11 research outputs found
Dualizability of automatic algebras
We make a start on one of George McNulty's Dozen Easy Problems: "Which finite
automatic algebras are dualizable?" We give some necessary and some sufficient
conditions for dualizability. For example, we prove that a finite automatic
algebra is dualizable if its letters act as an abelian group of permutations on
its states. To illustrate the potential difficulty of the general problem, we
exhibit an infinite ascending chain of finite automatic algebras that are alternately dualizable and
non-dualizable
Area-dependent quantum field theory
Area-dependent quantum field theory is a modification of two-dimensional topological quantum field theory, where one equips each connected component of a bordism with a positive real numberβinterpreted as areaβwhich behaves additively under glueing. As opposed to topological theories, in area-dependent theories the state spaces can be infinite-dimensional. We introduce the notion of regularised Frobenius algebras in Hilbert spaces and show that area-dependent theories are in one-to-one correspondence to commutative regularised Frobenius algebras. We also provide a state sum construction for area-dependent theories. Our main example is two-dimensional YangβMills theory with compact gauge group, which we treat in detail