Transgressive loop group extensions

Abstract

A central extension of the loop group of a Lie group is called transgressive, if it corresponds under transgression to a degree four class in the cohomology of the classifying space of the Lie group. Transgressive loop group extensions are those that can be explored by finite-dimensional, higher-categorical geometry over the Lie group. We show how transgressive central extensions can be characterized in a loop-group theoretical way, in terms of loop fusion and thin homotopy equivariance.

Appendix A: Regression of trivial fusion bundles

In this appendix we discuss the regression of trivial bundles with trivial fusion products (but non-trivial connections) over the loop space LX of a connected smooth manifold X. For this purpose, we restrict the constructions of [24, Sections 5 and 6] to that case; this has not yet been worked out explicitly.

For $x\in X$ we consider the diffeological space $P_xX$ of paths in X starting at x with sitting instants, equipped with the subduction (the diffeological analog of a surjective submersion). Two paths with the same end point compose to a loop via the smooth map .

Suppose $\epsilon \in \Omega ^1(LX)$ is a superficial connection on the trivial bundle $\mathbf {I}$ that is fusive with respect to the trivial fusion product. The regression $\mathscr {R}^{\nabla }_x(\mathbf {I}_\epsilon )$ is a bundle gerbe with connection over X, composed of the subduction , the principal $S^1$-bundle with connection $\cup ^{*}_x\mathbf {I}_{\epsilon }$, and the identity bundle gerbe product, which is connection-preserving because $\epsilon$ is fusive. The difficult part is to specify a curving: a 2-form $B_{\epsilon } \in \Omega ^2(P_xX)$ such that ${\mathrm {pr}}_2^{*}B_{\epsilon } - {\mathrm {pr}}_1^{*}B_{\epsilon } = \mathrm {curv}(\cup _x^{*}\mathbf {I}_{\epsilon }) = \cup _x^{*}\mathrm{d}\epsilon$.

Such a curving can be constructed because $\epsilon$ is superficial, see [24, Section 5.2]. The construction uses a bijection between the 2-forms on a diffeological space Y and certain smooth maps on the space $\mathcal {B}Y$ of bigons in Y. A bigon is a smooth fixed-ends homotopy $\Sigma$ between two paths in Y, and the correspondence between a smooth map and a 2-form $B \in \Omega ^2(Y)$ is established by the relation

$$\begin{aligned} G(\Sigma ) = \exp 2\pi \mathrm {i}\left( - \int _{\Sigma } B \right) \text {.} \end{aligned}$$

(A.1)

Suppose $\Sigma \in \mathcal {B}P_xX$ is a bigon between a path $\gamma _0 \in PP_xX$ and a path $\gamma _1\in PP_xX$. Thus, it is a smooth map such that $\Sigma (0,t)=\gamma _0(t)$ and $\Sigma (1,t)=\gamma _1(t)$. For each $t\in [0,1]$ we extract a loop $\gamma _{\Sigma }(t) \in LX$ defined by

$$\begin{aligned} \gamma _{\Sigma }(t) := (\Sigma ^{m}(t) \star \Sigma ^{o}(t)) \cup (\mathrm {id}\star \Sigma ^{u}(t))\text {,} \end{aligned}$$

(A.2)

where $\Sigma ^{m}(t), \Sigma ^{o}(t)$, and $\Sigma ^{u}(t)$ are the three paths depicted in Fig. 3. Thus, $\gamma _{\Sigma }$ is a path in LX that starts and ends at flat loops.

Figure 3

The picture on the left shows a bigon $\Sigma$ in $P_{\!x}X$: it can be regarded as a bigon in X that has for each of its points a chosen path connecting x with that point. The picture on the right shows the three paths associated to a bigon $\Sigma$ and $t \in [0,1]$

Using a smoothing function , we define a parameterized version $\Sigma _{\sigma }$ for $\sigma \in [0,1]$ by $\Sigma _{\sigma }(s,t) := \Sigma (\phi (s)\sigma ,t)$, i.e. $\Sigma _0$ is the identity bigon at the path $\gamma _0$, and $\Sigma _1=\Sigma$. Now we consider . By [24, Lemma 5.2.1] the curving $B_{\epsilon }$ corresponds to the smooth map

