Rigidity of maximal holomorphic representations of Kähler groups

We investigate representations of Kähler groups [Formula: see text] to a semisimple non-compact Hermitian Lie group [Formula: see text] that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor–Wood inequality similar to those found by Burger–Iozzi and Koziarz–Maubon. Thanks to the study of the case of equality in Royden’s version of the Ahlfors–Schwarz lemma, we can completely describe the case of maximal holomorphic representations. If [Formula: see text], these appear if and only if [Formula: see text] is a ball quotient, and essentially reduce to the diagonal embedding [Formula: see text]. If [Formula: see text] is a Riemann surface, most representations are deformable to a holomorphic one. In that case, we give a complete classification of the maximal holomorphic representations, which thus appear as preferred elements of the respective maximal connected components.

THE STRUCTURE OF REPRESENTATION FOR THE W(3) MINIMAL MODEL

Some exact results in the representation theory of the W(3) algebra are presented. The embedding structure of the completely degenerate representation is studied in detail. The character formula is obtained by the Feigen-Fuchs-Rocha-Caridi method. It is manifestly irreducible and coincides with the branching coefficients of diagonal embedding [Formula: see text] in the unitary case. The W(n) character for general n can also be obtained completely in parallel. Four-point functions and fusion rules are calculated explicitly for the Z3 Potts model as a W(3) minimal theory, which agree with the Verlinde formula.