Homotopy Algebras and PDEs · difﬁety: the algebra of horizontal cohomologies. Variational...

GeometryHomologyHomotopy

Homotopy Algebras and PDEs

LUCA VITAGLIANO

University of Salerno, Italy

Lie Pseudogroups: Old and NewDriebergen, October 2–4, 2017

Luca Vitagliano Homotopy Algebras and PDEs 1 / 43


Introduction: Geometry of PDEs

A systemP (x) = 0, P = (. . . , Pa, . . .)

of algebraic equations is encoded geometrically by an algebraic subva-riety X of an affine space. Moreover, the Pa’s generate an ideal in thealgebra of polynomials in x: the ideal I (of the lhs) of all algebraic con-sequences of P = 0. The zero locus of I coincides with X.

Similarly, a system

F (x, . . . ,uI , . . .) = 0, F = (. . . , Fa, . . .)

of differential equations (PDEs) is encoded geometrically by a smoothsubmanifold of a jet space. Moreover, the Fa’s generate a “differentialideal” in the algebra of functions of (x,u, . . . ,uI , . . .): the ideal I (ofthe lhs) of all differential consequences of F = 0. The zero locus of I is asubmanifold in an ∞-jet space called a diffiety.



Introduction: Homological Algebra of PDEs

A diffiety encodes most of the relevant information about the originalPDE E0. Moreover, there is a rich homological algebra attached to adiffiety: the algebra of horizontal cohomologies. Variational principles,conservation laws, symmetries, cosymmetries, recursion operators, etc. areall suitable horizontal cohomologies.

Horizontal cohomologies have a natural interpretation as functions,vector fields, differential forms, tensors, etc. on the space of solutions of E0and this interpretation is supported by the existence of the “right” al-gebraic structures in horizontal cohomologies. This apparatus fits nicelywith the homological structure of classical field theory (BRST-BV for-malism).



Introduction: Homotopy Algebra of PDEs

When cohomologies possess an algebraic structureA, there is a chancethat cochains possess the same algebraic structure up to homotopy (in-ducing A in cohomology).

RemarkHomotopy algebras appear in classical field theories:

as homological perturbations in the BRST-BV formalism,as homotopy algebras of observables in multisymplectic FT.

AimThe aim of this talk is twofold:

1 reviewing horizontal cohomologies of PDEs;2 showing that homotopy algebras appear already in the theory of

PDEs: the algebraic structures on horizontal cohomologies come fromhomotopy algebraic structures on horizontal cochains.



Plan of the Talk

1 Geometry

2 Homology

3 Homotopy



Plan of the Talk

1 Geometry

2 Homology

3 Homotopy



PDEs on Submanifolds

RemarkThere are interesting PDEs imposed on generic submanifolds:

Lagrangian submanifolds,leaves of a foliation,minimal surfaces,totally geodesic submanifolds,worldsheets,· · ·

It makes sense to develop a geometric theory of PDEs imposed onsubmanifolds (rather than on sections of bundles only)!

A Related RemarkIn PDEs, there are interesting transformations mixing independentand dependent variables.



Jets of Submanifolds

Let E be an (n + m)-dim manifold, and N ⊂ E an n-dim submanifold:

N : u = f (x), (x,u) = (x1, . . . , xn,u) a divided chart.

Definition

N0, N1 are tangent up to order k at e ≡ (x,u) ∈ N0 ∩ N1 if

∂|I|f0

∂xI (x) =∂|I|f1

∂xI (x), |I| ≤ k.

Definition

The k-jet space is the manifold Jk(E, n) of classes jke N of tangency up toorder k, coordinatized by (x, . . . ,uI , . . .):

uI

(jke N)=

∂|I|f

∂xI (x) |I| ≤ k.



Geometric Portraits of PDEs

Remark

An n-dim submanifold N ⊂ E can be prolonged to the n-dim submani-fold Nk ⊂ Jk(E, n) made of classes of tangency of N:

Nk := jke N | e ∈ N.

In coordinates

Nk : uI =∂|I|f

∂xI (x), |I| ≤ k.

Definition

A system of k-th order PDEs (on n-dim submanifolds of E) is a subman-ifold E0 ⊂ Jk(E, n). In coordinates

E0 : F (x, . . . ,uI , . . .) = 0, |I| ≤ k.

A solution of E0 is an n-dim submanifold N ⊂ E such that Nk ⊂ E0.



