Flat connections and cohomology invariants

TL;DR

This paper develops geometric invariants for the topology of flat connection spaces on principal bundles, using a generalized Chern-Weil approach that applies to both real and complex manifolds, revealing new gauge-invariant cohomological structures.

Contribution

It introduces a novel method to define cohomological invariants for flat connections, extending classical characteristic class theory to the moduli space of flat connections and complex manifolds.

Findings

Constructed gauge-invariant homomorphisms to cohomology groups

Extended invariants to complex manifolds using Dolbeault cohomology

Provided examples demonstrating applications of the invariants

Abstract

The main goal of this article is to construct some geometric invariants for the topology of the set $\mathcal{F}$ of flat connections on a principal $G$-bundle $P\,\longrightarrow\, M$. Although the characteristic classes of principal bundles are trivial when $\mathcal{F}\neq \emptyset$, their classical Chern-Weil construction can still be exploited to define a homomorphism from the set of homology classes of maps $S\longrightarrow \mathcal{F}$ to the cohomology group $H^{2r-k}(M,\mathbb{R})$, where $S$ is null-cobordant $(k-1)$-manifold, once a $G$-invariant polynomial $p$ of degree $r$ on $\text{Lie}(G)$ is fixed. For $S=S^{k-1}$, this gives a homomorphism $\pi_{k-1}(\mathcal{F})\longrightarrow H^{2r-k}(M,\mathbb{R})$. The map is shown to be globally gauge invariant and furthermore it descends to the moduli space of flat connections $\mathcal{F}/\mathrm{Gau}P$, modulo cohomology with integer coefficients. The construction is also adapted to complex manifolds. In this case, one works with the set $\mathcal{F}^{0,2}$ of connections with vanishing $(0,2)$-part of the curvature, and the Dolbeault cohomology. Some examples and applications are presented.

Equations

$$\mathrm{Map}(S,\mathcal{F})⟶H^{2r-k}(M,\mathbb{R})$$

$$\{\Pi \circ f\in \mathrm{Map}(S,\mathcal{F}/\mathrm{Gau}P)\mid f\in \mathrm{Map}(S,\mathcal{F})\},$$

$$d\pi :TP⟶{\pi }^{\ast }TM$$

$${\pi }_{\mathrm{ad}}:\mathrm{ad}P:=P{\times }^G\mathfrak{g}⟶M$$

$$0⟶\mathrm{ad}P⟶\text{At}(P)\stackrel{d\pi }{⟶}TM⟶0$$

$$\text{At}(P)\otimes T^{\ast }M\stackrel{d\pi \otimes \mathrm{Id}}{⟶}TM\otimes T^{\ast }M=\mathrm{End}(TM),$$

$$q:C⟶M$$

$$\mathcal{A}=\Gamma (M,C),$$

$${\tilde{B}}_{(x,g)}={\frac{d}{dt}|}_{t=0}(x,\exp (tB)g).$$

$$\frac{\partial }{\partial x^i}⟼\frac{\partial }{\partial x^i}-A_i^{\alpha }{\tilde{B}}_{\alpha }.$$

$$A^{\ast }F\in {\Omega }^2(M,A^{\ast }{q}^{\ast }\mathrm{ad}P)={\Omega }^2(M,\mathrm{ad}P)$$

$$F=(d{A}_i^{\alpha }\wedge d{x}^i+\frac{1}{2}{c}_{\beta \gamma }^{\alpha }{A}_i^{\beta }{A}_j^{\gamma }d{x}^{i}d{x}^j){\tilde{B}}_{\alpha },$$

$$p(F)=p(F,\stackrel{(r)}{\cdots },F)\in {\Omega }^{2r}(C).$$

$${\Phi }_C:C⟶C.$$

$${\Omega }^{\bullet }(C{)}^{\mathrm{Gau}P}={\Omega }^{\bullet }(M)[p_1(F),\cdots ,p_l(F)],$$

$${\beta }_k^p\in {\Omega }^k(\mathcal{A},{\Omega }^{2r-k}(M))$$

$${\beta }_k^p(A)({\xi }_1,\cdots ,{\xi }_k)=A^{\ast }(i_{{\xi }_1}\cdots i_{{\xi }_k}p(F)),$$

$${\beta }_k^p(A)({\xi }_1,\cdots ,{\xi }_k)=N_{r,k}p({\xi }_1,\cdots ,{\xi }_k,F^A,\stackrel{(r-k)}{\cdots },F^A),$$

$$i_{{\xi }_1}F=q^{\ast }{\xi }_1,i_{{\xi }_2}{i}_{{\xi }_1}F=0,$$

$$i_{\xi }p(F)=r\cdot p(q^{\ast }\xi ,F,\stackrel{(r-1)}{\cdots },F)$$

$${\beta }_k^p(A)({\xi }_1,\cdots ,{\xi }_k)$$

$$=A^{\ast }(N_{r,k}p(q^{\ast }{\xi }_1,\cdots ,q^{\ast }{\xi }_k,F,\stackrel{(r-k)}{\cdots },F)$$

$$=N_{r,k}p({\xi }_1,\cdots ,{\xi }_k,F^A,\stackrel{(r-k)}{\cdots },F^A).$$

$${\Phi }_{\mathcal{A}}^{\ast }({\beta }_k^p)={\beta }_k^p,\forall \Phi \in \mathrm{Gau}P,$$

$$({\Phi }_{\mathcal{A}}^{\ast }{\beta }_k^p)(A)({\xi }_1,\cdots ,{\xi }_k)$$

$$={\beta }_k^p({\Phi }_{\mathcal{A}}(A))({\Phi }_{\mathrm{ad}}\circ {\xi }_1,\cdots ,{\Phi }_{\mathrm{ad}}\circ {\xi }_k),$$

$$\Phi :P⟶P.$$

$$({\Phi }_{\mathcal{A}}^{\ast }{\beta }_k^p)(A)({\xi }_1,\cdots ,{\xi }_k)$$

$$=N_{r,k}p({\Phi }_{\mathrm{ad}}\circ {\xi }_1,\cdots ,{\Phi }_{\mathrm{ad}}\circ {\xi }_k,{\Phi }_{\mathrm{ad}}\circ F^A,\stackrel{(r-k)}{\cdots },{\Phi }_{\mathrm{ad}}\circ F^A)$$

$$=N_{r,k}p({\xi }_1,\cdots ,{\xi }_k,F^A,\stackrel{(r-k)}{\cdots },F^A)$$

$$={\beta }_k^p({\xi }_1,\cdots ,{\xi }_k),$$

$$d_{\mathcal{A}}{\beta }_k^p=(r-k)d\circ {\beta }_{k+1}^p$$

$$d_{\mathcal{A}}:{\Omega }^k(\mathcal{A},{\Omega }^{2r-k}(M))⟶{\Omega }^{k+1}(\mathcal{A},{\Omega }^{2r-k}(M))$$

