1 Introduction
5
FT of a modular operad
geom. vertices
alg. vertices
Chains on a moduli space
CFT
Endomorphism operad
Fig. 1.1 Construction of string field theory in terms of morphisms of (odd modular) operads
differential vector space V , the state space of the closed conformal field theory, i.e.,
of the fist quantized closed bosonic string.
As already mentioned above, the bosonic quantum open-closed string theory can
be constructed similarly. Furthermore, we believe that also superstring field theories
could in principle be constructed in this way, here the difficult part is the horizontal
arrow. It is not obvious how to mimic Zwiebach’s construction of geometric vertices
to the case of super Riemann surfaces. Finally, since any quantum field theory can
be formulated as a BV theory, it should allow for an interpretation/construction
according to our commutative triangle. Probably, the moduli spaces of tropical
curves and world line formulation of quantum field theory (QTF) should lead to
this as well.