As $\mathrm {curv}(\mathbf {I}_{\epsilon })=\mathrm{d}\epsilon$ we want to apply Stokes’ theorem. Along the boundary of $[0,1]^2$, the map h is as follows: $h(0,t) = \gamma _0(t)\cup \gamma _0(t), h(1,t)=\gamma _{\Sigma }(t), h(\sigma ,0)=\gamma _0(0) \cup \gamma _0(0)$, and $h(\sigma ,1)=\gamma _0(1)\cup \gamma _0(1)$. As $\epsilon$ is fusive, we have $\flat ^{*}\epsilon =0$, where is the inclusion of flat loops, see Sect. 3.1. Thus, we get

$$\begin{aligned} G_{\epsilon }(\Sigma )=\exp 2 \pi \mathrm {i}\left( - \int _{0}^1 \gamma _{\Sigma }^{*}\epsilon \right) \text {.} \end{aligned}$$

The regressed bundle gerbe $\mathscr {R}^{\nabla }_x(\mathbf {I}_{\epsilon })$ is not the trivial bundle gerbe (that one would have the identity subduction $\mathrm {id}_X$). It is trivializable, but not canonically trivializable. However, a trivialization $\mathcal {T}_{\kappa }$ can be obtained from a path splitting $\kappa \in \Omega ^1(PX)$ of $\epsilon$, see Definition 4.4.

The trivialization $\mathcal {T}_{\kappa }$ is composed of the principal ${\mathrm {U}}(1)$-bundle $\mathbf {I}_{-\kappa }$ over $P_xX$ and of the bundle isomorphism

over $P_xX^{[2]}$, which is connection-preserving due to the defining property of a path splitting. There exists a unique 2-form $\rho _{\kappa }\in \Omega ^2(X)$ such that is a connection-preserving isomorphism; this 2-form is characterized by the condition $\mathrm {ev}_1^{*}\rho _{\kappa } = B_{\epsilon }-\mathrm{d}\kappa$.

Lemma 6.8

Suppose $\rho \in \Omega ^2(X)$ and $\epsilon := \tau _{S^1}(\rho ) \in \Omega ^1(LX)$. Then, $\kappa := \tau _{[0,1]}(\rho )$ is a path splitting for $\epsilon$, and $B_{\epsilon } = \mathrm {ev}_1^{*}\rho +\mathrm{d}\kappa$. In particular, $\rho _{\kappa }=\rho$.

Proof

That $\kappa$ is a path splitting for $\epsilon$ has been checked in Example 4.5. Let $\Sigma$ be a bigon in $P_xX$ between a path $\gamma _0$ and a path $\gamma _1$. We have

$$\begin{aligned} G_{\epsilon }(\Sigma ) =\exp 2\pi \mathrm {i}\left( - \int _{0}^1 \gamma _{\Sigma }^{*}\epsilon \right) = \exp 2\pi \mathrm {i}\left( \int _{[0,1] \times S^1} h_{\gamma _{\Sigma }}^{*}\rho \right) \end{aligned}$$

with defined by $h_{\gamma _{\Sigma }}(t,z) := \gamma _{\Sigma }(t)(z)$. We obtain from the definition (A.2) of $\gamma _{\Sigma }$:

$$\begin{aligned} h_{\gamma _{\Sigma }}(t,z) = {\left\{ \begin{array}{ll} \gamma _0(t)(4z) &{} \text { if }0 \le z \le \frac{1}{4} \\ \mathrm {ev}_1(\Sigma (4z-1)(t)) &{} \text { if }\frac{1}{4} \le z \le \frac{1}{2} \\ \gamma _1(t)(3-4z) &{} \text { if }\frac{1}{2} \le z \le \frac{3}{4} \\ x &{} \text { if }\frac{3}{4} \le z \le 1 \\ \end{array}\right. } \end{aligned}$$

Splitting the domain of integration into those four parts and taking care with the involved orientations yields

$$\begin{aligned} \exp 2\pi \mathrm {i}\left( \int _{[0,1] \times S^1} h_{\gamma _{\Sigma }}^{*}\rho \right) = \exp 2\pi \mathrm {i}\left( - \int _{\Sigma }\mathrm {ev}_1^{*}\rho + \int _{\gamma _0} \kappa - \int _{\gamma _1} \kappa \right) \text {.} \end{aligned}$$

Finally, Stokes’ theorem gives

$$\begin{aligned} \int _{\Sigma }\mathrm{d}\kappa = -\int _{\gamma _0}\kappa + \int _{\gamma _1}\kappa \text {.} \end{aligned}$$

All together, we obtain

$$\begin{aligned} G_{\epsilon }(\Sigma ) = \exp 2\pi \mathrm {i}\left( - \int _{\Sigma }(\mathrm {ev}_1^{*}\rho +\mathrm{d}\kappa ) \right) \text {.} \end{aligned}$$

Using (A.1), we get the claimed equality.