Full text

1. Introduction

The classical Chern-Weil construction of the characteristic classes of a principal $G$-bundle $\pi\,:\,P\,\longrightarrow\,M$ on a manifold $M$ of dimension $n$ can be built in terms of the bundle of connections $C\,\longrightarrow\,M$ associated to $\pi$, that is, the affine bundle over $M$ whose sections over any open subset $U\subset M$ are the connection on the restriction $P|_{U}$. The bundle $\pi^{*}P\,\longrightarrow\,C$ is endowed with a canonical connection so that the evaluation of its curvature $\Omega$ by the $G$-invariant polynomials of $\mathfrak{g}\,=\,\text{Lie}(G)$ provides a sort of universal characteristic form $p(\Omega)$ inducing the characteristic classes under the isomorphism $H^{\bullet}(M)\,\simeq\,H^{\bullet}(C)$. One of the advantages of this approach relies on the fact that the dimension $(1+\dim G)n$ of $C$ is bigger than the dimension $n$ of $M$, so that polynomials $p$ of degree $r$ bigger than $n/2$ still provide forms of potential relevance. For example, in [7], these polynomials are used to define a form of degree $2r-n$ on the space of all connections $\mathcal{A}$. In this article, we construct forms $\beta^{p}_{k}$ of degree $2r-k$, for any $0\leq k\leq r$.

The topological information provided by characteristic classes is trivial if the bundle $\pi$ admits flat connections. However, there are many interesting instances of these special bundles, both in physics and in mathematics: bundles modelling quantum phases as that of the Aharonov-Bohm effect, topological Field Theories, moduli spaces of Yang-Mills solution on Riemann surfaces, stability of vector bundles over algebraic varieties, etcetera. Interestingly, we show that the above mentioned forms $\beta^{p}_{k}$ provide topological information by integrating along submanifolds contained in the subset $\mathcal{F}\,\subset\,\mathcal{A}$ of flat connections when $\mathcal{F}\,\neq\,\emptyset$. More precisely, we construct a map

[TABLE]

for any manifold $S$ of dimension $k-1$ which is trivially cobordant. Furthermore, it is proved that the above map behaves well with respect to homology relation in $\mathrm{Map}(S,\,\mathcal{F})$ and also it is invariant under the gauge group $\mathrm{Gau}P$ acting on the left on $\mathrm{Map}(S,\,\mathcal{F})$. These results are connected with previous articles in the literature (see [9] and [10]) where similar constructions were obtained, but from a different approach and dealing with homology chains in $\mathcal{F}$ and for $S=S^{1}$ respectively. The case of spheres $S=S^{k-1}$ is of special interest as it induces a homomorphism defined on higher homotopy groups $\pi_{k-1}(\mathcal{F})\,\longrightarrow\,H^{2r-k}(M,\,\mathbb{R})$.

The invariance under gauge transformation is not enough to guarantee a well defined object in the moduli space of flat connections $\mathcal{F}/\mathrm{Gau}P$. In fact, the moduli space involves actions of $\Phi_{s}\in\mathrm{Gau}P$ in $\mathcal{F}$ depending on $s\in S$ itself, a situation for which the invariance above does not hold true. Still, we prove that the our main construction descends to a map defined on the set

[TABLE]

where $\Pi:\mathcal{A}\longrightarrow\mathcal{A}/\mathrm{Gau}P$, and taking values in $H^{2r-k}(M,\,\mathbb{R})$ modulo entire cohomology $H^{2r-k}(M,\,\mathbb{Z})$. This construction is consistent with particular cases previously given in the literature (see [10] and [11]).

We also extend the study of these geometric invariants to complex bundles and manifolds. In particular, suitable adaptations of the main objects allow a definition of a map taking values in the Dolbeault cohomology $H^{0,2r-k}(M)$, from homotopy classes of maps in the space of connections with vanishing $(0,2)$-part of the curvature. We remark that this type of connections is of essential interest when studying holomorphic principal connections, so that a bridge between the problem of existence of them with the geometric objects constructed by invariant polynomials is presented.

Roughly, the contents of the paper are organized as follows. Section 2 builds the basic ingredients used in Section 3 to define the invariants induced by maps to the space of flat connections $\mathcal{F}$. Section 4 tackles the case of complex manifolds and the formulation of the geometric invariants in the language of Dolbeault cohomology. Finally, Section 5 is devoted to some particular examples and applications which may serve as a motivation for future further investigation of the invariants herein defined.

All the objects in this article are smooth and the Einstein notation on repeated indices is assumed.

2. Forms in the space of connections $\mathcal{A}$

Let $M$ be a $C^{\infty}$ manifold of dimension $n$, and let $G$ be a Lie group of dimension $m$. The Lie algebra of $G$ will be denoted by $\mathfrak{g}$. Let $\pi\,:\,P\,\longrightarrow\,M$ be a $C^{\infty}$ principal $G$-bundle with

[TABLE]

being the differential of $\pi$. For any $x\,\in\,M$, the fiber $\pi^{-1}(x)$ will be denoted by $P_{x}$. The vector bundle

[TABLE]

associated to $P$ for the adjoint action of $G$ on $\mathfrak{g}$ is known as the adjoint vector bundle of $P$. We note that $\mathrm{ad}P\,=\,(\text{kernel}(d\pi))/G$. Therefore, any $\xi\,\in\,(\mathrm{ad}P)_{x}$, $x\,\in\,M$, produces a vector field along the fiber $\pi^{-1}(x)$ which is preserved by the action of $G$ on $\pi^{-1}(x)$.

