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Linear perturbations of Hyperkahler metrics

Sergei Alexandrov, Boris Pioline, Frank Saueressig, Stefan Vandoren

TL;DR

This work develops a general twistorial framework to describe linear perturbations of 4d hyperkähler spaces constructed via the generalized Legendre transform. Perturbations are encoded by holomorphic functions of $2d+1$ variables on the twistor space, typically breaking all tri-holomorphic isometries, and are related to deformations of holomorphic symplectic transition functions through a Penrose-type transform that yields the perturbed Kähler potential. The authors derive explicit formulas for the deformed twistor lines and Kähler potential, present a comprehensive treatment for $\mathcal{O}(2n)$ manifolds (with explicit results for $\mathcal{O}(2)$), and illustrate the framework with Taub-NUT and Atiyah-Hitchin examples, including the leading exponential deviation of AH from Taub-NUT. The paper provides practical tools for instanton-corrected moduli spaces in field theory and string theory, and sets the stage for extending the approach to quaternionic-Kähler spaces in a companion work.

Abstract

We study general linear perturbations of a class of 4d real-dimensional hyperkahler manifolds obtainable by the (generalized) Legendre transform method. Using twistor methods, we show that deformations can be encoded in a set of holomorphic functions of 2d+1 variables, as opposed to the functions of d+1 variables controlling the unperturbed metric. Such deformations generically break all tri-holomorphic isometries of the unperturbed metric. Geometrically, these functions generate the symplectomorphisms which relate local complex Darboux coordinate systems in different patches of the twistor space. The deformed Kahler potential follows from these data by a Penrose-type transform. As an illustration of our general framework, we determine the leading exponential deviation of the Atiyah-Hitchin manifold away from its negative mass Taub-NUT limit. In a companion paper arXiv:0810.1675, we extend these techniques to quaternionic-Kahler spaces with isometries.

Linear perturbations of Hyperkahler metrics

TL;DR

This work develops a general twistorial framework to describe linear perturbations of 4d hyperkähler spaces constructed via the generalized Legendre transform. Perturbations are encoded by holomorphic functions of variables on the twistor space, typically breaking all tri-holomorphic isometries, and are related to deformations of holomorphic symplectic transition functions through a Penrose-type transform that yields the perturbed Kähler potential. The authors derive explicit formulas for the deformed twistor lines and Kähler potential, present a comprehensive treatment for manifolds (with explicit results for ), and illustrate the framework with Taub-NUT and Atiyah-Hitchin examples, including the leading exponential deviation of AH from Taub-NUT. The paper provides practical tools for instanton-corrected moduli spaces in field theory and string theory, and sets the stage for extending the approach to quaternionic-Kähler spaces in a companion work.

Abstract

We study general linear perturbations of a class of 4d real-dimensional hyperkahler manifolds obtainable by the (generalized) Legendre transform method. Using twistor methods, we show that deformations can be encoded in a set of holomorphic functions of 2d+1 variables, as opposed to the functions of d+1 variables controlling the unperturbed metric. Such deformations generically break all tri-holomorphic isometries of the unperturbed metric. Geometrically, these functions generate the symplectomorphisms which relate local complex Darboux coordinate systems in different patches of the twistor space. The deformed Kahler potential follows from these data by a Penrose-type transform. As an illustration of our general framework, we determine the leading exponential deviation of the Atiyah-Hitchin manifold away from its negative mass Taub-NUT limit. In a companion paper arXiv:0810.1675, we extend these techniques to quaternionic-Kahler spaces with isometries.

Paper Structure

This paper contains 26 sections, 191 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General framework
  3. Twistorial construction of general hyperkähler manifolds
  4. Local sections of $\mathcal{O}(m)$ on $\mathbb{C} P^1$
  5. Local Darboux coordinates and transition functions
  6. A remark on holomorphic quantization
  7. Twistor spaces of $\mathcal{O}(2n)$ hyperkähler manifolds
  8. Symplectomorphisms compatible with the group action
  9. Kähler potential and twistor lines for $\mathcal{O}(2n)$ manifolds, $n\geq 2$
  10. Kähler potential and twistor lines for $\mathcal{O}(2)$ manifolds
  11. Logarithmic branch cuts
  12. Example: $\mathbb{R}^4$
  13. Free tensor multiplet
  14. Improved tensor multiplet
  15. Linear deformations of $\mathcal{O}(2)$ hyperkähler spaces
  16. ...and 11 more sections

Figures (2)

  • Figure 1: Structure of branch cuts in case (i), $c^{[0-]}=c^{[0\infty]}$. The solid (black) line represents the branch cut in $\mu^{[0]}(\zeta)$, the semi-dotted (red) line is the branch cut in $H^{[0-]}$, and the dotted (blue) line is the contour $C$ along which $\mu^{[0]}(\zeta)$ is integrated.
  • Figure 2: Structure of branch cuts in case (ii), $c^{[0+]}=-c^{[0-]}$. The solid line represents the branch cut in $\mu^{[0]}(\zeta)$, and the semi-dotted line is the branch cut in $H^{[0+]}$. On the left, the dotted line is the contour along which $\mu^{[0]}(\zeta)$ is integrated. On the right, the dotted line is the contour along which the singular part $c^{[0+]} \log \eta$ of $H^{[0+]}$ is integrated. The part of the contour between 2 and 4 lies on the $n+1$-th Riemann sheet of the logarithm.