As regression is a monoidal functor, we want to make sure that the trivialization $\mathcal {T}_{\kappa }$ is compatible with that monoidal structure.

Lemma 6.9

Suppose $\epsilon _1,\epsilon _2$ are superficial connections on the trivial bundle $\mathbf {I}$ over LX, and fusive with respect to the trivial fusion product. Suppose $\kappa _1$ and $\kappa _2$ are path splittings for $\epsilon _1$ and $\epsilon _2$, respectively. Then, we have $\rho _{\kappa _1 + \kappa _2}=\rho _{\kappa _1} + \rho _{\kappa _2}$, and there exists a connection-preserving transformation

Proof

The isomorphism $\mathscr {R}^{\nabla }_x(\mathbf {I}_{\epsilon _1+\epsilon _2}) \cong \mathscr {R}^{\nabla }_x(\mathbf {I}_{\epsilon _1}) \otimes \mathscr {R}^{\nabla }_x(\mathbf {I}_{\epsilon _2})$ that implements that $\mathscr {R}^{\nabla }_x$ is monoidal is induced from the connection-preserving, fusion-preserving isomorphism . In particular, we have $B_{\epsilon _1+\epsilon _2}=B_{\epsilon _1}+B_{\epsilon _2}$. We calculate $\mathrm {ev}_1^{*}(\rho _{\kappa _1}+\rho _{\kappa _2}) = B_{\epsilon _1}-\mathrm{d}\kappa _1+B_{\epsilon _1}-\mathrm{d}\kappa _1 = B_{\epsilon _1+\epsilon _2} + \mathrm{d}(\kappa _1+\kappa _2)$; this shows that $\rho _{\kappa _1+\kappa _2} = \rho _{\kappa _1}+\rho _{\kappa _2}$. The announced connection-preserving transformation is now simply induced by the connection-preserving isomorphism . $\square$

The next two propositions describe the relation between the trivialization $\mathcal {T}_{\kappa }$ of the regressed bundle gerbe $\mathscr {R}^{\nabla }_x(\mathbf {I}_{\epsilon })$, the canonical trivialization $t_{\rho }$ of $\mathscr {T}^{\nabla }_{\mathcal {I}_{\rho }}$, and the two natural equivalences

that establish that the functors $\mathscr {R}^{\nabla }_x$ and $\mathscr {T}^{\nabla }$ form an equivalence of categories [24, Theorem A].

Proposition 6.10

Suppose $\rho \in \Omega ^2(X)$. Let be the canonical trivialization, with $\epsilon = \tau _{S^1}(\rho )\in \Omega ^1(LX)$. Let be the component of the natural equivalence $\mathcal {A}$ at $\mathcal {I}_{\rho }$. Let $\kappa := \tau _{[0,1]}(\rho )$ be the canonical path splitting of $\epsilon$ and let be the corresponding trivialization. Then, there exists a connection-preserving transformation

$$\begin{aligned} \mathcal {A}_{\mathcal {I}_{\rho }}\cong \mathcal {T}_{\kappa } \circ \mathscr {R}^{\nabla }_x(t_{\rho }) \text {.} \end{aligned}$$

Proof

The isomorphism is induced from the bundle isomorphism over $P_xX^{[2]}$. The composition $\mathcal {T}_{\kappa } \circ \mathscr {R}^{\nabla }_x(t_{\rho })$ is thus given by the $S^1$-bundle $\mathbf {I}_{-\kappa }$ and the isomorphism

(A.3)

Next we describe the connection-preserving isomorphism $\mathcal {A}_{\mathcal {I}_{\rho }}$ following [24, Section 6.1]. It consists of an $S^1$-bundle Q over $P_xX$ with connection, and of a connection-preserving bundle isomorphism over $P_xX^{[2]}$.

