Group Actions and Cohomology in the Calculus of Variations

Group Actions and Cohomology in the Calculus of Variations JUHA POHJANPELTO Oregon State and Aalto Universities Focused Research Workshop on Exterior Differential Systems and Lie Theory Fields Institute, Toronto, Canada, December 2013

EXAMPLE: INTEGRABLE SYSTEMS Potential Kadomtsev-Petviashvili (PKP) equation u tx + 3 2 u xu xx + 1 4 u xxxx + 3 4 s2 u yy = 0, s 2 = ±1. Admits an infinite dimensional algebra of distinguished symmetries g PKP involving 5 arbitrary functions of time t. (David, Kamran, Levi, Winternitz, Symmetry reduction for the Kadomtsev-Petviashvili equation using a loop algebra, J. Math. Phys. 27 (1986), 1225 1237.)

PKP EQUATION The symmetry algebra g PKP is spanned by the vector fields X f = f t + 2 3 yf y + (1 3 xf 2 9 s2 y 2 f ) x + ( 1 3 uf + 1 9 x 2 f 4 27 s2 xy 2 f + 4 243 y 4 f ) u, Y g = g y 2 3 s2 yg x + ( 4 9 s2 xyg + 8 81 y 3 g ) u, Z h = h x + (2 3 xh 4 9 s2 y 2 h ) u, W k = yk u, and U l = l u, where f = f (t), g = g(t), h = h(t), k = k(t) and l = l(t) are arbitrary smooth functions of t.

PKP EQUATION Locally variational with the Lagrangian L = 1 2 u tu x 1 4 u3 x + 1 8 u2 xx 3 8 s2 u 2 y. But the PKP equation admits no Lagrangian that is invariant under g PKP! To what extent do these properties characterize the PKP-equation?

EXAMPLE: VECTOR FIELD THEORIES One-form A = A b (x i ) dx b on R m satisfying T a = T a (x i, A b, A b,i1, A b,i1 i 2,..., A b,i1 i 2 i k ) = 0, a = 1, 2,..., m. SYMMETRIES S 1 : spatial translations S 2 : Gauge transformations CONSERVATION LAWS x i x i + a i, (a i ) R m. A a (x i ) A a (x i ) + φ x a (x i ), φ C (R m ). C 1 : There are functions tj i = tj i(x i, A a, A a,i1, A a,i1 i 2,..., A a,i1 i 2 i l ) such that, for each j = 1, 2,..., m, A a,j T a = D i (t i j ). C 2 : The divergence of T a vanishes identically, D a T a = 0.

VECTOR FIELD THEORIES THEOREM (ANDERSON, P.) Suppose that the differential operator T a admits symmetries S 1, S 2 and conservation laws C 1, C 2. Then T a arises from a variational principle, T a = E a (L) for some Lagrangian L, if (i) m = 2, and T a is of third order; (ii) m 3, and T a is of second order; (iii) the functions T a are polynomials of degree at most m in the field variables A a and their derivatives. NATURAL QUESTION: Can the Lagrangian L be chosen to be invariant under [S1], [S2]?

The goal is to reduce these type of questions into algebraic problems.

VARIATIONAL BICOMPLEX Smooth fiber bundle F E π M Adapted coordinates {(x 1, x 2,..., x m, u 1, u 2,..., u p )} = {(x i, u α )} such that π(x i, u α ) = (x i ).

A local section is a smooth mapping σ : U op M E such that In adapted coordinates π σ = id. σ(x 1, x 2,..., x m ) = (x 1, x 2,..., x m, f 1 (x 1, x 2,..., x m ),..., f p (x 1, x 2,..., x m )).

INFINITE JET BUNDLE OF SECTIONS J (E) π o E π π M

INFINITE JET BUNDLE Adapted coordinates = locally J (E) {(x i, u α, u α x j 1, uα x j 1 x j 2,..., uα x j 1 x j 2 x j k,... )}. Often write u α x j 1 x j 2 x j k = uα j 1 j 2 j k = u α J, where J = (j 1, j 2,..., j k ), 1 j l m, is a multi-index.

