# Tree algebras: An algebraic axiomatization of intertwining vertex operators

## Igor Kriz and Yang Xiu

Address:

Igor Kriz, Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109-1043, U.S.A.

Yang Xiu, Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000 U.S.A.

E-mail: kriz.igor@gmail.com

Abstract: We describe a completely algebraic axiom system for intertwining operators of vertex algebra modules, using algebraic flat connections, thus formulating the concept of a tree algebra. Using the Riemann-Hilbert correspondence, we further prove that a vertex tensor category in the sense of Huang and Lepowsky gives rise to a tree algebra over $\mathbb{C}$. We also show that the chiral WZW model of a simply connected simple compact Lie group gives rise to a tree algebra over $\mathbb{Q}$.

AMSclassification: primary 17B69; secondary 81T40, 35Q15.

Keywords: vertex algebra, Riemann-Hilbert correspondence, D-module, KZ-equations, WZW-model.

DOI: 10.5817/AM2012-5-353