Infinite Jets

There is a tower of fiber bundle projections:

E = J0(E, n) J1(E, n)oo · · ·oo Jk(E, n)oo Jk+1(E, n)oo · · ·oo

Definition

The ∞-jet space is the space J∞(E, n) of classes j∞e N of tangency up toorder ∞. Equivalently (Borel Lemma), it is the inverse limit

J∞(E, n) = lim←−k Jk(E, N)

and it is coordinatized by (x, . . . ,uI , . . .), |I| ≥ 0.

Remark

J∞(E, n) is a pro-finite dimensional manifold. Calculus on pro-finite di-mensional manifolds can be developed algebraically!



the Cartan distribution

Remark

Take a point z = j∞e N ∈ J∞(E, n). Then N∞ 3 z, and Cz := TzN∞ is

independent of N. SoC : z 7−→ Cz

is a well-defined n-dim distribution J∞(E, n).

Properties of the Cartan distribution1 C is involutive and it is spanned by total derivatives

Di =∂

∂xi + ∑|I|≥0 uIi∂

∂uI, i = 1, . . . , n;

2 C detects ∞-prolongations: n-dim integral submanifolds are thoselocally of the form N∞.



Prolonging PDEs

Remark

A system of k-th order PDEs E0 ⊂ Jk(E, n) can be prolonged by addingtotal derivatives.

Definition

The first prolongation is:

E (1)0 := jk+1e N | Nk is tangent to E0 ⊂ Jk+1(E, n).

In coordinates

E (1)0 :

F (x, . . . ,uI , . . .) = 0 : E0DiF (x, . . . ,uI , . . .) = 0 .

The higher prolongations are E (`)0 := E (1)···(1)0 ⊂ Jk+`(E, n).



Infinite Prolongation

Remark

I assume formal integrability, i.e.

1 E (`)0 is well-defined and smooth for all `, and

2 E0 E (1)0oo · · ·oo E (`)0

oo · · ·oo is a sequence of fibrations.

Definition

The ∞-prolongation of E0 is

E := lim←− ` E

(`)0 ⊂ J∞(E, n).

In coordinates

E : Di1 · · ·Di`F (x, . . . ,uI , . . .) = 0, ` ≥ 0.



Diffieties

Remark

Whatever E0, the Cartan distribution is tangent to E : C|E ⊂ TE .

Properties of the Cartan distribution on E1 C|E is involutive;2 C|E detects solutions of E0: n-dim integral submanifolds of C|E are

those locally of the form N∞, with N a solution of E0. Morally,

solutions of E0 = n-dim integral submanifolds of C|E.

Warning

Beware that the Frobenius Theorem fails in ∞ dimensions.

Definition

(E , C|E ) is a diffiety.



Foliation-like Geometry of Diffieties

Let (E , C) be a diffiety. ConsiderNC = TE/C - the normal bundle sections denoted X.∧p(NC)∗ - the bundle of normal p-forms sections denoted CΩp.

RemarkFrom involutivity C is a Lie algebroid. As for foliations C acts on

NC via the Bott connection:

X.(Y mod C) = [X, Y]mod C, X ∈ Γ(C), Y ∈ X(E).

∧p(NC)∗ via the dual connection:

X.ω = LXω, X ∈ Γ(C), ω ∈ CΩp.

· · ·Luca Vitagliano Homotopy Algebras and PDEs 15 / 43


Plan of the Talk

1 Geometry

2 Homology

3 Homotopy



Horizontal Cohomology of PDEs

Let (E , C) be a diffiety.

Definition

The horizontal de Rham algebra

(Ω, d)

is the de Rham algebra of the Lie algebroid C.Ω - horizontal forms;

d - horizontal differential;

H(E) := H(Ω, d) - horizontal cohomology.

In coordinates:

ω = fi1···iq dxi1 ∧ · · · ∧ dxiq ∈ Ωq, 0 ≤ q ≤ n.

d = dxi Di.



Interpreting Horizontal Cohomology

Example (Infinite Jet Space - deg = n)

Let E0 be the trivial equation 0 = 0, so that E = J∞(E, n). ThenHq(E) = Hq(J∞(E, n)) = Hq

dR(E), for 0 ≤ q < n;morally, a top cohomology class [L ] is a variational principle imposed

on n-dim submanifolds N of E :

[L ] : N 7−→∫

N∞L , L = L(x, . . . ,uI , . . .)dnx.

Example (Generic PDE - deg = n− 1)

Let E0 be any PDE. Then(n− 1)-cocycles are conserved currents. In coordinates:

J = Ji(x, . . . ,uI , . . .)dn−1xi, Di Ji|E = 0.

morally, an (n− 1)-cohomology [J] is a conservation law.