Let $\text{At}(P)\,:=\,(TP)/G\,\longrightarrow\,M$ be the Atiyah bundle for $P$. It fits in a short exact sequence

[TABLE]

which is known as the Atiyah exact sequence (see [1]). Consider

[TABLE]

and define $C\,:=\,(d\pi\otimes{\rm Id})^{-1}({\rm Id}_{TM})\,\subset\,\text{At}(P)\otimes T^{*}M$. Clearly,

[TABLE]

is an affine bundle for the vector bundle $(\mathrm{ad}P)\otimes T^{*}M$. This affine bundle $(C\,,q)$ is known as the bundle of connections for $P$. Any section $A\,:\,M\,\longrightarrow\,C$ of $q$ is thus a connection on $P$, that is

[TABLE]

where $\mathcal{A}$ is the space of all connections on $P$. The difference $A_{1}-A_{2}$ of two connections $A_{1},A_{2}\,\in\,\mathcal{A}$ is a $1$-form in $M$ taking values in $\mathrm{ad}P$; note that the kernel of $d\pi\otimes{\rm Id}$ is $\mathrm{ad}P\otimes T^{*}M$. The isomorphism in (2.2) is compatible with the affine space structures of $C$ and $\mathcal{A}$.

Any element of $q^{-1}(x)\,\subset\,C$, $x\,\in\,M$, provides a $G$-invariant homomorphism of vector bundles $P_{x}\times T_{x}M\,\longrightarrow\,(TP)|_{P_{x}}$ over $P_{x}$ whose pre-composition with $d\pi$ is the identity map of $P_{x}\times T_{x}M$; it is a horizontal lift of tangent vectors over $x$. We now consider a coordinate domain $U\,\subset\,M$, with coordinates $(x^{1},\cdots,x^{n})$, and such that $\pi^{-1}(U)\,\simeq\,U\times G$ (after choosing a trivialization of $P|_{U}$). For any $B\,\in\,\mathfrak{g}$, let $\widetilde{B}$ be the vertical $G$-invariant vector field on $P|_{U}\,\simeq\,U\times G$ defined as

[TABLE]

If $(B_{1}\,,\cdots\,,B_{m})$ is a basis of $\mathfrak{g}$, the system $(\widetilde{B}_{1},\cdots,\widetilde{B}_{m})$ is a basis of sections of $\mathrm{ad}P$ considered as a $C^{\infty}(U)$-module. In addition, the horizontal lift given by $A_{x}\,\in\,q^{-1}(x)$ has an expression like

[TABLE]

The functions $(x^{i},A^{\alpha}_{j})$, $i,j\,=\,1,\cdots,n$, $\alpha\,=\,1,\cdots,m$, define a coordinate system on $q^{-1}(U)\,\subset\,C$.

The bundle of connections is equipped with a canonical $2$-form $F$ taking values in the bundle $q^{\ast}\mathrm{ad}P\,\longrightarrow\,C$ called the universal curvature. It is the curvature form of a canonical connection defined on the principal $G$-bundle $q^{\ast}P\,\longrightarrow\,C$ (for example, see [8] as well as [3], [4], [5]) and satisfies the following condition: given a section $A\,\in\,\Gamma(C)$ of the bundle of connections, the pulled back form

[TABLE]

coincides with the curvature $F^{A}$ of the connection $A$. For a coordinate system on $C$ as described above, the expression of the form $F$ is

[TABLE]

where $c_{\beta\gamma}^{\alpha}$ are the structure constants of the basis $(B_{1},\cdots,B_{m})$ of $\mathfrak{g}$.

Let $p\,\in\,S^{r}(\mathfrak{g}^{\ast})^{G}$ be a symmetric polynomial of degree $r$ on $\mathfrak{g}$ which is invariant under the adjoint action of $G$ on $\mathfrak{g}$. We define a $2r$-form in $C$ in the usual way:

[TABLE]

From the Chern-Weil theory we know that the form $p(F)$ is closed. For any $A\,\in\,\Gamma(C)$, the pull-back $A^{\ast}p(F)\,\in\,\Omega^{2r}(M)$ is the characteristic form $p(F^{A})$ defined by the Chern-Weil theory. Note that the homomorphism $q^{*}\,:\,H^{\bullet}(M,\,\mathbb{R})\,\longrightarrow\,H^{\bullet}(C,\,\mathbb{R})$ is an isomorphism. The forms $p(F)$ produce the characteristic classes of the principal $G$-bundle $P$.

A gauge transformation of $P$ is a $G$-equivariant diffeomorphism $\Phi\,:\,P\,\longrightarrow\,P$ such that $\pi\circ\Phi\,=\,\pi$. The set of gauge transformations of $P$, which will be denoted by $\mathrm{Gau}P$, is a group under the composition of maps. This group $\mathrm{Gau}P$ acts on connections so that any $\Phi\,\in\,\mathrm{Gau}P$ defines a transformation $\Phi_{\mathcal{A}}\,:\,\mathcal{A}\,\longrightarrow\,\mathcal{A}$. At the level the bundle of connections, this action also induces an affine isomorphism

[TABLE]

It is easy to see that the characteristic forms $p(F)$ defined above are gauge invariant, that is, $\Phi_{C}^{*}p(F)\,=\,p(F)$, for all $\Phi\,\in\,\mathrm{Gau}P$. It is known that the algebra of gauge invariant forms in $C$ is precisely the ring

[TABLE]

where $p_{1},\cdots,p_{l}$ generate $S^{\bullet}(\mathfrak{g}^{\ast})^{G}$ (see [6]).

