Deriving projective hyperspace from harmonic

Abstract

We derive actions for projective N=2 superspace ("hyperspace") from those for harmonic hyperspace, including that for nonabelian Yang-Mills (a new result). The method uses Wick rotation of the sphere from complex conjugate coordinates to real, null ones, which can be treated as independent. The result can be considered "holographic" in that the dimension of the internal (R-symmetry) space is reduced from 2 to 1, by solving equations of motion or gauge conditions for dependence on the other coordinate. The auxiliary nature of the redundant dimension makes the hypergraph rules and evaluation almost identical.

Note on massive scalar hypermultiplet in projective hyperspace

“…These are almost the same as those given in reference [6]. The only difference is the following modified identity: The non-renormalization theorem for massless scalar hypermultiplet holds for the massive case also for straightforward reasons.…”

“…We use the conventions of [6] for the superspace coordinates and derivatives. The centrally extended algebra of covariant derivatives then reads 1 :…”

Supersymmetric Sigma Model Geometry

“…It is really a misnomer in that it is unrelated to the usual definition of projective spaces) [6][7][8][9][10][11][12][13][14][15][16][17][18][19] has been developed independent of harmonic superspace [69]. The relation between the two approaches was first discussed in [70] and more recently in [71]. A key reference for this section is [72] and the review [73].…”

Improved methods for hypergraphs

“…We note that the closely related [4,5] N=2 harmonic superspace ( ) [6] doesn't encounter this issue as the hyperfield, V (++) describing the SYM multiplet is itself a connection, Aȳ. In fact, background field formalism in harmonic superspace has quite a straightforward construction [7].…”