Interpreting Horizontal Cohomology

Example (Gauge Systems - deg = n− 2)

Let E0 be the PDE governing a gauge system: Maxwell, Yang-Mills, etc.Hn−2(E) consists of gauge charges (e.g. electric charge).

Example (Finite Dimensional Diffieties - deg = 0)

Let dim E < ∞. Then C is the tangent distribution to a honest foliation.H0(E) consists of leaf-wise constant functions.

In all these cases horizontal cohomologies are morallyfunctions on the space of solutions!

Supporting fact

H(E) is a (graded) commutative algebra.



Coefficients for H(E): Normal Differential Forms

Let (E , C) be a diffiety.CΩ := ⊕pCΩp - normal differential forms

Remark

There is no natural differential on CΩ: CΩp −→/ CΩp+1

Remark

As C acts on normal forms, CΩ can serve as coefficients for H(E):

(Ω⊗ CΩ, d) H(E , CΩ) := H(Ω⊗ CΩ, d).

Proposition

There is a natural (p-increasing) differential on H(E , CΩ):

· · · // H(E , CΩp)d // H(E , CΩp+1)

d // · · ·



The C-Spectral Sequence

The best way to construct d is noticing that H(E , CΩ) is the first page ofa natural spectral sequence (cf. the spectral sequence of a foliation).

RemarkThe (standard) de Rham complex of E is filtered:

Ω(E) = F0Ω ⊃ F1Ω ⊃ · · · ⊃ FpΩ ⊃ · · · ,

whereF1Ω := ω ∈ Ω(E) |ω vanishes on Γ(C);FpΩ := (F1Ω)∧p.

Definition

The C-spectral sequence is the spectral sequence CE = (CEr, dr)r de-termined by filtration FpΩp.



The 0th Page of the C-Spectral Sequence

Ωn Ωn ⊗ CΩ1 · · · Ωn ⊗ CΩp · · ·

Ωn−1

d

OO

Ωn−1 ⊗ CΩ1

d

OO

· · · Ωn−1 ⊗ CΩp

d

OO

· · ·

· · ·

OO

· · ·

OO

· · · · · ·

OO

· · ·

Ω1

d

OO

Ω1 ⊗ CΩ1

d

OO

· · · Ω1 ⊗ CΩp

d

OO

· · ·

C∞(E)

d

OO

CΩ1

d

OO

· · · CΩp

d

OO

· · ·



The 1st Page of the C-Spectral Sequence

Hn(E) d // Hn(E , CΩ1)d // · · · // Hn(E , CΩp)

d // · · ·

Hn−1(E) d // Hn−1(E , CΩ1)d // · · · // Hn−1(E , CΩp)

d // · · ·

· · · · · · · · · · · · · · ·

H1(E) d // H1(E , CΩ1)d // · · · // H1(E , CΩp)

d // · · ·

H0(E) d // H0(E , CΩ1)d // · · · // H0(E , CΩp)

d // · · ·



The 1st Page of the C-Spectral Sequence of J∞(E, n)

Hn(J∞)d // Hn(J∞, CΩ1)

d // · · · // Hn(J∞, CΩp)d // · · ·

Hn−1(E) 0 · · · 0 · · ·

· · · · · · · · · · · · · · ·

H1(E) 0 · · · 0 · · ·

H0(E) 0 · · · 0 · · ·



Interpreting Horizontal Cohomology II

Focus on the non-trivial row.

Hn(J∞)d // Hn(J∞, CΩ1)

d // Hn(J∞, CΩ2)d // · · ·

Hn(J∞)d−→ Hn(J∞, CΩ1), variational

principle7−→ (lhs of) associated

Euler-Lagrange eqs.

Hn(J∞, CΩ1)d−→ Hn(J∞, CΩ2), square system

of PDEs7−→ (lhs of) associated

Helmoltz conds.

Horizontal cohomologies with coefficients in normal forms are morallydifferential forms on the space of solutions!

Supporting Fact

H(E , CΩ) is a DG algebra.



More Coefficients for H(E): Normal Vector Fields

Let (E , C) be a diffiety.X := Γ(NC) - normal vector fields

Remark

There is no natural Lie bracket on X.

Remark

As C acts on normal vector fields, X can serve as coefficients for H(E):

(Ω⊗X, d) H(E ,X) := H(Ω⊗X, d).

Proposition

There is a natural Lie bracket on H(E ,X).



