Let G be a simple Lie group and let G(ℂ((t))) be its loop group.

The Lie algebra g[[t]][t^-1] has a well known central extension (see e.g. Wikipedia) given by the cocycle
c(f,g) = Res₀ < f dg >. Here, < > : g⊗g→ℂ denotes some invariant bilinear form on g, and f dg is the (g⊗g)-valued differential given by multiplying f and dg.

Question: It there a similarly concrete cocycle for the central extension of G(ℂ((t))) by ℂ*?

To give you an idea of what I'm looking for, let me show you a cocycle for central extension by S¹ of the smooth loop group $LG = \mathit{Map} _ {C^\infty} (S^1,G)$ of a compact Lie group $G$.

Pick a bounding disc D_γ : D² → G for each element γ ∈ LG. The cocycle is then given by

$$c(\gamma,\delta) = exp\big(i\cdot\big(\quad\int \langle D_\gamma^*\theta_L,D_\delta^*\theta_R\rangle +\int H^*\eta\quad \big)\big)$$

where $\theta_L,\theta_R\in\Omega(G,\mathfrak{g})$ are the Maurer-Cartan 1-forms, $\eta\in\Omega^3(G)$ is the Cartan 3-form,
and $H:D^3\to G$ in a homotopy between $D_\gamma D_\delta$ and $D _ {\gamma\delta}$.

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  References:
The cocycle for the smooth loop group can be found on page 19 of the paper
From Loop groups to 2-groups, by Baez, Crans, Schreiber, and Stevenson,
and also on page 8 of Mickelsson's paper [From Gauge anomalies to Gerbes and Gerbal actions][1].

Let $G$ be a simple Lie group and let $G(\mathbb{C}((t)))$ be its loop group.

The Lie algebra $\mathfrak{g}[[t]][t^{-1}]$ has a well known central extension (see e.g. Wikipedia) given by the cocycle
$c(f,g) = Res_0\langle f,dg\rangle$. Here, $\langle\ ,\ \rangle \colon \mathfrak{g}\otimes \mathfrak{g}\to\mathbb{C}$ denotes some invariant bilinear form on $\mathfrak{g}$, and $f dg$ is the $\mathfrak{g}\otimes \mathfrak{g}$-valued differential given by multiplying $f$ and dg.

Question: It there a similarly concrete cocycle for the central extension of $G(\mathbb{C}((t)))$ by $\mathbb{C}^\ast$?

To give you an idea of what I'm looking for, let me show you a cocycle for central extension by $S^1$ of the smooth loop group $LG = \mathop{Map} _ {C^\infty} (S^1,G)$ of a compact Lie group $G$.

Pick a bounding disc $D_\gamma$ : $D^2 \to G$ for each element $\gamma\in LG$. The cocycle is then given by

$$ c(\gamma,\delta) = \exp\left(i\int \langle D_\gamma^*\theta_L,D_\delta^*\theta_R\rangle +i\int H^*\eta\right) $$

where $\theta_L,\theta_R\in\Omega(G,\mathfrak{g})$ are the Maurer-Cartan 1-forms, $\eta\in\Omega^3(G)$ is the Cartan 3-form,
and $H:D^3\to G$ in a homotopy between $D_\gamma D_\delta$ and $D _ {\gamma\delta}$.

 

---

References:
The cocycle for the smooth loop group can be found on page 19 of the paper
From Loop groups to 2-groups, by Baez, Crans, Schreiber, and Stevenson,
and also on page 8 of Mickelsson's paper From Gauge anomalies to Gerbes and Gerbal actions.

Let G be a simple Lie group and let G(ℂ((t))) be its loop group.

The Lie algebra g[[t]][t^-1] has a well known central extension (see e.g. Wikipedia) given by the cocycle
c(f,g) = Res₀ < f dg >. Here, < > : g⊗g→ℂ denotes some invariant bilinear form on g, and f dg is the (g⊗g)-valued differential given by multiplying f and dg.

Question: It there a similarly concrete cocycle for the central extension of G(ℂ((t))) by ℂ*?

To give you an idea of what I'm looking for, let me show you a cocycle for central extension by S¹ of the smooth loop group $LG = \mathit{Map} _ {C^\infty} (S^1,G)$ of a compact Lie group $G$.

Pick a bounding disc D_γ : D² → G for each element γ ∈ LG. The cocycle is then given by

$$c(\gamma,\delta) = exp\big(i\cdot\big(\quad\int \langle D_\gamma^*\theta_L,D_\delta^*\theta_R\rangle +\int H^*\eta\quad \big)\big)$$

where $\theta_L,\theta_R\in\Omega(G,\mathfrak{g})$ are the Maurer-Cartan 1-forms, $\eta\in\Omega^3(G)$ is the Cartan 3-form,
and $H:D^3\to G$ in a homotopy between $D_\gamma D_\delta$ and $D _ {\gamma\delta}$.

