Finite Subgroups Of SU Research Articles

A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds

Let G G be a finite subgroup of S U ( 4 ) \mathrm {SU}(4) such that its elements have age at most one. In the first part of this paper, we define K K -theoretic stable pair invariants on a crepant resolution of the affine quotient C 4 / G {\mathbb {C}}^4/G , and conjecture a closed formula for their generating series in terms of the root system of G G . In the second part, we define degree zero Donaldson-Thomas invariants of Calabi-Yau 4-orbifolds, develop a vertex formalism that computes the invariants in the toric case, and conjecture closed formulae for their generating series for the quotient stacks [ C 4 / Z r ] [{\mathbb {C}}^4/{\mathbb {Z}}_r] , [ C 4 / Z 2 × Z 2 ] [{\mathbb {C}}^4/{\mathbb {Z}}_2\times {\mathbb {Z}}_2] . Combining these two parts, we formulate a crepant resolution correspondence which relates the above two theories.

ADHM in 8d, coloured solid partitions and Donaldson-Thomas invariants on orbifolds

We study the moduli space of SU(4) invariant BPS conditions in supersymmetric gauge theory on non-commutative C4 by means of an ADHM-like quiver construction and we classify the invariant solutions under the natural toric action in terms of solid partitions. In the orbifold case C4/G, G being a finite subgroup of SU(4), the classification is given in terms of coloured solid partitions. The statistical weight for their counting is defined through the associated equivariant cohomological gauge theory. We explicitly compute several terms of the partition function expansion in the instanton counting parameter on C4 and C2×(C2/Z2), which conjecturally provides the corresponding orbifold Donaldson-Thomas invariants.

Anticoherent subspaces

We extend the notion of anticoherent spin states to anticoherent subspaces. An anticoherent subspace of order t is a subspace whose unit vectors are all anticoherent states of order at least t. We use Klein’s description of algebras of polynomials which are invariant under finite subgroups of SU(2) on C2 to provide constructions of anticoherent subspaces. We discuss applications of this idea to the entanglement of n qubit symmetric states. Furthermore, we show a connection between the existence of these subspaces and the properties of the higher-rank numerical range for certain products of spin observables. We also note that these constructions give us subspaces of spin states all of whose unit vectors have Majorana representations which are spherical designs of order at least t.

Nonperturbative spectral action of round coset spaces of SU(2)

We compute the spectral action of SU(2)/ \Gamma with the trivial spin structure and the round metric and find it in each case to be equal to \frac{1}{\vert \Gamma \vert}(\Lambda^3 \hat{f}^{(2)}(0) - \frac{1}{4}\Lambda \hat{f}(0) )+ O(\Lambda^{-\infty}) . We do this by explicitly computing the spectrum of the Dirac operator for SU(2)/ \Gamma equipped with the trivial spin structure and a selection of metrics. Here \Gamma is a finite subgroup of SU(2). In the case where \Gamma is cyclic, or dicyclic, we consider the one-parameter family of Berger metrics, which includes the round metric, and when \Gamma is the binary tetrahedral, binary octahedral or binary icosahedral group, we only consider the case of the round metric.

A new set of generators and a physical interpretation for the $$SU (3)$$ finite subgroup $$D(9,1,1;2,1,1)$$

After 100 years of effort, the classification of all the finite subgroups of SU(3) is yet incomplete. The most recently updated list can be found in P.O. Ludl, J. Phys. A: Math. Theor. 44 255204 (2011), where the structure of the series (C) and (D) of SU(3)-subgroups is studied. We provide a minimal set of generators for one of these groups which has order 162. These generators appear up to phase as the image of an irreducible unitary braid group representation issued from the Jones-Kauffman version of SU(2) Chern-Simons theory at level 4. In light of these new generators, we study the structure of the group in detail and recover the fact that it is isomorphic to the semidirect product Z_9 \times Z_3 \rtimes S_3 with respect to conjugation.

Topological quantum gate construction by iterative pseudogroup hashing

We describe the hashing technique for obtaining a fast approximation of a target quantum gate in the unitary group SU(2) represented by a product of the elements of a universal basis. The hashing exploits the structure of the icosahedral group (or other finite subgroups of SU(2)) and its pseudogroup approximations to reduce the search within a small number of elements. One of the main advantages of the pseudogroup hashing is the possibility of iterating to obtain more accurate representations of the targets in the spirit of the renormalization group approach. We describe the iterative pseudogroup hashing algorithm using the universal basis given by the braidings of Fibonacci anyons. An analysis of the efficiency of the iterations based on the random matrix theory indicates that the runtime and braid length scale poly-logarithmically with the final error, comparing favorably to the Solovay–Kitaev algorithm.

Root systems and the quantum cohomology of ADE resolutions

We compute the ℂ∗-equivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity ℂ2∕G where G is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADE root system canonically associated to G. We generalize the resulting Frobenius manifold to nonsimply laced root systems to obtain an n parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Gromov–Witten potential of [ℂ2∕G].

Flavor group Δ(3n2)

The large neutrino mixing angles have generated interest in finite subgroups of SU(3), as clues towards understanding the flavor structure of the standard model. In this work, we study the mathematical structure of the simplest non-Abelian subgroup, Δ(3n2). Using simple mathematical techniques, we derive its conjugacy classes and character table, and build its irreducible representations, their Kronecker products, and its invariants.