The fibre of Q over $\gamma \in P_xX$ consists of triples $(\mathcal {T},t_0,t)$, where is a trivialization (in turn consisting of an $S^1$-bundle T with connection over [0, 1] and of a connection-preserving bundle isomorphism which here is necessarily the identity $\tau =\mathrm {id}_T$), $t_0 \in T_{0}$ and $t \in T_1$. Two triples $(\mathcal {T},t_0,t)$ and $(\mathcal {T}',t_0',t')$ are identified if there exists a connection-preserving transformation $\varphi :\mathcal {T} \Rightarrow \mathcal {T}'$ such that $\varphi (t_0)=t_0'$ and $\varphi (t)=t'$. The $S^1$-action on $S^1$ is $(\mathcal {T},t_0,t)\cdot z := (\mathcal {T},t_0,t\cdot z)$. In our situation, Q has a canonical section , using that $\gamma ^{*}\mathcal {I}_{\rho }=\mathcal {I}_{\gamma ^{*}\rho }=\mathcal {I}_0$. The bundle isomorphism $\alpha$ is over a point $(\gamma _1,\gamma _2)\in P_xX^{[2]}$ a map

and it is characterized by $\alpha (t_{\rho }(\gamma _1\cup \gamma _2) \otimes s(\gamma _2))=s(\gamma _1)$. The connection on Q is defined via its parallel transport. Using the section , it suffices to define a 1-form on $P_xX$, and we do this by defining a smooth map . This map is given by

$$\begin{aligned} F(\gamma ) = \int _{h_\gamma }\rho \text {,} \end{aligned}$$

where is defined by $h_\gamma (s,t)=\gamma (s)(t)$. However, this map characterizes precisely the parallel transport of the 1-form $-\kappa = -\tau _{[0,1]}(\rho )\in \Omega ^1(P_xX)$.

Summarizing, s defines a connection-preserving transformation between $\mathcal {A}_{\mathcal {I}_0}$ and the the trivialization consisting of the trivial bundle $\mathbf {I}_{-\kappa }$ and of the isomorphism (A.3). $\square$

Proposition 6.11

Suppose $\epsilon \in \Omega ^1(LX)$ is a superficial connection on the trivial ${\mathrm {U}}(1)$-bundle over LX, and fusive with respect to the trivial fusion product. Let be the component of the natural equivalence $\varphi$ at $\mathbf {I}_{\epsilon }$. Let $\kappa \in \Omega ^1(PX)$ be a contractible path splitting for $\epsilon$, and let be the corresponding trivialization. Let be the canonical trivialization with $\epsilon _{\kappa } = \tau _{S^1} (\rho _{\kappa })$. Then,

$$\begin{aligned} \varphi _{\mathbf {I}_{\epsilon }} = t_{\rho _{\kappa }} \circ \mathscr {T}^{\nabla }_{\mathcal {T}_{\kappa }}\text {;} \end{aligned}$$

(A.4)

in particular, $\epsilon _{\kappa }=\epsilon$.

Proof

Note that (A.4) is an equality between two connection-preserving bundle isomorphisms going from $\mathscr {T}^{\nabla }_{\mathscr {R}^{\nabla }_x(\mathbf {I}_{\epsilon })}$ to $\mathbf {I}_{\epsilon }$ and $\mathbf {I}_{\epsilon _{\kappa }}$, respectively. This implies $\epsilon =\epsilon _{\kappa }$. For $\beta \in \textit{LG}$ a loop, $\beta ^{*}\mathcal {T}_{\kappa }$ is a trivialization of $\beta ^{*}\mathscr {R}^{\nabla }_x(\mathbf {I}_{\epsilon })$, and thus an element of $\mathscr {T}^{\nabla }_{\mathscr {R}^{\nabla }_x(\mathbf {I}_{\epsilon })}$ over $\beta$. We have

$$\begin{aligned} \mathscr {T}^{\nabla }_{\mathcal {T}_{\kappa }}(\beta ^{*}\mathcal {T}_{\kappa }) = \beta ^{*}\mathcal {T}_{\kappa } \circ \beta ^{*}\mathcal {T}_{\kappa }^{-1} = \mathrm {id}_{\mathcal {I}_{\beta ^{*}\rho _{\kappa }}}\text {,} \end{aligned}$$

considered as an element of $\mathscr {T}^{\nabla }_{\mathcal {I}_{\rho _{\kappa }}}$. Under the canonical trivialization $t_{\rho _{\kappa }}$, this element is equal to $(\beta ,1)\in LX\times {\mathrm {U}}(1)$.