Interpreting Horizontal Cohomology III

Example

H0(E ,X) consists of higher symmetries of E0, i.e. infinitesimal automor-phisms of (E , C) up to vector fields in C. Let Y mod C ∈ X be a cocycle.If Y integrates to a flow, then it induces a flow on the space of solutions!

Horizontal cohomologies with coefficients in normal vector fields are morallyvector fields on the space of solutions!

Supporting Fact

(H(E), H(E ,X)) is a (graded) Lie-Rinehart algebra.

More Supporting Facts

Cartan calculus has a horizontal cohomology analogue!



An Application: the Covariant Phase Space

Let E0 be an Euler-Lagrange equation and (E , C) the associated diffiety.

Theorem [E. Witten, G. J. Zuckermann 87]

There is a canonical “pre-symplectic form on the space of solutions of E0”.More precisely there is a canonical element

Ω ∈ Hn−1(E , CΩ2), such that dΩ = 0.

By contraction Ω maps two infinitesimal symmetries to a conservation law!

Remark

Basically, the diffiety (E , C) and the “pre-symplectic form” Ω are theinitial data used by physicists in the BV approach to classical (andquantum) field theory.



Symplectic Noether in Lagrangian Field Theory

Theorem (Symplectic Noether I-II for Field Theory) [L. V. 09]

1 Let X ∈ H0(E ,X) (an infinitesimal symmetry) and let f ∈ Hn−1(E)be the conservation law associated to X via Noether I. Then

iXΩ = df .

2 Let Ω[ : H0(E ,X)→ Hn−1(E , CΩ1) be contraction with Ω. Then

ker Ω[ = infinitesimal gauge symmetries.

3 Ω induces a (degree 1 − n) Poisson bracket on gauge invariant hori-zontal cohomologies, agreeing with the Dickey bracket on conservationlaws.



Plan of the Talk

1 Geometry

2 Homology

3 Homotopy



Homotopy Algebras from PDEs

Summary

PDE diffiety horizontal cohomology space P of solutions

algebraic structure interpretation

H(E) commutative algebra functions on P

H(E ,X) Lie algebra vector fields on P

H(E , CΩ) differential algebra differential forms on P

Question

Do these algebraic structures come from homotopy algebras on cochains?

Theorem [L. V. 14]

Ω ⊗ X is a homotopy Lie-Rinehart algebra and Ω ⊗ CΩ is its Chevalley-Eilenberg DG algebra.



Homotopy Lie-Rinehart Algebras

Definition [L. Kjeseth 01, up to decalage]

An LR∞[1]-algebra is a pair (A, L) whereA is a graded commutative algebra, and L is an A-module,L is an L∞[1]-algebra acting on A by derivations,

compatibility conditions hold: for a, b ∈ A and v1, . . . , vk ∈ L

ρ(a · v1, v2, . . . , vk−1|b) = ±a · ρ(v1, . . . , vk−1|b),v1, . . . , vk−1, avk = ±a · v1, . . . , vk+ ρ(v1, . . . , vk−1|a) · vk.

RemarkToday example: from a diffiety. More examples: from representa-tions up to homotopy, complex submanifolds, coisotropic submani-folds, BRST, ...



Higher Chevalley-Eilenberg Construction

Proposition

An LR∞[1]-algebra (A, L) determines a (formal) homological derivation

D = D1 + D2 + · · ·

in SymA(L, A) via:

(Dkω)(v1, . . . , vr+k)

:= ∑±ρ(vσ(1), . . . , vσ(k) |ω(vσ(k+1), . . . , vσ(k+r)))

−∑±ω(vσ(1), . . . , vσ(k+1), vσ(k+2), . . . , vσ(k+r)),

ω ∈ SymrA(L, A), v1, . . . , vr+k ∈ L.

RemarkIf L is projective and finitely generated as A-module, then the CE con-struction is one-to-one.



Auxiliary Geometric Data on a Diffiety

Let (E , C) be a diffiety. A splitting

0 // Γ(C) // X(E) // X //ee

0 (∗)

determines a decomposition

X(E) = Γ(C)⊕X,

and an associated projector

PC : X(E) −→ Γ(C), PC ∈ X(E)⊗Ω1(E).

Definition

The curvature of the splitting (∗) is the vector valued 1-form

R := 12 [PC , PC ]FN .

Dually, a splitting (∗) determines a factorization

Ω(E) = Ω⊗ CΩ.



An LR∞[1]-Algebra from a Diffiety

Theorem [J. Huebschmann 05], [L. V. 14] (see also [X. Ji 14])

A splitting (∗) determines an LR∞[1]-algebra structure on Ω⊗X with:1 X = dX2 X, Y = [X, Y]FN ± [[R, X]NR, Y]NR

3 X, Y, Z = ∓[[[R, X]NR, Y]NR, Z]NR

4 no higher brackets!

