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We develop a new method for representing Hilbert series based on the highest
weight Dynkin labels of their underlying symmetry groups. The method draws on
plethystic functions and character generating functions along with Weyl
integration. We give explicit examples showing how the use of such highest
weight generating functions (HWGs) permits an ef...
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In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a $1 \times \infty$ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, wh...
Citations
... Among them, the pionless EFT is also included by removing all the pion fields. To construct the most general effective operator basis, we adopt the Hilbert series [86][87][88][89][90][91][92][93][94][95] and the Young tensor method [96][97][98], ...
... In this paper, we adopt the Young tensor method [96][97][98], which will be discussed in Sec. 4, to efficiently establish the complete and independent operators of these sectors. In addition, we utilize the Hilbert series method [86][87][88][89][90][91][92][93][94][95] to count the invariant operators conserving the C and P symmetries for complementation, which will be discussed in Sec. 3. Despite the modifications above, the formula in Eq. (2.61) is not applicable in the 2-nucleon sector L N N , since it excludes any bound states at LO, thus can not explain the shallow deuteron bound state. Thus the LO nucleon-nucleon scattering must be non-perturbative, and all the 2PR diagrams should be resumed. ...
... Before writing down all the independent operators in the Lagrangian, the Hilbert series [86][87][88][89][90][91][92][93][94][95] can be used to count the numbers of them. To reach this goal, let us consider the general structure of the operators. ...
The chiral effective field theory (ChEFT) is an extension of the chiral perturbation theory that includes the nuclear forces and weak currents at the hadronic and nuclear scales. We propose a systematic framework of parametrising the pion-nucleon and nucleon-nucleon Lagrangian via the Weinberg power counting rules. We enumerate the operator bases of ChEFT by extending the Hilbert series of the pure meson sector to the nucleon sector with the CP symmetries. The Young tensor method is utilized to obtain the complete sets of the nucleon-nucleon, three-nucleon operators with/without pions (-EFT). Then we use the spurion technique to reconstruct the ChEFT Lagrangian by taking the adjoint spurion and leptonic fields as the building blocks, without the need of external sources. These operators can be applied to both the strong dynamics, and the nucleon/nuclear weak current processes.
... Indeed, Hilbert polynomials and Hilbert series have played important roles in physics. For instance, Hanany and Kalveks [15] (2014) found an efficient encoding and analysis of the Hilbert series of the vacuum moduli spaces of instantons. Cremonesi, Mekareeya and Zaffaroni [6] (2016) presented a formula for the Hilbert series that counts gauge invariant chiral operators in a large class of 3d N ≥ 2 Yang-Mills-Chern-Simons theories. ...
By applying Fourier transformations to the natural orthogonal oscillator representations of special linear Lie algebras, Luo and the second author (2013) obtained a large family of infinite-dimensional irreducible representations of the algebras on the homogeneous solutions of the Laplace equation. In our earlier work, we proved that the associated varieties of these irreducible representations are the intersection of determinantal varieties. In this paper, we find the Hilbert polynomial of these associated varieties. Moreover, we show that the Hilbert polynomial of such an irreducible module M with respect to any generating subspace satisfies for sufficiently large positive integer k and find a necessary and sufficient condition that the equality holds. Furthermore, we explicitly determine the leading term of , which is independent of the choice of .
... (3 2 ,1 2g−4 ) is non-normal [78] as is the S lodowy intersection from this orbit to the O Cg+1 (2,1 2g ) .It is convenient to convert the Hilbert series for these one parameter family of moduli spaces to a Highest Weight Generating funciton (HWG) [79] which is given as HWG S Cg+1 ...
This paper classifies all Higgs branch Higgsing patterns for simply-laced unitary quiver gauge theories with eight supercharges (including multiple loops) and introduces a Higgs branch subtraction algorithm. All possible minimal transitions are given, identifying differences between slices that emerge on the Higgs and Coulomb branches. In particular, the algorithm is sensitive to global information including monodromies and Namikawa-Weyl groups. Guided by symplectic duality, the algorithm further determines the global symmetry on the Coulomb branch, and verifies the exclusion of C type or global symmetry for (simply-laced) unitary quiver gauge theories. The Higgs branches of some unitary quivers are verified to give slices in the nilpotent cones of exceptional simple Lie algebras.