For any $p\,\in\,S^{r}(\mathfrak{g}^{\ast})^{G}$ and $0\,\leq\,k\,\leq\,2r$, we construct a $k$-form

[TABLE]

on $\mathcal{A}$ (defined in (2.2)) taking values in the vector space $\Omega^{2r-k}(M)$ as follows:

[TABLE]

for $A\,\in\,\mathcal{A}\,=\,\Gamma(C)$, and $\xi_{1},\cdots,\xi_{k}\,\in\,T_{A}\mathcal{A}\,=\,\Omega^{1}(M,\mathrm{ad}P)$; by $i_{\xi_{j}}$ we denote the contraction of differential forms by the vector field $\xi_{j}$. As $C\,\longrightarrow\,M$ is an affine bundle modelled on the vector bundle $T^{\ast}M\otimes\mathrm{ad}P\,\longrightarrow\,M$, we consider $\xi_{i}$ as a vertical vector field, with respect to the projection $q$, along $A(M)$. If $k\,=\,0$, then $\beta_{0}^{p}\,\in\,C^{\infty}(\mathcal{A},\,\Omega^{2r}(M))$ is just the characteristic form $\beta_{0}^{p}(A)\,=\,p(F^{A})$.

We assume in the following that $k\,>\,0$.

Lemma 2.1**.**

For $k\,\leq\,r$, we have

[TABLE]

with $N_{r,k}\,=\,r(r-1)\cdots(r-k+1)\,=\,\frac{r!}{(r-k)!}$, where $\xi_{1},\cdots,\xi_{k}\,\in\,T_{A}\mathcal{A}\,=\,\Omega^{1}(M,\,\mathrm{ad}P)$.

If $r\,<\,k\,\leq\,2k$, then we have $\beta_{k}^{p}\,=\,0$.

Proof.

Using formula (2.3), it is easy to see that for $\xi_{1}\,,\xi_{2}\,\in\,\Omega^{1}(M,\mathrm{ad}P)$, we have

[TABLE]

where $q$ is the projection in (2.1). Therefore, we have

[TABLE]

and then, for $k\,\leq\,r$,

[TABLE]

For $k\,>\,r$, we have $\beta_{k}^{p}(\xi_{1},\cdots,\xi_{k})\,=\,0$. ∎

Proposition 2.2**.**

The form $\beta_{k}^{p}$ is invariant under the action of the gauge group in $\mathcal{A}$, in other words,

[TABLE]

where $\Phi_{\mathcal{A}}\,:\,\mathcal{A}\,\longrightarrow\,\mathcal{A}$ is the automorphism induced by $\Phi_{C}$ in (2.4).

Proof.

For any choices of $\Phi\,\in\,\mathrm{Gau}P$ and $\xi_{1},\cdots,\xi_{k}\,\in\,T_{A}\mathcal{A}$, we have

[TABLE]

where $\Phi_{\mathrm{ad}}\,:\,\mathrm{ad}P\,\longrightarrow\,\mathrm{ad}P$ is the automorphism of the adjoint bundle induced by

[TABLE]

Then

[TABLE]

by taking into account the invariance of $p$ under the adjoint action. ∎

3. Cohomology defined by maps to $\mathcal{F\subset A}$

3.1. Definition of the invariant

Proposition 3.1**.**

The identity**

[TABLE]

holds for all $0\,\leq\,k\,\leq\,r$, where

[TABLE]

is the de Rham differential of forms in $\mathcal{A}$ taking values in the vector space $\Omega^{2r-k}(M)$, and

[TABLE]

is the standard de Rham differential of forms in $M$.

In particular, the form $d_{\mathcal{A}}\beta_{k}^{p}$ takes values in the subspace of exact forms $B^{2r-k}(M)\,:=\,d\Omega^{2r-k-1}(M)\,\subset\,\Omega^{2r-k}(M)$.

Proof.

We have

[TABLE]

for $\xi_{0},\cdots,\xi_{k}\,\in\,T_{A}\mathcal{A}$, where $[\xi_{i}\,,\xi_{j}]\,=\,0$ as we take $\xi_{i}$ to be the constant vector fields in the affine space $\mathcal{A}$; by $\widehat{\xi}_{i}$ it is meant that ${\xi}_{i}$ is omitted. Hence

[TABLE]

we have used the fact that

[TABLE]

for $\xi\,\in\,\Omega^{1}(M,\,\mathrm{ad}P)$ and $A\,\in\,\mathcal{A}$. Then

[TABLE]

Recall that $C(P)$ is endowed with a derivation law

[TABLE]

given by the universal connection of the principal $G$-bundle $q^{\ast}P\,\longrightarrow\,C$. This derivation satisfies

[TABLE]

where the last step is obtained by applying the Bianchi identity $\nabla F\,=\,0$. Hence

[TABLE]

and then

[TABLE]

proving the proposition. ∎

Remark 3.2*.*

In particular, for $k=r$ we have

[TABLE]

A connection on $P$ gives a $1$-forms in $P$ with values in the Lie algebra $\mathfrak{g}$. Therefore, we can equip $\mathcal{A}$ with the structure of a Fréchet manifold. Let $\mathcal{F}\,\subset\,\mathcal{A}$ be the closed subspace defined by the flat connections on $P$. This subspace may be empty, but we are interested in the case where $\mathcal{F}\,\neq\,\varnothing$. Take a compact oriented manifold $S$, with $\dim S\,=\,k-1$, together with a smooth map

[TABLE]

We assume that there is another compact oriented manifold $T$ of dimension $k$ such that

[TABLE]

and the inclusion $S\,\hookrightarrow\,T$ is compatible with the orientations of $S$ and $T$. Since $\mathcal{A}$ is contractible, there is a smooth extension

[TABLE]

We now define the main object of the work: given $S$, $f$, $T$ and $\overline{f}$ as above, consider the integral of the pull-back by $\overline{f}$ of the form $\beta_{k}^{p}\,\in\,\Omega^{k}(\mathcal{A},\,\Omega^{2r-k}(M))$

[TABLE]

There is an alternative way of defining $\Lambda_{k}^{p}(f)$ which is close to the one used in [9]. Consider the extended principal bundle $T\times P\,\longrightarrow\,T\times M$, endowed with the canonical connection $\widehat{A}$ defined as follows: The horizontal lift of tangent vectors to $T$ is trivial, whereas the horizontal lift of $v\,\in\,T_{x}M$ at a point $(z,\,x)\,\in\,T\times M$ is the one given by the connection $\overline{f}(z)\,\in\,\mathcal{A}$. Let $\widehat{F}$ be the curvature form of this connection, and let $p(\widehat{F})\,\in\,\Omega^{2r}(T\times M)$ be the associated characteristic form defined by the invariant polynomial $p$.

