Logarithmic Connections, WZNW Action, and Moduli of Parabolic Bundles on the Sphere

Moduli spaces of stable parabolic bundles of parabolic degree 0 over the Riemann sphere are stratified according to the Harder–Narasimhan filtration of underlying vector bundles. Over a Zariski open subset $$\mathscr {N}_{0}$$ N 0 of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function $$\mathscr {S}$$ S is defined as the regularized critical value of the non-compact Wess–Zumino–Novikov–Witten action functional. The definition of $$\mathscr {S}$$ S depends on a suitable notion of parabolic bundle ‘uniformization map’ following from the Mehta–Seshadri and Birkhoff–Grothendieck theorems. It is shown that $$-\mathscr {S}$$ - S is a primitive for a (1,0)-form $$\vartheta $$ ϑ on $$\mathscr {N}_{0}$$ N 0 associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that $$-\mathscr {S}$$ - S is a Kähler potential for $$(\Omega -\Omega _{\mathrm {T}})|_{\mathscr {N}_{0}}$$ ( Ω - Ω T ) | N 0 , where $$\Omega $$ Ω is the Narasimhan–Atiyah–Bott Kähler form in $$\mathscr {N}$$ N and $$\Omega _{\mathrm {T}}$$ Ω T is a certain linear combination of tautological (1, 1)-forms associated with the marked points. These results provide an explicit relation between the cohomology class $$[\Omega ]$$ [ Ω ] and tautological classes, which holds globally over certain open chambers of parabolic weights where $$\mathscr {N}_{0} = \mathscr {N}$$ N 0 = N .

The symplectic structure for renormalization of circle diffeomorphisms with breaks

The main goal of this paper is to reveal the symplectic structure related to renormalization of circle maps with breaks. We first show that iterated renormalizations of [Formula: see text] circle diffeomorphisms with [Formula: see text] breaks, [Formula: see text], with given size of breaks, converge to an invariant family of piecewise Möbius maps, of dimension [Formula: see text]. We prove that this invariant family identifies with a relative character variety [Formula: see text] where [Formula: see text] is a [Formula: see text]-holed torus, and that the renormalization operator identifies with a sub-action of the mapping class group [Formula: see text]. This action allows us to introduce the symplectic form which is preserved by renormalization. The invariant symplectic form is related to the symplectic form described by Guruprasad et al. [Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J. 89(2) (1997) 377–412], and goes back to the earlier work by Goldman [The symplectic nature of fundamental groups of surfaces, Adv. Math. 54(2) (1984) 200–225]. To the best of our knowledge the connection between renormalization in the nonlinear setting and symplectic dynamics had not been brought to light yet.

Finite Generation of the Algebra of Type A Conformal Blocks via Birational Geometry

We study the birational geometry of the moduli space of parabolic bundles over a projective line, in the framework of Mori’s program. We show that the moduli space is a Mori dream space. As a consequence, we obtain the finite generation of the algebra of type A conformal blocks. Furthermore, we compute the H-representation of the effective cone that was previously obtained by Belkale. For each big divisor, the associated birational model is described in terms of moduli space of parabolic bundles.

Metric Properties of Parabolic Ample Bundles

We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern–Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Griffiths conjecture in the parabolic setting and prove some results that provide evidence in its favor for certain kinds of parabolic bundles. For these kinds of parabolic structures, we prove that the conjecture holds on Riemann surfaces. We also prove that a Berndtsson-type result holds and that there are metrics on stable bundles over surfaces whose Schur forms are positive.