... The most well-known rephasing invariant is the Jarlskog invariant [8,9], that is proportional to multiple higher-dimensional CP-violating operators, making the task of classifying basis invariants challenging. Tools like the Hilbert-Poincaré series and its Plethystic logarithm (PL) offer some aid toward identifying the number of invariants [7,17,[23][24][25][26][27][28][29][30][31][32][33][34]. However, the current methods fall short in finding a "primary invariant", especially when it comes to clearly identifying CP-even and CP-odd invariants, which are essential for a complete understanding of CPV. ...
In this study, we introduce a transformative, automated framework for classifying basis invariants in generic field theories. Utilizing a novel ring-diagram methodology accompanied by the well-known Cayley-Hamilton theorem, our approach uniquely enables the identification of basic invariants and their CP-property characterization. Critically, our framework also unveils previously concealed attributes of established techniques reliant on the Hilbert-Poincaré series and its associated Plethystic logarithm. This paradigm shift has broad implications for the deeper understanding and more accurate classification of CP invariants in generic field theories.
... From a refined HS one can derive the Highest Weight Generating function (HWG) that enumerates the irreps (identified by Dynkin labels) of the global symmetry that appear at each order in the HS [3]. Sometimes, an HS or HWG can be expressed in a compact form using the Plethystic Exponential [4]. ...
We develop the diagrammatic technique of quiver subtraction to facilitate the identification and evaluation of the SU(n) hyper-K\"ahler quotient (HKQ) of the Coulomb branch of a 3d unitary quiver theory. The target quivers are drawn from a wide range of theories, typically classified as ''good'' or ''ugly'', which satisfy identified selection criteria. Our subtraction procedure uses quotient quivers that are ''bad'', differing thereby from quiver subtractions based on Kraft-Procesi transitions. The procedure identifies one or more resultant quivers, the union of whose Coulomb branches corresponds to the desired HKQ. Examples include quivers whose Coulomb branches are moduli spaces of free fields, closures of nilpotent orbits of classical and exceptional type, and slices in the affine Grassmanian. We calculate the Hilbert Series and Highest Weight Generating functions for HKQ examples of low rank. For certain families of quivers, we are able to conjecture HWGs for arbitrary rank. We examine the commutation relations between quotient quiver subtraction and other diagrammatic techniques, such as Kraft-Procesi transitions, quiver folding, and discrete quotients.
... Hilbert series are a useful tool for determining the number of group invariants [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47]. Formally, Hilbert series are a power series H = n c n t n where c n is the number of invariants at order t n . ...
A bstract
Following a recent publication, in this paper we count the number of independent operators at arbitrary mass dimension in N = 1 supersymmetric gauge theories and derive their field and derivative content. This work uses Hilbert series machinery and extends a technique from our previous work on handling integration by parts redundancies to vector superfields. The method proposed here can be applied to both abelian and non-abelian gauge theories and for any set of (chiral/antichiral) matter fields. We work through detailed steps for the abelian case with single flavor chiral superfield at mass dimension eight, and provide other examples in the appendices.
... The following fugacity map, 1 ,y 2 ,y 3 ) . The highest weight generating function [39] takes the form, y 1 ,y 2 ,y 3 ) . In terms of characters of irreducible representations of the global symmetry, the plethystic logarithm of the refined Hilbert series of Irr F ♭ takes the form, From the plethystic logarithm, we identify the generators of the master space Irr F ♭ as, ...
... The master space Irr F ♭ is then given by the following quotient, [n] SU (2)y . We can write the character expansion in (4.60) as a highest weight generating function [39]. The corresponding highest weight generating function takes the form, ...
