scholarly journals Coulomb branch Hilbert series and Hall-Littlewood polynomials

2014 ◽  
Vol 2014 (9) ◽  
Author(s):  
Stefano Cremonesi ◽  
Amihay Hanany ◽  
Noppadol Mekareeya ◽  
Alberto Zaffaroni
Keyword(s):  
Hilbert Series ◽  
Coulomb Branch ◽  
Littlewood Polynomials

scholarly journals Magnetic lattices for orthosymplectic quivers

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Antoine Bourget ◽  
Julius F. Grimminger ◽  
Amihay Hanany ◽  
Rudolph Kalveks ◽  
Marcus Sperling ◽  
...  
Keyword(s):  
Gauge Group ◽  
Moduli Spaces ◽  
Hilbert Series ◽  
Matter Content ◽  
Coulomb Branch ◽  
Gauge Groups ◽  
Orbit Closures ◽  
Littlewood Polynomials ◽  
Monopole Operators ◽  
Crucial Ingredient

Abstract For any gauge theory, there may be a subgroup of the gauge group which acts trivially on the matter content. While many physical observables are not sensitive to this fact, the choice of the precise gauge group becomes crucial when the magnetic lattice of the theory is considered. This question is addressed in the context of Coulomb branches for 3d $$ \mathcal{N} $$ N = 4 quiver gauge theories, which are moduli spaces of dressed monopole operators. We compute the Coulomb branch Hilbert series of many unitary-orthosymplectic quivers for different choices of gauge groups, including diagonal quotients of the product gauge group of individual factors, where the quotient is by a trivially acting subgroup. Choosing different such diagonal groups results in distinct Coulomb branches, related as orbifolds. Examples include nilpotent orbit closures of the exceptional E-type algebras and magnetic quivers that arise from brane physics. This includes Higgs branches of theories with 8 supercharges in dimensions 4, 5, and 6. A crucial ingredient in the calculation of exact refined Hilbert series is the alternative construction of unframed magnetic quivers from resolved Slodowy slices, whose Hilbert series can be derived from Hall-Littlewood polynomials.

scholarly journals Discrete gauging and Hasse diagrams

2021 ◽  
Vol 11 (2) ◽  
Author(s):  
Guillermo Arias Tamargo ◽  
Antoine Bourget ◽  
Alessandro Pini
Keyword(s):  
Gauge Theories ◽  
Hilbert Series ◽  
Higgs Mechanism ◽  
Coulomb Branch ◽  
Gauge Groups ◽  
Hasse Diagrams

We analyse the Higgs branch of 4d \mathcal{N}=2𝒩=2 SQCD gauge theories with non-connected gauge groups \widetilde{\mathrm{SU}}(N) = \mathrm{SU}(N) \rtimes_{I,II} \mathbb{Z}_2SŨ(N)=SU(N)⋊I,IIℤ2 whose study was initiated in . We derive the Hasse diagrams corresponding to the Higgs mechanism using adapted characters for representations of non-connected groups. We propose 3d \mathcal{N}=4𝒩=4 magnetic quivers for the Higgs branches in the type II discrete gauging case, in the form of recently introduced wreathed quivers, and provide extensive checks by means of Coulomb branch Hilbert series computations.

scholarly journals Coulomb branch Hilbert series and three dimensional Sicilian theories

2014 ◽  
Vol 2014 (9) ◽  
Author(s):  
Stefano Cremonesi ◽  
Amihay Hanany ◽  
Noppadol Mekareeya ◽  
Alberto Zaffaroni
Keyword(s):  
Hilbert Series ◽  
Three Dimensional ◽  
Coulomb Branch

scholarly journals Orthosymplectic implosions

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Antoine Bourget ◽  
Andrew Dancer ◽  
Julius F. Grimminger ◽  
Amihay Hanany ◽  
Frances Kirwan ◽  
...  
Keyword(s):  
Hilbert Series ◽  
Unitary Groups ◽  
Symplectic Groups ◽  
Coulomb Branch

