Holomorphic volume forms on representation varieties of surfaces with boundary

Heusener, Michael; Porti, Joan

Annales Henri Lebesgue, Volume 3 (2020), pp. 341-380.

For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the other one is the power of the Atiyah–Bott–Goldman–Narasimhan symplectic form. We introduce an holomorphic volume form on the space of representations of the circle, so that, for surfaces with boundary, it appears as peripheral term in the generalization of Witten’s formula. We compute explicit volume and symplectic forms for some simple surfaces and for the Lie group ${\mathrm{SL}}_N\left(ℂ\right)$.

Pour les surfaces hyperboliques fermées et orientées, une formule de Witten établit une égalité entre deux formes de volume sur l’espace de représentations des groupes de surface dans un groupe de Lie semi-simple. Une de ces formes est une torsion de Reidemeister, l’autre est la forme de volume canoniquement associée à la forme symplectique d’Atiyah–Bott–Goldman–Narasimhan. Nous introduisons une forme de volume holomorphe sur l’espace des représentations du cercle, de sorte que, pour les surfaces à bord, elle apparaisse comme terme périphérique dans la généralisation de la formule de Witten. Pour certaines surfaces simples et pour le groupe de Lie ${\mathrm{SL}}_N\left(ℂ\right)$ nous calculons explicitement les formes volume et les formes symplectiques.

Classification: 53D30, 57M99
Keywords: representation varieties, volume forms

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