The tropical Abel–Prym map

We prove that, under mild assumptions, the tropical Abel–Prym map <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ψ</mml:mi> <mml:mo lspace="0pt">:</mml:mo> <mml:mover accent="true"> <mml:mi>Γ</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mo>→</mml:mo> <mml:mo form="prefix">Prym</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mover accent="true"> <mml:mi>Γ</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mo>/</mml:mo> <mml:mi>Γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> associated with a free double cover <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>π</mml:mi> <mml:mo lspace="0pt">:</mml:mo> <mml:mover accent="true"> <mml:mi>Γ</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mo>→</mml:mo> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:math> is harmonic of degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> if and only if the source graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover accent="true"> <mml:mi>Γ</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> </mml:math> is hyperelliptic. This is in accordance with the already established algebraic result. In this case, the Abel–Prym graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ψ</mml:mi> <mml:mo>(</mml:mo> <mml:mover accent="true"> <mml:mi>Γ</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is hyperelliptic of genus <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>Γ</mml:mi> </mml:msub> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and its Jacobian is isomorphic, as a pptav, to the Prym variety of the cover. We further show that the Abel–Prym graph coincides with a connected component of the tropical bigonal construction. En route, we count the number of distinct free double covers by hyperelliptic metric graphs.