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T-duality for toric manifolds in $\mathcal{N}=(2, 2)$ superspace

Dmitri Bykov, Savva Kutsubin, Andrew Kuzovchikov

TL;DR

The paper establishes that T-duality for toric Kähler manifolds in $ ext{N}=(2,2)$ superspace generically yields generalized Kähler geometries in the dual frame when multiple isometries act in a way that is not a symmetry of the Kähler potential. To gauge these isometries, the authors introduce an enlarged gauge multiplet $(oldsymbol{V},oldsymbol{X})$ that includes a semi-chiral field, enabling a coherent dualization and accounting for cross-terms that encode nontrivial topology, such as a fixed imaginary period (e.g., $oldsymbol{T} o oldsymbol{T}+i/4$). Applying the construction to the $oldsymbol{ ext{C}}^ imes imes oldsymbol{ ext{C}}^ imes$ model and to the $ ext{CP}^{n-1}_{oldsymbol{ abla}}$ eta-deformed model, the dual geometries are explicit: the dual of $ ext{CP}^{n-1}_{oldsymbol{ abla}}$ is Kähler, while the original is generalized Kähler, with the Legendre transform yielding the symplectic potential governing the dual geometry. In the $oldsymbol{ ho} o0$ (undeformed) limit, both frames reduce to Kähler manifolds, connecting with familiar toric/Hilbert-space dualities and providing a concrete, algorithmic approach to construct dual generalized Kähler backgrounds in 2D supersymmetric sigma models.

Abstract

We study the situation when the T-dual of a toric Kähler geometry is a generalized Kähler geometry involving semi-chiral fields. We explain that this situation is generic for polycylinders, tori and related geometries. Gauging multiple isometries in this case requires the introduction of semi-chiral gauge fields on top of the standard ones. We then apply this technology to the generalized Kähler geometry of the $η$-deformed $\mathbb{CP}^{n-1}$ model, relating it to the Kähler geometry of its T-dual.

T-duality for toric manifolds in $\mathcal{N}=(2, 2)$ superspace

TL;DR

The paper establishes that T-duality for toric Kähler manifolds in superspace generically yields generalized Kähler geometries in the dual frame when multiple isometries act in a way that is not a symmetry of the Kähler potential. To gauge these isometries, the authors introduce an enlarged gauge multiplet that includes a semi-chiral field, enabling a coherent dualization and accounting for cross-terms that encode nontrivial topology, such as a fixed imaginary period (e.g., ). Applying the construction to the model and to the eta-deformed model, the dual geometries are explicit: the dual of is Kähler, while the original is generalized Kähler, with the Legendre transform yielding the symplectic potential governing the dual geometry. In the (undeformed) limit, both frames reduce to Kähler manifolds, connecting with familiar toric/Hilbert-space dualities and providing a concrete, algorithmic approach to construct dual generalized Kähler backgrounds in 2D supersymmetric sigma models.

Abstract

We study the situation when the T-dual of a toric Kähler geometry is a generalized Kähler geometry involving semi-chiral fields. We explain that this situation is generic for polycylinders, tori and related geometries. Gauging multiple isometries in this case requires the introduction of semi-chiral gauge fields on top of the standard ones. We then apply this technology to the generalized Kähler geometry of the -deformed model, relating it to the Kähler geometry of its T-dual.
Paper Structure (27 sections, 110 equations, 1 figure)

This paper contains 27 sections, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Kähler potential and flat space isometries
  3. Gauging with chiral & twisted chiral fields
  4. Only chiral fields.
  5. General case.
  6. Examples.
  7. T-duality
  8. Chiral fields.
  9. Periodicity of $\mathbf{T}$.
  10. Chiral and twisted chiral fields.
  11. Periodicity of $\mathbf{Y}$.
  12. Application to the $\mathbb{C}^\ast \times \mathbb{C}^\ast$ model
  13. The deformed $\mathbb{CP}^{n-1}_{\eta}$ model
  14. Gauging isometries.
  15. The Legendre transform.
  16. ...and 12 more sections

Figures (1)

  • Figure 1: The moment polytope of $\mathbb{CP}^2$. Regularity of the metric at the encircled corner is used to identify the periodicities of the angles.