Abstract We propose quivers for Coulomb branch constructions of universal implosions for orthogonal and symplectic groups, extending the work on special unitary groups in [1]. The quivers are unitary-orthosymplectic as opposed to the purely unitary quivers in the A-type case. Where possible we check our proposals using Hilbert series techniques.

scholarly journals Magnetic quivers from brane webs with O7+-planes

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Mohammad Akhond ◽  
Federico Carta
Keyword(s):  
Fixed Points ◽  
Hilbert Series ◽  
Coulomb Branch ◽  
Unitary Gauge

Abstract We consider the Higgs branch of 5d fixed points engineered using brane webs with an O7+-plane. We use the brane construction to propose a set of rules to extract the corresponding magnetic quivers. Such magnetic quivers are generically framed non-simply-laced quivers containing unitary as well as special unitary gauge nodes. We compute the Coulomb branch Hilbert series of the proposed magnetic quivers. In some specific cases, an alternative magnetic quiver can be obtained either using an ordinary brane web or a brane web with an O5-plane. In these cases, we find a match at the level of the Hilbert series.

scholarly journals Factorised 3d $$ \mathcal{N} $$ = 4 orthosymplectic quivers

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Mohammad Akhond ◽  
Federico Carta ◽  
Siddharth Dwivedi ◽  
Hirotaka Hayashi ◽  
Sung-Soo Kim ◽  
...  
Keyword(s):  
Moduli Space ◽  
Gauge Theories ◽  
Hilbert Series ◽  
Point Of View ◽  
Coulomb Branch ◽  
Dual Pairs ◽  
Superconformal Field Theories ◽  
Quiver Gauge Theory ◽  
Highest Weight ◽  
Exceptional Sequences

Abstract We study the moduli space of 3d $$ \mathcal{N} $$ N = 4 quiver gauge theories with unitary, orthogonal and symplectic gauge nodes, that fall into exceptional sequences. We find that both the Higgs and Coulomb branches of the moduli space factorise into decoupled sectors. Each decoupled sector is described by a single quiver gauge theory with only unitary gauge nodes. The orthosymplectic quivers serve as magnetic quivers for 5d $$ \mathcal{N} $$ N = 1 superconformal field theories which can be engineered in type IIB string theories both with and without an O5 plane. We use this point of view to postulate the dual pairs of unitary and orthosymplectic quivers by deriving them as magnetic quivers of the 5d theory. We use this correspondence to conjecture exact highest weight generating functions for the Coulomb branch Hilbert series of the orthosymplectic quivers, and provide tests of these results by directly computing the Hilbert series for the orthosymplectic quivers in a series expansion.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals The Hilbert Series of a Linear Symplectic Circle Quotient

2014 ◽  
Vol 23 (1) ◽  
pp. 46-65 ◽  
Author(s):  
Hans-Christian Herbig ◽  
Christopher Seaton
Keyword(s):  
Hilbert Series

scholarly journals 2, 12, 117, 1959, 45171, 1170086, …: a Hilbert series for the QCD chiral Lagrangian

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lukáš Gráf ◽  
Brian Henning ◽  
Xiaochuan Lu ◽  
Tom Melia ◽  
Hitoshi Murayama
Keyword(s):  
Hilbert Series ◽  
Charge Conjugation ◽  
External Fields ◽  
Dynkin Diagrams ◽  
Non Linear ◽  
Very High ◽  
Chiral Lagrangian

Abstract We apply Hilbert series techniques to the enumeration of operators in the mesonic QCD chiral Lagrangian. Existing Hilbert series technologies for non-linear realizations are extended to incorporate the external fields. The action of charge conjugation is addressed by folding the $$ \mathfrak{su}(n) $$ su n Dynkin diagrams, which we detail in an appendix that can be read separately as it has potential broader applications. New results include the enumeration of anomalous operators appearing in the chiral Lagrangian at order p8, as well as enumeration of CP-even, CP-odd, C-odd, and P-odd terms beginning from order p6. The method is extendable to very high orders, and we present results up to order p16.(The title sequence is the number of independent C-even and P-even operators in the mesonic QCD chiral Lagrangian with three light flavors of quarks, at chiral dimensions p2, p4, p6, …)

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