Being confused about differential cohomology I

In this post I want to talk about differential cohomologies, which are ``differential refinement’’ of cohomology theories. I am mainly confused because there are two major approaches, one is the classical differential Hexagons, which we will get to, the other is sheaves on spectra, which makes a lot more sense to me.

In this post, Ww are going to focus ordinary differential cohomology. In the classical case we will review Cheegar-Simons, then we will move to Delign cohomology to define sheaves on spectra. A good reference is this.

Baby example I: \(\hatH^1(X; \mbb{Z})\)

Let us first review how differential cohomology is different from ordinary topological cohomology theory. Let \(X\) be a topological space, then the first cohomology group \(H^1(X; \mbb{Z}) = \Hom(H_1(X; \mbb{Z}); \mbb{Z})\) is the set of linear maps from the first homology \(H_1(X; \mbb{Z})\) to \(\mbb{Z}\).

Now let’s consider the differential cohomology counterpart: now we take \(X\) to be a smooth manifold; we define \(\hatH^1(X; \mbb{Z})\) to be smooth maps from \(X\) to \(S^1\). It is clear that there is a map \(\hatH^1(X; \mbb{Z}) \to H^1(X; \mbb{Z})\) given by taking a map \(f \colon X \to S^1\) to the image of the generator \(\nu \in H^1(S^1, \mbb{Z})\) under the pullback map \(f^* \colon H^1(S^1; \mbb{Z}) \to H^1(X; \mbb{Z})\). Explicitly, given a \(1\)-cycle \(\gamma \in X\), then \(f\) restricted to \(\gamma\) is a map from \(S^1 \to S^1\). Then \(f^*(\nu)[\gamma]\) is simply the winding number of this map.

Let us also remark that the map \(\hatH^1(X; \mbb{Z}) \to H^1(X; \mbb{Z})\) is surjective: this follows from the de Rham theorem. Given any class \(a \in H^1(X; \mbb{Z})\), then by de Rham theorem there exists a closed differential form \(\alpha \in \Omega^1(X)\) whose class in de Rham cohomology corresponds to \(a\) in \(H^1(X; \mbb{R})\). By constructions, \(\alpha\) has integral periods. We can define a smooth map \(f \colon X \to S^1\) from \(\alpha\) has follows: let’s say that \(X\) is connected, and we fix \(x \in X\). Then for any \(y \in X\) we take \(f(y) \coloneqq \int_\gamma \alpha \mod \mbb{Z}\), where \(\gamma\) is a path from \(x\) to \(y\). Note that this is well-defined as two different path form a closed loop, whose integral is an integer as \(\alpha\) has integral periods.

It is clear that \(\hatH^1(X; \mbb{Z})\) contains more information then just \(H^1(X; \mbb{Z})\), in particular it contains differential informations: given \(f \in \hatH^1(X; \mbb{Z})\), we have a one-form \(f^*\alpha\) where the one-form \(\alpha \in \Omega^1(S^1)\) is simply flat one \(d_x\), when we write \(S^1\) as \(\mbb{R}/\mbb{Z}\). Moreoever, the de Rham class of \(\alpha\) is equal to \(f^*(\nu)\), viewed as a class in \(H^1(X; \mbb{R})\).

This is a part of the ``differential cohomology hexagon’’:

\[\begin{equation}\label{eq:diff-hex-0} \begin{array}{ccccccccc} & & H^0(X; \mbb{R}/\mbb{Z}) & & & & H^1(X; \mbb{R}) & & \\\ & \nearrow& & \searrow & & \nearrow & & \searrow & \\\ H^0(X; \mbb{R}) & & & & \hatH^1(X; \mbb{Z}) & & & & H^1(X; \mbb{R})\\\ & \searrow& & \nearrow & & \searrow & & \nearrow & \\\ & & \Omega^0(X)/\Omega^0_{\cl}(X)_{\mbb{Z}} & & & & \Omega^1_{\cl}(X)_{\mbb{Z}} & & \end{array} \end{equation}\]

Here \(\Omega^i_{\cl}\) means closed forms, and \(\Omega^i_{\cl}(X)_{\mbb{Z}}\) means closed forms with integral periods. Note that the diagonals are in fact exact, and the top maps is the Bockstein homomorphism (which is \(0\) here but not in general), while the lower map is the de Rham differential. \eqref{eq:diff-hex-0} is called the differential Hexagon, one of the two approaches to diff cohomology we will discuss in this post.

Let us shed light on the other approach too. Let \(\Mfld\) be the category of manifold, which we equipped with the Grothendieck topology consisting of open sets and open covering. We are going to consider sheaves on \(\Mfld\).

