We prove a canonical identifications between the spaces of generalized theta functions on the moduli spaces of anti-invariant vector bundles in the ramified case and the conformal blocks associated to twisted Kac-Moody affine algebras. We also show a strange duality on level one in the unramiffied case, this gives the dimensions of the spaces of generalized theta functions of level one.

Let X be a smooth irreducible projective curve with an involution σ. A vector bundle E over X is called anti-invariant if there exists an isomorphism σ * E → E * . In this paper, we give a construction of the moduli spaces of anti-invariant vector bundles over X.Date: October 28, 2019.

Let X be a smooth irreducible projective curve. In this notes, we generalize the main result of [PPN18] to principal G−bundles for any semisimple linear algebraic group G. After defining very stability of principal G−bundles, we show that this definition is equivalent to the fact that the Hitchin fibration restricted to the space of Higgs fields on that principal bundle is finite. We also study the relation between very stability and other stability conditions in the case of SL 2 −bundles.

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