# nLab Hodge star operator

Contents

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

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

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 $(X,g)$, the Hodge star operator “completes” a $k$-differential form to the volume form of $(X,g)$.

## Definition

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

### Hodge inner product

The metric $g$ naturally induces a nondegenerate symmetric bilinear form

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

If $X$ is compact then the integral of this against the volume form $vol_g$ exists. This is the Hodge inner product

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

### Hodge star operator

The Hodge star operator is the unique linear function

${\star}\colon \Omega^k (X) \to \Omega^{n-k} (X)$

defined by the identity

$\alpha \wedge \star\beta = (\alpha \mid \beta) vol_g, \qquad \forall \alpha,\beta \in \textstyle{\bigwedge^k} X \,,$

where $vol_g \in \Omega^n X$ is the volume form induced by $g$.

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

$\langle \alpha , \beta\rangle = \textstyle{\int_X} \alpha \wedge \star \beta \,.$

### Generalizations

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

## Properties

### Component expression

Let $X$ be a (pseudo-)Riemannian manifold of dimension $D$, and locally, on some open subset $U \subset X$, let

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

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

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

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

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

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

(1)\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

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

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

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

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

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

### Basic properties

Let $(X,g)$ be a (pseudo-) Riemannian manifold of dimension $D$ and let $\omega,\lambda \in \Omega^k(X)$. Then the following holds:

(2)$\star(\star\omega) \;=\; \left\{ \begin{array}{rc} (-1)^{k(D+1)} \omega = (-1)^{k(D-k)} \omega & \text{(Riemannian)} \\ -(-1)^{k(D+1)} \omega = - (-1)^{k(D-k)} \omega & \text{(pseudo-Riemannian)} \end{array} \right.$
(3)$\langle\star\omega , \star\lambda\rangle \;=\; \langle \omega | \lambda \rangle$
(4)$\star 1 = dvol \,,$

where $dvol$ denotes the volume form.

## Examples

### Hodge star operator on a Kähler manifold

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

$\star \;\colon\; \Omega^{p,q}(X) \longrightarrow \Omega^{n-q,n-p}(X) \,.$

(notice the exchange of the role of $p$ and $q$). 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 $D = d+1$-dimensional Minkowski spacetime.

#### Conventions

We use Einstein summation convention throughout. With this convention, a generic differential p-form reads

$\alpha \;=\; \tfrac{1}{p!} \alpha_{ \color{green} \mu_1 \cdots \mu_p } d x^{ \color{green} \mu_1} \wedge \cdots \wedge d x^{\color{green} \mu_p} \,.$

Here $p! \coloneqq 1 \cdot 2 \cdot 3 \cdots p \,\in \mathbb{N} \subset \mathbb{R}$ denotes the factorial of $p \in \mathbb{N}$.

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

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

We normalize the Levi-Civita symbol as

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

which means that

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

We normalize the sign of the volume form as

(7)\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)$\delta^{ \mu_1 \cdots \mu_p }_{ \nu_1 \cdots \nu_p } \;\coloneqq\; \left\{ \array{ sgn(\sigma) &\vert& \underset{ \sigma \in Sym(p) }{\exists} \left( \underset{1 \leq i \leq p}{\forall} \left( \nu_{\sigma(i)} = \mu_i \right) \right) \\ 0 &\vert& \text{otherwise} } \right.$

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)$\epsilon_{ { \color{green} \mu_1 \cdots \mu_p } {\color{blue} \mu_{p+1} \cdots \mu_{D} } } \epsilon^{ { \color{orange} \nu_1 \cdots \nu_p } { \color{blue} \mu_{p+1} \cdots \mu_D } } \;=\; { \color{magenta} - } (D-p)! \; \delta_{ \color{green} \mu_1 \cdots \mu_p }^{ \color{orange} \nu_1 \cdots \nu_p }$

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

#### Definition

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

$\big\{ \iota_\mu, d x^\nu \wedge \big\} \;=\; \delta_\mu^\nu$

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

(10)$\alpha^{ \color{green} \mu_1 \cdots \mu_p } \iota_{\color{green} \mu_p} \cdots \iota_{ \color{green} \mu_1} dvol \;=\; \epsilon_{ { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \nu_1 \cdots \nu_{(D-p)} } } d x^{\color{orange} \nu_1} \wedge \cdots \wedge d x^{\color{orange} \nu_{(D-p)}}$
###### Definition

For a differential p-form

$\alpha \;\coloneqq\; \tfrac{1}{ \color{green} p! } \alpha_{ \color{green} \mu_1 \cdots \mu_p} d x^{ \color{green} \mu_1 } \wedge \cdots \wedge d x^{ \color{green} \mu_p }$

its Hodge dual is:

(11)\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 $\alpha \;\coloneqq\; \tfrac{1}{p!} \alpha_{\mu_1 \cdots \mu_p} d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_p}$ on $D$-dimensional Minkowski spacetime its wedge product with its Hodge dual (11) is

(12)$\alpha \wedge \star \alpha \;=\; \tfrac{ \color{magenta} -1 }{ { p! } } \alpha_{ \mu_1 \cdots \mu_p } \alpha^{ \mu_1 \cdots \mu_p } \, dvol \,.$
###### Proof

We compute as follows:

\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 $\alpha \;=\; \tfrac{1}{p!} \alpha_{\mu_1 \cdots \mu_p} d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_p}$ on $D$-dimensional Minkowski spacetime, its double Hodge dual (11) is

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

We compute as follows:

\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 $\alpha = \tfrac{1}{p!} \alpha_{\mu_1 \cdots \mu_p} d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_p}$ be a differential p-form on $D$-dimensional Minkowski spacetime such that

$\partial^\nu \alpha_{\nu \mu_1 \cdots \mu_{p-1}} \;=\; 0$

(i.e. Lorenz gauge).

Then the Laplace-Beltrami operator

$\star d \star d \alpha \;=\; { \color{magenta} - } \partial^\nu \partial_\nu \alpha$

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

###### Proof

We compute as follows:

\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:

A unified perspective in terms of Berezin integration:

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:

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 June 28, 2024 at 11:48:29. See the history of this page for a list of all contributions to it.