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 0≤k≤n0 \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)→ℝ \langle - , - \rangle \;\colon\; \Omega^k(X)\otimes \Omega^k(X) \to \mathbb{R}

⟨α,β⟩:=∫ X(α∣β)vol. \langle \alpha, \beta \rangle := \textstyle{\int_X} (\alpha\mid \beta) vol \,.

Hodge star operator

The Hodge star operator is the unique linear function

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

defined by the identity

α∧⋆β=(α∣β)vol g,∀α,β∈⋀ kX, \alpha \wedge \star\beta = (\alpha \mid \beta) vol_g, \qquad \forall \alpha,\beta \in \textstyle{\bigwedge^k} X \,,

where vol g∈Ω 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⟩=∫ Xα∧⋆β. \langle \alpha , \beta\rangle = \textstyle{\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 (n−k)(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 (n−k)(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 (n−k)(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 U⊂XU \subset X, let

e 1,…,e D∈Ω 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 i≔dx ie^i \coloneqq d x^i is such a frame.

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

α=1p!α i 1…i pe i 1∧⋯∧e 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 1⋯i 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!(D−p)!|det((g ij))|α j 1…j pg j 1i 1⋯g j pi pϵ i 1…i pi p+1⋯i De i p+1∧⋯∧e i D =1p!(D−p)!|det((g ij))|α i 1…i pϵ i 1…i pi p+1⋯i De i p+1∧⋯∧e 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 } {\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!, (D−p)!(D-p)! are the factorials of pp and (D−p)(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 i⊗e 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 (pseudo-) Riemannian manifold of dimension DD and let ω,λ∈Ω k(X)\omega,\lambda \in \Omega^k(X). Then the following holds:

(2)⋆(⋆ω)={(−1) k(D+1)ω=(−1) k(D−k)ω (Riemannian) −(−1) k(D+1)ω=−(−1) k(D−k)ω (pseudo-Riemannian) \star(\star\omega) \;=\; \left\{ \begin{array}{rc} (-1)^{k(D+1)} \omega = (-1)^{k(D-k)} \omega & \text{(Riemannian)} \\ -(-1)^{k(D+1)} \omega = - (-1)^{k(D-k)} \omega & \text{(pseudo-Riemannian)} \end{array} \right.

(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.

(e.g. Frankel 1997 (14.9), (14.5); Padmanabhan 2010 (11.61))

Examples

Hodge star operator on a Kähler manifold

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

⋆:Ω p,q(X)⟶Ω n−q,n−p(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. With this convention, a generic differential p-form reads

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

Here p!≔1⋅2⋅3⋯p∈ℕ⊂ℝp! \coloneqq 1 \cdot 2 \cdot 3 \cdots p \,\in \mathbb{N} \subset \mathbb{R} denotes the factorial of p∈ℕp \in \mathbb{N}.

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 012⋯d≔+1 \epsilon_{0 1 2 \cdots d} \;\coloneqq\; + 1

which means that

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

We normalize the sign of the volume form as

(7)dvol ≔dx 0∧dx 1∧⋯∧dx d =1D!ϵ μ 1⋯μ Ddx μ 1∧⋯∧dx 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 1⋯ν p μ 1⋯μ p≔{sgn(σ) | ∃σ∈Sym(p)(∀1≤i≤p(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 1⋯μ pμ p+1⋯μ Dϵ ν 1⋯ν pμ p+1⋯μ D=−(D−p)!δ μ 1⋯μ p ν 1⋯ν 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 1⋯μ pι μ p⋯ι μ 1dvol=ϵ μ 1⋯μ pν 1⋯ν (D−p)dx ν 1∧⋯∧dx ν (D−p) \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

α≔1p!α μ 1⋯μ pdx μ 1∧⋯∧dx 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!(D−p)!α μ 1⋯μ pι μ p⋯ι μ 1dvol =1p!(D−p)!α μ 1⋯μ pϵ μ 1⋯μ pμ p+1⋯μ Ddx μ p+1∧⋯∧dx 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 α≔1p!α μ 1⋯μ pdx μ 1∧⋯∧dx 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)α∧⋆α=−1p!α μ 1⋯μ pα μ 1⋯μ pdvol. \alpha \wedge \star \alpha \;=\; \tfrac{ \color{magenta} -1 }{ { p! } } \alpha_{ \mu_1 \cdots \mu_p } \alpha^{ \mu_1 \cdots \mu_p } \, dvol \,.

