Seiberg-Witten curves of $$ \hat{D} $$-type Little Strings

A bstract Little Strings are a type of non-gravitational quantum theories that contain extended degrees of freedom, but behave like ordinary Quantum Field Theories at low energies. A particular class of such theories in six dimensions is engineered as the world-volume theory of an M5-brane on a circle that probes a transverse orbifold geometry. Its low energy limit is a supersymmetric gauge theory that is described by a quiver in the shape of the Dynkin diagram of the affine extension of an ADE-group. While the so-called $$ \hat{A} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> -type Little String Theories (LSTs) are very well studied, much less is known about the $$ \hat{D} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> -type, where for example the Seiberg-Witten curve (SWC) is only known in the case of the $$ {\hat{D}}_4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mn>4</mml:mn> </mml:msub> </mml:math> theory. In this work, we provide a general construction of this curve for arbitrary $$ {\hat{D}}_M $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mi>M</mml:mi> </mml:msub> </mml:math> that respects all symmetries and dualities of the LST and is compatible with lower-dimensional results in the literature. For M = 4 our construction reproduces the same curve as previously obtained by other methods. The form in which we cast the SWC for generic $$ {\hat{D}}_M $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mi>M</mml:mi> </mml:msub> </mml:math> allows to study the behaviour of the LST under modular transformations and provides insights into a dual formulation as a circular quiver gauge theory with nodes of Sp( M − 4) and SO(2 M ).