Madhusudhan Raman

Here we discuss how to construct forms of higher degree.

Higher-Degree Forms

We may generalise the notion of a $ 1 $-form as follows: just as a $ 1 $-form “eats” a vector and spits out a scalar, a $p$-form $\alpha$ eats a collection of $p$ tangent vectors and returns a number: \begin{equation} \alpha(X _ {1},\cdots,X _ {p})\in\mathbb{R} \ . \end{equation} This map is linear in every argument and antisymmetric under the exchange of arguments: \begin{equation} \alpha(\cdots,X _ {i},\cdots,X _ {j},\cdots) =-\alpha(\cdots,X _ {j},\cdots,X _ {i},\cdots) \ . \end{equation} It follows immediately that a form vanishes whenever two of its arguments coincide. The space of smooth $p$-forms on $M$ is denoted $\Omega^{p}(M)$. Functions may be regarded as $ 0 $-forms, so that $\Omega^{0}(M)$ is simply the space of smooth functions on $M$.

In a coordinate basis, a $p$-form can be written as \begin{equation} \alpha=\frac{1}{p!}\alpha _ {\mathsf{a} _ {1}\cdots\mathsf{a} _ {p}} \,\mathrm{d}y^{\mathsf{a} _ {1}}\wedge\cdots\wedge \mathrm{d}y^{\mathsf{a} _ {p}} \ , \label{eq:p-form-components} \end{equation} where the components $\alpha _ {\mathsf{a} _ {1}\cdots\mathsf{a} _ {p}}$ are completely antisymmetric. The symbol $\wedge$ denotes the exterior or wedge product. For a $p$-form $\alpha$ and a $q$-form $\beta$, their wedge product is a $(p+q)$-form that satisfies a graded version of commutativity: \begin{equation} \alpha\wedge\beta=(-1)^{pq}\beta\wedge\alpha \ . \label{eq:graded-commutativity} \end{equation}

For example, $\mathrm{d}y^{\mathsf{a}}\wedge\mathrm{d}y^{\mathsf{b}}=-\mathrm{d}y^{\mathsf{b}}\wedge\mathrm{d}y^{\mathsf{a}}$, and consequently $\mathrm{d}y^{\mathsf{a}}\wedge\mathrm{d}y^{\mathsf{a}}=0$. More generally, forms of odd degree anticommute, forms of even degree commute, and forms of odd and even degree commute with each other. This should be reminiscent of bosonic and fermionic operators and is not a coincidence! Perhaps we will develop this correspondence later in this series of notes.

For two one-forms $\alpha$ and $\beta$, the definition of the wedge product is equivelant to: \begin{equation} (\alpha\wedge\beta)(X,Y) =\alpha(X)\beta(Y)-\alpha(Y)\beta(X) \ , \end{equation} and the definition for forms of higher degree is the corresponding completely antisymmetrised product on the right-hand side. On an $n$-dimensional manifold, it follows from the definition that there are no forms of degree greater than $n$.

Under a change of coordinates, the components of a $p$-form transform as \begin{equation} \alpha’ _ {\mathsf{a} _ {1}\cdots\mathsf{a} _ {p}} =\frac{\partial y^{\mathsf{b} _ {1}}}{\partial y’^{\mathsf{a} _ {1}}} \cdots \frac{\partial y^{\mathsf{b} _ {p}}}{\partial y’^{\mathsf{a} _ {p}}} \alpha _ {\mathsf{b} _ {1}\cdots\mathsf{b} _ {p}} \ . \end{equation} Thus, the components of a differential form make up a completely antisymmetric tensor field. Depending on the context, we will sometimes find it more convenient to discuss a form in terms of its components. The component expression in \eqref{eq:p-form-components} is often more useful for calculations, but it can occasionally obscure the underlying geometric structures. The index-free notation often offers structural clarity. For example, if \begin{equation} \alpha=\frac{1}{2}\alpha _ {\mathsf{ab}}\, \mathrm{d}y^{\mathsf{a}}\wedge\mathrm{d}y^{\mathsf{b}} \ , \end{equation} then \begin{equation} \alpha(X,Y)=\alpha _ {\mathsf{ab}}X^{\mathsf{a}}Y^{\mathsf{b}} \ . \end{equation} The two descriptions contain precisely the same information. We will move freely between them, using indices when they make a calculation transparent and suppressing them when the geometric structure is clearer without them.

