Highest weight generating functions for Hilbert series. (English) Zbl 1333.81401
Summary: We develop a new method for representing Hilbert series based on the highest weight Dynkin labels of their underlying symmetry groups. The method draws on plethystic functions and character generating functions along with Weyl integration. We give explicit examples showing how the use of such highest weight generating functions (”HWGs”) permits an efficient encoding and analysis of the Hilbert series of the vacuum moduli spaces of classical and exceptional SQCD theories and also of the moduli spaces of instantons. We identify how the HWGs of gauge invariant operators of a selection of classical and exceptional SQCD theories result from the interaction under symmetrisation between a product group and the invariant tensors of its gauge group. In order to calculate HWGs, we derive and tabulate character generating functions for low rank groups by a variety of methods, including a general character generating function, based on the Weyl Character Formula, for any classical or exceptional group.
MSC:
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
Software:
LieARTReferences:
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