Proof

We compute as follows:

α∧⋆α =1p!p!(D−p)!α μ 1⋯μ pdx μ 1∧⋯∧dx μ p∧α ν 1⋯ν pι ν p⋯ι ν 1dvol =1p!p!(D−p)!α μ 1⋯μ pα ν 1⋯ν pϵ ν 1⋯ν pν p+1⋯ν Ddx μ p∧⋯∧dx μ 1∧dx ν p+1∧⋯dx ν D =1p!p!(D−p)!α μ 1⋯μ pα ν 1⋯ν pϵ ν 1⋯ν pν p+1⋯ν Dϵ μ p⋯μ 1ν p+1⋯ν Ddvol =−1p!p!α μ 1⋯μ pα ν 1⋯ν pδ ν 1⋯ν p μ 1⋯μ pdvol =−1p!α μ 1⋯μ pα μ 1⋯μ 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 \\ & = \tfrac{ \color{magenta} -1 }{ { \color{green} p! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \alpha^{ \color{green} \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 1⋯μ pdx μ 1∧⋯∧dx 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(D−p)α. \star \star \alpha \;=\; {\color{magenta} -} (-1)^{ p (D - p) } \, \alpha \,.

Proof

We compute as follows:

⋆⋆1p!α μ 1⋯μ pdx μ 1∧⋯∧dx μ p =⋆1p!(D−p)!α μ 1⋯μ pι μ p⋯ι μ 1dvol =⋆1p!(D−p)!α μ 1⋯μ pϵ μ 1⋯μ pμ p+1⋯μ Ddx μ p+1∧⋯dx μ d =1p!(D−p)!p!α μ 1⋯μ pϵ μ 1⋯μ pμ p+1⋯μ Dϵ μ p+1⋯μ Dν 1⋯ν pdx ν 1∧⋯∧dx ν D =(−1) p(D−p)p!(D−p)!p!α μ 1⋯μ pϵ μ 1⋯μ pμ p+1⋯μ Dϵ ν 1⋯ν pμ p+1⋯μ Ddx ν 1∧⋯∧dx ν D =−(−1) p(D−p)p!p!α μ 1⋯μ pδ ν 1⋯ν p μ 1⋯μ pdx ν 1∧⋯∧dx ν D =−(−1) p(D−p)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).

Proof

We compute as follows:

⋆d⋆d1p!α μ 1⋯μ pdx μ 1∧⋯∧dx μ p =⋆d⋆1p!∂ να μ 1⋯μ pdx ν∧dx μ 1∧⋯∧dx μ p =⋆d1p!(D−(p+1))!∂ να μ 1⋯μ pϵ νμ 1⋯μ pμ p+2⋯μ Ddx μ p+2∧⋯∧dx μ D =⋆1p!(D−(p+1))!∂ ν′∂ να μ 1⋯μ pϵ νμ 1⋯μ pμ p+2⋯μ Ddx ν′∧dx μ p+2∧⋯∧dx μ D =1p!(D−(p+1))!p!∂ ν′∂ να μ 1⋯μ pϵ νμ 1⋯μ pμ p+2⋯μ Dϵ ν′μ p+2⋯μ Dκ 1⋯κ pdx κ 1∧⋯dx κ p =−1p!p!∂ ν′∂ να μ 1⋯μ pδ ν′κ 1⋯κ p νμ 1⋯μ pdx κ 1∧⋯dx κ p =−1p!p!∂ ν∂ να μ 1⋯μ pδ κ 1⋯κ p μ 1⋯μ pdx κ 1∧⋯dx 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

Lecture notes:

Hodge theory on Riemannian manifolds , lecture notes (pdf)

Textbook accounts:

Theodore Frankel, §14.1a in: The Geometry of Physics - An Introduction, Cambridge University Press (1997, 2004, 2012) [doi:10.1017/CBO9781139061377]
Thanu Padmanabhan, §11.5 in: Thermodynamical Aspects of Gravity: New insights, Rep. Prog. Phys. 73 (2010) 046901 [arXiv:0911.5004]

A unified perspective in terms of Berezin integration:

Leonardo Castellani, Roberto Catenacci, Pietro Antonio Grassi, The Hodge Operator Revisited (arXiv:1511.05105)

Discussion in complex geometry:

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

With an eye towards application in supergravity and string theory:

Igor Bandos, Alexei Nurmagambetov, Dmitri Sorokin, Appendix A of Various Faces of Type IIA Supergravity, Nucl. Phys. B676 (2004) 189-228 (arXiv:hep-th/0307153)

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

Leonardo Castellani, Roberto Catenacci, Pietro Antonio Grassi, Hodge Dualities on Supermanifolds, Nuclear Physics B Volume 899, October 2015, Pages 570-593 (arXiv:1507.01421)

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