scholarly journals Random Unitary Representations of Surface Groups I: Asymptotic Expansions

2021 ◽  
Author(s):  
Michael Magee
Keyword(s):  
Moduli Space ◽  
Symplectic Form ◽  
Expected Value ◽  
Random Model ◽  
Unitary Groups ◽  
Irreducible Representations ◽  
Fundamental Groups ◽  
Topological Surface ◽  
Fixed Order ◽  
Sigma G

AbstractIn this paper, we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott and Goldman. Let $$\Sigma _{g}$$ Σ g denote a topological surface of genus $$g\ge 2$$ g ≥ 2 . We establish the existence of a large n asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of $$\pi _{1}(\Sigma _{g})$$ π 1 ( Σ g ) under a random representation of $$\pi _{1}(\Sigma _{g})$$ π 1 ( Σ g ) into $$\mathsf {SU}(n)$$ SU ( n ) . Each such expected value involves a contribution from all irreducible representations of $$\mathsf {SU}(n)$$ SU ( n ) . The main technical contribution of the paper is effective analytic control of the entire contribution from irreducible representations outside finite sets of carefully chosen rational families of representations.

scholarly journals Exploring the landscape of CHL strings on Td

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Anamaría Font ◽  
Bernardo Fraiman ◽  
Mariana Graña ◽  
Carmen A. Núñez ◽  
Héctor Parra De Freitas
Keyword(s):  
Gauge Group ◽  
Moduli Space ◽  
Dynkin Diagram ◽  
Fundamental Groups ◽  
Computer Algorithms ◽  
Abelian Gauge ◽  
Reduced Rank ◽  
Systematic Exploration ◽  
String Models ◽  
Simply Connected

Abstract Compactifications of the heterotic string on special Td/ℤ2 orbifolds realize a landscape of string models with 16 supercharges and a gauge group on the left-moving sector of reduced rank d + 8. The momenta of untwisted and twisted states span a lattice known as the Mikhailov lattice II(d), which is not self-dual for d > 1. By using computer algorithms which exploit the properties of lattice embeddings, we perform a systematic exploration of the moduli space for d ≤ 2, and give a list of maximally enhanced points where the U(1)d+8 enhances to a rank d + 8 non-Abelian gauge group. For d = 1, these groups are simply-laced and simply-connected, and in fact can be obtained from the Dynkin diagram of E10. For d = 2 there are also symplectic and doubly-connected groups. For the latter we find the precise form of their fundamental groups from embeddings of lattices into the dual of II(2). Our results easily generalize to d > 2.

scholarly journals QUANTIZATION OF A MODULI SPACE OF PARABOLIC HIGGS BUNDLES

2004 ◽  
Vol 15 (09) ◽  
pp. 907-917 ◽  
Author(s):  
INDRANIL BISWAS ◽  
AVIJIT MUKHERJEE
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Symplectic Form ◽  
Dense Subset ◽  
Projective Structure ◽  
Open Dense Subset ◽  
Higgs Bundles ◽  
Smooth Variety ◽  
Structure Formula

Let [Formula: see text] be a moduli space of stable parabolic Higgs bundles of rank two over a Riemann surface X. It is a smooth variety defined over [Formula: see text] equipped with a holomorphic symplectic form. Fix a projective structure [Formula: see text] on X. Using [Formula: see text], we construct a quantization of a certain Zariski open dense subset of the symplectic variety [Formula: see text].

On particle-like representations of the complete homogeneous Lorentz group

1973 ◽  
Vol 74 (1) ◽  
pp. 149-160 ◽  
Author(s):  
J. A. de Wet
Keyword(s):  
Lorentz Group ◽  
Wave Functions ◽  
Unitary Groups ◽  
Irreducible Representations ◽  
Electron Wave ◽  
Precise Form ◽  
Homogeneous Lorentz Group ◽  
The Many ◽  
Matrix Contraction

In two previous papers (1, 2) representations of the unitary groups U4, U2 were found which described some of the properties of nucleons and electrons. In particular, the many electron wave functions were constructed from the irreducible representations of U2 restricted to the proper orthochronous Lorentz group Lp. In this paper the irreducible representations of U4 found in (1) will be shown to be also irreducible representations of the complete homogeneous Lorentz group L0 and the techniques of matrix contraction employed in (2) will be used to find the precise form of the matrices of the infinitesimal ring.

scholarly journals Commuting elements in central products of special unitary groups

