On classification of conformal vectors in vertex operator algebra and the vertex algebra automorphism group

The correspondence between four-dimensional $${\mathcal {N}}=2$$ N = 2 superconformal field theories and vertex operator algebras, when applied to theories of class $${\mathcal {S}}$$ S , leads to a rich family of VOAs that have been given the monicker chiral algebras of class$${\mathcal {S}}$$ S . A remarkably uniform construction of these vertex operator algebras has been put forward by Tomoyuki Arakawa in Arakawa (Chiral algebras of class $${\mathcal {S}}$$ S and Moore–Tachikawa symplectic varieties, 2018. arXiv:1811.01577 [math.RT]). The construction of Arakawa (2018) takes as input a choice of simple Lie algebra $${\mathfrak {g}}$$ g , and applies equally well regardless of whether $${\mathfrak {g}}$$ g is simply laced or not. In the non-simply laced case, however, the resulting VOAs do not correspond in any clear way to known four-dimensional theories. On the other hand, the standard realisation of class $${{{\mathcal {S}}}}$$ S theories involving non-simply laced symmetry algebras requires the inclusion of outer automorphism twist lines, and this requires a further development of the approach of Arakawa (2018). In this paper, we give an account of those further developments and propose definitions of most chiral algebras of class $${{{\mathcal {S}}}}$$ S with outer automorphism twist lines. We show that our definition passes some consistency checks and point out some important open problems.

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“…Note that with and is conical—so is of CFT type. As a result, Lemma 4.1 of [ 58 ] applies. Namely, for any , if and then .…”

Section: Twisted Trinions From Mixed Feigin–frenkel Gluingmentioning

confidence: 99%

“…Now we wish to show that is the unique conformal vector whose -grading agrees with . The argument from Proposition 4.6 using Lemma 4.1 of [ 58 ] still works, with minor alteration, since are conical.…”

Section: Twisted Trinions From Mixed Feigin–frenkel Gluingmentioning

confidence: 99%

Twisted Chiral Algebras of Class $${\mathcal {S}}$$ and Mixed Feigin–Frenkel Gluing

Beem

Nair

2022

Commun. Math. Phys.

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Twisted chiral algebras of class $\mathcal{S}$ and mixed Feigin-Frenkel gluing

Beem¹,

Nair²

2022

Preprint

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The correspondence between four-dimensional N = 2 superconformal field theories and vertex operator algebras, when applied to theories of class S, leads to a rich family of VOAs that have been given the monicker chiral algebras of class S. A remarkably uniform construction of these vertex operator algebras has been put forward by Tomoyuki Arakawa in [1]. The construction of [1] takes as input a choice of simple Lie algebra g, and applies equally well regardless of whether g is simply laced or not. In the non-simply laced case, however, the resulting VOAs do not correspond in any clear way to known four-dimensional theories. On the other hand, the standard realisation of class S theories involving non-simply laced symmetry algebras requires the inclusion of outer automorphism twist lines, and this requires a further development of the approach of [1]. In this paper, we give an account of those further developments and propose definitions of most chiral algebras of class S with outer automorphism twist lines. We show that our definition passes some consistency checks and point out some important open problems. Contents 1 Introduction 2 Review of (twisted) class S 2.1 Untwisted and twisted punctures 2.2 Trinions, gluing, and duality 2.3 The superconformal index 2.4 Residual gauge symmetry and derived structures 3 Chiral algebras of class S after Arakawa 3.1 Critical level current algebras and the Kazhdan-Lusztig category 3.2 The Feigin-Frenkel centre 3.3 BRST and semi-infinite cohomology 3.4 Drinfel'd-Sokolov reduction 3.5 Feigin-Frenkel gluing 3.6 Vertex algebras at genus zero 4 Twisted trinions from mixed Feigin-Frenkel gluing 4.1 The (un)twisted Feigin-Frenkel centre and the mixed trinion 4.2 Opers and the Feigin-Frenkel centre 4.3 Properties of the mixed trinion 4.4 Rearrangment lemmas 4.5 Mixed vertex algebras at genus zero 4.6 Properties of the genus zero mixed vertex algebras 4.7 Generalised S-duality and 4-moves 4.8 The Z 3 twist of d 4 5 Further observations 5.1 Three dimensional mirrors and the non-simply laced case 5.2 Unexpected Feigin-Frenkel gluings 5.3 A six dimensional perspective on the index A Spectral sequences B Miura opers and a proof of Theorem 4.1 B.1 Miura opers and the no-monodromy condition B.2 Proof of Theorem 4.1 B.3 Infinitesmal coordinate changes C The proof of Theorem 4.3 D Rewriting Arakawa's character E Drinfel'd-Sokolov reduction of the free hypermultiplet E.1 The d 2 Case E.2 The d 3 Case

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On classification of conformal vectors in vertex operator algebra and the vertex algebra automorphism group

Cited by 2 publications

References 13 publications

Twisted Chiral Algebras of Class $${\mathcal {S}}$$ and Mixed Feigin–Frenkel Gluing

Twisted Chiral Algebras of Class $${\mathcal {S}}$$ and Mixed Feigin–Frenkel Gluing

Twisted chiral algebras of class $\mathcal{S}$ and mixed Feigin-Frenkel gluing

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