Quantum affine Cartan matrices, Poincaré series of binary polyhedral groups, and reflection representations

We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a, b, h; p, q, r) that one usually associates with such a group and hence with a simply-laced Coxeter–Dynkin diagram have a meaningful definition for the non-simply-laced diagrams, too, and as a byproduct we extend Saito’s formula for the determinant of the Cartan matrix to all cases. Returning to invariant theory we show that for each irreducible representation i of a binary tetrahedral, octahedral, or icosahedral group one can find a homomorphism into a finite complex reflection group whose defining reflection representation restricts to i.

On the Quantum Invariants for the Spherical Seifert Manifolds

We study the Witten--Reshetikhin--Turaev SU(2) invariant for the Seifert manifold $S^3/\Gamma$ where $\Gamma$ is a finite subgroup of SU(2). We show that the WRT invariants can be written in terms of the Eichler integral of the modular forms with half-integral weight, and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the Eichler integral. We further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to $\Gamma$.

Euclidean Subspaces of the Complex Spaces ℓpn Constructed by Orbits of Finite Subgroups of SU(m)

A group representation method is used to construct minimal isometric embeddings l22 into l810 and l1012 over C. The second of them yields a tight 5-design in C2. The corresponding angle set contains some irrational numbers.

Brane cube realization of three-dimensional nonabelian orbifolds

We study D-branes on three-dimensional orbifolds ${\bf C}^3/\Gamma$ where $\Gamma$ are finite subgroups of SU(3). The quiver diagram of $\ZnZn \in SU(3)$ can be expressed in three-dimensional form. According to the correspondence between quiver diagrams and brane configurations, we construct a brane configuration for ${\bf C}^3/\ZnZn$ which has essentially three-dimensional structrue. Brane configurations for nonabelian orbifolds $\C^3/\Delta(3n^2)$ and $\C^3/\Delta(6n^2)$ are obtained from that for $\C^3/\ZnZn$ by certain quotienting procedure.

D-branes on three-dimensional nonabelian orbifolds

We study D-branes on a three complex dimensional nonabelian orbifold ${\bf C}^3/\Gamma$ with $\Gamma$ a finite subgroup of SU(3). We present general formulae necessary to obtain quiver diagrams which represent the gauge group and the spectrum of the D-brane worldvolume theory for dihedral-like subgroups $\Delta(3n^2)$ and $\Delta(6n^2)$. It is found that the quiver diagrams have a similar structure to webs of branes.

SU(n)-CHERN–SIMONS INVARIANTS OF SEIFERT FIBERED 3-MANIFOLDS

We derive the formula for the Chern–Simons invariants of irrreducible SU(n)-flat connections on the Seifert fibered 3-manifolds. As an example, we calculate the values explicitly for the Seifert fibered 3-manifold S3/Γ, where Γ is a finite subgroup of SU(2).

Fibonacci orbits and SU(2)-dynamics

Motivated by an interpretation of the Fibonacci sequence as a subset of the free group F2 we describe orbits on finite subgroups of SU(2). A prerequisite for the calculation of these orbits is the knowledge of all generating sets of finite subgroups of SU(2). In order to facilitate this problem we introduce an equivalence relation between generating sets which can be extended to another one between Fibonacci orbits. Additionally, we describe a possible experiment with quasiperiodic dynamics.

PARTITION FUNCTIONS, INTERTWINERS AND THE COXETER ELEMENT

The partition functions of Pasquier models on the cylinder, and the associated intertwiners, are considered. It is shown that earlier results due to Saleur and Bauer can be rephrased in a geometrical way, reminiscent of formulae found in certain purely elastic scattering theories. This establishes the positivity of these intertwiners in a general way and elucidates connections between these objects and the finite subgroups of SU(2). It also offers the hope that analogous geometrical structures might lie behind the so-far mysterious results found by DiFrancesco and Zuber in their search for generalisations of these models.

MINIMAL-RANK MODELS FROM NON-ABELIAN ORBIFOLDS

All space-time supersymmetric non-Abelian orbifolds produced by finite subgroups of SU(2) whose gauge groups are of minimal rank are systematically classified for the [Formula: see text] heterotic string. Also, we discuss a novel method of removing global anomalies in the operator formalism.

Curiosities at c = 1

We consider conformal field theories on a torus with central charge c = 1, and in particular models based upon modding out string propagation on the SU(2) group manifold by its finite subgroups. We find that the partition functions for these models coincide with the continuum limit partition functions of a recently introduced class of RSOS models, defined in terms of the extended Dynkin diagrams of simply-laced Lie algebras, thus giving an alternative interpretation for the primary fields in these latter theories. Three of the models have no massless moduli and thus do not lie on the same line of critical points with the rest. The particular correspondence between simply-laced Lie algebras and finite subgroups of SU(2) that emerges coincides with that which has already appeared in other mathematical contexts.

On finite subgroups of SU(2), simple Lie algebras, and the McKay correspondence

The question is considered of how the restriction pi(n)Gamma, where pi(n) is irreducible for all n and Gamma is a finite subgroup of SU(2), decomposes into Gamma-irreducibles for any arbitrary n in [unk](+). It is announced that, using the McKay correspondence, the problem has an elegant solution in terms of the Coxeter element for the associated Lie algebra and that the numbers involved come in a beautiful way from the root structure.

Monte Carlo simulations of lattice models with finite subgroups of SU(3) as gauge groups

A Monte Carlo study of four-dimensional gauge theory on five crystal-like subgroups of SU(3) is carried out using the Wilson action. All these models are found to have a first-order freezing (order-disorder) phase transition due to the discrete nature of the groups. This transition occurs before the continuum, scaling limit is approached. Hence, it is not possible to use any of these groups to approximate the continuum limit of the SU(3) gauge theory with the Wilson action.