1 ρ(|ω) = dω

2 ρ(X|ω) = LXω± [R, X]NR ω

3 ρ(X, Y|ω) = ∓[[[R, X]NR, Y]NR ω

4 no higher anchors!

X, Y, Z ∈ Ω⊗X, and ω ∈ Ω.



The CE Algebra of Ω⊗X

Proposition [J. Huebschmann 05], [L. V. 14]

The CE algebra of Ω⊗ X is Ω⊗ CΩ equipped with the sequence of deriva-tions D1, D2, D3, . . .

1 D1 = d2 D2 = ddR − d + iR3 D3 = −iR4 no higher derivations!

In particular D = D1 + D2 + D3 = ddR.



Alternative Constructions of the LR∞[1]-Algebra

There are several ways to construct the LR∞[1]-algebra of a PDE:

by hands,inverting the CE construction,via higher derived brackets,via homotopy transfer.

Proposition (Homotopy Transfer of Lie-Rinehart Algebras)

Let (A, L) be a DG Lie-Rinehart algebra, let L be a DG A-module, and let

Lh##

p// L

joo

be contraction data such that p, j, h are A-linear. Then there is an LR∞[1]-algebra structure on (A, L) which can be computed in terms of (A, L) andthe contraction data.



The LR∞[1]-algebra of a PDE via Homotopy Transfer

Let (E , C) be a diffiety.

Remark

A splitting of 0 // Γ(C) // X(E) // X // 0 determines Ω-linearcontraction data (

Der Ω, [d,−])

h p

//(Ω⊗X, d)

joo .

Proposition [L. V. 14]

The LR∞[1]-algebra of (E , C) is induced via homotopy transfer.

This suggests how to construct more homotopy algebras from PDEs!



Even More Coefficients for Horizontal Cohomology

Let (E , C) be a diffiety. C acts on the algebra D(E) of linear DOs

∆ : C∞(E)→ C∞(E)

from both the left and the right.D := D(E)/D(E) · Γ(C) - normal DOs

Remark

There is no natural associative product on D.

RemarkC acts on normal DOs:

(Ω⊗D, d) H(E ,D) := H(Ω⊗D, d).

Proposition

There is a natural associative product on H(E ,D).Luca Vitagliano Homotopy Algebras and PDEs 39 / 43


An A∞-algebra from a PDE

Theorem [L.V. 15]

There is an A∞-algebra structure on Ω⊗D uniquely determined by suitableauxiliary geometric data on (E , C).

Proof. A splitting of 0 // Γ(C) // X(E) // X // 0 and a connec-tion in TE determine (via homological perturbations) contraction data

(D(Ω), [d,−])h p

//(Ω⊗D, dD)j

oo .

Now, notice that (D(Ω), [d,−]) is a DG associative algebra and usehomotopy transfer.

Ω⊗D should be interpreted as the universal enveloping A∞-algebra of Ω⊗X.



Conclusions and Perspectives

Homotopy algebras naturally appear from PDEs and account for alge-braic structures in horizontal cohomologies. What are the Massey prod-ucts in horizontal de Rham cohomology? Formality analysis of PDEs?

The field equations of a gauge theory are Euler-Lagrange equations.The covariant phase space possesses a canonical pre-symplectic 2-forminducing Poisson brackets on gauge-invariant functionals. Such Pois-son bracket plays a prominent role in the BV-formalism and should beunderstood as a shifted Poisson bracket or Poisson bracket up to homotopy.Derived geometry of PDEs?



References

J. HUEBSCHMANN,Higher homotopies and Maurer-Cartan algebras: Quasi-Lie-Rinehart, Gerstenhaber, andBatalin-Vilkovisky algebras,in: The Breadth of Symplectic and Poisson Geometry, Progr. in Math. 232 (2005)237–302.

L. V.,Secondary calculus and the covariant phase space,J. Geom. Phys. 59 (2009) 426–447.

X. JI,Simultaneous deformation of a Lie algebroids and its Lie subalgebroids,J. Geom. Phys. 84 (2014) 8–29.

L. V.,On the strong homotopy Lie-Rinehart algebra of a foliation,Commun. Contemp. Math. 16 (2014) 1450007 (49 pages)

L. V.,On the strong homotopy associative algebra of a foliation,Commun. Contemp. Math. 17 (2015) 1450026 (34 pages).



Thank you!


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