---

Reference:
The cocycle for the smooth loop group can be found on page 19 of the paper
From Loop groups to 2-groups, by Baez, Crans, Schreiber, and Stevenson.

Let G be a simple Lie group and let G(ℂ((t))) be its loop group.

The Lie algebra g[[t]][t^-1] has a well known central extension (see e.g. Wikipedia) given by the cocycle
c(f,g) = Res₀ < f dg >. Here, < > : g⊗g→ℂ denotes some invariant bilinear form on g, and f dg is the (g⊗g)-valued differential given by multiplying f and dg.

Question: It there a similarly concrete cocycle for the central extension of G(ℂ((t))) by ℂ*?

To give you an idea of what I'm looking for, let me show you a cocycle for central extension by S¹ of the smooth loop group $LG = \mathit{Map} _ {C^\infty} (S^1,G)$ of a compact Lie group $G$.

Pick a bounding disc D_γ : D² → G for each element γ ∈ LG. The cocycle is then given by

$$c(\gamma,\delta) = exp\big(i\cdot\big(\quad\int \langle D_\gamma^*\theta_L,D_\delta^*\theta_R\rangle +\int H^*\eta\quad \big)\big)$$

where $\theta_L,\theta_R\in\Omega(G,\mathfrak{g})$ are the Maurer-Cartan 1-forms, $\eta\in\Omega^3(G)$ is the Cartan 3-form,
and $H:D^3\to G$ in a homotopy between $D_\gamma D_\delta$ and $D _ {\gamma\delta}$.

---

References:
The cocycle for the smooth loop group can be found on page 19 of the paper
From Loop groups to 2-groups, by Baez, Crans, Schreiber, and Stevenson,
and also on page 8 of Mickelsson's paper [From Gauge anomalies to Gerbes and Gerbal actions][1].

Let G be a simple Lie group and let G(ℂ((t,t ^-1))) be its loop group.

The Lie algebra g((t,t  ^-1)) has a well known central extension (see e.g. Wikipedia) given by the cocycle
c(f,g) = Res₀ < f dg >. Here, < > : g⊗g→ℂ denotes some invariant bilinear form on g, and f dg is the (g⊗g)-valued differential given by multiplying f and dg.

Question: It there a similarly concrete cocycle for the central extension of G(ℂ((t,t ^-1))) by ℂ*?

To give you an idea of what I'm looking for, let me show you a cocycle for central extension by S¹ of the smooth loop group $LG = \mathit{Map} _ {C^\infty} (S^1,G)$ of a compact Lie group $G$.

Pick a bounding disc D_γ : D² → G for each element γ ∈ LG. The cocycle is then given by

$$c(\gamma,\delta) = exp\big(i\cdot\big(\quad\int \langle D_\gamma^*\theta_L,D_\delta^*\theta_R\rangle +\int H^*\eta\quad \big)\big)$$

where $\theta_L,\theta_R\in\Omega(G,\mathfrak{g})$ are the Maurer-Cartan 1-forms, $\eta\in\Omega^3(G)$ is the Cartan 3-form,
and $H:D^3\to G$ in a homotopy between $D_\gamma D_\delta$ and $D _ {\gamma\delta}$.

---

Reference:
The cocycle for the smooth loop group can be found on page 19 of the paper
From Loop groups to 2-groups, by Baez, Crans, Schreiber, and Stevenson.

Let G be a simple Lie group and let G(ℂ((t))) be its loop group.

The Lie algebra g[[t]][t^-1] has a well known central extension (see e.g. Wikipedia) given by the cocycle
c(f,g) = Res₀ < f dg >. Here, < > : g⊗g→ℂ denotes some invariant bilinear form on g, and f dg is the (g⊗g)-valued differential given by multiplying f and dg.

Question: It there a similarly concrete cocycle for the central extension of G(ℂ((t))) by ℂ*?

To give you an idea of what I'm looking for, let me show you a cocycle for central extension by S¹ of the smooth loop group $LG = \mathit{Map} _ {C^\infty} (S^1,G)$ of a compact Lie group $G$.

Pick a bounding disc D_γ : D² → G for each element γ ∈ LG. The cocycle is then given by

$$c(\gamma,\delta) = exp\big(i\cdot\big(\quad\int \langle D_\gamma^*\theta_L,D_\delta^*\theta_R\rangle +\int H^*\eta\quad \big)\big)$$

where $\theta_L,\theta_R\in\Omega(G,\mathfrak{g})$ are the Maurer-Cartan 1-forms, $\eta\in\Omega^3(G)$ is the Cartan 3-form,
and $H:D^3\to G$ in a homotopy between $D_\gamma D_\delta$ and $D _ {\gamma\delta}$.

---

Reference:
The cocycle for the smooth loop group can be found on page 19 of the paper
From Loop groups to 2-groups, by Baez, Crans, Schreiber, and Stevenson.