4d S-duality wall and SL(2, ℤ) relations

Abstract

<jats:title>A<jats:sc>bstract</jats:sc> </jats:title><jats:p>In this paper we present various 4<jats:italic>d</jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> = 1 dualities involving theories obtained by gluing two <jats:italic>E</jats:italic>[USp(2<jats:italic>N</jats:italic>)] blocks via the gauging of a common USp(2<jats:italic>N</jats:italic>) symmetry with the addition of 2<jats:italic>L</jats:italic> fundamental matter chiral fields. For <jats:italic>L</jats:italic> = 0 in particular the theory has a quantum deformed moduli space with chiral symmetry breaking and its index takes the form of a delta-function. We interpret it as the Identity wall which identifies the two surviving USp(2<jats:italic>N</jats:italic>) of each <jats:italic>E</jats:italic>[USp(2<jats:italic>N</jats:italic>)] block. All the dualities are derived from iterative applications of the Intriligator-Pouliot duality. This plays for us the role of the fundamental duality, from which we derive all others. We then focus on the 3<jats:italic>d</jats:italic> version of our 4<jats:italic>d</jats:italic> dualities, which now involve the <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> = 4 <jats:italic>T</jats:italic>[SU(<jats:italic>N</jats:italic>)] quiver theory that is known to correspond to the 3<jats:italic>d S</jats:italic>-wall. We show how these 3<jats:italic>d</jats:italic> dualities correspond to the relations <jats:italic>S</jats:italic><jats:sup>2</jats:sup> = <jats:italic>−</jats:italic>1, <jats:italic>S</jats:italic><jats:sup><jats:italic>−</jats:italic>1</jats:sup><jats:italic>S</jats:italic> = 1 and <jats:italic>STS</jats:italic> = <jats:italic>T</jats:italic><jats:sup><jats:italic>−</jats:italic>1</jats:sup><jats:italic>S</jats:italic><jats:sup><jats:italic>−</jats:italic>1</jats:sup><jats:italic>T</jats:italic><jats:sup><jats:italic>−</jats:italic>1</jats:sup> for the <jats:italic>S</jats:italic> and <jats:italic>T</jats:italic> generators of SL(2<jats:italic>,</jats:italic> ℤ). These observations lead us to conjecture that <jats:italic>E</jats:italic>[USp(2<jats:italic>N</jats:italic>)] can also be interpreted as a 4<jats:italic>d S</jats:italic>-wall.</jats:p>