Hadron-quark hybrid model, modular transformation, and Roberge-Weiss transition

In the framework of modular transformations, we reformulate the recently proposed hadron-quark hybrid model when the imaginary baryonic chemical potential is introduced. In this case, the number density of the hybrid model is obtained by the modular transformation of the complex number densities of the baryons (antibaryons) and the quarks (antiquarks). We can regard these number densities as the basis in the complex plane. As a result, we can consider the torus, which is characterized by the basis. Since the complex structure of the torus is invariant under the modular transformation, we can extract the topological property of the hadron-quark system using the untransformed baryon (antibaryon) and the quark (antiquark) number densities. We apply this model to analyze the Roberge-Weiss transition. It is shown that the torus vanishes at the baryonic chemical potential where the Roberge-Weiss transition appears because the number density of baryons (antibaryons) is not linearly independent of the number density of quarks (antiquarks). When the temperature <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>T</a:mi></a:math> is lower than the Roberge-Weiss transition temperature <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>T</c:mi><c:mrow><c:mi>RW</c:mi></c:mrow></c:msub></c:math>, the torus shrinks smoothly to the one-dimensional object at the Roberge-Weiss transition point, but the discontinuity does not appear. On the other hand, the discontinuity of the geometrical object appears when <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>T</e:mi><e:mo>&gt;</e:mo><e:msub><e:mi>T</e:mi><e:mrow><e:mi>RW</e:mi></e:mrow></e:msub></e:math>. We also calculate the modulus of the torus and transform it into the fundamental region. The transformed moduli are symmetric below <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>T</g:mi><g:mrow><g:mi>RW</g:mi></g:mrow></g:msub></g:math>, but the symmetry is broken above <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>T</i:mi><i:mrow><i:mi>RW</i:mi></i:mrow></i:msub></i:math>.