On the other hand, we compute the element $p := \varphi _{\mathbf {I}_{\epsilon }}(\beta ^{*}\mathcal {T}_{\kappa }) \in \mathbf {I}_{\epsilon }$ following the definition of $\varphi$ given in [24, Section 6.2]. We have to consider the space $Z:=S^1 {{}_{\beta }\times _{\mathrm {ev}_1}} P_xX$ as the subduction of $\beta ^{*} \mathscr {R}^{\nabla }_x(\mathbf {I}_{\epsilon })$ and over Z the bundle $\mathbf {I}_{-\kappa }$, pulled back along the projection . Over $Z \times _{S^1} Z$ the trivialization $\beta ^{*}\mathcal {T}_{\kappa }$ has the identity morphism

also pulled back along . We represent the loop $\beta$ by a path $\gamma \in P_xX$ with $\gamma (1)=\beta (0)$ and paths $\gamma _k \in PX$ with $\gamma _k(0)=\beta (0)$ and $\gamma _k(1)=\beta (\frac{1}{2})$, related via a thin homotopy . In Z we consider the retracting paths $\alpha _i$ with $\alpha _i(0)=(0,\mathrm {id}\star \gamma )$ and $\alpha _i(1)=(\frac{1}{2},\gamma _i \star \gamma )$. Then, the prescription is

$$\begin{aligned} p =\left( \beta , \exp 2\pi \mathrm {i}\left( - \int _{\alpha _2 \star \overline{\alpha _1}} {\mathrm {pr}}^{*}\kappa \right) \right) =\left( \beta , \exp 2\pi \mathrm {i}\left( \int _{\alpha _1} {\mathrm {pr}}^{*}\kappa - \int _{\alpha _2 } {\mathrm {pr}}^{*}\kappa \right) \right) \text {.} \end{aligned}$$

Since the paths $\alpha _i$ are retractions and the path splitting $\kappa$ is contractible, both integrals vanish separately. Thus, we have $p=(\beta ,1)$; this yields the claimed equality. $\square$

Table of terminology

Fusion product		Definition 2.1 (a)
—Multiplicative	$\lambda$ is a group homomorphism	Definition 2.1 (b)
Thin homotopy equivariant structure
	for thin homotopic loops $\tau _0,\tau _1$	Definition 2.2 (a)
—Multiplicative	d is a group homomorphism	Definition 2.2 (b)
—Compatible	d is connection-preserving	Definition 2.4 (a)
—Symmetrizing	Rotation by an angle of $\pi$ switches factors under fusion	Definition 2.4 (b)
—Fusive	Compatible and symmetrizing	Definition 2.4 (c)
Bundle morphism
—Fusion-preserving	$\varphi (\lambda (p\otimes q))=\lambda '(\varphi (p) \otimes \varphi (q))$	Definition 2.1 (b)
—Thin	$\varphi (d_{\tau _0,\tau _1}(p))=d'_{\tau _0,\tau _1}(\varphi (p))$	Definition 2.2 (b)
Connection
—Thin	Induces a thin homotopy equivariant structure	Definition 4.1 (a)
—Superficial	Thin and its holonomy is thin homotopy invariant	Definition 4.1 (b)

—Compatible	Fusion product is connection-preserving	Definition 4.2 (a)
—Symmetrizing	Induced thin homotopy equivariant structure is symmetrizing	Definition 4.2 (b)
—Fusive	Compatible and symmetrizing	Definition 4.2 (c)
Path splitting of $\epsilon \in \Omega ^k(\textit{LG} \times \textit{LG} )$
	$\kappa \in \Omega ^k(\textit{PG} \times \textit{PG})$ with $\epsilon _{\gamma _1\cup \gamma _2,\gamma _1' \cup \gamma _2'} = \kappa _{\gamma _1,\gamma _1'} - \kappa _{\gamma _2,\gamma _2'}$	Definition 4.4
—Multiplicative	$\kappa _{\gamma _1\gamma _1',\gamma _2\gamma _2'} = \kappa _{\gamma _1,\gamma _2}+\kappa _{\gamma _1',\gamma _2'}$	Definition 4.4
—Contractible	$\int _{\phi _\gamma } \kappa = 0$, where $\phi _\gamma$ is the contraction of a path $\gamma$	Definition 4.4
Thin structure	Thin homotopy equivariant structure induced by a superficial connection	Definition 4.3
—Fusive	An inducing connection is fusive	Definition 4.3
—Multiplicative	The error 1-form of an inducing connection admits a multiplicative and contractible path splitting
—Fusive and multiplicative	There is an inducing connection with both of above properties	Definition 4.6
Central extension of $\textit{LG}$
—Thin fusion	Equipped with a multiplicative fusion product and a fusive and multiplicative thin structure	Definition 4.7