Homotopy Algebras and the Geometry of PDEs · 2015-09-11 · difﬁety: the algebra of horizontal...

GeometryHomologyHomotopy

Homotopy Algebrasand the Geometry of PDEs

Luca Vitagliano

University of Salerno, Italy

Differential Geometry and its ApplicationsBrno, August 19–23, 2013

Luca Vitagliano Homotopy Algebras and PDEs 1 / 29


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 algeba 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 a homotopy algebraic structure inducing A.

Let A be a type of algebra. A homotopy A algebra structure on acochain complex is a set of operations that satisfy the axioms of Aonly up to homotopy (in fact, a coherent system of higher homotopies).

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 to show that homotopy algebras appear alreadyin the theory of PDEs: the algebraic structures on horizontal cohomologiescome from homotopy algebraic structures on horizontal cochains.



Outline

1 Geometry

2 Homology

3 Homotopy



Outline

1 Geometry

2 Homology

3 Homotopy



Geometric Portraits of PDEs

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

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

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

∂|I|f0∂xI (x) = ∂|I|f1

∂xI (x), |I| ≤ k.

Jk(E, n) is the manifold of classes of tangency up to the order k, with co-ordinates (x, . . . ,uI , . . .), |I| ≤ k.

Nk : uI =∂|I|f∂xI (x), |I| ≤ k.

Definition

A system of k-th order PDEs (on n-dim submfs of E) is E0 ⊂ Jk(E, n):E0 : F (x, . . . ,uI , . . .) = 0, |I| ≤ k.

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



Prolongations of PDEs

J∞(E, n) is the (countable dimensional) manifold of classes of tangency upto the order ∞, with coordinates (x, . . . ,uI , . . .), |I| ≥ 0.

E J1(E, n)oo · · ·oo Jk(E, n)oo Jk+1(E, n)oo · · ·oo J∞(E, n)oo

A system of k-th order PDEs E0 ⊂ Jk(E, n) can be prolonged by addingtotal derivatives. The prolongation is E ⊂ J∞(E, n):

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

∂xi + ∑I uIi∂

∂uI, |I|, ` ≥ 0.

Remark

E is a diffiety. It comes equipped with the canonical, involutive dis-tribution C = 〈. . . , Di, . . .〉. n-dim integral submfs of (E , C) can becharacterized as submfs of the form N∞, N ⊂ E a solution of E0. Thus

{solutions of E0} ' {n-dim integral manifolds of (E , C)}.



Differential Geometry of Diffieties

Let (E , C) be a diffiety. The distribution C determinesA C∞(E)-module CX of vector fields in C,

C involutive =⇒ CX is a Lie algebra.A C∞(E)-module X := X/CX of vector fields transversal to C.A C∞(E)-module CΛ1 of 1-forms annihilating CX,C involutive =⇒ CΛ1 generates an ideal I ⊂ Λ(E) closed under

d.A graded algebra Λ := Λ(E)/I of forms longitudinal to C.

RemarkThe Lie algebra CX

1 acts on X via the Bott (partial) connection,2 acts on CΛ, the exterior algebra of CΛ1, via Lie derivative.

Transversal vector fields which are flat with respect to the Bott connectionidentify with non-trivial infinitesimal symmetries of the initial PDE E0.



Outline

1 Geometry

2 Homology

3 Homotopy



Horizontal Cohomology of PDEs

A diffiety (E , C) defines a rich homological algebra.

RemarkSince the ideal I of differential forms vanishing on C is closed un-der the exterior differential d, then the quotient algebra Λ = Λ(E)/Icomes equipped with and induced differential d. Locally d = dxi Di.

Definition

The DG algebra (Λ, d) is the horizontal de Rham algebra, and its coho-mology H(E) := H(Λ, d) is the horizontal de Rham cohomology.

A CX-module M defines a differential module (Λ ⊗ M, dM), and its coho-mology H(E , M) := H(Λ⊗M, dM) is the horizontal de Rham cohomol-ogy with coefficients in M.

There are canonical CX-modules: C∞(E), X, CΛ, etc. Their horizontalcohomologies all have nice mathematical and physical interpretations.



Canonical Horizontal Cohomologies

Let (E , C) be a diffiety and P the space of solutions of E0. When E0 is anEuler-Lagrange PDE, P is naturally interpretad as the phase space.

degmathematicalinterpretation

“physical”interpretation notation

H(E) nn− 1

var. principlescons. laws functions on P C∞

H(E ,X)012

symmetriesinf. deformations

obstructionsvect. fields on P X

H(E , CΛ1) n− 1 cosymmetries diff. forms on P Λ1

Remark

Similarly, Euler-Lagrange equations, Helmoltz conditions, recursion opera-tors, Hamiltonian structures, etc. are horizontal cohomologies.