COTANGENT BUNDLE OF J (E) Horizontal forms: dx 1, dx 2,..., dx m. Contact forms: θ α J = duα J uα Jk dx k. The space of differential forms Λ (J (E)) on J (E) splits into a direct sum of spaces of horizontal degree r and vertical (or contact) degree s: Λ (J (E)) = Λ r,s (J (E)). r,s 0 Here ω Λ r,s (J (E)) is a finite sum of terms of the form f (x i, u α, u α j,..., u α J ) dx k 1 dx kr θ α 1 L 1 θ αs L s.

HORIZONTAL AND VERTICAL DIFFERENTIALS The horizontal connection generated by the total derivative operators D i = is flat = x i + u α i The exterior derivative splits as where u α + uα ij 1 uj α + u α ij 1 j 2 1 uj α + 1 j 2 d = d H + d V, d H : Ω r,s Ω r+1,s, d V : Ω r,s Ω r,s+1.

HORIZONTAL AND VERTICAL DIFFERENTIALS m d H f (x i, u α,..., uj α ) = D j f (x i, u α,..., uj α )dx j, j=1 p d V f (x i, u α,..., uj α ) = β=1 u β K 0 K f (x i, u α,..., uj α )θβ K. d 2 = 0 = d 2 H = 0, d 2 V = 0, d Hd V + d V d H = 0.

d V d V d V d V d H d H d H 0 Λ 0,1 Λ 1,1 Λ m 1,1 Λ m,1 d V d V d V d V d H d H d H R Λ 0,0 Λ 1,0 Λ m 1,0 Λ m,0 π π π π R Λ 0 d M Λ 1 d M Λ m 1 d M Λ m M

FUNCTIONAL FORMS Define I αu β J = { δ β αδ (i 1 j 1 δ i k ) j k, if I = J, 0, otherwise. Interior Euler operator Fα I : Λ r,s Λ r,s 1, s 1, Fα(ω) I = ( ) I + J ( D) J ( α IJ I J 0 ω). Integration-by-parts operator I : Λ m,s Λ m,s, s 1, I(ω) = 1 s θα F α (ω). Spaces of functional s-forms F s = I(Λ m,s ), s 1. Differentials δ V = I d V : F s F s+1. Then δv 2 = 0.

FREE VARIATIONAL BICOMPLEX d V d V d V d V δ V d H d H d H I 0 Λ 0,2 Λ 1,2 Λ m 1,2 Λ m,2 F 2 d V d V d V d V δ V d H d H d H I 0 Λ 0,1 Λ 1,1 Λ m 1,1 Λ m,1 F 1 d V d V d V d V E d H d H d H R Λ 0,0 Λ 1,0 Λ m 1,0 Λ m,0 π π π π R Λ 0 d M Λ 1 d M Λ m 1 d M Λ m M

EULER-LAGRANGE COMPLEX Columns are locally exact Interior rows are globally exact! Horizontal homotopy operator where c I = h r,s H (ω) = 1 s I +1 n r+ I +1. I 0 c I D I [ θ α F Ij α(d j ω)], s 1,

EULER-LAGRANGE COMPLEX The edge complex R Λ 0,0 d H Λ m 1,0 d H Λ 1,0 d H d H Λ m,0 δ V F 1 δ V F 2 δ V Div E H is called the Euler-Lagrange complex E (J (E)).

CANONICAL REPRESENTATIONS Then ω = V i (x i, u [k] )( x i ν) Λ m 1,0, λ = L(x i, u [k] )ν Λ m, = α (x i, u [k] )θ α ν F 1, H = 1 2 HI αβ (x i, u [k] )θ α θ β I. λ = d H ω L = D i V i, = δ V λ α = E α (L), H = δ V H I αβ = I β α + ( 1) I E I α( β ), where E I α(f) = J 0 ( I + J ) ( D)J ( IJ I α F).

COHOMOLOGY Associated cohomology spaces: H r (E (J (E))) = ker δ V : E r E r+1 im δ V : E r 1 E r. This complex is locally exact and its cohomology H (E (J (E)) is isomorphic with the de Rham cohomology of E singular cohomology of E.

GROUP ACTIONS A Lie pseudo-group G consists a collection of local diffeomorphisms on E satisfying 1. id G; 2. If ψ 1, ψ 2 G, then ψ 1 (ψ 2 ) 1 G where defined; 3. There is k o such that the pseudo-group jets G k = {j k z ψ ψ G, z dom ψ}, k k o, form a smooth bundle. 4. A local diffeomorphism ψ G jz k ψ G k, k k o, for all z dom ψ. EXAMPLE: Symmetry groups of differential equations, gauge groups,....

