nLab
Hodge star operator

Contents

Context

Riemannian geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

Given a finite dimensional (pseudo)-Riemannian manifold (X,g)(X,g), the Hodge star operator “completes” a kk-differential form to the volume form of (X,g)(X,g).

Definition

Let (X,g)(X,g) be an oriented nn-dimensional smooth manifold XX endowed with a (pseudo)-Riemannian metric gg. For 0kn0 \leq k \leq n, write Ω k(X)\Omega^k(X) for the vector space of kk-forms on XX.

Hodge inner product

The metric gg naturally induces a nondegenerate symmetric bilinear form

():Ω k(X)Ω k(X)Ω 0(X). (-\mid-) \;\colon\; \Omega^k(X) \otimes \Omega^k(X) \to \Omega^0(X) \,.

If XX is compact then the integral of this against the volume form vol gvol_g exists. This is the Hodge inner product

,:Ω k(X)Ω k(X) \langle - , - \rangle \;\colon\; \Omega^k(X)\otimes \Omega^k(X) \to \mathbb{R}
α,β:= X(αβ)vol. \langle \alpha, \beta \rangle := \int_X (\alpha\mid \beta) vol \,.

Hodge star operator

The Hodge star operator is the unique linear function

:Ω k(X)Ω nk(X) {\star}\colon \Omega^k (X) \to \Omega^{n-k} (X)

defined by the identity

αβ=(αβ)vol g,α,β kX, \alpha \wedge \star\beta = (\alpha \mid \beta) vol_g, \qquad \forall \alpha,\beta \in \bigwedge^k X \,,

where vol gΩ nXvol_g \in \Omega^n X is the volume form induced by gg.

Therefore in terms of the Hodge operator the Hodge inner product reads

α,β= Xαβ. \langle \alpha , \beta\rangle = \int_X \alpha \wedge \star \beta \,.

Generalizations

The metric gg is used in two places in the specification of the Hodge operator: in the inner product on forms and in the volume form. If XX is equipped only with a volume form (not necessarily coming from a metric), then the Hodge operator still takes kk-forms to (nk)(n-k)-vector fields. If the manifold is not oriented, then the metric only gives a volume pseudoform, but the Hodge operator still takes kk-forms to (nk)(n-k)-pseudoforms. Finally, if XX is equipped with only a volume pseudoform (which is equivalent to an absolutely continuous Radon measure on XX), then the Hodge operator takes kk-forms to (nk)(n-k)-pseudovector fields. (Of course, in every case, one might apply the operator to pseudoforms or multivector fields to begin with.)

Properties

Component expression

Let XX be a (pseudo-)Riemannian manifold of dimension DD, and locally, on some open subset UXU \subset X, let

e 1,,e DΩ 1(U) e^1, \dots, e^D \;\in\; \Omega^1(U)

be a frame of differential 1-forms (a vielbein). For example if {x i}\{x^i\} is a coordinate chart on UU, then e idx ie^i \coloneqq d x^i is such a frame.

With this choice, any differential p-form αΩ p(U)\alpha \in \Omega^p(U) has a component expansion

α=1p!α i 1i pe i 1e i p \alpha \;=\; \frac{1}{p!} \alpha_{i_1 \dots i_p} \, e^{i_1} \wedge \cdots \wedge e^{i_p}

for smooth function-components {α i 1i p}\{\alpha_{i_1 \cdots i_p}\} (where here and in the following we use the Einstein summation convention).

In terms of these components, the Hodge dual α\star \alpha of α\alpha is expressed by the following formula:

(1)α =1p!(Dp)!|det((g ij))|α j 1j pg j 1i 1g j pi pϵ i 1i pii p+1i De i p+1e i D =1p!(Dp)!|det((g ij))|α i 1i pϵ i 1i pi p+1i De i p+1e i D \begin{aligned} \star \alpha & = \; \frac{1}{ p! (D-p)! } \sqrt{ \left\vert det\big((g_{i j})\big) \right\vert } \, \alpha_{ \color{green} j_1 \dots j_p } g^{ {\color{green} j_1 } {\color{cyan} i_1 } } \cdots g^{ {\color{green} j_p } {\color{cyan} i_p } } \epsilon_{ {\color{cyan} i_1 \dots i_p i } {\color{orange} i_{p+1} \cdots i_D } } e^{ \color{orange} i_{p+1} } \wedge \cdots \wedge e^{ \color{orange} i_D } \\ & = \frac{1}{ p! (D-p)! } \sqrt{ \left\vert det\big((g_{i j})\big) \right\vert } \, \alpha^{ \color{green} i_1 \dots i_p } \epsilon_{ { \color{green} i_1 \dots i_p } { \color{orange} i_{p + 1} \cdots i_D } } e^{ \color{orange} i_{p + 1} } \wedge \cdots \wedge e^{ \color{orange} i_{D} } \end{aligned}

Here

  • p!p!, (Dp)!(D-p)! are the factorials of pp and (Dp)(D-p), respectively,

  • ϵ i 1,,i n{+1,,1}\epsilon_{i_1,\dots,i_n} \in \{+1, ,-1\} (the Levi-Civita symbol) is the signature of the permutation (1,2,,D)(i 1,i 2,,i D)(1,2,\dots,D) \mapsto (i_1,i_2,\dots,i_D)

  • (g ij)(g_{i j}) is the square matrix of components of the metric tensor in the chosen basis, i.e. such that

    g=g ije ie j g \;=\; g_{i j} e^i \otimes e^j
  • det(g)det(g) is the determinant of (g ij)(g_{i j})

  • |g|\left\vert g \right\vert is the absolute value of the determinant.

Basic properties

Let (X,g)(X,g) be a Riemannian manifold of dimension nn and let ω,λΩ k(X)\omega,\lambda \in \Omega^k(X). Then the following holds:

(2)(ω)=(1) k(n+1)ω=(1) k(nk)ω \star(\star\omega) = (-1)^{k(n+1)} \omega = (-1)^{k(n-k)} \omega
(3)ω,λ=ω|λ \langle\star\omega , \star\lambda\rangle = \langle\omega | \lambda\rangle
(4)1=dvol, \star 1 = dvol \,,

where dvoldvol denotes the volume form.

Examples

Hodge star operator on a Kähler manifold

On a Kähler manifold Σ\Sigma of dimension dim (Σ)=ndim_{\mathbb{C}}(\Sigma) = n the Hodge star operator acts on the Dolbeault complex as

:Ω p,q(X)Ω nq,np(X). \star \;\colon\; \Omega^{p,q}(X) \longrightarrow \Omega^{n-q,n-p}(X) \,.

(notice the exchange of the role of pp and qq). See e.g. (Biquerd-Höring 08, p. 79). See also at Serre duality.

Hodge star operator on Minkowski spacetime

We spell out component expressions for the Hodge star operator on D=d+1D = d+1-dimensional Minkowski spacetime.

Conventions

We use Einstein summation convention throughout. Hence a generic differential p-form is

α=1p!α μ 1μ pdx μ 1dx μ p. \alpha \;=\; \tfrac{1}{p!} \alpha_{ \mu_1 \cdots \mu_p } d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_p} \,.

Here p!123pp! \coloneqq 1 \cdot 2 \cdot 3 \cdots p is the factorial of pp.

We take the Minkowski metric to be the D×DD \times D diagonal matrix of the form

η=(η μν)=(η μν)diag(1,+1,+1,,+1). \eta \;=\; (\eta_{\mu \nu}) \;=\; (\eta^{\mu \nu}) \;\coloneqq\; diag(-1,+1, +1 , \cdots , +1) \,.

We normalize the Levi-Civita symbol as

(5)ϵ 012d+1 \epsilon_{0 1 2 \cdots d} \;\coloneqq\; + 1

which means that

(6)ϵ 012d=1. \epsilon^{0 1 2 \cdots d} \;=\; - 1 \,.

