[Paper Review] Hyper-Kahler geometries and nonlinear supermultiplets

Č. Burdík, S. Krivonos+1 more|ArXiv.org|Sep 30, 2006

Advanced Differential Geometry Research3 citations

TL;DR

This paper presents a dualization procedure for coupling constants in one-dimensional $N=4$ and $N=8$ supersymmetric sigma-models, treating the constancy of the coupling as a dynamical constraint via a Lagrange multiplier. By applying this method to nonlinear off-shell supermultiplets, the authors construct general $N=4$ and $N=8$ supersymmetric sigma-models with hyper-Kähler target spaces possessing one triholomorphic isometry, yielding the most general four-dimensional hyper-Kähler geometries of Gibbons–Hawking type.

ABSTRACT

It is presented a method of construction of sigma-models with target space geometries different from conformally flat ones. The method is based on a treating of a constancy of a coupling constant as a dynamical constraint following as an equation of motion. In this way we build N=4 and N=8 supersymmetric four-dimensional sigma-models in d=1 with hyper-Kahler target space possessing one isometry, which commutes with supersymmetry.

Motivation & Objective

• To construct $N=4$ and $N=8$ supersymmetric sigma-models with target spaces that are not conformally flat, overcoming limitations of prior approaches.
• To resolve the gap in existing models that fail to describe hyper-Kähler geometries arising from dimensional reduction of $N=2$ four-dimensional sigma-models.
• To develop a systematic method for generating nonlinear off-shell supermultiplets in one dimension, crucial for realizing non-conformally-flat target space geometries.
• To demonstrate that dualizing a coupling constant as a dynamical constraint leads to hyper-Kähler manifolds with one triholomorphic isometry, particularly in the $N=8$ case.

Proposed method

• Introduce a Lagrange multiplier to treat the constancy of a coupling constant $g$ as a dynamical constraint, replacing $g = \text{const}$ with $\dot{g} = 0$ as an equation of motion.
• Extend the action to include the coupling constant $g(t)$ as a dynamical field, leading to a new action where $g$ is no longer a constant but a function of time.
• Use the $N=2$ superfield formalism with a real superfield $V$, chiral superfields $\Phi$, $\bar{\Phi}$, and a potential term $gH$ with harmonic $H$ satisfying $H_{VV} + H_{\Phi\bar{\Phi}} = 0$.
• Apply the dualization procedure to the $N=4$ supermultiplet (3,4,1), promoting the coupling constant $g$ to a physical scalar field, thereby generating a nonlinear realization of supersymmetry.
• Derive the component action after eliminating auxiliary fields and the Lagrange multiplier, resulting in a kinetic term and potential that define a hyper-Kähler metric.
• Show that the resulting metric takes the Gibbons–Hawking form, corresponding to a four-dimensional hyper-Kähler manifold with one triholomorphic isometry, realized as a shift in the $y$-coordinate.

Experimental results

Research questions

• RQ1 How can one construct $N=4$ and $N=8$ supersymmetric sigma-models with hyper-Kähler target spaces that are not conformally flat?
• RQ2 What is the role of nonlinear off-shell supermultiplets in realizing non-conformally-flat geometries in one-dimensional supersymmetric mechanics?
• RQ3 Can the constancy of a coupling constant be reformulated as a dynamical constraint to generate new target space geometries?
• RQ4 How does dualizing the coupling constant lead to the emergence of a Gibbons–Hawking-type metric in $N=8$ supersymmetric mechanics?
• RQ5 What are the transformation properties of the resulting nonlinear supermultiplets, and are they defined off-shell?

Key findings

• The dualization of the coupling constant $g$ as a dynamical field leads to a new action where $g(t)$ becomes a physical scalar, resulting in a four-dimensional hyper-Kähler target space.
• The resulting bosonic metric takes the Gibbons–Hawking form: $ds^2 = F_{,vv}(dv^2 + 4d\varphi d\bar{\varphi}) + \frac{1}{F_{,vv}}(dy - iF_{,v\varphi}d\varphi + iF_{,v\bar{\varphi}}d\bar{\varphi})^2$, which describes the most general four-dimensional hyper-Kähler manifold with one triholomorphic isometry.
• The construction yields a nonlinear realization of $N=4$ supersymmetry, where the supermultiplet transformations depend nontrivially on a harmonic function $F$, making the multiplet off-shell and nonlinear.
• For $N=8$, the method produces a general $N=8$ supersymmetric sigma-model with a hyper-Kähler target space, though the off-shell nature of the resulting multiplet requires further investigation.
• The kinetic term in the final action is expressed in terms of the second derivative of the function $F$, with the metric structure fully determined by $F_{,vv}$, $F_{,v\varphi}$, and $F_{,v\bar{\varphi}}$, confirming the hyper-Kähler property.
• The potential term in the action encodes the full nonlinear structure of the fermionic interactions, with coefficients involving higher-order derivatives of $F$, confirming the nontrivial supersymmetry algebra.