Proposition 3.3**.**

The form $\Lambda_{k}^{p}(f)\,\in\,\Omega^{2r-k}(M)$ is the partial integration on $T\times M$

[TABLE]

Proof.

Given coordinates $(z^{1}\,,\cdots\,,z^{k})$ on an open subset of $T$, it is easy to check that

[TABLE]

Then

[TABLE]

and, taking into account Lemma 2.1, this integral is precisely $\int_{T}\overline{f}^{\ast}\beta_{k}^{p}$. ∎

Remark 3.4*.*

Furthermore, in the same vein of Proposition 3.3, we can provide a third approach to the form $\Lambda^{p}_{k}(f)$. Let $A_{0}$ be any fixed connection on $P\to M$. We use the same notation $A_{0}$ for the pull-back connection on $T\times P\to T\times M$ defined by the projection $T\times M\to M$. The transgression formula provides a form $\vartheta(\widehat{A},A_{0})\in\Omega^{2r-1}(T\times M)$ such that

[TABLE]

Then, from formula (3.2) we have

[TABLE]

where the integral of $p(F_{0})$ along $T$ vanishes as it is the pull-back of the characteristic form of the bundle $P\to M$ by the projection $T\times M\to M$. This definition of $\Lambda^{p}_{k}(f)$ may be used for some of the proofs of the following. However, note that the form $\vartheta(\widehat{A},A_{0})$ is not gauge invariant as it depends on the chosen connection $A_{0}$.

Lemma 3.5**.**

The differential form $\Lambda_{k}^{p}(f)\,\in\,\Omega^{2r-k}(M)$, $k\leq r$, constructed in (3.1) is closed.

Proof.

The case $k=0$ being trivial, we assume $k>0$. As $d$ is a linear operator, we have

[TABLE]

where we have taken into account Proposition 3.1. The last integral is zero as $\beta_{k-1}^{p}$ vanishes along $\mathcal{F}$. ∎

Lemma 3.5 produces a map

[TABLE]

which is also denoted by $\Lambda_{k}^{p}$ for notational convenience. Furthermore, from the definition of $\beta^{p}_{k}$ given in Proposition 3.3 we have the following:

Lemma 3.6**.**

The cohomology class $\Lambda^{p}_{k}$ is the cap product

[TABLE]

where $q_{M}\,:\,T\times M\,\longrightarrow\,M$ is the natural projection, and $p(q^{*}_{M}P)\,\in\,H^{2r}(T\times M,\,\mathbb{R})$ is the characteristic class of the pull-back principal bundle $q^{*}_{M}P\to T\times M$, defined by the invariant polynomial $p$.

Although the construction of $\Lambda_{k}^{p}(f)$ depends a priori on the choice of $T$ and the extension $\overline{f}$, the following proposition shows that $\Lambda_{k}^{p}(f)$ is actually independent of them.

Proposition 3.7**.**

The element $\Lambda_{k}^{p}(f)\,\in\,H^{2r-k}(M,\,\mathbb{R})$ does not depend on either the choice of $T$ or the choice of the extension $\overline{f}\,:\,T\,\longrightarrow\,\mathcal{A}$.

Proof.

Let $\overline{f}_{1}\,:\,T_{1}\,\longrightarrow\,\mathcal{A}$ and $\overline{f}_{2}\,:\,T_{2}\,\longrightarrow\,\mathcal{A}$ be two extensions. We consider

[TABLE]

(the orientation of $T_{2}$ is reversed) and we glue the maps $\overline{f}_{1}$ and $\overline{f}_{2}$ to a map $\widehat{f}\,:\,D\,\longrightarrow\,\mathcal{A}$. The $(2r-k)$–form on $M$

[TABLE]

is closed by setting $T$ in Lemma 3.5 to be $D$, because the boundary of $D$ is empty. As in Lemma 3.6, the cohomology class $\widetilde{\theta}\,\in\,H^{2r-k}(M,\,\mathbb{R})$ given by the form $\theta$ is the cap product $(p(q^{*}_{M}P))\cap[D]\,\in\,H^{2r-k}(M,\,\mathbb{R})$. But

[TABLE]

so $(p(q^{*}_{M}P))\cap[D]\,=\,0$. Therefore, we have $\widetilde{\theta}\,=\,0$. The proposition follows from this. ∎

Proposition 3.8**.**

Fix $k\,<\,r$. Let $f_{1}\,:\,S\,\longrightarrow\,\mathcal{F}$ and $f_{2}\,:\,S\,\longrightarrow\,\mathcal{F}$ be homologous maps in $\mathcal{F}$. Then

[TABLE]

in $H^{2r-k}(M,\,\mathbb{R})$.

Proof.

Let $D\subset\mathcal{F}$ be a $k$-dimensional chain such that $\partial D=f_{1}(S)\cup(-f_{2}(S))$. The (oriented) set $D\cup(-\overline{f}_{1}(T))\cup\overline{f}_{2}(T)$ is a cycle, and hence it is a border of a $(k+1)$-chain $C$ in the affine space $\mathcal{A}$. Then we have

[TABLE]

where the integral along $D$ vanishes as $\beta^{p}_{k}$ restricts to [math] on $\mathcal{F}$ if $k<r$. Hence, $\Lambda^{p}_{k}(f_{1})$ and $\Lambda^{p}_{k}(f_{2})$ define the same element in $H^{2r-k}(M,\,\mathbb{R})$ because using Proposition 3.1,

[TABLE]

∎

3.2. Gauge invariance and the moduli space

Let $C(S,\mathcal{F})$ denote the space of all smooth maps from $S$ to $\mathcal{F}$. Recall that a gauge transformations $\Phi\in\mathrm{Gau}P$ induces a transformation $\Phi_{\mathcal{A}}:\mathcal{A}\to\mathcal{A}$ in the space of principal connections. In particular, this action leaves the subset $\mathcal{F}$ of flat connection invariant.

