scholarly journals K-theory and phase transitions at high energies

2016 ◽  
Vol 9 (1) ◽  
Author(s):  
T. V. Obikhod
Keyword(s):  
Vector Bundles ◽  
Grothendieck Group ◽  
Soliton Solutions ◽  
Complex Vector ◽  
Supergravity Theory ◽  
Isomorphism Classes ◽  
High Energies ◽  
Complete Set ◽  
K Theory ◽  
Calabi Yau Manifolds

The duality between E8xE8 heteritic string on manifold K3xT2 and Type IIA string compactified on a Calabi-Yau manifold induces a correspondence between vector bundles on K3xT2 and Calabi-Yau manifolds. Vector bundles over compact base space K3xT2 form the set of isomorphism classes, which is a semi-ring under the operation of Whitney sum and tensor product. The construction of semi-ring V ect X of isomorphism classes of complex vector bundles over X leads to the ring KX = K(V ect X), called Grothendieck group. As K3 has no isometries and no non-trivial one-cycles, so vector bundle winding modes arise from the T2 compactification. Since we have focused on supergravity in d = 11, there exist solutions in d = 10 for which space-time is Minkowski space and extra dimensions are K3xT2. The complete set of soliton solutions of supergravity theory is characterized by RR charges, identified by K-theory. Toric presentation of Calabi-Yau through Batyrev's toric approximation enables us to connect transitions between Calabi-Yau manifolds, classified by enhanced symmetry group, with K-theory classification.

scholarly journals Non-abelian Quantum Statistics on Graphs

2019 ◽  
Vol 371 (3) ◽  
pp. 921-973 ◽  
Author(s):  
Tomasz Maciążek ◽  
Adam Sawicki
Keyword(s):  
Quantum Computing ◽  
General Framework ◽  
Vector Bundles ◽  
Quantum Statistics ◽  
Configuration Spaces ◽  
Topological Invariants ◽  
Complex Vector ◽  
Homology Groups ◽  
Isomorphism Classes ◽  
Effective Models

Abstract We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space X. The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of X which can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non-abelian anyons on networks that are used in quantum computing. These conclusions are based on our solution of the so-called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs.

A Cancellation Theorem for Modules Over the Group C*-Algebras of Certain Nilpotent Lie Groups

1987 ◽  
Vol 39 (2) ◽  
pp. 365-427 ◽  
Author(s):  
Albert Jeu-Liang Sheu
Keyword(s):  
Vector Bundles ◽  
Grothendieck Group ◽  
Positive Cone ◽  
Point Of View ◽  
Order Structure ◽  
Topological Spaces ◽  
Projective Modules ◽  
Finitely Generated ◽  
C Algebras ◽  
Isomorphism Classes

In recent years, there has been a rapid growth of the K-theory of C*-algebras. From a certain point of view, C*-algebras can be treated as “non-commutative topological spaces”, while finitely generated projective modules over them can be thought of as “non-commutative vector bundles”. The K-theory of C*-algebras [30] then generalizes the classical K-theory of topological spaces [1]. In particular, the K0-group of a unital C*-algebra A is the group “generated” by (or more precisely, the Grothendieck group of) the commutative semigroup of stable isomorphism classes of finitely generated projective modules over A with direct summation as the binary operation. The semigroup gives an order structure on K0(A) and is usually called the positive cone in K0(A).Around 1980, the work of Pimsner and Voiculescu [18] and of A. Connes [4] provided effective ways to compute the K-groups of C*-algebras. Then the classification of finitely generated projective modules over certain unital C*-algebras up to stable isomorphism could be done by computing their K0-groups as ordered groups. Later on, inspired by A. Connes's development of non-commutative differential geometry on finitely generated projective modules [2], the deeper question of classifying such modules up to isomorphism and hence the so-called cancellation question were raised (cf. [21] ).

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 Heterotic models from vector bundles on toric Calabi-Yau manifolds

2010 ◽  
Vol 2010 (5) ◽  
Author(s):  
Yang-Hui He ◽  
Seung-Joo Lee ◽  
André Lukas
Keyword(s):  
Vector Bundles ◽  
Calabi Yau Manifolds

scholarly journals The group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra

2019 ◽  
Vol 30 (11) ◽  
pp. 1950057 ◽  
Author(s):  
M. Izumi ◽  
T. Sogabe
Keyword(s):  
Automorphism Group ◽  
Vector Bundles ◽  
Group Structure ◽  
Cuntz Algebra ◽  
Classifying Space ◽  
Cuntz Algebras ◽  
K Theory ◽  
Continuous Fields

We determine the group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra [Formula: see text] for finite [Formula: see text] in terms of K-theory. We show that there is an example of a space for which the homotopy set is a noncommutative group, and hence, the classifying space of the automorphism group of the Cuntz algebra for finite [Formula: see text] is not an H-space. We also make an improvement of Dadarlat’s classification of continuous fields of the Cuntz algebras in terms of vector bundles.

K-Theory of Vector Bundles, of Modules, and of Idempotents

2007 ◽  
pp. 45-54
Author(s):  
D. Husemöller ◽  
M. Joachim ◽  
B. Jurčo ◽  
M. Schottenloher
Keyword(s):  
Vector Bundles ◽  
K Theory

Moduli of wild Higgs bundles on with -actions

2021 ◽  
pp. 1-34
Author(s):  
LAURA FREDRICKSON ◽  
ANDREW NEITZKE
Keyword(s):  
Fixed Points ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Central Charge ◽  
Higgs Field ◽  
Vertex Algebra ◽  
Higgs Bundles ◽  
Irregular Singularity ◽  
Isomorphism Classes ◽  
Type Decomposition

Abstract We study a set $\mathcal{M}_{K,N}$ parameterising filtered SL(K)-Higgs bundles over $\mathbb{C}P^1$ with an irregular singularity at $z = \infty$ , such that the eigenvalues of the Higgs field grow like $\vert \lambda \vert \sim \vert z^{N/K} \mathrm{d}z \vert$ , where K and N are coprime. $\mathcal{M}_{K,N}$ carries a $\mathbb{C}^\times$ -action analogous to the famous $\mathbb{C}^\times$ -action introduced by Hitchin on the moduli spaces of Higgs bundles over compact curves. The construction of this $\mathbb{C}^\times$ -action on $\mathcal{M}_{K,N}$ involves the rotation automorphism of the base $\mathbb{C}P^1$ . We classify the fixed points of this $\mathbb{C}^\times$ -action, and exhibit a curious 1-1 correspondence between these fixed points and certain representations of the vertex algebra $\mathcal{W}_K$ ; in particular we have the relation $\mu = {k-1-c_{\mathrm{eff}}}/{12}$ , where $\mu$ is a regulated version of the L 2 norm of the Higgs field, and $c_{\mathrm{eff}}$ is the effective Virasoro central charge of the corresponding W-algebra representation. We also discuss a Białynicki–Birula-type decomposition of $\mathcal{M}_{K,N}$ , where the strata are labeled by isomorphism classes of the underlying filtered vector bundles.

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