5D and 6D SCFTs from $\mathbb{C}^3$ orbifolds
Jiahua Tian, Yi-Nan Wang
Abstract
We study the orbifold singularities X=\mathbb{C}^3/\Gamma <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℂ</mml:mi> </mml:mstyle> <mml:mn>3</mml:mn> </mml:msup> <mml:mi>/</mml:mi> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:math> where \Gamma <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Γ</mml:mi> </mml:math> is a finite subgroup of SU(3) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> . M-theory on this orbifold singularity gives rise to a 5d SCFT, which is investigated with two methods. The first approach is via 3d McKay correspondence which relates the group theoretic data of \Gamma <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Γ</mml:mi> </mml:math> to the physical properties of the 5d SCFT. In particular, the 1-form symmetry of the 5d SCFT is read off from the McKay quiver of \Gamma <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Γ</mml:mi> </mml:math> in an elegant way. The second method is to explicitly resolve the singularity X <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>X</mml:mi> </mml:math> and study the Coulomb branch information of the 5d SCFT, which is applied to toric, non-toric hypersurface and complete intersection cases. Many new theories are constructed, either with or without an IR quiver gauge theory description. We find that many resolved Calabi-Yau threefolds, \widetilde{X} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mover> <mml:mi>X</mml:mi> <mml:mo accent="true">̃</mml:mo> </mml:mover> </mml:math> , contain compact exceptional divisors that are singular by themselves. Moreover, for certain cases of \Gamma <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Γ</mml:mi> </mml:math> , the orbifold singularity \mathbb{C}^3/\Gamma <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℂ</mml:mi> </mml:mstyle> <mml:mn>3</mml:mn> </mml:msup> <mml:mi>/</mml:mi> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:math> can be embedded in an elliptic model and gives rise to a 6d (1,0) SCFT in the F-theory construction. Such 6d theory is naturally related to the 5d SCFT defined on the same singularity. We find examples of rank-1 6d SCFTs without a gauge group, which are potentially different from the rank-1 E-string theory.