The role of Sp(2n) duality in quantum theory

Abstract

D-dimensional maximal supergravities type I with G/H coset spaces have global G-symmetry and local H symmetry, which can be gauge-fixed in symmetric or Iwasawa-type gauges. Maximal D-dimensional supergravities type II derived from higher dimensions without dualization have less local and global symmetries. In 4D, Gaillard-Zumino duality group Sp(56) enhances U-duality symmetry group $E_{7(7)}$. Using the Hamiltonian path integral, we show how extra symmetries beyond $E_{7(7)}$, serve to establish S-matrix equivalence of supergravities I and II in different gauges. Enhanced dualities are not available in D $> 4$. This is consistent with the existence of local H symmetry and global G symmetry anomalies and UV divergences in D $> 4$ supergravities, and with the absence of these anomalies and UV divergences, so far, in 4D ${\cal N}\geq 5$ supergravities.