Algebraic Structures on Horizontal Cohomologies

The “physical” interpretation of horizontal cohomologies is supportedby the existence of the “right” algebraic structures on them.

“physical”interpretation algebraic structure notation

H(E) functions on P commutative algebra C∞

H(E ,X) vect. fields on P Lie algebra X

H(E , CΛ) diff. forms on P differential algebra Λ

RemarkCartan calculus has a horizontal cohomology analogue!

Remark

The algebraic structures on horizontal cohomologies do not come fromsimilar structures on cochains, in general. However, they come fromstructures up to homotopy, i.e. homotopy algebraic structures!



Outline

1 Geometry

2 Homology

3 Homotopy



Homotopy Algebras

Consider a cochain complex of vector spaces (V, δ) and let A be analgebraic structure (Lie, differential, associative algebra, etc.).

Rough Definition

A homotopy A-structure in (V, δ) is a linear operation in (V, δ) which1 is compatible with δ,2 is of the type A only up to coherent (higher) homotopies.

Rough Motivation

Let (A, d) be a differential algebra of typeA and f : (A, d) � (V, δ) : ga pair of homotopy equivalences. The algebra structure in A can betransferred to V along ( f , g), but the transferred structure is of thetype A only up to higher homotopies. On the other hand

homotopy algebras are homotopy invariant!



Homotopy Lie Algebras

Let V be a graded vector space.

Example (Homotopy Lie algebra)

An L∞-algebra structure in V is a sequence of operations:

• δ : V −→ V• [−,−] : V∧2 −→ V• [−,−,−] : V∧3 −→ V• · · ·satisfying the following coherence conditions

• δ2(x) = 0• δ[x, y] = [δx, y]± [x, δy]• [x, [y, z]]± [y, [z, x]]± [z, [x, y]] =

δ[x, y, z]− [δx, y, z]∓ [x, δy, z]∓ [x, y, δz]• · · ·

H(V, δ) is a genuine (graded) Lie algebra!



Homotopy Differential Algebras

Let A be a (bi-)graded algebra.

Example (Homotopy Differential Algebra)

A homotopy diff. algebra structure in A is a sequence of derivations:

• δ : A −→ A• d : A −→ A• d3 : A −→ A• · · ·satisfying the following coherence conditions

• δ2 = 0• [δ, d] = 0• d2 = −[δ, d3]• · · ·Equivalently, (δ + d + d3 + · · · )2 = 0.

H(A, δ) is a genuine differential algebra!Luca Vitagliano Homotopy Algebras and PDEs 17 / 29


Auxiliary Geometric Data on a Diffiety

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

0 // CX // X(E) // X //ee

0 (∗)

determines a decomposition

X(E) = CX⊕X,

and an associated projector

PC : X(E) −→ CX, PC ∈ X(E)⊗Λ1(E).

Definition

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

R := 12 [[PC , PC ]].

Dually, a splitting (∗) determines a factorization

Λ(E) = Λ⊗ CΛ.



An L∞-Algebra from a PDE

Let (E , C) be a diffiety and P the space of solutions of the underlyingPDE. The “defining complex of vector fields on P ” is: (Λ⊗X, dX)

Proposition [Huebschmann 05], [Ji 12], [LV 12])

A splitting

0 // CX // X(E) // X //ee

0

determines an L∞-algebra structure on Λ⊗X with structure maps

1) δX := dXX2) [X, Y] := [[X, Y]]± [[R, X]nr, Y]nr3) [X, Y, Z] := ∓[[[R, X]nr, Y]nr, Z]nrk) no higher operations!

Any two such L∞-algebras are canonically L∞-isomorphic, and the inducedLie algebra structure on X = H(E ,X) is the standard one.



A Homotopy Differential Algebra from a PDE

Let (E , C) be a diffiety and P the space of solutions of the underlyingPDE. The “defining complex of diff. forms on P ” is: (Λ⊗ CΛ, dCΛ)

Proposition [Huebschmann 05], [LV 12]

A splitting

0 // CX // X(E) // X //ee

0

determines a homotopy differential algebra structure {δ, d, d3, · · · } on Λ⊗CΛ with

1) δ := dCΛ2) d := d− dCΛ + iR3) d3 := −iRk) no higher derivations!

Any two such homotopy differential algebras are canonically isomorphic, andthe induced differential algebra in Λ = H(E , CΛ) is the standard one.



Contraction Data

A standard way to produce homotopy algebras is via homotopy trans-fer along contraction data. Let (K, δ) and (K, δ) be cochain complexes.Consider the following data

(K, δ)h%% p // (K, δ)

joo .

