Central extensions of gauge transformation groups of

higher gerbes

Kiyonori GOMI

§Introduction

Purpose

M ⋯ an oriented compact smooth manifold

(dim M = 2n + 1, ∂M = ∅)

G := H^{n+1} (M, ℤ(n + 1)_{D}^{∞}) ⋯ the smooth Deligne cohomology

(the gauge transformation group of an “n-gerbe” over M)

[figure]

We construct a central extension Ĝ of G by 𝕋 = {u ∈ ℂ| |u| = 1}:

1 ⟶ 𝕋 ⟶ Ĝ ⟶ G ⟶ 1. [figure]

We use cup product and integration for smooth Deligne cohomology groups.

Contents The program is as follows: §Smooth Deligne cohomology §Cup product and integration §Central extension Ĝ §Example §When Ĝ is non-trivial?

§Smooth Deligne cohomology

Definition

X ⋯ a smooth manifold
A^{q} ⋯ the sheaf of q-forms on X
[figure]

The smooth Deligne cohomology group (order p and degree m)

H^{m}(X, ℤ(p)_{D}^{∞})

is defined to be the mth hypercohomology group of the complex of sheaves:

ℤ →^{i} A^{0} →^{d} A^{1} →^{d} ⋯ →^{d} A^{p−1} → 0 → · · · . [figure]

Basic properties

H^{0}(X, ℤ(0)_{D}^{∞}) ≅ H^{0}(X, ℤ),
H^{1}(X, ℤ(1)_{D}^{∞}) ≅ C^{∞}(X, 𝕋),
H^{2}(X, ℤ(2)_{D}^{∞}) ≅ the group of isomorphism

classes of 𝕋-bundles with connection over X , [Brylinski]

H^{3}(X, ℤ(3)_{D}^{∞}) ≅ the group of isomorphism

classes of gerbes with connection over X , [Brylinski]

[figure]

H^{n+2} (X, ℤ(n + 2)_{D}^{∞}) is isomorphic to the group of isomorphism classes of “n-gerbes with connection” over X.

[figure]

⋯ a geometric interpretation We have an exact sequence:

H^{n+1} (X, ℤ(n + 1)_{D}^{∞})

↓ A^{n+1} (X)

↓

H^{n+2} (X, ℤ(n + 2)^{∞})

↓ H^{n+2} (X, ℤ)

↓ 0

We remark:

H^{n+2} (X, ℤ) ≅ {“n-gerbes” over X }/iso ,
A^{n+1} (X) := the group of (n + 1)-forms
≅ \{ \begin{matrix}“connections” on an \\ “n-gerbe” over X \end{matrix}  \} .
[figure]

We can regard H^{n+1} (X, ℤ(n + 1)_{D}^{∞}) as the gauge transformation group of an “ngerbe” over X.

[figure]

⋯ another interpretation

§Cup product and integration

Analogies of the cup product and the integration for the ordinary cohomology groups.

Cup product ⋯ a homomorphism

H^{p}(X, ℤ(p)_{D}^{∞}) ⊗_{ℤ} H^{q}(X, ℤ(q)_{D}^{∞})
∪↓
H^{p+q} (X, ℤ(p + q)_{D}^{∞}).

graded commutative: f ∪ g = (−1)^{pq}g ∪ f. associative: (f ∪ g) ∪ h = f ∪ (g ∪ h).

Integration ⋯ a homomorphism

∫_{X} : H^{d+1} (X, ℤ(d + 1)_{D}^{∞}) ⟶ ℝ/ℤ,

where X is an oriented compact smooth ddimensional manifold without boundary.

§Central extension

M ⋯ an oriented compact smooth manifold

(dim M = 2n + 1, ∂M = ∅) We put G := H^{n+1} (M, ℤ(n + 1)_{D}^{∞}).

[figure]

We define γ : G × G → 𝕋 by

γ(f, g) = exp 2π√−1 ∫_{M} f ∪ g. [figure]

H^{n+1} (M, ℤ(n + 1)_{D}^{∞}) ⊗_{ℤ} H^{n+1} (M, ℤ(n + 1)_{D}^{∞})

↓^{∪}
H^{2n+2} (M, ℤ(2n + 2)_{D}^{∞})
↓_{∫M}
ℝ/ℤ
↓_{exp 2π√−1}
𝕋

γ(f, g) = exp 2π√−1 ∫_{M} f ∪ g. Theorem 1. γ is a 2-cocycle of G:

γ(f, g)γ(f + g, h) = γ(f, g + h)γ(g, h).

Proof. Because ∫_{M} is a homomorphism, the apparent formula

f ∪ g + (f + g) ∪ h = f ∪ (g + h) + g ∪ h. gives Theorem 1.

[figure]

We define our central extension

1 ⟶ 𝕋 ⟶ Ĝ ⟶ G ⟶ 1

by twisting the trivial product of G × 𝕋:

\{ \begin{matrix}Ĝ = G × 𝕋, \\ (f, u) · (g, v) = (f + g, γ(f, g)uv). \end{matrix}
[figure]

§Example

The case of n = 0 and M = S^{1} .

G = H^{1}(S^{1}, ℤ(1)_{D}^{∞}) = C^{∞}(S^{1}, 𝕋) = L𝕋 ⋯ The loop group of 𝕋.

L̂𝕋 ⋯ The universal central extension of L𝕋. [Pressley-Segal ]

Proposition 1. [Brylinski-McLaughlin ]

Ĝ ≅ L̂𝕋/ℤ_{2}.
ℤ_{2} = {±1} ⊂ 𝕋 ⊂ L̂𝕋

For n > 1, Ĝ is thought of as a generalization of the central extension of the loop group, though may be trivial.

§When Ĝ is non-trivial?

Our central extension (dim M = 2n + 1):

\begin{matrix}Ĝ = H^{n+1} (M, ℤ(n + 1)_{D}^{∞}) × 𝕋 \\ (f, u) · (g, v) = (f + g, γ(f, g)uv) \end{matrix}

Theorem 2. n : odd ⇒ Ĝ is trivial .

\begin{aligned}1 & ⟶ & 𝕋 & ⟶ & Ĝ & ⟶ & G & ⟶ & 1 \\ & & ∥ & & _{∃}↓≅ & & ∥ \\ 1 & ⟶ & 𝕋 & ⟶ & G × 𝕋 & ⟶ & G & ⟶ & 1 \end{aligned} ; (f, u) · (g, v) = (f + g, uv)

The proof of Theorem 2 uses a recent result of Hopkins-Singer (AT/0211216).

Our central extension (dim M = 2n + 1):

\begin{matrix}Ĝ = H^{n+1} (M, ℤ(n + 1)_{D}^{∞}) × 𝕋 \\ (f, u) · (g, v) = (f + g, γ(f, g)uv) \end{matrix}

Theorem 3. n : even ⇒ Ĝ is non-trivial .

Proof. We can show Ĝ is non-abelian. ⇒ Ĝ is not isomorphic to G×𝕋 as a group.

Non-trivial Ĝ for \{ \begin{matrix}1-dimensional \\ 5-dimensional \\ 9-dimensional \\ ⋯ \end{matrix} manifolds.

[figure]

Possible application: TQFT, M-theory

(Representation theory of Ĝ)

[figure]