The Coulomb and Higgs branches of $$ \mathcal{N} $$ = 1 theories of Class $$ {\mathcal{S}}_k $$

Thomas Bourton; Alessandro Pini; Elli Pomoni

doi:10.1007/jhep02(2021)137

The Coulomb and Higgs branches of $$ \mathcal{N} $$ = 1 theories of Class $$ {\mathcal{S}}_k $$

Journal of High Energy Physics ◽

10.1007/jhep02(2021)137 ◽

2021 ◽

Vol 2021 (2) ◽

Author(s):

Thomas Bourton ◽

Alessandro Pini ◽

Elli Pomoni

Keyword(s):

String Theory ◽

Special Class ◽

Hilbert Series ◽

Superconformal Index

Abstract Even though for generic $$ \mathcal{N} $$ N = 1 theories it is not possible to separate distinct branches of supersymmetric vacua, in this paper we study a special class of $$ \mathcal{N} $$ N = 1 SCFTs, these of Class $$ {\mathcal{S}}_k $$ S k for which it is possible to define Coulomb and Higgs branches precisely as for the $$ \mathcal{N} $$ N = 2 theories of Class $$ \mathcal{S} $$ S from which they descend. We study the BPS operators that parameterise these branches of vacua using the different limits of the superconformal index as well as the Coulomb and Higgs branch Hilbert Series. Finally, with the tools we have developed, we provide a check that six dimensional (1, 1) Little String theory can be deconstructed from a toroidal quiver in Class $$ {\mathcal{S}}_k $$ S k .

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K-RATIONAL D-BRANE CRYSTALS

International Journal of Modern Physics A ◽

10.1142/s0217751x12501126 ◽

2012 ◽

Vol 27 (22) ◽

pp. 1250112

Author(s):

ROLF SCHIMMRIGK

Keyword(s):

String Theory ◽

Algebraic Number ◽

Special Class ◽

Building Blocks ◽

Number Fields ◽

Central Point ◽

Algebraic Number Fields ◽

Special Values ◽

Toroidal Compactifications ◽

Base Points

In this paper the problem of constructing space–time from string theory is addressed in the context of D-brane physics. It is suggested that the knowledge of discrete configurations of D-branes is sufficient to reconstruct the motivic building blocks of certain Calabi–Yau varieties. The collections of D-branes involved have algebraic base points, leading to the notion of K-arithmetic D-crystals for algebraic number fields K. This idea can be tested for D0-branes in the framework of toroidal compactifications via the conjectures of Birch and Swinnerton-Dyer. For the special class of D0-crystals of Heegner type these conjectures can be interpreted as formulae that relate the canonical Néron–Tate height of the base points of the D-crystals to special values of the motivic L-function at the central point. In simple cases the knowledge of the D-crystals of Heegner type suffices to uniquely determine the geometry.

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Enumerating projective reflection groups

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2898 ◽

2011 ◽

Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽

Author(s):

Riccardo Biagioli ◽

Fabrizio Caselli

Keyword(s):

Representation Theory ◽

Generating Functions ◽

Special Class ◽

Hilbert Series ◽

Weyl Groups ◽

Reflection Groups ◽

One Dimensional ◽

Complex Reflection Groups ◽

Major Index ◽

International Audience

International audience Projective reflection groups have been recently defined by the second author. They include a special class of groups denoted G(r,p,s,n) which contains all classical Weyl groups and more generally all the complex reflection groups of type G(r,p,n). In this paper we define some statistics analogous to descent number and major index over the projective reflection groups G(r,p,s,n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r,p,s,n), as distribution of one-dimensional characters and computation of Hilbert series of some invariant algebras, are also treated. Les groupes de réflexions projectifs ont été récemment définis par le deuxième auteur. Ils comprennent une classe spéciale de groupes notée G(r,p,s,n), qui contient tous les groupes de Weyl classiques et plus généralement tous les groupes de réflexions complexes du type G(r,p,n). Dans ce papier on définit des statistiques analogues au nombre de descentes et à l'indice majeur pour les groupes G(r,p,s,n), et on calcule plusieurs fonctions génératrices. Certains aspects de la théorie des représentations de G(r,p,s,n), comme la distribution des caractères linéaires et le calcul de la série de Hilbert de quelques algèbres d'invariants, sont aussi abordés.

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