<p>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 <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_26114_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mover accent="true"> <mi>A</mi> <mo stretchy="true">̂</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">\( \hat{A} \)</EquationSource> </InlineEquation>-type Little String Theories (LSTs) are very well studied, much less is known about the <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_26114_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mover accent="true"> <mi>D</mi> <mo stretchy="true">̂</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">\( \hat{D} \)</EquationSource> </InlineEquation>-type, where for example the Seiberg-Witten curve (SWC) is only known in the case of the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_26114_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <msub> <mover accent="true"> <mi>D</mi> <mo stretchy="true">̂</mo> </mover> <mn>4</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">\( {\hat{D}}_4 \)</EquationSource> </InlineEquation> theory. In this work, we provide a general construction of this curve for arbitrary <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_26114_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <msub> <mover accent="true"> <mi>D</mi> <mo stretchy="true">̂</mo> </mover> <mi>M</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">\( {\hat{D}}_M \)</EquationSource> </InlineEquation> that respects all symmetries and dualities of the LST and is compatible with lower-dimensional results in the literature. For <i>M</i> = 4 our construction reproduces the same curve as previously obtained by other methods. The form in which we cast the SWC for generic <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_26114_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <msub> <mover accent="true"> <mi>D</mi> <mo stretchy="true">̂</mo> </mover> <mi>M</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">\( {\hat{D}}_M \)</EquationSource> </InlineEquation> 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(<i>M</i> − 4) and SO(2<i>M</i>).</p>

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

  • Baptiste Filoche,
  • Stefan Hohenegger,
  • Taro Kimura

摘要

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 A ̂ \( \hat{A} \) -type Little String Theories (LSTs) are very well studied, much less is known about the D ̂ \( \hat{D} \) -type, where for example the Seiberg-Witten curve (SWC) is only known in the case of the D ̂ 4 \( {\hat{D}}_4 \) theory. In this work, we provide a general construction of this curve for arbitrary D ̂ M \( {\hat{D}}_M \) 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 D ̂ M \( {\hat{D}}_M \) 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(2M).