$$\sqrt{T\overline{T}}$$-deformed oscillator inspired by ModMax
J. Antonio Garcı́a, R. Abraham Sánchez-Isidro
Abstract
Abstract Inspired by a recently proposed Duality and Conformal invariant modification of Maxwell theory (ModMax), we construct a one-parameter family of two-dimensional dynamical systems in classical mechanics that share many features with the ModMax theory. It consists of a couple of $$\sqrt{T\overline{T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mrow> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:msqrt> </mml:math> -deformed oscillators that nevertheless preserves duality $$(q \rightarrow p,p \rightarrow -q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>→</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>→</mml:mo> <mml:mo>-</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and depends on a continuous parameter $$\gamma$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> , as in the ModMax case. Despite its nonlinear features, the system is integrable. Remarkably, it can be interpreted as a pair of two coupled oscillators whose frequencies depend on some basic invariants that correspond to the duality symmetry and rotational symmetry. Based on the properties of the model, we can construct a nonlinear map dependent on $$\gamma$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> that maps the oscillator in 2D to the nonlinear one, but with parameter $$2\gamma$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>γ</mml:mi> </mml:mrow> </mml:math> . The reason behind the existence of such map can be revealed through a construction of two Lax pairs associated with the system. The dynamics also shows the phenomenon of energy transfer and we calculate a Hannay angle associated to geometric phases and holonomies.