Proposition 3.9**.**

The mapping

[TABLE]

is invariant under the action of the gauge group in $\mathcal{F}$, that is, for every $\Phi\in\mathrm{Gau}P$ and $f\,\in\,C(S,\mathcal{F})$,

[TABLE]

where $\Phi\cdot f\in C(S,\mathcal{F})$ is defined as $(\Phi\cdot f)(x)=\Phi_{\mathcal{A}}(f(x))$, $x\in S$.

Proof.

It is a direct consequence of Proposition 2.2 and the fact that $\mathcal{F}$ is invariant under $\Phi_{\mathcal{A}}$. ∎

The following proposition studies the behavior of $\Lambda^{p}_{k}$ under pointwise gauge transformations for invariant polynomials defining integral characteristic classes. By pointwise we mean that the gauge transformation is not fixed and may depend on the $S$.

Proposition 3.10**.**

Take any $\bar{\Phi}\,:\,S\,\longrightarrow\,\mathrm{Gau}P$. For any two maps $f_{1}\,:\,S\,\longrightarrow\,\mathcal{F}$ and $f_{2}\,:\,S\,\longrightarrow\,\mathcal{F}$ with

[TABLE]

the following holds:

[TABLE]

that is, the invariant polynomial $p$ produces characteristic class for $G$-bundles with values in integral cohomology classes.

Proof.

Take any $T$ with $\partial T\,=\,S$, and also take two extensions $\overline{f}_{1}\,:\,T\,\longrightarrow\,\mathcal{A}$ and $\overline{f}_{2}\,:\,T\,\longrightarrow\,\mathcal{A}$ of $f_{1}$ and $f_{2}$ respectively. We consider two copies of the principal $G$-bundle

[TABLE]

equipped with the following connections: both the connections are trivial along the direction of $T$, and the connection $\overline{f}_{1}(z)$ (respectively, $\overline{f}_{2}(z)$) on $P$ coincides the connection on $\{z\}\times P\,\longrightarrow\,\{z\}\times M$, $z\,\in\,T$, in the first (respectively, second) copy. The orientation of the second $T$ is reversed so that we can glue along the boundary to get an oriented manifold $D\,=\,T\cup_{S}T$. As for the $G$-bundles, we glue the two copies of $\overline{P}$ using $\bar{\Phi}$, that is, $(z\,,p)\,\in\,S\times P$ of the first copy is glued with $(z\,,\bar{\Phi}(z)(p))\,\in\,S\times P$ of the second copy. The connections of the two copies of $G$-bundles glue compatibly to give a connection on the $G$-bundle over $D\times M$. We consider any $(2r-k)$-dimensional cycle $C\,\subset\,M$, and the restriction of the new bundle to $D\times C$. The integral of the characteristic class gives an integer, but this integral is precisely

[TABLE]

and the proof is complete. ∎

Notation 3.11**.**

Let $[C(S,\,\mathcal{F}/\mathrm{Gau}P)]_{\mathrm{lift}}$ be the homotopy classes of maps from $S$ the quotient space $\mathcal{F}/\mathrm{Gau}P$ that lift to a map from $S$ to $\mathcal{F}$.**

Corollary 3.12**.**

There is a well defined map

[TABLE]

Proof.

Given any $\varphi\,\in\,C(S,\,\mathcal{F}/\mathrm{Gau}P)$, take any map $f\,:\,S\,\longrightarrow\,\mathcal{F}$ that projects to $\varphi$, and consider the corresponding $\Lambda^{p}_{k}(f)$. In view of Proposition 3.10 it does not depend on the choice of $f$ up to integral cohomology. ∎

Remark 3.13*.*

Taking into account Proposition 3.8, the presentation of the previous Corollary could be given for the set of homology classes of maps from $S$ to $\mathcal{F}/\mathrm{Gau}P$ that lift to a map from $S$ to $\mathcal{F}$. We’d rather follow the homotopy formulation instead.

3.3. Application to homotopy groups

Here, we set $k\,\geq\,2$, $S\,=\,S^{k-1}$ (the $(k-1)$-sphere) and $T\,=\,\overline{B}^{k}\,\subset\,\mathbb{R}^{k}$ (the unit closed ball). We also set $2r-k\,\leq\,n$ and $k\,<\,r$ so that Proposition 3.8 applies. Homotopy classes of maps

[TABLE]

are elements of the homotopy group $\pi_{k-1}(\mathcal{F})$. Given an invariant polynomial $p$ of degree $r$, we thus have a map

[TABLE]

given by

[TABLE]

where $\overline{f}\,:\,\overline{B}^{k}\,\longrightarrow\,\mathcal{A}$ is any extension of $f$.

The following is straight-forward to check.

Proposition 3.14**.**

The above map $\Lambda^{p}\,:\,\pi_{k-1}(\mathcal{F})\,\longrightarrow H^{2r-k}\,(M,\mathbb{R})$ is a homomorphism of groups.

4. Invariants in Dolbeault cohomology

In this section we assume that $M$ is a compact complex manifold and $G$ is a complex Lie group. Let

[TABLE]

be the projections associated to the decomposition

[TABLE]

of complex $m$-forms on $M$. Given an invariant polynomial $p\,\in\,S^{r}(\mathfrak{g}^{\ast})^{G}$ of degree $r$ and an integer $0\,\leq\,k\,\leq\,r$, we define

[TABLE]

as the $(0,2r-k)$-component of the form $\beta_{k}^{p}$ constructed in (2.5), more precisely,

[TABLE]

Proposition 4.1**.**

The following holds:

[TABLE]

where

[TABLE]

is the differential of forms on $\mathcal{A}$ taking values in the vector space $\Omega^{2r-k}(M)$, and

[TABLE]

is the Dolbeault differential on $M$.

