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.
In @cite_52 both bosonic and fermionic T-duality in the context of pure spinor heterotic superstring are studied. @cite_40 attempts to treat fermionic T-duality in a background independent matter by defining a supersymmetric sigma-model which is globally invariant with respect to a super-duality group. In @cite_59 it is shown that fermionic T-duality, like its bosonic counterpart, can be viewed as a canonical transformation in phase space. @cite_12 explores extensions of fermionic T-duality beyond the classical approximation. In @cite_47 a connection between fermionic T-duality and the Morita equivalence @cite_58 @cite_16 for noncommutative supertori is established. Noncommutativity in momenta can also be shown to arise as a consequence of fermionic T-duality @cite_55 . Finally, aspects of Dirichlet boundary conditions in the context of fermionic T-duality are touched upon in @cite_42 .
@cite_55 @cite_42 @cite_52 @cite_16 @cite_40 @cite_59 @cite_47 @cite_58 @cite_12
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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. In this article we investigate the relation between consequences of Dirichlet boundary conditions (momenta noncommutativity and parameters of the effective theory) and background fields of fermionic T-dual theory. We impose Dirichlet boundary conditions on the endpoints of the open string propagating in background of type IIB superstring theory. We showed that on the solution of the boundary conditions the momenta become noncommutative, while the coordinates commute. Fermionic Tduality is also introduced and its relation to noncommutativity is considered. We use compact notation so that type IIB superstring formally gets the form of the bosonic one with Grassman variables. Then momenta noncommutativity parameters are fermionic T-dual fields. The effective theory, the initial theory on the solution of boundary conditions, is bilinear in the effective coordinates, odd under world-sheet parity transformation. The effective metric is equal to the initial one and terms with the effective Kalb-Ramond field vanish. In this note we study the preservation of the classical pure spinor BRST constraints under super T-duality transformations. We also determine the invariance of the one-loop conformal invariance and of the local gauge and Lorentz anomalies under the super T-dualities. It was shown by Connes, Douglas, Schwarz [hep-th 9711162] that one can compactify M(atrix) theory on a non-commutative torus To. We prove that compactifications on Morita equivalent tori are in some sense physically equivalent. This statement can be considered as a generalization of non-classical SL(2,Z)N duality conjectured by Connes, Douglas and Schwarz for compactifications on two-dimensional non-commutative tori. Abstract We study the dualities for sigma models with fermions and bosons. We found that the generalization of the SO ( m , m ) duality for D = 2 sigma models and the Sp ( 2 n ) duality for D = 4 sigma models is the orthosymplectic duality OSp ( m , m | 2 n ) . We study the implications of this and we derive the most general D = 2 sigma model, coupled to fermionic and bosonic one-forms, with such dualities. To achieve this we generalize Gaillard–Zumino analysis to orthosymplectic dualities, which requires to define embedding of the superisometry group of the target space into the duality group. We finally discuss the recently proposed fermionic dualities as a by-product of our construction. 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. 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 ). One can describe an @math -dimensional noncommutative torus by means of an antisymmetric @math -matrix @math . We construct an action of the group @math on the space of antisymmetric matrices and show that, generically, matrices belonging to the same orbit of this group give Morita equivalent tori. Some applications to physics are sketched. 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.