Quantum Field Theory Applications of Heun Type Functions

After a brief introduction to Heun type functions we note that the actual solutions of the eigenvalue equation emerging in the calculation of the one loop contribution to QCD from the Belavin-Polyakov-Schwarz-Tyupkin instanton and the similar calculation for a Dirac particle coupled to a scalar $CP^1$ model in two dimensions can be given in terms of confluent Heun equation in their original forms. These equations were previously modified to be solved by more elementary functions. We also show that polynomial solutions with discrete eigenvalues are impossible to find in the unmodified equations.