Definition

The data (p, j, h) are contraction data for (K, δ) over (K, δ) if1 j is a right inverse of p, i.e., pj = id,2 h is a contracting homotopy, i.e., [h, δ] = id− jp,3 the side conditions h2 = 0, hj = 0, ph = 0 are satisfied.



Homotopy Transfer

Homotopy Transfer Theorem

Let (K, δ) be a DG (Lie, associative, etc.) algebra and let

(K, δ)h%% p // (K, δ)

joo

be contraction data. There is a homotopy (Lie, associative, etc.) algebra struc-ture on (K, δ) wich can be computed in terms of the contraction data.

Remark

Let (K, δ) be a DG algebra. There exist always contraction data for(K, δ) over its cohomology (H(K, δ), 0) =⇒ H(K, δ) is a homotopyalgebra. The latter structure characterizes the homotopy type of (K, δ).

Example (Massey products)

Let (K, δ) = (C•(X), d) be singular cochains of a topological space X. TheMassey products in H•(X) are obtained via homotopy transfer.



The L∞-Algebra of a PDE via Homotopy Transfer

RemarkA splitting

0 // CX // X(E) // X //ee

0

determines contraction data

(DerΛ, [d,−])h�� p //

(Λ⊗X, dX)joo .

Proposition [LV 12]

The L∞-algebra of a PDE is induced via homotopy transfer.

This suggests how to produce more homotopy algebras from PDEs!



More Coefficients for Horizontal Cohomology

Let (E , C) be a diffiety. The distribution C determinesA C∞(E)-module D of DOs transversal to C as follows.

CX acts on D(E), the ring of linear DOs C∞(E) −→ C∞(E) from boththe left and the right. Define: D := D(E)/D(E) · CX.

Remark

The Lie algebra CX acts on D, so that D is a system of coefficients forhorizontal cohomology =⇒ H(E ,D).

“physical interpretation” algebraic structure notation

H(E ,D) DOs on P associative algebra D

Remark

The associative product on H(E ,D) do not come from an associativeproduct on cochains, in general. But it comes from an associative prod-uct up to homotopy, i.e. an homotopy associative algebra structure!



Homotopy Associative Algebras

Let V be a graded vector space.

Example (Homotopy Associative Algebra)

An A∞-algebra structure in V is a sequence of operations:

• δ : V −→ V• ? : V⊗2 −→ V• α : V⊗3 −→ V• · · ·satisfying the following coherence conditions

• δ2(x) = 0• δ(x ? y) = δx ? y± x ? δy• x ? (y ? z)− (x ? y) ? z =

−δα(x, y, z)− α(δx, y, z)∓ α(x, δy, z)∓ α(x, y, δz)• · · ·

H(V, δ) is a genuine (graded) associative algebra!



An A∞-algebra from a PDE

The “defining complex of DOs on P ” is: (Λ⊗D, dD)

Theorem [LV 12]

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

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

(D(Λ), [d,−])h�� p //

(Λ⊗D, dD)joo .

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



Conclusions and Perspectives

Homotopy algebras naturally appear in the homological theory of par-tial differential equations and account for algebraic structures in hor-izontal cohomologies. It would be interesting to compute the Masseyproducts in horizontal cohomology.

The field equations of a gauge theory are Euler-Lagrange equations.The diffiety encoding Euler-Lagrange equations possesses a canoni-cal, closed, variational 2-form, i.e., an element ω ∈ Λ2 := H(E , CΛ2).In its turn, ω induces Poisson brackets on gauge-invariant function-als. Such Poisson bracket plays a prominent role in the BV-formalismand should be understood as a Poisson bracket up to homotopy. It islikely that the homotopy algebras described so far can be made inter-acting nicely with those coming from the gauge structure in the BV-formalism.



References

J. Huebschmann, Higher homotopies and Maurer-Cartan alge-bras: Quasi-Lie-Rinehart, Gerstenhaber, and Batalin-Vilkoviskyalgebras, in: The Breadth of Symplectic and Poisson Geometry,Progr. in Math. 232 (2005) 237–302; e-print: arXiv: math/0311294.L. V., Secondary calculus and the covariant phase space, J. Geom.Phys. 59 (2009) 426–447; e-print: arXiv:0809.4164.X. Ji, Simultaneous deformation of Lie algebroids and Lie subal-gebroids, (2012); e-print: arXiv:1207.4263.L. V., On the strong homotopy Lie-Rinehart algebra of a foliation,(2012); e-print: arXiv:1204.2467.L. V., On the strong homotopy associative algebra of a foliation,(2012); e-print: arXiv:1212.1090.



Thank you!


Date post:	28-Jun-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Homotopy Algebras and the Geometry of PDEs · 2015-09-11 · difﬁety: the algebra of horizontal...

Documents