2012 ◽  
Vol 56 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Alejandro Adem ◽  
F. R. Cohen ◽  
José Manuel Gómez
Keyword(s):  
Prime Number ◽  
Moduli Space ◽  
Unitary Group ◽  
Unitary Groups ◽  
Connected Components ◽  
Flat Connections ◽  
Isomorphism Classes ◽  
Central Product ◽  
Central Products ◽  
Path Connected

AbstractWe study the space of commuting elements in the central product Gm,p of m copies of the special unitary group SU(p), where p is a prime number. In particular, a computation for the number of path-connected components of these spaces is given and the geometry of the moduli space Rep(ℤn, Gm,p) of isomorphism classes of flat connections on principal Gm,p-bundles over the n-torus is completely described for all values of n, m and p.

scholarly journals THE DUAL STRUCTURE OF CROSSED PRODUCT -ALGEBRAS WITH FINITE GROUPS

2013 ◽  
Vol 88 (2) ◽  
pp. 243-249 ◽  
Author(s):  
FIRUZ KAMALOV
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Orbit Space ◽  
Crossed Product ◽  
Irreducible Representations ◽  
Natural Action ◽  
Dual Structure ◽  
Projective Representations ◽  
A Finite Group ◽  
Sigma G

AbstractWe study the space of irreducible representations of a crossed product ${C}^{\ast } $-algebra ${\mathop{A\rtimes }\nolimits}_{\sigma } G$, where $G$ is a finite group. We construct a space $\widetilde {\Gamma } $ which consists of pairs of irreducible representations of $A$ and irreducible projective representations of subgroups of $G$. We show that there is a natural action of $G$ on $\widetilde {\Gamma } $ and that the orbit space $G\setminus \widetilde {\Gamma } $ corresponds bijectively to the dual of ${\mathop{A\rtimes }\nolimits}_{\sigma } G$.

scholarly journals Combinatorial Reid's recipe for consistent dimer models

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Alastair Craw ◽  
Liana Heuberger ◽  
Jesus Tapia Amador
Keyword(s):  
Moduli Space ◽  
Hilbert Scheme ◽  
Abelian Subgroup ◽  
Main Tool ◽  
Derived Category ◽  
Irreducible Representations ◽  
Crepant Resolution ◽  
Mckay Correspondence ◽  
Dimer Model ◽  
Dimer Models

Reid's recipe for a finite abelian subgroup $G\subset \text{SL}(3,\mathbb{C})$ is a combinatorial procedure that marks the toric fan of the $G$-Hilbert scheme with irreducible representations of $G$. The geometric McKay correspondence conjecture of Cautis--Logvinenko that describes certain objects in the derived category of $G\text{-Hilb}$ in terms of Reid's recipe was later proved by Logvinenko et al. We generalise Reid's recipe to any consistent dimer model by marking the toric fan of a crepant resolution of the vaccuum moduli space in a manner that is compatible with the geometric correspondence of Bocklandt--Craw--Quintero-V\'{e}lez. Our main tool generalises the jigsaw transformations of Nakamura to consistent dimer models. Comment: 29 pages, published version

scholarly journals Local Langlands correspondence for unitary groups via theta lifts

2021 ◽  
Vol 25 (30) ◽  
pp. 861-896
Author(s):  
Rui Chen ◽  
Jialiang Zou
Keyword(s):  
Unitary Groups ◽  
Irreducible Representations ◽  
Theta Correspondence ◽  
Langlands Correspondence ◽  
Alternative Approach ◽  
Local Langlands Correspondence

Using the theta correspondence, we extend the classification of irreducible representations of quasi-split unitary groups (the so-called local Langlands correspondence, which is due to Mok) to non quasi-split unitary groups. We also prove that our classification satisfies some good properties, which characterize it uniquely. In particular, this paper provides an alternative approach to the works of Kaletha-Mínguez-Shin-White and Mœglin-Renard.

scholarly journals GENERALIZED CALABI–YAU STRUCTURES, K3 SURFACES, AND B-FIELDS

2005 ◽  
Vol 16 (01) ◽  
pp. 13-36 ◽  
Author(s):  
DANIEL HUYBRECHTS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Symplectic Form ◽  
Weak Form ◽  
K3 Surfaces ◽  
Standard Theory ◽  
General Notion ◽  
Symplectic Structures ◽  
Special Cases ◽  
Torelli Theorem

Generalized Calabi–Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are both special cases of this general notion. The moduli space of generalized Calabi–Yau structures admits a canonical symplectic form with respect to which the moduli space of symplectic structures is Lagrangian. The standard theory of K3 surfaces implies surjectivity of the period map and a weak form of the Global Torelli theorem.

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