Through a self-dual mapping of the geometry AdS5 ×S5, fermionic T-duality provides a beautiful geometric interpretation of hidden symmetries for scattering amplitudes in super-Yang–Mills. Starting with Green–Schwarz sigma-models, we consolidate developments in this area into this small review. In particular, we discuss the translation of fermionic T-duality into the supergravity fields via pure spinor formalism and show that a general class of fermionic transformations can be identified directly in the supergravity. In addition to discussing fermionic T-duality for the geometry AdS4 × ℂP3, dual to ABJM theory, we review work on other self-dual geometries. Finally, we present a short round-up of studies with a formal interest in fermionic T-duality.
Unrelated to whether geometries exhibit a self-duality transformation incorporating fermionic T-duality, a small body of works studying some formal aspects of fermionic T-duality have appeared in the literature @cite_52 @cite_40 @cite_59 @cite_12 @cite_47 @cite_55 @cite_42 . We now present brief summary of these papers.
@cite_55 @cite_42 @cite_52 @cite_40 @cite_59 @cite_47 @cite_12
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In this paper we investigate the relationship between the so-called fermionic T-duality and the Morita equivalence of noncommutative supertori. We first get an action satisfying the BRST invariance under nonvanishing constant R-R and NS-NS backgrounds in the hybrid formalism. We investigate the effect of bosonic T-duality transformation together with fermionic T-duality transformation in this background and look for the resultant symmetry of transformations. We find that the duality transformations correspond to Morita equivalence of noncommutative supertori. In particular, we obtain the symmetry group ( SO ( 2,2, V _ Z ^0 ) ) in two dimensions, where ( V _ Z ^0 ) denotes Grassmann even number whose body part belongs to ( Z ). We establish that the recently discovered fermionic T-duality can be viewed as a canonical transformation in phase space. This requires a careful treatment of constrained Hamiltonian systems. Additionally, we show how the canonical transformation approach for bosonic T-duality can be extended to include Ramond–Ramond backgrounds in the pure spinor formalism. We study two aspects of fermionic T-duality: the duality in purely fermionic sigma models exploring the possible obstructions and the extension of the T-duality beyond classical approximation. We consider fermionic sigma models as coset models of supergroups divided by their maximally bosonic subgroup OSp(m|n) SO(m) × Sp(n). Using the non-abelian T-duality and a non-conventional gauge fixing we derive their fermionic T-duals. In the second part of the paper, we prove the conformal invariance of these models at one and two loops using the Background Field Method and we check the Ward Identities. In this article we establish the relationship between fermionic T-duality and momenta noncommuativity. This is extension of known relation between bosonic Tduality and coordinate noncommutativity. The case of open string propagating in background of the type IIB superstring theory has been considered. We perform T-duality with respect to the fermionic variables instead to the bosonic ones. We also choose Dirichlet boundary conditions at the string endpoints, which lead to the momenta noncommutativity, instead Neumann ones which lead to the coordinates noncommutativity. Finally, we establish the main result of the article that momenta noncommutativity parameters are just fermionic T-dual fields.