The graph Γ σ E of a local section σ of E M is the set Γ σ = {σ(x i ) (x i ) dom σ}. Let ψ G. Define the transform ψ σ of σ under ψ by Γ ψ σ = ψ(γ σ ). The prolonged action of G on J (E) is then defined by j x o σ pr ψ j ψ(x o ) (ψ σ) σ ψ ψ σ

A function F defined on a G-invariant open U J (E) is called a differential invariant of G if F pr ψ = F for all ψ G. A k-form ω Λ k (U) is G invariant if (pr ψ) ω = ω for all ψ G.

The prolongation pr V of a local vector field V on E is defined by Φ V t pr Φ V t d dt V pr V A local vector field V on E is a G vector field, V g, if the flow Φ V t G for all fixed t on some interval about 0.

Suppose that G consists of projectable transformations. Then the actions of G and g both preserve the spaces Λ r,s (J (E)) and commute with the horizontal and vertical differentials d H, d V, and the integration-by-parts operator I. = The differentials d H, d V, δ V map G- and g-invariant forms to G- and g-invariant forms, respectively.

g-invariant VARIATIONAL BICOMPLEX: d V d V d V d V δ V d 0 Λ 0,2 H d g Λ 1,2 H d g Λ m 1,2 H I g Λ m,2 g F 2 g d V d V d V d V δ V d 0 Λ 0,1 H d g Λ 1,1 H d g Λ m 1,1 H I g Λ m,1 g F 1 g d V d V d V d V E R Λ 0,0 g Λ 1,0 g d H d H d H Λ m 1,0 g Λ m,0 g π π π π R Λ 0 d M,g Λ 1 d M,g Λ m 1 d M,g Λ m M,g

g-invariant EULER-LAGRANGE COMPLEX E g (J (E)): R Λ 0,0 g d H Λ 1,0 g d H d H Λ m 1,0 g d H Λ m,0 g Div δ V E F 1 g δ V H F 2 g δ V Associated cohomology spaces: H r (Eg (J (E))) = ker δ V : Eg r Eg r+1 im δ V : Eg r 1. Eg r

EXACTNESS OF THE INTERIOR HORIZONTAL ROWS THEOREM Let g be a pseudo-group of projectable transformations acting on E M, and let ω i and θ α be g invariant horizontal frame and zeroth order contact frame defined on some G-invariant open set U J (E) contained in an adapted coordinate system. Then the interior rows of the g-invariant augmented variational bicomplex restricted to U are exact, H (Λ,s g (U), d H ) = {0}, s 1. COROLLARY: Under the above hypothesis H (E g (U), δ V ) = H (Λ g(u), d).

COMPUTATIONAL TECHNIQUES EXPLICIT DESCRIPTION OF THE INVARIANT VARIATIONAL BICOMPLEX. Given a local cross section K (k) J k (E) to the action of G k on J k (E), let H k K (k) = {(g k, z k ) z k K (k), g k, z k based at the same point}, and let µ k : H k K (k) J k (E), µ k (g k, z k ) = g k z k. Then, if the action is locally free, µ k will be a G-equivariant local diffeomorphism with the action of G on H k K (k) given by ϕ (g k, z k ) = (ϕ g k, z k ).

COMPUTATIONAL TECHNIQUES Upshot: Locally one can find a complete set of differential invariants {I α } and a coframe on U J k (E) consisting of {di α } and g-invariant 1-forms {ϑ β } such that the algebra A generated by {ϑ β } is closed under d = H g (U, d) = H (A, d). (Apply the g-equivariant homotopy I α t I α, di α t di α, ϑ β ϑ β, 0 t 1.)

GELFAND-FUKS COHOMOLOGY Formal power series vector fields on R m : { m } W m = a l x l a l R[[x 1,..., x m ]]. l=1 Lie bracket [, ]: W m W m W m. Give W m a topology relative to the ideal m =< x 1, x 2,..., x m >. Λ c(w m ): continuous alternating functionals on W m. Λ c(w m ) is generated by δ i j 1 j 2 j k, where δj i 1 j 2 j k (a l x l ) = k a i x j 1 x j 2 x j k (0).