Proof.

As $\rho^{0,2r-k}$ is a fiberwise linear,

[TABLE]

Then, from Proposition 3.1,

[TABLE]

completing the proof. ∎

According to the Chern-Weil construction of characteristic classes, for any $A\,\in\,\mathcal{A}$ the form $\widetilde{\beta}_{0}^{p}(A)\,\in\,C(\mathcal{A},\,\Omega^{0,2r}(M))$ is the $(0,r)$-part of the characteristic polynomial $p(F^{A})$.

Below we assume that $k\,>\,0$.

Let $\mathcal{F}^{0,2}$ denote the subset of $\mathcal{A}$ consisting of all connections $A$ satisfying the condition that the $(0,2)$-part of the curvature of $A$ vanishes, that is

[TABLE]

We note that any $A\,\in\,\mathcal{F}^{0,2}$ produces a complex structure on the principal $G$-bundle $P$. Let $S$ be a $(k-1)$-dimensional compact oriented manifold, and let

[TABLE]

be a smooth map.

Lemma 4.2**.**

Take any $k\,<\,r$. The form $\widetilde{\beta}_{k}^{p}\,\in\,\Omega^{k}(\mathcal{A},\,\Omega^{2r-k}(M))$ vanishes along $\mathcal{F}^{0,2}$.

Proof.

We have

[TABLE]

for $\xi_{1},\cdots,\xi_{k}\in T_{A}\mathcal{A}$. Since $r-k\,>\,0$, and $\rho^{0,2}(F^{A})\,=\,0$, the top $(0,2r-k)$-part of $p(\xi_{1},\cdots,\xi_{k},F^{A},\overset{(r-k)}{\cdots},F^{A})$ is zero. ∎

Assume that $S\,=\,\partial T$, where $T$ as before is a $k$-dimensional oriented manifold such that the inclusion $S\,\hookrightarrow\,T$ is compatible with the orientations. Choose an extension $\overline{f}\,:\,T\,\longrightarrow\,\mathcal{A}$. Let $\widetilde{\Lambda}_{k}^{p}(f)\,\in\,\Omega^{0,2r-k}(M)$ be the element defined by

[TABLE]

Proposition 4.3**.**

The form $\widetilde{\Lambda}_{k}^{p}(f)\in\Omega^{0,2r-k}(M)$ is $\overline{\partial}$-closed.

Proof.

From Proposition 4.1, we have

[TABLE]

This last integral vanished by virtue of Lemma 4.2. ∎

Therefore, we have an element

[TABLE]

of the Dolbeault cohomology.

The following three Propositions can be proved by adapting the proofs of Propositions 3.7, 3.8 and 3.9 respectively; this in particular involves substituting the de Rham differential $d$ by the Dolbeault differential $\overline{\partial}$ and taking into account Lemma 4.2.

Proposition 4.4**.**

Take any $1\,<\,k\,<\,r$. The Dolbeault cohomology class $\widetilde{\Lambda}_{k}^{p}(f)$ does not depend on either the choice of $T$ or the choice of the extension $\overline{f}$.

Proposition 4.5**.**

The Dolbeault class $\widetilde{\Lambda}_{k}^{p}(f)$ is gauge invariant, that is, given any $\Phi\,\in\,\mathrm{Gau}P$,

[TABLE]

where $\Phi$ acts on connection in the natural way.

Proposition 4.6**.**

Take any $k\,<\,r$. If $f_{1}\,:\,S\,\longrightarrow\,\mathcal{F}^{0,2}$ and $f_{2}\,:\,S\,\longrightarrow\,\mathcal{F}^{0,2}$ are homologous in $\mathcal{F}^{0,2}$, then

[TABLE]

in $H^{0,2r-k}(M)$.

As in Section 3.3, for any $1\,<\,k\,<\,r$, setting $S\,=\,S^{k-1}$ and $T\,=\,\overline{B}^{k}$, a map from the homotopy groups to the Dolbeault cohomology

[TABLE]

is obtained.

Define

[TABLE]

Proposition 4.7**.**

Take two maps $\Phi\,:\,S\,\longrightarrow\,\mathrm{Gau}P$ and $f_{1}\,:\,S\,\longrightarrow\,\mathcal{F}$, and define

[TABLE]

Then

[TABLE]

Proof.

As in the proof of Proposition 3.10, for any $(2r-k)$-cycle $C$, we have

[TABLE]

and hence

[TABLE]

Then

[TABLE]

completing the proof. ∎

Corollary 4.8**.**

There is a well defined homomorphism

[TABLE]

defined on the set of homotopy classes of maps from $S$ to $\mathcal{F}^{0,2}/\mathrm{Gau}P$ that can be lifted to $\mathcal{F}^{0,2}$.

5. Examples

5.1. Surfaces

Recall that the form $\beta_{k}^{p}$ is not necessarily trivial for

[TABLE]

Furthermore, the homomorphism $\Lambda^{p}\,:\,\pi_{k-1}(\mathcal{F)}\,\longrightarrow\,H^{2r-k}(M)$ defined above only makes sense for $2\,\leq\,k\,<\,r$. If $M$ is a surface ($n\,=\,2$), we have the following:

• •

Assume that $r\,=\,1$ and $k\,=\,0$. Then $\beta_{k}^{p}\,\in\,C^{\infty}(\mathcal{A},\,\Omega^{2}(M))$ is the Chern-Weil characteristic form map.

• •

Assume that $r\,=\,1$ and $k\,=\,1$. Then $\beta_{1}^{p}\,\in\,\Omega^{1}(\mathcal{A},\,\Omega^{1}(M))$ does not depend on $A\,\in\,\mathcal{A}$

[TABLE]

We do not have homotopy invariance as $k\,=\,r$ so Proposition 3.8 does not apply.