Let us view \(\mbb{Z}\) as a discrete abelian group manifold, hence \(\Hom(-, \mbb{Z})\) is a sheaf on the category \(\Mfld\) of manifolds with valued in \(\Ab\). Furthermore, we have sheaves \(\Omega^i\) parametrizing \(i\)-th forms on \(X\). In particular, the sheave \(\Omega^0\) is represented by the abelian group smooth manifold \(\mbb{R}\). The canonical inclusion \(\mbb{Z} \to \mbb{R}\) can be viewed as a map of sheaf. Let \(\mbb{Z}(1)\) denote the chain complex \(\mbb{Z} \to \Omega^0\), with \(\mbb{Z}\) in degree \(0\). Note that it is actually equivalent to \(\Omega \mbb{T}\), where we view \(\mbb{T} = S^1\) toghether with its abelian group structure.

Now its straightforward to define \(\hatH^1(X; \mbb{Z})\), it is simply the value of \(\mbb{T}\) on \(X\), equivalently it is the first hyper-cohomology \(H^1(X; \mbb{Z}(1))\). Later on we will define all cohomology that way.

Baby example II: \(\hatH^2(X; \mbb{Z})\)

Alright, let’s do the second cohomology. Recall that \(H^2(X; \mbb{Z})\) is the isomorphism classes of (topological) line bundles on \(X\). Let us define \(\hatH^2(X; \mbb{Z})\) is the isomorphism classes of smooth line bundles with connections. Once again there is a map \(\hatH^2(X; \mbb{Z}) \to H^2(X; \mbb{Z})\) that forgets the line bundle. Once again, it additionally has differential information, which is \(\hatH^2(X; \mbb{Z}) \to \Omega^2_{\cl}(X)_{\mbb{Z}}\), taking a connection \(\nabla\) to its curvature \(F_{\nabla}\). Furthermore, there is a Hexagon similar to \eqref{eq:diff-hex-0}. I will skip writing it since we are going to do the general case very soon!

Moving on to the sheaf perspective. There is actually a lot of options Definition: for \(n \geq 0\), let \(\mbb{Z}(n)\) be the sheaves of chain complex \begin{equation}\label{eq:Deligne} \mbb{Z} \to \Omega^0 \to \Omega^1 \to \cdots \to \Omega^{n-1} \end{equation} Note that \(\mbb{Z}(0) = \mbb{Z}\) and there are maps \(\mbb{Z}(n) \to \mbb{Z}(n-1)\).

Let us consider their second cohomology:

1. \(H^2(X; \mbb{Z})\) is the ordinary topological cohomology.
2. \(H^2(X; \mbb{Z}(1))\) is the group of isomorphism classes of smooth \(\mbb{T}\) bundles.
3. \(H^2(X; \mbb{Z}(2))\) is \(\hatH^2(X; \mbb{Z})\).

Well, what does \(H^2(X; \mbb{Z}(n))\) represent in general? First note that \(H^n(X; \mbb{Z}(n+1)) \simeq H^n(X; \mbb{Z}(n+1+k))\) for any \(k \geq 0\). This is because those higher differential forms does not matter. Secondly, they are all equivalent to \(H^N(X; \mbb{R}/\mbb{Z}_{\disc})\). This is because by de Rham theorem, the chain complex \(\Omega^0 \to \cdots \to \Omega^n \to \cdots\) is equivalent to the discrete sheaf \(\mbb{R}_{\disc}\), where we view \(\mbb{R}\) as a discrete group manifold. Thus \(\mbb{Z}(\infty) \simeq \mbb{Z} \to \mbb{R}_{\disc} \simeq 0 \to \mbb{T}_{\disc}\). Therefore we see that \(H^2(X; \mbb{Z}(n))\) is \(H^1(X; \mbb{T}_{\disc})\), which is equivalent to line bundles with \emph{flat} connections! This is easy to see if we consider the hyper-Cech-cohomology, where we are asking for \(d(\alpha_i) = 0\) for each local \(1\)-form \(\alpha_i\) on the open set \(U_i\) of \(X\).