... Note that even if we set the fugacities for the extra GLSM fields to u 1 = 1 and u 2 = 1, the corresponding fugacity map in (4.82) will allow us to rewrite the refined Hilbert series in (4.77) in terms of the same characters of irreducible representations of the global symmetry of the master space Irr F ♭ . The corresponding highest weight generating function [39] is given by ...
We systematically study the master space of brane brick models that represent a large class of 2d (0,2) quiver gauge theories. These 2d (0,2) theories are worldvolume theories of D1-branes that probe singular toric Calabi-Yau 4-folds. The master space is the freely generated space of chiral fields subject to the J- and E-terms and the non-abelian part of the gauge symmetry. We investigate several properties of the master space for abelian brane brick models with U(1) gauge groups. For example, we calculate the Hilbert series, which allows us by using the plethystic programme to identify the generators and defining relations of the master space. By studying several explicit examples, we also show that the Hilbert series of the master space can be expressed in terms of characters of irreducible representations of the full global symmetry of the master space.
... Hilbert series are a useful tool for determining the number of group invariants [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45]. Formally, Hilbert series are a power series H = n c n t n where c n is the number of invariants at order t n . ...
Following a recent publication, in this paper we count the number of independent operators at arbitrary mass dimension in N=1 supersymmetric gauge theories and derive their field and derivative content. This work uses Hilbert series machinery and extends a technique from our previous work on handling integration by parts redundancies to vector superfields. The method proposed here can be applied to both abelian and non-abelian gauge theories and for any set of (chiral/antichiral) matter fields. We work through detailed steps for the abelian case with single flavor chiral superfield at mass dimension eight, and provide other examples in the appendices.
... The Hilbert series [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50] is a useful way to count the number of independent group invariants. In a field theory consisting of N fields, the Hilbert series has the following form: JHEP04(2023)097 partial derivative. ...
... One technique to calculate the Hilbert series is via the plethystic exponential (PE) [38,[42][43][44]50]. The plethystic exponential generates all (symmetric or anti-symmetric) products of its arguments. ...
A bstract
In this paper we introduce a Hilbert series approach to build the operator basis for a N = 1 supersymmetry theory with chiral superfields. We give explicitly the form of the corrections that remove redundancies due to the equations of motion and integration by parts. In addition, we derive the maps between the correction spaces. This technique allows us to calculate the number of independent operators involving chiral and antichiral superfields to arbitrarily high mass dimension. Using this method, we give several illustrative examples.
... The Poisson bracket is a structure defined on the variety, independent of any gauge theory construction, and thus the magnetic charge m of the Poisson bracket itself should be zero. The magnetic charge of the result of a Poisson bracket between two Coulomb branch operators O 1 and O 2 should therefore be 27) and therefore, using (2.25), the Poisson bracket between any two generators from (2.23) must have conformal dimension ...
... In this case, (7.1) morphs into the affine Dynkin diagram of E 8 and hence its Coulomb branch is the corresponding minimal nilpotent orbit closure, e 8 . Since here the global symmetry enhances to E 8 , we could write the HWG [27] in terms of representations of E 8 -as P E[m 7 t 2 ] for m 1 , . . . , m 8 highest weight fugacities of E 8 -however it will be more useful to write it as the n = 0 case of (7.2) -i.e. in terms of SO(16) ⊂ E 8 representations -so that we can use our results to generalise to higher n: ...
... 8) 27 In the 16d analogy to the Lorentz transformations, µ, ν = 1, . . . , 16 are antisymmetrised. ...
A bstract
We construct Poisson bracket relations between the operators which generate the chiral ring of the Coulomb branch of certain 3 d N = 4 quiver gauge theories. In the case where the Coulomb branch is a free space, ADE Klein singularity, or the minimal A 2 nilpotent orbit, we explicitly compute the Poisson brackets between the generators using either inherited properties of the abstract Coulomb branch variety, or the expected charges of these operators under the global symmetry (known through use of the monopole formula). We also conjecture Poisson brackets for Higgs branches that originate from 6 d theories with tensionless strings or 5 d theories with massless instantons for which the HWG is known, based on representation theoretic and operator content constraints known from the study of their magnetic quiver.