• •

Assume that $r\,=\,2$ and $k\,=\,2$. Then $\beta_{2}^{p}\,\in\,\Omega^{2}(\mathcal{A},\,\Omega^{2}(M))$ does not depend on $A\,\in\,\mathcal{A}$. Again, as $k=r$, we do not have homotopy invariance.

In any case, for $G\,=\,{\rm U}(m)$ and $p(X)\,=\,\mathrm{tr}(X^{2})$, integrating $\beta_{2}^{p}$ along $M$ gives the canonical symplectic form on $\mathcal{A}$ (see [2]) given by

[TABLE]

5.2. Case of $k=1$

We consider a set of two points $S\,=\,\{0\,,1\}$ and $T\,=\,[0\,,1]$. A map $f\,:\,S\,\longrightarrow\,\mathcal{F}$ just gives two flat connections whereas the extension $\overline{f}\,:\,[0\,,1]\,\longrightarrow\,\mathcal{A}$ is a path connecting them. The invariant $\Lambda^{p}(f)\,\in\,H^{2r-1}(M,\,\mathbb{R})$ defined as

[TABLE]

is independent of the extension $\overline{f}$ and the homotopy class of $f$. If the two points $A_{1}\,,A_{2}\,\in\,\mathcal{F}$ are in the same arc-connected component of $\mathcal{F}$, then the extension $\overline{f}$ can be defined entirely contained in $\mathcal{F}$, so that $\Lambda^{p}([f])\,=\,0$. The map $\Lambda^{p}([f])$ thus gives information of points in different connected components of $\mathcal{F}$. Fixing $A_{1}$, we can define

[TABLE]

where $f\,:\,\{0\,,1\}\,\longrightarrow\,\mathcal{F}$, $f(0)\,=\,A_{1}$, $f(1)\,\in\,K$, with $K$ being an arc-connected component of $\mathcal{F}$.

We develop the definition of $\Lambda^{p}([f])$. Given $A_{1}$ and $A_{2}$, the extension $\overline{f}$ can be taken as the line

[TABLE]

so that

[TABLE]

where $F_{t}$ is the curvature of $A_{t}\,=\,(1-t)A_{1}+tA_{2}$. This is precisely the expression of the transgression formula for $p$, $A_{1}$ and $A_{2}$. In fact, writing $\xi\,=\,A_{2}-A_{1}$, we have

[TABLE]

and as $F^{A_{1}}\,=\,0$ and $F^{A_{1}+\xi}\,=\,\nabla^{A_{1}}\xi+[\xi,\xi]\,=\,0$, we have

[TABLE]

Then the invariant $\Lambda^{p}([f])$ is

[TABLE]

For Abelian gauge theories, this expression is trivial, but when $G$ is non-Abelian, we get a non-vanishing invariant. For example, consider $P\,=\,M\times{\rm SU}(2)$, and let $p$ be the inner product on $\mathfrak{su}(2)\,\simeq\,\mathbb{R}^{3}$. In this case,

[TABLE]

$\xi\,\in\,\Omega^{1}(M,\,\mathfrak{su}(2))\,=\,\Omega^{1}(M,\,\mathbb{R}^{3})$. If we put $\xi\,=\,\xi^{i}B_{i}$ for an orthonormal basis (for instance, the Pauli matrices) $B_{1},B_{2},B_{3}$, then

[TABLE]

As $d\xi+[\xi,\xi]\,=\,0$, this is equivalent to

[TABLE]

which is a Chern-Simons type formula.

5.3. Case of $k=2$

For $r>2$, we then have morphisms

[TABLE]

and

[TABLE]

which make sense if $2r-2\,\leq\,\dim M$. The first relevant case is $\dim M=4$ and $r=3$. For example, when $G\,=\,{\rm U}(2)$, the set of invariant polynomials is generated by the trace $p_{1}$ and the determinant $p_{2}$. For $n=4,r=3$, we can consider either $p\,=\,p_{1}^{3}$ or $p\,=\,p_{2}p_{1}$. We then have

[TABLE]

Given a loop $A(s)$ in $\mathcal{F}$, we can choose a simple extension as $\overline{f}\,:\,[0\,,1]\times[0,1]\,\longrightarrow\,\mathcal{A}$

[TABLE]

where $A_{0}\,=\,A(0)$. Then

[TABLE]

where $\dot{A}(s)\,=\,\partial_{s}A(s)$. As both $A_{0}$ and $A(s)$ are flat connections, it is easy to see that

[TABLE]

Then, setting $\xi(s)\,=\,A(s)-A_{0}$, we have

[TABLE]

where $\dot{\xi}$ stands for $d\xi/ds$; for the last step we have taken into account that $F^{A_{0}+\xi(s)}\,=\,\nabla^{A_{0}}\xi(s)+[\xi(s),\xi(s)]\,=\,0$. In particular, this expression has a simpler form when the bundle $P$ is trivial, $A_{0}$ is chosen to be the trivial connection and connections are seen as $1$-forms in $M$ taking values in $\mathfrak{g}$, that is

[TABLE]

If $A(s)=(\Phi(s))_{\mathcal{A}}(A_{0})$, for a loop $\Phi(s)\subset\mathrm{Gau}P$, $0\leq s\leq 1$, the definition of degree of the loop is

[TABLE]

for a metric $p$ (that is, $r=2$) in $\mathfrak{g}$ defining integer cohomology class (see, for example, [11]). We can understand the expression (5.1) as a higher order generalization of the degree. Of course, the integral class vanishes when considered in $[S^{1},\mathcal{F}/\mathrm{Gau}P]_{\mathrm{lift}}$. The expression for other loops in $A(s)$ in $\mathcal{F}$ is to be analyzed.

Acknowledgements

We thank the referee for very helpful comments. The first author is supported by a J. C. Bose Fellowship. The second author thanks the TATA Institufe of Fundamental Research for its hospitality during a visit in which a part of this work was developed. The second author was partially funded by MINECO (Spain) Project MTM2015-63612-P.