The pullback introduced earlier extends to forms of arbitrary degree. If $\alpha\in\Omega^{p}(N)$, then \begin{equation} (\Phi^{\star}\alpha)(X _ {1},\cdots,X _ {p}) =\alpha(\Phi _ {\star}X _ {1},\cdots,\Phi _ {\star}X _ {p}) \ . \label{eq:pullback-p-form} \end{equation} Additionally, pullbacks act as a homomorphism on wedge products: \begin{equation} \Phi^{\star}(\alpha\wedge\beta) =\Phi^{\star}\alpha\wedge\Phi^{\star}\beta \ . \label{eq:pullback-wedge} \end{equation} This identity will, later, ensure that preserving the symplectic form also preserves the volume form constructed from it.

Exterior Derivatives

In an earlier section, we saw that the differential $ \text{d}f $ of a function $ f $ is a $ 1 $-form, and that $ f $ can be thought of as a $ 0 $-form, i.e. a smooth function on the manifold. It is natural to ask, then: does this generalise to forms of higher degree? We define the exterior derivative $ \text{d} $ that sends $ p $-forms to $ (p+1) $-forms: \begin{equation} \mathrm{d}:\Omega^{p}(M)\longrightarrow\Omega^{p+1}(M) \ , \end{equation} If $\alpha$ is the $p$-form in \eqref{eq:p-form-components}, then \begin{equation} \mathrm{d}\alpha =\frac{1}{p!}\partial _ {\mathsf{b}} \alpha _ {\mathsf{a} _ {1}\cdots\mathsf{a} _ {p}} \,\mathrm{d}y^{\mathsf{b}}\wedge \mathrm{d}y^{\mathsf{a} _ {1}}\wedge\cdots\wedge \mathrm{d}y^{\mathsf{a} _ {p}} \ . \label{eq:exterior-derivative-components} \end{equation}

This exterior derivative is nilpotent: \begin{equation} \mathrm{d}^{2}=0 \ . \label{eq:d-squared-zero} \end{equation} Additionally, it obeys a graded Leibniz rule. If $\alpha\in\Omega^{p}(M)$, then \begin{equation} \mathrm{d}(\alpha\wedge\beta) =\mathrm{d}\alpha\wedge\beta +(-1)^{p}\alpha\wedge\mathrm{d}\beta \ . \label{eq:d-leibniz} \end{equation} It helps to think of the exterior derivative as “jumping over” forms when writing down the Liebniz rule. Whenever you jump over an odd-degree form, you pick up a sign, and when you jump over an even-degree form, you don’t pick up a sign.

A form $\alpha$ satisfying $\mathrm{d}\alpha=0$ is called closed, while one that can be written as $\alpha=\mathrm{d}\beta$ is called exact. Equation \eqref{eq:d-squared-zero} tells us that every exact form is closed. The converse is always true locally, but need not be true globally. This kind of obstruction is typically a signal of some non-trivial topology on $ M $.

Despite the appearance of partial derivatives in \eqref{eq:exterior-derivative-components}, this definition is independent of coordinates. One way to see this is to observe how this definition interacts with the pullback. We find: \begin{equation} \Phi^{\star}(\mathrm{d}\alpha) =\mathrm{d}(\Phi^{\star}\alpha) \ . \label{eq:pullback-exterior-derivative} \end{equation} Therefore, a form that is closed or exact in one system of coordinates, remains so in every other.

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