[Paper Review] Tinkertoys for the Twisted D-Series

Oscar Chacaltana, Jacques Distler+1 more|arXiv (Cornell University)|Sep 9, 2013

Black Holes and Theoretical Physics29 references21 citations

TL;DR

This paper classifies 4D $σ=2$ superconformal field theories (SCFTs) arising from compactifying 6D $σ=(2,0)$ $D_N$ theories on Riemann surfaces with $π_2$-twisted punctures, extending the class S construction to include $π_2$ outer-automorphism twists. It provides explicit classifications for $D_4$, $D_5$, and $D_6$, identifying new realizations of $Spin(8)$, $Spin(7)$, and $Sp(3)$ gauge theories with matter in vector, spinor, and 3-index antisymmetric representations, all with vanishing $β$-function.

ABSTRACT

We study 4D N=2 superconformal field theories that arise from the compactification of 6D N=(2,0) theories of type D_N on a Riemann surface, in the presence of punctures twisted by a Z_2 outer automorphism. Unlike the untwisted case, the family of SCFTs is in general parametrized, not by M_{g,n}, but by a branched cover thereof. The classification of these SCFTs is carried out explicitly in the case of the D_4 theory, in terms of three-punctured spheres and cylinders, and we provide tables of properties of twisted punctures for the D_5 and D_6 theories. We find realizations of Spin(8) and Spin(7) gauge theories with matter in all combinations of vector and spinor representations with vanishing beta-function, as well as Sp(3) gauge theories with matter in the 3-index traceless antisymmetric representation.

Motivation & Objective

• To extend the class S construction of 4D $σ=2$ SCFTs to include $π_2$ outer-automorphism twists in $D_N$ theories.
• To classify the resulting SCFTs arising from compactification on Riemann surfaces with twisted punctures, particularly for $D_4$, $D_5$, and $D_6$.
• To identify new realizations of gauge theories with vanishing $β$-function, including those not accessible in the untwisted $D_N$ sector.
• To compute key physical invariants such as global symmetry, central charges $(a,c)$, pole structures, and Seiberg-Witten curves for twisted punctures.
• To provide a systematic Tinkertoy-like construction using three-punctured spheres and cylinders with twisted punctures as building blocks.

Proposed method

• Classify twisted punctures in $D_N$ theories via $π_2$-twisted compactifications, using cohomology classes in $H^1(C - \{p_i\}, π_2)$ to distinguish twisted vs. untwisted punctures.
• Compute local data such as pole structures, constraints, and central charges $(a,c)$ from the twisted puncture data to determine the 4D SCFT's global symmetry and Coulomb branch geometry.
• Construct building blocks—three-punctured spheres and cylinders—with twisted punctures as fundamental elements for pair-of-pants decompositions of higher-genus Riemann surfaces.
• Use Hall-Littlewood limit of the superconformal index to compute and verify global symmetry enhancements and central charges.
• Apply the classification to identify gauge theory fixtures, including $Sp(3)$ with 3-index antisymmetric matter, and $Spin(8)$/$Spin(7)$ with mixed vector and spinor representations.
• Provide explicit tables of twisted sector data for $D_5$ and $D_6$, including pole structures, central charges, and global symmetry groups.

Experimental results

Research questions

• RQ1 What is the classification of 4D $σ=2$ SCFTs arising from $π_2$-twisted compactification of $D_N$ (2,0) theories on Riemann surfaces?
• RQ2 How do twisted punctures modify the Coulomb branch geometry, global symmetries, and central charges compared to the untwisted case?
• RQ3 Which gauge theories with vanishing $β$-function can be realized in the twisted $D_N$ sector that were not accessible in the untwisted $D_N$ theory?
• RQ4 What are the precise pole structures and constraints for twisted punctures in $D_5$ and $D_6$?
• RQ5 Can new realizations of $Sp(N)$ gauge theories with exotic matter representations (e.g., 3-index antisymmetric) be constructed via this twisted class S framework?

Key findings

• The paper constructs a complete classification of $π_2$-twisted $D_4$ SCFTs using three-punctured spheres and cylinders with twisted punctures as building blocks.
• It identifies new realizations of $Spin(8)$ gauge theory with matter in the $6(8_v)$ and $5(8_v)+1(8_s)$ representations, both with vanishing $β$-function.
• It realizes $Spin(7)$ gauge theory with matter in the $5(7)$ and $1(8)+4(7)$ representations, including previously inaccessible combinations.
• It provides a realization of $Sp(3)$ gauge theory with matter in the $\tfrac{11}{2}(6)+\tfrac{1}{2}(14')$ and $3(6)+1(14')$ representations, where $14'$ is the 3-index traceless antisymmetric tensor.
• For $D_5$ and $D_6$, the paper tabulates twisted puncture data including pole structures, central charges $(a,c)$, and global symmetry groups, enabling systematic construction of SCFTs.
• The study reveals that the moduli space of twisted $D_N$ SCFTs is a branched cover of $\mathcal{M}_{g,n}$, not $\mathcal{M}_{g,n}$ itself, due to the presence of twisted punctures.