We normalize the sign of the volume form as

(7)dvol dx 0dx 1dx d =1D!ϵ μ 1μ Ddx μ 1dx μ D \begin{aligned} dvol & \coloneqq\; d x^0 \wedge d x^1 \wedge \cdots \wedge d x^d \\ & = \tfrac{1}{D!} \epsilon_{ \color{green} \mu_1 \cdots \mu_D } d x^{\color{green}\mu_1} \wedge \cdots \wedge d x^{\color{green}\mu_D} \end{aligned}

We write

(8)δ ν 1ν p μ 1μ p{sgn(σ) | σSym(p)(1ip(ν σ(i)=μ i)) 0 | otherwise \delta^{ \mu_1 \cdots \mu_p }_{ \nu_1 \cdots \nu_p } \;\coloneqq\; \left\{ \array{ sgn(\sigma) &\vert& \underset{ \sigma \in Sym(p) }{\exists} \left( \underset{1 \leq i \leq p}{\forall} \left( \nu_{\sigma(i)} = \mu_i \right) \right) \\ 0 &\vert& \text{otherwise} } \right.

for the generalized Kronecker delta, whose value is the signature of the permutation that takes the upper indices to the lower indices, if any such exists, and zero otherwise.

This appears whenever the Levi-Civita symbol is contracted with itself:

(9)ϵ μ 1μ pμ p+1μ Dϵ ν 1ν pμ p+1μ D=(Dp)!δ μ 1μ p ν 1ν p \epsilon_{ { \color{green} \mu_1 \cdots \mu_p } {\color{blue} \mu_{p+1} \cdots \mu_{D} } } \epsilon^{ { \color{orange} \nu_1 \cdots \nu_p } { \color{blue} \mu_{p+1} \cdots \mu_D } } \;=\; { \color{magenta} - } (D-p)! \; \delta_{ \color{green} \mu_1 \cdots \mu_p }^{ \color{orange} \nu_1 \cdots \nu_p }

Notice the minus sign in (9), which comes, via (6), from the Minkowski signature.

Definition

We write ι μ\iota_\mu for the operator of contraction of differential forms with the vector field d/dx μd/d x^\mu, hence the linear operator on differential forms with anticommutator

{ι μ,dx ν}=δ μ ν \big\{ \iota_\mu, d x^\nu \wedge \big\} \;=\; \delta_\mu^\nu

With the volume form as in (7) it follows that (notice the reversion of the index ordering in the contraction operators ι\iota)

(10)α μ 1μ pι μ pι μ 1dvol=ϵ μ 1μ pν 1ν (Dp)dx ν 1dx ν (Dp) \alpha^{ \color{green} \mu_1 \cdots \mu_p } \iota_{\color{green} \mu_p} \cdots \iota_{ \color{green} \mu_1} dvol \;=\; \epsilon_{ { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \nu_1 \cdots \nu_{(D-p)} } } d x^{\color{orange} \nu_1} \wedge \cdots \wedge d x^{\color{orange} \nu_{(D-p)}}
Definition

For a differential p-form

α1p!α μ 1μ pdx μ 1dx μ p \alpha \;\coloneqq\; \tfrac{1}{ \color{green} p! } \alpha_{ \color{green} \mu_1 \cdots \mu_p} d x^{ \color{green} \mu_1 } \wedge \cdots \wedge d x^{ \color{green} \mu_p }

its Hodge dual is:

(11)α 1p!(Dp)!α μ 1μ pι μ pι μ 1dvol =1p!(Dp)!α μ 1μ pϵ μ 1μ pμ p+1μ Ddx μ p+1dx μ D, \begin{aligned} \star \alpha & \coloneqq \tfrac{1}{ { \color{green} p! } { \color{orange} (D-p)! } } \, \alpha^{ \color{green} \mu_1 \cdots \mu_p } \iota_{ \color{green} \mu_p } \cdots \iota_{ \color{green} \mu_1 } \, dvol \\ & = \tfrac{1}{ { \color{green} p! } { \color{orange} (D-p)! } } \, \alpha^{ \color{green} \mu_1 \cdots \mu_p } \epsilon_{ { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \mu_{p+1} \cdots \mu_D } } \, d x^{ \color{orange} \mu_{p+1} } \wedge \cdots \wedge d x^{ \color{orange} \mu_D } \,, \end{aligned}

where in the second line we used (10).

