Hyperkähler sigma models on cotangent bundles of Hermitian symmetric spaces using projective superspace

Abstract

Kähler manifolds have a natural hyperkähler structure associated with (part of) its cotangent bundle. Using projective superspace, we construct four-dimensional N = 2 models on the tangent bundles of some classical Hermitian symmetric spaces (specifically, the four regular series of irreducible compact symmetric Kähler manifolds, and their non-compact versions). A further dualization yields the Kähler potential for the hyperkähler metric on the cotangent bundle.

Cotangent bundle over all the compact Hermitian symmetric spaces and projective superspace

“…We checked that the results obtained by this formula for all the compact Hermitian symmetric spaces perfectly coincide with the previous results in [9,10,12,13].…”

“…(6.36). In the former methods in [9,10,12,13], the Legendre transform is used to obtain a cotangent bundle action. This procedure, although in principle applicable to the E 7 /(E 6 ×U (1)) case, is not straightforward and one often must find a suitable ansatz to solve involved algebraic equations.…”

“…With the use of this formalism, first the SUSY NLSMs on the tangent bundles over the compact and non-compact classical Hermitian symmetric as well as, using the generalized Legendre transform [4], the cotangent bundles corresponding to the hyperkähler metrics have been constructed [9,10,11,12]. Developed the results in [9,10,11,12], the cotangent bundle over the compact type of E 6 /SO(10) × U (1) have been also constructed [13]. After these works, the closed form expression of the cotangent bundle over any compact Hermitian symmetric space was obtained [14], which gives a form of the cotangent bundle presented in a matrix form.…”

Supersymmetric Sigma Model Geometry

“…This section contains an introduction to [26] where the generalized Legendre transform described in the previous section is used to find metrics on the Hermitian symmetric spaces listed in the following table:…”

“…Over the years it has been developed and refined in, e.g., [6][7][8][9][10][11][12][13][14][15][16][17][18][19]. In this article we report on some of that development along with some more recent development, such as projective superspace for supergravity [20][21][22][23][24][25], and applications such as the construction of certain classes of hyperkähler metrics [26][27][28].…”