## The definition of conformal field theory.(English)Zbl 0657.53060

Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165-171 (1988).
[For the entire collection see Zbl 0646.00009.]
A category $${\mathcal C}$$ is introduced with a sequence $$C_0, C_1, C_2,\ldots,$$. where $$C_n$$ is a disjoint union of $$n$$ oriented circles. A morphism $$C_n\to C_m$$ is a Riemann surface bounded by the corresponding circles (which geometrically looks like a surface connecting $$n$$ circles with $$m$$ circles by a surface – or connecting $$n$$ closed strings with $$m$$ closed strings – in physical application to string theory). The conformal field theory in dimension 2 (one space and one time dimension), which is related to string theory, is then defined as a representation of $${\mathcal C}$$ in a complex Hilbert space.

### MSC:

 53C80 Applications of global differential geometry to the sciences 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 30F99 Riemann surfaces

### Keywords:

Riemann surface; conformal field theory; string theory

Zbl 0646.00009