Properties

Proposition

(Hodge pairing)

For a differential p-form α1p!α μ 1μ pdx μ 1dx μ p \alpha \;\coloneqq\; \tfrac{1}{p!} \alpha_{\mu_1 \cdots \mu_p} d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_p} on DD-dimensional Minkowski spacetime its wedge product with its Hodge dual (11) is

(12)αα=α μ 1μ pα μ 1μ pdvol. \alpha \wedge \star \alpha \;=\; { \color{magenta} - } \alpha_{ \mu_1 \cdots \mu_p } \alpha^{ \mu_1 \cdots \mu_p } \, dvol \,.
Proof

We compute as follows:

αα =1p!p!(Dp)!α μ 1μ pdx μ 1dx μ pα ν 1ν pι ν pι ν 1dvol =1p!p!(Dp)!α μ 1μ pα ν 1ν pϵ ν 1ν pν p+1ν Ddx μ pdx μ 1dx ν p+1dx ν D =1p!p!(Dp)!α μ 1μ pα ν 1ν pϵ ν 1ν pν p+1ν Dϵ μ pμ 1ν p+1ν Ddvol =1p!p!α μ 1μ pα ν 1ν pδ ν 1ν p μ 1μ pdvol =α μ 1μ pα μ 1μ pdvol \begin{aligned} \alpha \wedge \star \alpha & = \tfrac{1}{ { \color{green} p! } { \color{orange} p! } { \color{blue} (D-p)! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } d x^{ \color{green} \mu_1 } \wedge \cdots \wedge d x^{ \color{green} \mu_p } \wedge \alpha^{ \color{orange} \nu_1 \cdots \nu_p } \iota_{ \color{orange} \nu_p } \cdots \iota_{ \color{orange} \nu_1 } dvol \\ & = \tfrac{1}{ { \color{green} p! } { \color{orange} p! } { \color{blue} (D-p)! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \alpha^{ \color{orange} \nu_1 \cdots \nu_p } \epsilon_{ { \color{orange} \nu_1 \cdots \nu_p } { \color{blue} \nu_{p+1} \cdots \nu_D } } d x^{ \color{green} \mu_p } \wedge \cdots \wedge d x^{ \color{green} \mu_1 } \wedge d x^{ \color{blue} \nu_{p+1} } \wedge \cdots d x^{ \color{blue} \nu_{D} } \\ & = \tfrac{1}{ { \color{green} p! } { \color{orange} p! } { \color{blue} (D-p)! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \alpha^{ \color{orange} \nu_1 \cdots \nu_p } \epsilon_{ { \color{orange} \nu_1 \cdots \nu_p } { \color{blue} \nu_{p+1} \cdots \nu_D } } \epsilon^{ { \color{green} \mu_p \cdots \mu_1 } { \color{blue} \nu_{p+1} \cdots \nu_{D} } } \, dvol \\ & = \tfrac{ \color{magenta} -1 }{ { \color{green} p! } { \color{orange} p! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \alpha^{ \color{orange} \nu_1 \cdots \nu_p } \delta^{ \color{green} \mu_1 \cdots \mu_p }_{ \color{orange} \nu_1 \cdots \nu_p } \, dvol \\ & = {\color{magenta} -} \alpha_{ \mu_1 \cdots \mu_p } \alpha^{ \mu_1 \cdots \mu_p } \, dvol \end{aligned}

Here the sign in the last lines arises from the Minkowski signature via (9).

Proposition

(double Hodge dual)

For a differential p-form α=1p!α μ 1μ pdx μ 1dx μ p \alpha \;=\; \tfrac{1}{p!} \alpha_{\mu_1 \cdots \mu_p} d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_p} on DD-dimensional Minkowski spacetime, its double Hodge dual (11) is

(13)α=(1) p(Dp)α. \star \star \alpha \;=\; {\color{magenta} -} (-1)^{ p (D - p) } \, \alpha \,.
Proof

We compute as follows:

