[Paper Review] Projective unitary representations of smooth Deligne cohomology groups

Kiyonori Gomi|ArXiv.org|Oct 10, 2005

Advanced Algebra and Geometry6 references4 citations

TL;DR

This paper constructs and classifies projective unitary representations of smooth Deligne cohomology groups for compact oriented $(4k+1)$-dimensional Riemannian manifolds, generalizing positive energy representations of loop groups. It establishes that admissible representations decompose into finite direct sums of irreducible representations, with the number of equivalence classes of irreducible representations being $2^b r$, where $b$ is the $(2k+1)$st Betti number and $r$ is the number of 2-torsion elements in $H^{2k+1}(M,\mathbb{Z})$. The construction relies on harmonic splitting and central extensions via the cup product and integration in Deligne cohomology.

ABSTRACT

We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4k+1, generalizing positive energy representations of the loop group of the circle. We also classify such representations under a certain condition. The number of the equivalence classes of irreducible representations is finite, and is determined by the cohomology of the manifold.

Motivation & Objective

• To generalize positive energy representations of loop groups to higher-dimensional smooth Deligne cohomology groups.
• To construct projective unitary representations of the smooth Deligne cohomology group $\mathcal{G}(M) = H^{2k+1}(M,\mathbb{Z}(2k+1)^\infty_D)$ for compact oriented $(4k+1)$-manifolds.
• To classify admissible representations under a specific condition, determining the number of equivalence classes of irreducible representations.
• To establish a correspondence between the cohomology of the manifold and the structure of these representations.

Proposed method

• Uses the cup product and integration in smooth Deligne cohomology to define a natural group 2-cocycle $S_M: \mathcal{G}(M) \times \mathcal{G}(M) \to \mathbb{R}/\mathbb{Z}$, yielding a central extension $\tilde{\mathcal{G}}(M)$.
• Applies harmonic splitting of differential forms to decompose $\mathcal{G}(M)$ into a free part $\mathcal{G}^0(M) = A^{2k}(M)/A^{2k}(M)_{\mathbb{Z}}$ and a torsion part.
• Constructs irreducible projective unitary representations $\rho_\lambda$ of $\mathcal{G}^0(M)$ for each $\lambda \in \mathcal{X}(M) = \mathrm{Hom}(\mathbb{H}^{2k}(M)/\mathbb{H}^{2k}(M)_{\mathbb{Z}}, \mathbb{R}/\mathbb{Z})$, using a Heisenberg-type group construction.
• Defines admissible representations as those equivalent to a Hilbert direct sum of finitely many copies of the $\rho_\lambda$'s, with multiplicity $m(\lambda) \in \mathbb{Z}_{\geq 0}$.
• Employs Schur's lemma and spectral theory to prove that irreducible representations are uniquely determined by their central character and that intertwiners are scalar multiples.
• Uses the topology on the central extension induced by the $L^2$-norm on differential forms to ensure continuity of the representations.

Experimental results

Research questions

• RQ1 How can positive energy representations of loop groups be generalized to higher-dimensional smooth Deligne cohomology groups?
• RQ2 What is the structure of projective unitary representations of $\mathcal{G}(M)$ for a compact oriented $(4k+1)$-dimensional manifold $M$?
• RQ3 How many equivalence classes of irreducible admissible representations exist, and what determines their number?
• RQ4 What role does the harmonic splitting of differential forms play in constructing and classifying these representations?

Key findings

• Admissible representations of $\mathcal{G}(M)$ decompose into finite direct sums of irreducible admissible representations.
• The number of equivalence classes of irreducible admissible representations is $2^b r$, where $b = b_{2k+1}(M)$ is the $(2k+1)$st Betti number and $r = \#\{t \in H^{2k+1}(M,\mathbb{Z}) \mid 2t = 0\}$.
• For $k=0$ and $M = S^1$, the number of irreducible admissible representations is 2, matching the number of irreducible positive energy representations of level 2 for $L\mathbb{T}$.
• Irreducible representations $\rho_\lambda$ exist for each $\lambda \in \mathcal{X}(M)$, and are inequivalent, with the cocycle $e^{2\pi i S_M}$ realized via harmonic splitting.
• The classification is complete: every admissible representation is equivalent to a finite direct sum of these irreducible representations.
• The central extension $\tilde{\mathcal{G}}(M)$ is topologized via the $L^2$-norm on differential forms, ensuring continuity of the representations.