Narrow Results

In this paper we study the noncommutative description of the DBI Lagrangian and its T-dual counterpart. We restrict the freedoms of the noncommutativity parameters of these Lagrangians. Therefore the noncommutativity parameter, the effective metric, the effective coupling constant of the string and the extra modulus of the effective T-dual theory, can be expressed in terms of the closed string variables g, B, g_s and the noncommutativity parameter of the effective theory of open string.

In this paper we study the noncommutativity of a moving membrane with background fields. The open string variables are analyzed. Some scaling limits are studied. The equivalence of the magnetic and electric noncommutativities is investigated. The conditions for equivalence of noncommutativity of the T-dual theory in the rest frame and noncommutativity of the original theory in the moving frame are obtained.

In an earlier paper, Alvarez, Alvarez-Gaumé, Barbón and Lozano pointed out, that the only way to "flatten" negative curvature by means of a T-duality is by introducing an appropriate, non-constant NS–NS two-form B. In this paper, we are investigating this further and ask, whether it is possible to T-dualize AdS_d space to flat space with some suitably chosen B. To answer this question, we derive a relationship between the original curvature tensor and the one of the T-dualized metric involving the B-field. It turns out that there is one component which is independent of B. By inspection of this component, we show that it is not possible to dualize AdS_d to flat space irrespective of the choice of B. Finally, we examine the extension of AdS to an AdS₅ × S⁵ geometry and propose a chain of S- and T-dualities together with an SL(2, ℤ) coordinate transformation, leading to a dual D9-brane geometry.

Starting from the classification of real Manin triples we look for those that are isomorphic as six-dimensional Drinfeld doubles i.e. Lie algebras with the ad-invariant form used for construction of the Manin triples. We use several invariants of the Lie algebras to distinguish the nonisomorphic structures and give the explicit form of maps between Manin triples that are decompositions of isomorphic Drinfeld doubles. The result is a complete list of six-dimensional real Drinfeld doubles. It consists of 22 classes of nonisomorphic Drinfeld doubles.

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.

We present a procedure for application of T-duality transformation on scattering amplitudes of closed bosonic stringy states. These states arise due to compactification of closed string to lower space–time dimensions through dimensional reduction. The amplitude, in the first quantized formalism, is computed by introducing vertex operators. The amplitude is constructed by the standard prescription and the vertex operators are required to respect conformal invariance. Such vertex operators are constructed in the weak field approximation. Therefore, the vertex operators of the stringy states of our interest are to be defined accordingly. We propose a prescription to implement T-duality on the three-point functions and N-point functions. We argue that it is possible to generate new amplitudes through the transformations on a given amplitude just as T-duality transformations can take us to a new set of string vacuum when acted upon an initial set. Explicit examples are given for three-point and four-point functions.

We study T-duality transformations in canonical formalism for Nambu–Goto action. Then we investigate the relation between worldsheet double Wick rotation and sequence of target space T-dualities and Wick rotation in case of fundamental string and D1-brane.

We study the dissipative Hofstadter model on a triangular lattice, making use of the $O(2,2;R)$ T-dual transformation of string theory. The $O(2,2;R)$ dual transformation transcribes the model in a commutative basis into the model in a noncommutative basis. In the zero-temperature limit, the model exhibits an exact duality, which identifies equivalent points on the two-dimensional parameter space of the model. The exact duality also defines magic circles on the parameter space, where the model can be mapped onto the boundary sine-Gordon on a triangular lattice. The model describes the junction of three quantum wires in a uniform magnetic field background. An explicit expression of the equivalence relation, which identifies the points on the two-dimensional parameter space of the model by the exact duality, is obtained. It may help us to understand the structure of the phase diagram of the model.

Recently, it has been suggested that the elements of the S-matrix on the worldvolume of an Abelian D-brane might be in accordance with the Ward identity associated with the T-duality. This shows that by applying linear T-duality, a group of S-matrix elements could be found invariant under such transformations. In this work, we apply the T-duality transformations on the S-matrix elements of one B-field and three NS Abelian gauge fields and find some other Abelian S-matrix elements of one closed and three open string states. Also, we will show that the predicted S-matrix elements are reproduced exactly by explicit calculations.

Given the $\beta$ functions of the closed string sigma model up to one loop in ${\alpha }^{\prime }$, the effective action implements the condition $\beta =0$ to preserve conformal symmetry at quantum level. One of the more powerful and striking results of string theory is that this effective action contains Einstein gravity as an emergent dynamics in space–time. We show from the $\beta$ functions and its relation with the equations of motion of the effective action that the differential identities are the Noether identities associated with the effective action and its gauge symmetries. From here, we reconstruct the gauge and space–time symmetries of the effective action. In turn, we can show that the differential identities are the contracted Bianchi identities of the field strength $H$ and Riemann tensor $R$. Next, we apply the same ideas to DFT. Taking as starting point that the generalized $\beta$ functions in DFT are proportional to the equations of motion, we construct the generalized differential identities in DFT. Relating the Noether identities with the contracted Bianchi identities of DFT, we were able to reconstruct the generalized gauge and space–time symmetries. Finally, we recover the original $\beta$ functions, effective action, differential identities, and symmetries when we turn off the $\tilde{x}$ space–time coordinates from DFT.

These notes present an introduction to the method of geometric quantization. We discuss the main theorems in a style suitable for a theoretical physicist with an eye towards the physical motivation and the interpretation of the geometric construction as providing a solution to Dirac’s axioms of quantization. We provide in detail the examples of free relativistic particles, their corresponding quantum fields, and the bosonic string using formalism of double field theory. This paper is based on lectures written by Gabriel Cardoso.

In this paper, we describe non-Abelian T-dualities for a two-dimensional (2D) gauged linear sigma model (GLSM) with $\mathcal{𝒩}=2$ supersymmetry (SUSY). We start with the case of left and right SUSY $(2,2)$, $U(1)$ gauge group, and global non-Abelian symmetries. Our analysis applies to the specialization of the GLSM with the global group $SU(2)\times SU(2)$, whose original model is the resolved conifold. We analyze the dual model and compute the periods on the dual geometry, matching the effective potential for the $U(1)$ gauge field, and discussing the coincident singularity at the conifold point. We further present the non-Abelian T-dualization of models with $(0,2)$ SUSY, analyzing the case of global symmetry $SU(2)$. In this case, the full non-Abelian duality can be solved. The dual geometries with and without nonperturbative corrections to the superpotential coincide. The structure of the dual nonperturbative corrections is determined based on symmetry arguments. The non-Abelian T-dualities reviewed here lead to potential new symmetries between physical theories.

The holographic principle and its realization as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence leads to the existence of the so-called precursor operators. These are boundary operators that carry nonlocal information regarding events occurring deep inside the bulk and which cannot be causally connected to the boundary. Such nonlocal operators can distinguish nonvacuum-like excitations within the bulk that cannot be observed by any local gauge invariant operators in the boundary. The boundary precursors are expected to become increasingly nonlocal the further the bulk process is from the boundary. Such phenomena are expected to be related to the extended nature of the strings. Standard gauge invariance in the boundary theory equates to quantum error correction which furthermore establishes localization of bulk information. I show that when double field theory quantum error correction prescriptions are considered in the bulk, gauge invariance in the boundary manifests residual effects associated to stringy winding modes. Also, an effect of double field theory quantum error correction is the appearance of positive cosmological constant. The emergence of space–time from the entanglement structure of a dual quantum field theory appears in this context to generalize for de Sitter space–times as well.