Boundary vertex algebras for 3d $\mathcal{N}=4$ rank-0 SCFTs

Abstract

We initiate the study of boundary Vertex Operator Algebras (VOAs) of topologically twisted 3d \mathcal{N}=4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> rank-0 SCFTs. This is a recently introduced class of \mathcal{N}=4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> SCFTs that by definition have zero-dimensional Higgs and Coulomb branches. We briefly explain why it is reasonable to obtain rational VOAs at the boundary of their topological twists. When a rank-0 SCFT is realized as the IR fixed point of a \mathcal{N}=2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> Lagrangian theory, we propose a technique for the explicit construction of its topological twists and boundary VOAs based on deformations of the holomorphic-topological twist of the \mathcal{N}=2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> microscopic description. We apply this technique to the B <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>B</mml:mi> </mml:math> twist of a newly discovered family of 3d \mathcal{N}=4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> rank-0 SCFTs {\mathcal T}_r <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>𝒯</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:math> and argue that they admit the simple affine VOAs L_r(\mathfrak{osp}(1|2)) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mrow> <mml:mi>𝔬</mml:mi> <mml:mi>𝔰</mml:mi> <mml:mi>𝔭</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> at their boundary. In the simplest case, this leads to a novel level-rank duality between L_1(\mathfrak{osp}(1|2)) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mrow> <mml:mi>𝔬</mml:mi> <mml:mi>𝔰</mml:mi> <mml:mi>𝔭</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and the minimal model M(2,5) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . As an aside, we present a TQFT obtained by twisting a 3d \mathcal{N}=2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> QFT that admits the M(3,4) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mrow>

Topics & Concepts