GELFAND-FUKS COHOMOLOGY The differential d GF : Λ r c(w m ) Λ r+1 c (W m ) is induced by Lie bracket of vector fields so that d 2 GF = 0! d GF ω(x, Y ) = ω([x, Y ]), ω Λ 1 c(w m ). Let g o g W m be subalgebras. Define Λ c(g) = Λ c(w m ) g, Λ c(g, g o ) = {ω Λ c(g) X ω = 0, X d GF ω = 0, for all X g o }. The Gelfand-Fuks cohomology H GF (g, g o) of g relative to g o is the cohomology of the complex (Λ c(g, g o ), d GF ).

EVALUATION MAPPING Pick σ J (E). For a given infinitesimal transformation group g acting on E, let g o = {X g pr X(σ ) = 0}. Define ρ: Λ g(j (E)) Λ c(g, g o ) by ρ(ω)(x 1,..., X r ) = ( 1) r ω(pr X 1,..., pr X r )(σ ). Then ρ is a cochain mapping, that is, it commutes with the application of d and d GF, and thus induces a mapping ρ: H (Λ g(j (E)), d) H GF (g, g o). Goal is to show that ρ is an isomorphism (moving frames!).

EQUIVARIANT DEFORMATIONS Construct a submanifold P U J (E) such that 1. pr g acts transitively on P, and 2. P is pr g-equivariant strong deformation retract of U, that is, there is a smooth map H : U [0, 1] U such that H(σ, 0) = σ, for all σ U, H(σ, 1) P, for all σ U, H(σ, t) = σ, for all (σ, t) P [0, 1], (H t ) (pr V σ ) = pr V H(σ,t), for all V g, (σ, t) U [0, 1].

EQUIVARIANT DEFORMATIONS Under these circumstances the inclusion map ι: P U and the evaluation map ρ: Λ g(p ) Λ c(g, g o ) induce isomorphisms in cohomology.

PKP EQUATION AGAIN The symmetry algebra g PKP of the PKP equation u tx + 3 2 u xu xx + 1 4 u xxxx + 3 4 s2 u yy = 0. is spanned by the vector fields X f = f t + 2 3 yf y + (1 3 xf 2 9 s2 y 2 f ) x + ( 1 3 uf + 1 9 x 2 f 4 27 s2 xy 2 f + 4 243 y 4 f ) u, Y g = g y 2 3 s2 yg x + ( 4 9 s2 xyg + 8 81 y 3 g ) u, Z h = h x + (2 3 xh 4 9 s2 y 2 h ) u, W k = yk u, and U l = l u, where f = f (t), g = g(t), h = h(t), k = k(t) and l = l(t) are arbitrary smooth functions of t.

PKP equation Now E = {(t, x, y, u)} {(t, x, y)}. The PKP source form PKP = (u tx + 32 u xu xx + u xxxx + 34 ) s2 u yy θ dt dx dy generates non-trivial cohomology in H 4 (E gpkp (J (E)))! The characterization problem of the PKP-equation by its symmetry algebra amounts to the computation of H 4 (E g PKP (U)). For a suitable U J (U), H (E g PKP (U)) can be computed by an explicit description of differential invariants and an invariant coframe arising from the moving frames construction.

The Gelfand-Fuks complex for g PKP admits a basis α n, β n, γ n, υ n, ϑ n, n = 0, 1, 2,..., of invariant forms so that ( ) n dα n n = α k α n k+1, k k=0 ( ) n dβ n n {α = k β n k+1 2 k 3 αk+1 β n k}, k=0 ( ) n dγ n n {α = k γ n k+1 1 k 3 αk+1 γ n k 2 3 s2 β k β n k+1}, k=0 ( ) n+1 dυ n n + 1 {α = k υ n k+1 + 4 k 9 s2 (β k+1 γ n k+1 dϑ n = k=0 ( ) n n {α k ϑ n k+1 + 1 k 3 αk+1 ϑ n k k=0 2β k γ n k+2 ) }, + β k υ n k + 2 3 γk γ n k+1}. The complex splits into a direct sum of simultaneous eigenspaces of 2 Lie derivative operators.

