0 Fix a fiber bundle p : E → B and a (first-order) Lagrangian density L ∈ Ω∞ n with associated Lepage form 𝛼 ∈ Ω∞ and variational n-multisymplectic form n+1 𝜔 ∈ Ω∞ . Consider a Lie algebra 𝔥 and a variational action of 𝔥 𝜌 : 𝔥 → Der(O∞ ) such that L (𝜌)𝜔 = 0. Definition ∗ ) of A variational homotopy moment map for 𝜌 is an element 𝜈 ∈ C∗ (𝔥, Ω∞ total degree n − 1 such that (𝛿 + d )𝜈 = 𝜔. Homotopy moment maps 11 / 16 The Chern–Simons classical field theory Let M be a closed oriented 3-manifold. Let 𝔤 be a reductive Lie algebra, with invariant inner product (−, −). Let G be a connected compact Lie group with Lie algebra 𝔤. Let P be a G-principal bundle over M . We consider a field theory over M , where the fields (dependent coordinates) are the components of a connection over P . Over a chart U ⊂ M where P is trivialized, the Lepage form of the Chern–Simons Lagrangian theory equals 𝛼 = 12 ( A, dA) + 16 ( A, [ A, A]). If G is simply connected, then P is a trivial bundle, and the Lepage form is globally defined. In general, P is not trivial, and the Lepage form is a Čech ∗ ) of total degree 3. cochain 𝛼 ∈ Č∗ (U, Ω∞ Homotopy moment maps 12 / 16 The 3-multisymplectic form of Chern–Simons field theory The 3-multisymplectic form equals 𝜔 = 12 (F, F) = ( F , d 0,1 A) + 12 ( d 0,1 A, d 0,1 A), where F = d 1,0 A + 12 [ A, A] is the curvature, and F = dA + 12 [ A, A] = d 0,1 A + F is its analogue in the variational bicomplex. From this point, the slides from the talk contained an error, which is corrected here: the multisymplectic form of Chern–Simons theory is only nondegenerate on evolutionary vector fields. This 3-form is nondegenerate on evolutionary vector fields: those of the form     ∑︁ I 𝜕 𝜕 pr X , = 𝜕 X, , 𝜕A 𝜕(𝜕 I A) I 1,0 where X ∈ Ω∞ ⊗ 𝔤. Homotopy moment maps 13 / 16 The Atiyah Lie algebra The Atiyah Lie algebra A ( P ) of a principal bundle P is spanned by covariant derivatives ∇𝜉A , and gauge transformations 𝜂 ∈ P ×G 𝔤. In adapted coordinates associated to a local trivialization
of P over an open subset U ⊂ M , we have ∇𝜉A + 𝜂 = 𝜉 𝜇 𝜕𝜇 + ad 𝜄 (𝜉) A + 𝜂 . The evolutionary vector fields  

𝜕 A A 𝜌 ∇𝜉 + 𝜂 = pr −L (𝜉) A + ad ∇ 𝜄 (𝜉) A + 𝜂 , 𝜕A   𝜕 = pr −𝜄 (𝜉)F + ad ∇A𝜂, 𝜕A yield an action of the Atiyah Lie algebra on the variational de Rham complex. Homotopy moment maps 14 / 16 The (local) homotopy moment map We have 𝜄F = ∇A𝜂, and hence e𝜄 F = F + ∇A𝜂 Since 𝛿 0 A = 0, we see that (𝛿 0 + d ) A + 12 [ A, A] = F. Applying e𝜄 to both sides, and using e𝜄 A = A, we see that (𝛿 + d ) A + 12 [ A, A] = e𝜄 F. Hence (𝛿 + d ) ( A − 𝜂) + 12 [ A − 𝜂, A − 𝜂] = F. Theorem 𝜈 = 12 ( A − 𝜂, (𝛿 + d ) ( A − 𝜂)) + 16 ( A − 𝜂, [ A − 𝜂, A − 𝜂]) is a variational homotopy moment map for the Chern–Simons theory. Homotopy moment maps 15 / 16 The (global) homotopy moment map The point of this talk is that when P is not a trivial bundle, the Chern–Simons action, and Lepage form, are Čech cochains for a cover U of M . This forces the variational homotopy moment map ∗ 𝜈 ∈ C∗ (𝔤, Č∗ (U, Ω∞ )) to be a Čech cochain of total degree 3, satisfying the equation (𝛿 + 𝛿ˇ + d )𝜈 = 𝜔. The formulas for the higher degree Čech cochains components of 𝜈 are left as an exercise. ∗ ), we have an example of a Since 𝜇 = e−𝜄 𝜈 − 𝛼 is not an element of C∗ (𝔤, Ω∞ homotopy moment map that is a differential character, but not a globally defined differential form. This phenomenon only arises because n > 1. Homotopy moment maps 16 / 16