1p!α μ 1μ pdx μ 1dx μ p =1p!(Dp)!α μ 1μ pι μ pι μ 1dvol =1p!(Dp)!α μ 1μ pϵ μ 1μ pμ p+1μ Ddx μ p+1dx μ d =1p!(Dp)!p!α μ 1μ pϵ μ 1μ pμ p+1μ Dϵ μ p+1μ Dν 1ν pdx ν 1dx ν D =(1) p(Dp)p!(Dp)!p!α μ 1μ pϵ μ 1μ pμ p+1μ Dϵ ν 1ν pμ p+1μ Ddx ν 1dx ν D =(1) p(Dp)p!p!α μ 1μ pδ ν 1ν p μ 1μ pdx ν 1dx ν D =(1) p(Dp)α \begin{aligned} & \star \star \tfrac{1}{ \color{green} p! } \alpha_{ \color{green} \mu_1 \cdots \mu_p} d x^{ \color{green} \mu_1 } \wedge \cdots \wedge d x^{ \color{green} \mu_p } \\ & = \star \tfrac{1}{ { \color{green} p! } { \color{orange} (D-p)! } } \alpha^{ \color{green} \mu_1 \cdots \mu_p} \iota_{ \color{green} \mu_p} \cdots \iota_{ \color{green} \mu_1} dvol \\ & = \star \tfrac{1}{ { \color{green} p! } { \color{orange} (D-p)! } } \alpha^{ \color{green} \mu_1 \cdots \mu_p } \epsilon_{ { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \mu_{p+1} \cdots \mu_D } } d x^{\color{orange} \mu_{p+1}} \wedge \cdots d x^{ \color{orange} \mu_d} \\ & = \tfrac{1}{ { \color{green} p! } { \color{orange} (D-p)! } { \color{blue} p! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \epsilon^{ { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \mu_{p+1} \cdots \mu_D } } \epsilon_{ { \color{orange} \mu_{p+1} \cdots \mu_D } { \color{blue} \nu_1 \cdots \nu_p } } \, d x^{ \color{blue} \nu_1} \wedge \cdots \wedge d x^{ \color{blue} \nu_D } \\ \\ & = \tfrac{ (-1)^{ {\color{green} p} { \color{orange} (D-p) } } }{ { \color{green} p! } { \color{orange} (D-p)! } { \color{blue} p! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \epsilon^{ { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \mu_{p+1} \cdots \mu_D } } \epsilon_{ { \color{blue} \nu_1 \cdots \nu_p } { \color{orange} \mu_{p+1} \cdots \mu_D } } \, d x^{ \color{blue} \nu_1} \wedge \cdots \wedge d x^{ \color{blue} \nu_D } \\ & = {\color{magenta} -} \tfrac{ (-1)^{ {\color{green}p} {\color{orange} (D-p) } } }{ { \color{green} p! } { \color{blue} p! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \delta^{ { \color{green} \mu_1 \cdots \mu_p } }_{ { \color{blue} \nu_1 \cdots \nu_p } } \, d x^{ \color{blue} \nu_1} \wedge \cdots \wedge d x^{ \color{blue} \nu_D } \\ & = {\color{magenta} -} (-1)^{ {\color{green}p} {\color{orange} (D-p) } } \, \alpha \end{aligned}

Here the sign in the last lines arises from the Minkowski signature via (9).

Proposition

(Laplace operator/wave operator)

Let α=1p!α μ 1μ pdx μ 1dx μ p \alpha = \tfrac{1}{p!} \alpha_{\mu_1 \cdots \mu_p} d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_p} be a differential p-form on DD-dimensional Minkowski spacetime such that

να νμ 1μ p1=0 \partial^\nu \alpha_{\nu \mu_1 \cdots \mu_{p-1}} \;=\; 0

(i.e. Lorenz gauge).

Then the Laplace-Beltrami operator

ddα= ν να \star d \star d \alpha \;=\; { \color{magenta} - } \partial^\nu \partial_\nu \alpha

is the wave operator acting on the components of α\alpha.

Proof

We compute as follows:

dd1p!α μ 1μ pdx μ 1dx μ p =d1p! να μ 1μ pdx νdx μ 1dx μ p =d1p!(D(p+1))! να μ 1μ pϵ νμ 1μ pμ p+2μ Ddx μ p+2dx μ D =1p!(D(p+1))! ν να μ 1μ pϵ νμ 1μ pμ p+2μ Ddx νdx μ p+2dx μ D =1p!(D(p+1))!p! ν να μ 1μ pϵ νμ 1μ pμ p+2μ Dϵ νμ p+2μ Dκ 1κ pdx κ 1dx κ p =1p!p! ν να μ 1μ pδ νκ 1κ p νμ 1μ pdx κ 1dx κ p =1p!p! ν να μ 1μ pδ κ 1κ p μ 1μ pdx κ 1dx κ p = ν να \begin{aligned} & \star d \star d \tfrac{1}{ \color{green} p! } \alpha_{ \color{green} \mu_1 \cdots \mu_p } d x^{ \color{green} \mu_1 } \wedge \cdots \wedge d x^{ \color{green} \mu_p } \\ & = \star d \star \tfrac{1}{ \color{green} p! } \partial_{ \color{magenta} \nu } \alpha_{ \color{green} \mu_1 \cdots \mu_p } d x^{ \color{magenta} \nu } \wedge d x^{ \color{green} \mu_1 } \wedge \cdots \wedge d x^{ \color{green} \mu_p } \\ & = \star d \tfrac{1}{ { \color{green} p! } { \color{orange} (D-(p+1))! } } \partial^{ \color{magenta} \nu } \alpha^{ \color{green} \mu_1 \cdots \mu_p } \epsilon_{ { \color{magenta} \nu } { \color{green} \mu_1 \cdots \mu_{p} } { \color{orange} \mu_{p+2} \cdots \mu_D } } \, d x^{ \color{orange} \mu_{p+2} } \wedge \cdots \wedge d x^{ \color{orange} \mu_D } \\ & = \star \tfrac{1}{ { \color{green} p! } { \color{orange} (D-(p+1))! } } \partial_{ \color{red} \nu' } \partial^{ \color{magenta} \nu } \alpha^{ \color{green} \mu_1 \cdots \mu_p } \epsilon_{ { \color{magenta} \nu } { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \mu_{p+2} \cdots \mu_D } } \, d x^{ \color{red} \nu' } \wedge d x^{ \color{orange} \mu_{p+2} } \wedge \cdots \wedge d x^{ \color{orange} \mu_D } \\ & = \tfrac{1}{ { \color{green} p! } { \color{orange} (D-(p+1))! } { \color{blue} p! } } \partial^{ \color{red} \nu' } \partial_{ \color{magenta} \nu } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \epsilon^{ { \color{magenta} \nu } { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \mu_{p+2} \cdots \mu_D } } \epsilon_{ { \color{red} \nu' } { \color{orange} \mu_{p+2} \cdots \mu_D } { \color{blue} \kappa_1 \cdots \kappa_p } } \, d x^{\color{blue} \kappa_1} \wedge \cdots d x^{\color{blue}\kappa_p} \\ & = { \color{magenta} - } \tfrac{1}{ { \color{green} p! } { \color{blue} p! } } \partial^{ \color{red} \nu' } \partial_{ \color{magenta} \nu } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \delta^{ { \color{magenta} \nu } { \color{green} \mu_1 \cdots \mu_p } } _{ { \color{red} \nu' } { \color{blue} \kappa_1 \cdots \kappa_p } } \, d x^{\color{blue} \kappa_1} \wedge \cdots d x^{\color{blue}\kappa_p} \\ & = { \color{magenta} - } \tfrac{1}{ { \color{green} p! } { \color{blue} p! } } \partial^{ \color{magenta} \nu } \partial_{ \color{magenta} \nu } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \delta^{ { \color{green} \mu_1 \cdots \mu_p } } _{ { \color{blue} \kappa_1 \cdots \kappa_p } } \, d x^{\color{blue} \kappa_1} \wedge \cdots d x^{\color{blue}\kappa_p} \\ & = { \color{magenta} - } \partial^{ \color{magenta} \nu } \partial_{ \color{magenta} \nu } \alpha \end{aligned}

Here the sign in the last lines arises from the Minkowski signature via (9).

References

Some useful basic formulas are listed in

  • Hodge theory on Riemannian manifolds , lecture notes (pdf)

A unified perspective in terms of Berezin integration:

Discussion in complex geometry includes

  • O. Biquard, A. Höring, Kähler geometry and Hodge theory, 2008 (pdf)

With an eye towards application in supergravity and string theory:

Discussion of the Hodge star operator on supermanifolds (in terms of picture changing operators and integral top-forms for integration over supermanifolds):

Last revised on May 6, 2020 at 09:04:19. See the history of this page for a list of all contributions to it.