Note that the map \(H^1(X; \mbb{R}/\mbb{Z}_{\disc}) = H^2(X; \mbb{Z}(3)) \to H^2(X; \mbb{Z}(2))\) appears in the Hexagon axioms. cite[Yamashita-Yonekura, 1.3], cite[Cheeger-Simons]

Cheegar-Simons differential group

Let us now review the Cheegar-Simons differential character group: Cheeger-Simons differential character: Let \(X\) be a manifold, then \(\hatH^n_{\CS}\) consists of pairs \((\omega, k)\), where

• \(\omega\) is a closed form \(\omega \in \Omega^n_{\cl}(X)\),
• \(k\) is a group homomorphism \(k \colon Z_{\infty, n-1}(X; \mbb{Z}) \to \mbb{R}/\mbb{Z}\),
• \(\omega\) and \(k\) satisfy the following compatibility condition: for any \(c \in C_{\infty, n}(X; \mbb{Z})\), we have \begin{equation} k(\partial c) = <c, \omega>_{X} \mod \mbb{Z}. \end{equation}

Here \(Z_{\infty, *}\) and \(C_{\infty, *}\) is the groups of smooth singular cycles and chains. Let’s check that \(\hatH^n_{\CS}\) agrees with our \(\hatH^1\) and \(\hatH^2\): for \(n = 1\), given a map \(f \colon X \to S^1\), then the associated Cheegar-Simon character is \((\omega = f^* \alpha, a \mapsto \sum_{x \in a} f(x))\). As for the other way, \(f (x)\) is simply just \(a(\{x\})\). The compatibility with \(\omega\) implies that \(f\) is smooth.

Similary, for \(n = 2\), then given a connection the associated Cheegar-Simon character is \((d\nabla, \gamma \mapsto \mathrm{hol}_{\nabla}\gamma)\).

Let’s check that the Cheegar-Simons character satsfies the differential Hexagon: \(\begin{equation}\label{eq:diff-hex-n} \begin{array}{ccccccccc} & & H^{n-1}(X; \mbb{R}/\mbb{Z}) & & & & H^n(X; \mbb{R}) & & \\\ & \nearrow& & \searrow & & \nearrow & & \searrow & \\\ H^{n-1}(X; \mbb{R}) & & & & \hatH^n(X; \mbb{Z}) & & & & H^n(X; \mbb{R})\\\ & \searrow& & \nearrow & & \searrow & & \nearrow & \\\ & & \Omega^{n-1}(X)/\Omega^{n-1}_{\cl}(X)_{\mbb{Z}} & & & & \Omega^{n}_{\cl}(X)_{\mbb{Z}} & & \end{array} \end{equation}\) Let’s describe the maps:

• \(H^{n-1}(X; \mbb{R}/\mbb{Z}) \to \hatH^n(X; \mbb{Z})\) is given by \(\alpha \mapsto (0, c \mapsto <c, \alpha>_{X})\).
• \(\Omega^{n-1}(X) \to \hatH^n(X; \mbb{Z})\) is given by \(\beta \mapsto (d\beta, c \mapsto \int_{c}\beta \mod \mbb{Z})\).
• \(\hatH^n(X; \mbb{Z}) \to \Omega^n_{\cl}(X)_{\mbb{Z}}\) is given by \((\omega, h) \mapsto \omega\).
• \(\hatH^n(X; \mbb{Z}) \to\hatH^n(X; \mbb{Z})\) is given by \((\omega, h) \mapsto (c \mapsto \int_c \omega - \tilde{h}(\partial c))\). Here \(\tilde{h}\) is a lift of \(h\) to a morphism \(Z_{\infty, n-1}(X) \to \mbb{R}\). Note that such map always exist as \(Z_{\infty, n-1}(X)\) is free. It is straightforward to check that this is closed, and a different choice of \(\tilde{h}\) differ by an exact cycle.

In fact, there is a statement that map that satisfies the differential Hexagon are isomorphic to each other. I think its Simons-Sullivan.

Deligne cohomology

We have already met Deligne cohomologies, they are the \(\mbb{Z}(n)\) groups before. The important thing is the the top degree recovers differential cohomology: Theorem : Let \(X\) be a manifold, then there exists an equivalence: \begin{equation} H^n(X; \mbb{Z}(n)) \simeq \hatH^n_{\CS}(X; \mbb{Z}). \end{equation}

However, we see that there is a vast family of cohomology in between \(H^n(X; \mbb{Z})\) and \(H^n(X; \mbb{Z}(n))\). It turns out that the intermediate cohomology groups are also interesting, in particular when evaluating not at a manifold \(X\), but rather a stack. See this paper that I wrote with Arun and Christoph a while back for how middle dimension cohomology groups are related to the family of Virasoro central extensions.

We end the blog with the following question: Question : why does cohomology in \(\mbb{Z}(n)\) satisfies the differential Hexagon axioms \eqref{eq:diff-hex-n}?

Appearantly this is answered in this paper, which we will visit next time. Hopefully we will also check out some generalized differential cohomology theories, such as differential \(KO, K\) and bordism theories.

References:

• https://arxiv.org/abs/2109.12250.
• https://math.mit.edu/juvitop/pastseminars/notes_2019_Fall/cheeger-simons.pdf.
• https://arxiv.org/abs/math/0701077