Differential Forms

In General > s.a. exterior algebra and calculus; integration on manifolds.
* Idea: Antisymmetric tensors, for which Cartan developed a special formalism, terminology and notation.
$ Def: A \(p\)-form is a totally antisymmetric covariant tensor field of order \(p\).
* Notation: The space of one-forms on M is usually denoted by T*\(M\), that of \(p\)-forms by Λ^p T*M, Λ^pM or Ω^pM.
* Chiral p-forms: Abelian p-form potentials with self-dual field strengths.
* And other structure: Forms are a submodule of the algebra of all tensor fields on M; Together with the exterior product, this forms the exterior algebra or Grassmann algebra of M; dim \(\Lambda^p{}_x(M) = n!/p!(n-p)!\)
@ General references: Flanders 63; Cartan 67; Von Westenholz 81; Bott & Tu 82; Darling 94; Jancewicz gq/98 [visualization]; Morita 01 [geometry]; Ivancevic & Ivancevic a0807-ln; Lessig a1206 [primer]; Guillemin & Haine 19.
@ Related topics: in de Rham 60 [of odd kind]; Hitchin m.DG/00 [3-forms in 6D and 7D]; > s.a. grassmann structures; types of cohomology theories.
@ Chiral p-forms: Bengtsson & Kleppe IJMPA(97)ht/96; Pasti et al PRD(97) [covariant actions]; Bonetti et al PLB(13) [Kaluza-Klein inspired action]; Buratti et al PLB(19)-a1909 [self-interacting chiral 2n-form in 4n+2 dimensions, Lagrangian].
@ And electrodynamics: Schleifer AJP(83)dec; Dappiaggi et al a1908 [manifold with timelike boundary]; Fumeron et al a2009 [in the classroom]
@ And other physics: Petrova mp/01 [conservation laws], mp/05 [equations of mathematical physics]; Rodrigues AFLB(07)-a0712 [detailed]; Perot & Zusi JCP(14) [review]; Alfaro & Riquelme PRD(14)-a1402 [bosonic (p–1)-forms as surces of torsion]; Estabrook a1405/JMP [for field theories]; Dray 14 [and general relativity]; Pommaret a1707 [in mathematical physics]; > s.a. field theories; lattice field theory.

(Hodge) Dual
$ Def: Given a p-form f on an n-manifold M with volume form ε and metric g (with p ≤ n), its dual is the (n−p)-form

*f_{c.. d} := (1/p!) f ^{a.. b} ε_{a.. bc.. d} .

* Properties: It follows that **f = (−1)^s+p(n−p) f.
* For a curvature tensor: The left and right duals are, respectively,

*R_abcd := \(1\over2\)ε_ab^mn R_mncd    and    R*_abcd:= \(1\over2\)R_abmn ε^mn_cd .

* And physics: The duality operation applied to the electromagnetic (Faraday) field tensor (2-form) interchanges the electric and magnetic fields; > s.a. duality.
@ And conformal structure: Dray et al JMP(89); Harnett JMP(91).
@ Related topics: Saa JGP(95) [for affine manifold with torsion]; Klinker JGP(11) [generalized definition]; Sen a1903 [self-dual forms, dynamics].

Volume Form > s.a. types of field theories.
$ Def: For an n-dimensional manifold M, a nowhere-vanishing n-form ω.
* And connections: A volume form is compatible with a connection if for all X ∈ TM, \(\cal L\)_X ω = (D_a X^a) ω (for the divergence theorem to apply).
* Example: The volume form ε defined by a metric g is ε = |g|^1/2 dx¹ ∧ ... ∧ dxⁿ; It is compatible with the Christoffel symbols, but not with a Riemann-Cartan connection, \(\cal L\)_X ω = (D_a X^a − 2 T_a X^a) ω, where T_a:= T_ba^b.
@ General references: Cartier et al in(01)mp/00 [characterization].
@ As a dynamical variable: Guendelman & Kaganovich a0811-conf; Guendelman et al a1505-conf, Benisty et al a1905-conf [in gravity and cosmology]; Benisty et EPJP-a2006 [non-canonical volume forms and modified gravity, rev].

Lie-Algebra Valued Forms
* Canonical form: Given a Lie group G, the canonical (Maurer-Cartan) form on G is the Lie-algebra-valued 1-form ω: TG → \(\cal G\) = T_eG defined by ω(v_g) = L_g⁻¹' v_g; Theorem: R_g* ω = Ad(g⁻¹) · ω.
* Maurer-Cartan structure equation: If {θ^I} is a basis for the dual \(\cal G\)* of the Lie algebra of a group G, i.e. for the left-invariant 1-forms on G, then

dθ^I = −\(1\over2\)C^I_JK θ^J ∧ θ^K .

Other Special Types and Generalizations > s.a. deformation quantization; Star Product; superspace.
* Closed form: A form ω whose exterior derivative vanishes, dω = 0.
* Exact form: A form ω which can be expressed as the exterior derivative of another form, ω = dθ.
* Poincaré's lemma: On a contractible space, any closed form is exact (e.g., not in \({\mathbb R}^2\setminus\{0\}\), non-contractible).
@ General references: in Flanders 63; in Nash & Sen 83.
@ Closed forms: Torre CQG(95)gq/94 [classification]; Farber 04; Farber & Schütz Top(06) [closed 1-forms with at most one zero]; Burns a1906 [generalization of Poincaré lemma to divergence-free multivector fields].
@ Fractional order: Cottrill-Shepherd & Naber JMP(01)mp/03, mp/03; Tarasov JPA(05).
@ Discrete: Richter et al CQG(07)gq/06 [and spherical symmetry in general relativity]; Dowker JGP(07) [on sphere tessellations]; Dolotin et al TMP(08)-a0704 [based on simplicial complexes]; Richter & Frauendiener JSC(10)-a0805 [Gowdy solutions, numerical].
@ Other generalizations: Nurowski & Robinson CQG(01) [and spacetime geometry]; Robinson IJTP(03) [and gauge theories], JMP(03) [and general relativity], JPA(07) [rev, and applications], CQG(09) [integral calculus, Stokes' theorem, Chern-Simons and Einstein-Yang-Mills theories]; Chatterjee et al IJGMP(08)-a0706 [negative forms, and path-space forms]; Gallego Torromé a1207 [higher-order, and applications to electrodynamics]; Robinson a1312 [and gravitation]; > s.a. clifford calculus; formulations of electromagnetism [pair and impair].
@ Related topics: Frauendiener talk(03).

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