An Introduction to Generalised Dualities and their Applications to Holography and Integrability

Abstract

These pedagogical lectures given at the Corfu Summer Institute 2018 review two generalised notions of T-duality, non-Abelian T-duality and Poisson-Lie duality, and their applications. We explain how each of these has seen recent application in the context of holography. Non-Abelian T-duality has been used to construct new holographic dual geometries. Poisson-Lie duality has been used to construct new integrable string sigma-models including the η-and λ -deformations of the AdS 5 × S 5 superstring thought to encode quantum group deformations of holography. We also comment on the doubled worldsheet description that makes such dualities manifest. duality serves to interchange a conservation law, that associated to the U(1) global symmetry of the compact boson θ of radius R, with the Bianchi identity for the dual bosonθ :( 1.3) Just as with bosonisation the connection between the two variables involves derivatives, i.e. the map between variables is a non-local one. Also akin to the Abelian bosonisation, the dual theory can be established through an analogous path integral manipulation 1 known as the Buscher procedure [22,23]. Given the close analogy between T-duality and Abelian bosonisation one might wonder if there is a T-duality equivalent to non-Abelian bosonisation. More specifically, in string theory we are immediately prompted to ask if there is an extension of T-duality to the case of a non-Abelian set of isometries, or even if there are still more exotic notions of T-duality? A secondary question is then, if such dualities exist, what are they useful for? The remainder of these lectures seek to outline some answers to these two questions.Our journey will involve exploring a hierarchy of possible dualities in the following string theory scenarios:

Integrable deformation of ℂPn and generalised Kähler geometry

“…The η-deformation of the PCM on G falls into this class [8] with the Drinfel'd double given by the complexification of the Lie algebra, d = g C . Relaxing the requirement that d be a Drinfel'd double allows the E-model to also describe the λ-deformation of WZW CFTs [9,10].…”

Poisson-Lie U-duality in exceptional field theory

“…Whilst the status of Non-Abelian T-duality in terms of the string genus expansion remains unclear, recent evidence [2] at two-loops provides confidence that the duality could remain robust to quantum (α ′ ) corrections on the worldsheet. What is absolutely clear is that in the context of holography at large N , where both string genus and α ′ corrections are suppressed, Non-Abelian T-duality can be a powerful solution generating technique as advocated first in [3][4][5][6] (see [7] for a review and further references) .…”

Exploring exceptional Drinfeld geometries

“…Less standard is the application of T-duality to backgrounds whose isometry group is non-Abelian [3]. While its status as a precise duality in either α and g s expansions is not fully resolved, at the very least non-Abelian T-duality (NATD) is a useful tool as a solution generating symmetry of Type II supergravity (for a review see [4]). More exotic still are applications of T-duality to backgrounds which have no isometries at all.…”

“…Note that a systematic construction of generalised frames corresponding to a given set of generalised fluxes was set out in[26] in which a similar condition plays a necessary role: it really just ensures that the section condition of Exceptional Field Theory is satisfied 4. In DFT this would actually be a spinor representation, in ExFT it is not generically spinorial but will obey a purity constraint projecting out certain representations in the tensor product of Λ with itself.…”