PKP EQUATION Let A be a non-vanishing differential function on some open set U J (E) satisfying pr X f (A) + 1 3 Af (t) = 0, A y = 0, for every smooth f (t), and let B be a differential function on U satisfying pr X f (B)+ 2 3 ya 1 f (t) = 0, For example, one can choose B y = 0, for every smooth f (t). A = (u x n) 1 n+1 and B = 3 2 s2 u x n 1 y (u n+2 x n) n+1, n 3, on U = {u x n > 0}.

PKP EQUATION THEOREM Suppose that differential functions A and B, defined on an open U J (E), are chosen as above. Then the dimensions of the cohomology spaces H r (E g PKP (U), δ V ) are r 1 2 3 4 5 6 7 8 dim 0 1 1 3 3 2 3 0

REPRESENTATIVES OF THE COHOMOLOGY CLASSES Let {α 0, β 0, γ 0 } be the g PKP invariant horizontal frame defined by α 0 = A 3 dt, β 0 = A 2 dy + A 3 Bdt, γ 0 = Adx 2 3 s2 A 2 Bdy + A 3 Cdt, where C = 3 2 u xa 2 1 3 s2 B 2, and let K be the g PKP differential invariant K = (u tx + 3 4 s2 u yy + 3 2 u xu xx )A 5.

REPRESENTATIVES OF THE COHOMOLOGY CLASSES Let 1, 2 E 4 g PKP (U) be the source forms 1 = (u tx + 3 2 u xu xx + 3 4 s2 u yy ) dt dx dy du, 2 = u xxxx dt dx dy du, and let 3 E 4 g PKP (U) be the source form which is the Euler-Lagrange expression 3 = E(BK α 0 β 0 γ 0 ). Then H 4 (E (U), δ V ) =< 1, 2, 3 >. Note that the PKP source form is the sum PKP = 1 + 2.

COROLLARY: Let E 4 g PKP (U) be a g PKP invariant source form that is the Euler-Lagrange expression of some Lagrangian 3-form λ E 3 (U). Then there are constants c 1, c 2, c 3 and a g PKP -invariant Lagrangian 3-form λ 0 E 3 g PKP (U) such that = c 1 1 + c 2 2 + c 3 3 + E(λ 0 ).

VECTOR FIELD THEORIES Here E = T R m = {(x i, A j )} {(x i )}. Now the infinitesimal transformation group g is spanned by T i = x i, V φ = φ,i A i, where φ is an arbitrary smooth function on R m. Need to compute H m+1 (E g (J (T R m )))! The standard horizontal homotopy operator for the free variational bicomplex commutes with the action of g = H,s (Λ, g (J (E)), d H, I) = {0}, s 1. So it suffices to compute H (Λ g(j (E)), d).

VECTOR FIELD THEORIES Parametrize J (T R m ) by (x i, A a, A (a,b1 ), F ab1, A (a,b1 b 2 ), F a(b1,b 2 ), A (a,b1 b 2 b 3 ), F a(b1,b 2 b 3 ),...), where F ab = A a,b A b,a. Now the variables F a(b1,b 2 b r ) are invariant under the action of g = P = { σ J (T R m ) F ij (σ ) = 0, F i(j,h) (σ ) = 0,... } is a g-equivariant strong deformation retract of J (T M) on which g acts transitively.

VECTOR FIELD THEORIES In conclusion, H (E g (J (T M))) = H GF ( g), where the Lie algebra of formal vector fields g is spanned by the vector fields T i and V j 1j 2...j k = x (j 1 x j2 x j k 1 j k ) A, j A = A j.

VECTOR FIELD THEORIES A basis for H (E g (J (T M))) is given by dx i 1 dx i k F l Λ r,0 g (J (T M)), k + 2l = r, dx i 1 dx i k F l (d V A) s F s g (J (T M)), k + 2l + s = m. (A = A i dx i, F = F ij dx i dx j.) Generators for H m+1 (E g (J (T M))) i 1i 2 i k = dx i 1 dx i 2 dx i k F l d V A, k + 2l = m 1, dim H m+1 (E g (J (T M))) = 2 m 1. Note that when m = 2r + 1, = F r d V A is the Chern-Simons mass term with components i = ɛ ij 1k 1 j 2 k 2 j r k r F j1 k 1 F j2 k 2 F jr k r.