The Grothendieck group of invariant rings and of simple hypersurface singularities

1987 ◽  
pp. 134-149
Author(s):  
Jürgen Herzog ◽  
Herbert Sanders
Keyword(s):  
Grothendieck Group ◽  
Invariant Rings ◽  
Hypersurface Singularities

scholarly journals 5d and 4d SCFTs: canonical singularities, trinions and S-dualities

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Cyril Closset ◽  
Simone Giacomelli ◽  
Sakura Schäfer-Nameki ◽  
Yi-Nan Wang
Keyword(s):  
String Theory ◽  
Field Theories ◽  
Flavor Symmetry ◽  
Crepant Resolution ◽  
Superconformal Field Theories ◽  
Hypersurface Singularities ◽  
Canonical Singularities ◽  
Type Iib String Theory ◽  
M Theory

Abstract Canonical threefold singularities in M-theory and Type IIB string theory give rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. In this paper, we study canonical hypersurface singularities whose resolutions contain residual terminal singularities and/or 3-cycles. We focus on a certain class of ‘trinion’ singularities which exhibit these properties. In Type IIB, they give rise to 4d $$ \mathcal{N} $$ N = 2 SCFTs that we call $$ {D}_p^b $$ D p b (G)-trinions, which are marginal gaugings of three SCFTs with G flavor symmetry. In order to understand the 5d physics of these trinion singularities in M-theory, we reduce these 4d and 5d SCFTs to 3d $$ \mathcal{N} $$ N = 4 theories, thus determining the electric and magnetic quivers (or, more generally, quiverines). In M-theory, residual terminal singularities give rise to free sectors of massless hypermultiplets, which often are discretely gauged. These free sectors appear as ‘ugly’ components of the magnetic quiver of the 5d SCFT. The 3-cycles in the crepant resolution also give rise to free hypermultiplets, but their physics is more subtle, and their presence renders the magnetic quiver ‘bad’. We propose a way to redeem the badness of these quivers using a class $$ \mathcal{S} $$ S realization. We also discover new S-dualities between different $$ {D}_p^b $$ D p b (G)-trinions. For instance, a certain E8 gauging of the E8 Minahan-Nemeschansky theory is S-dual to an E8-shaped Lagrangian quiver SCFT.

Characterization of isolated homogeneous hypersurface singularities in ℂ4

2006 ◽  
Vol 49 (11) ◽  
pp. 1576-1592 ◽  
Author(s):  
Kepao Lin ◽  
Zhenhan Tu ◽  
Stephen S. T. Yau
Keyword(s):  
Hypersurface Singularities ◽  
Homogeneous Hypersurface

scholarly journals On the semicontinuity of the mod 2 spectrum of hypersurface singularities

2015 ◽  
Vol 24 (2) ◽  
pp. 379-398 ◽  
Author(s):  
Maciej Borodzik ◽  
András Némethi ◽  
Andrew Ranicki
Keyword(s):  
Hypersurface Singularities

Generic hypersurface singularities

1997 ◽  
Vol 107 (2) ◽  
pp. 139-154 ◽  
Author(s):  
James Alexander ◽  
André Hirschowitz
Keyword(s):  
Generic Hypersurface ◽  
Hypersurface Singularities

scholarly journals CATEGORIFICATION OF QUANTUM GENERALIZED KAC–MOODY ALGEBRAS AND CRYSTAL BASES

2012 ◽  
Vol 23 (11) ◽  
pp. 1250116 ◽  
Author(s):  
SEOK-JIN KANG ◽  
SE-JIN OH ◽  
EUIYONG PARK
Keyword(s):  
Crystal Structures ◽  
Grothendieck Group ◽  
Integral Form ◽  
Cartan Matrix ◽  
Algebra Homomorphism ◽  
Finitely Generated ◽  
Moody Algebra ◽  
Graded Modules ◽  
Isomorphism Classes ◽  
Highest Weight

We construct and investigate the structure of the Khovanov-Lauda–Rouquier algebras R and their cyclotomic quotients Rλ which give a categorification of quantum generalized Kac–Moody algebras. Let U𝔸(𝔤) be the integral form of the quantum generalized Kac–Moody algebra associated with a Borcherds–Cartan matrix A = (aij)i, j ∈ I and let K0(R) be the Grothendieck group of finitely generated projective graded R-modules. We prove that there exists an injective algebra homomorphism [Formula: see text] and that Φ is an isomorphism if aii ≠ 0 for all i ∈ I. Let B(∞) and B(λ) be the crystals of [Formula: see text] and V(λ), respectively, where V(λ) is the irreducible highest weight Uq(𝔤)-module. We denote by 𝔅(∞) and 𝔅(λ) the isomorphism classes of irreducible graded modules over R and Rλ, respectively. If aii ≠ 0 for all i ∈ I, we define the Uq(𝔤)-crystal structures on 𝔅(∞) and 𝔅(λ), and show that there exist crystal isomorphisms 𝔅(∞) ≃ B(∞) and 𝔅(λ) ≃ B(λ). One of the key ingredients of our approach is the perfect basis theory for generalized Kac–Moody algebras.

scholarly journals A note on strongly invariant rings

1978 ◽  
Vol 50 (2) ◽  
pp. 297-298
Author(s):  
S.M Bhatwadekar
Keyword(s):  
Invariant Rings
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