Example 2.2.11. **Noncommutative** **tori**. The **noncommutative** **tori** A n θ are perhaps the most popular platform for examining the methods of **noncommutative** geometry. The origin of these **noncommutative** algebras can be traced back to Heisenberg formu- lation of quantum mechanics. It was proposed by Hermann Weyl (see [34]) that the Heisenberg commutation relations should be replaced by their exponential form in or- der to obtain a bounded Hilbert space realization. The C ∗ -algebra generated by these exponential elements are in fact the **noncommutative** **tori**. From another point of view, the **noncommutative** **tori** can be thought as the strict deformation quantization of alge- bra of functions on **tori** ([32]) There are two main spectral triples on **noncommutative** **tori**, each reflecting a diﬀerent aspect of their geometry. The Spin spectral triples on **noncommutative** **tori** (see [18]) are basically the deformation of the spin spectral triple on commutative **tori**. The other class of spectral triples on **noncommutative** **tori** are the so called Dolbeault spectral triples (see [15]) which are meant to reflect the complex geometry of **tori**.

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Since the introduction of **noncommutative** geometry by Alain Connes in [3.3] (see also [3.4]), **noncommutative** **tori** have proved to be an invaluable tool to understand and test many aspects of **noncommutative** geometry that are not present in the commutative case. The results are simply too many to be cited here. The present paper should be seen as a step in understanding aspects of measure theory and analysis on noncomutative **tori** that have been largely untouched so far. The combinatorial challenges one faces in extending the logarithmic Sobolev inequality, at least in the form that we understand it, seemed to us as a very interesting problem by itself.

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The elliptic theory of differential operators and pseudodifferential calculus operate perfectly in **noncommutative** settings as developed in [12], and analyzed in further detail for **noncommutative** **tori** in [41, 42, 67]. Moreover, there is a crucial need for local index formulas for twisted spectral triples as we explained in §2.3. Therefore, in the present paper we take the heat expansion approach to find a local formula for the index of the Dirac operator of the twisted spectral triple described in §2.3 with coefficients (or twisted by) an auxiliary finitely generated projective module on the **noncommutative** two torus, playing the role of a general vector bundle, cf. [56, 63]. We follow closely the setup provided in [21, 60], and the work has intimate connections with [49].

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A **noncommutative** 2-torus is one of the main toy models of **noncommutative** geome- try, and a **noncommutative** n -torus is a straightforward generalization of it. In 1980, Pimsner and Voiculescu in [17] described a 6-term exact sequence, which allows for the computation of the K -theory of non-commutative **tori**. It follows that both even and odd K -groups of n -dimensional **noncommutative** **tori** are free abelian groups on 2 n−1 generators. In 1981, the PowersRieel projector was described [19], which, together with the class of identity, generates the even K -theory of non-commutative 2-**tori**. In 1984, Elliott [10] computed trace and Chern character on these K -groups. According to Rieel [20], the odd K -theory of a **noncommutative** n -torus coincides with the group of connected components of the elements of the algebra. In particu- lar, generators of K -theory can be chosen to be invertible elements of the algebra. In Chapter 1, we derive an explicit formula for the rst non-trivial generator of the odd K -theory of **noncommutative** **tori**. This gives the full set of generators for the odd K -theory of **noncommutative** 3-**tori** and 4-**tori**.

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There have been several approaches to constructing T-dual pairs, each with their particular successes. For example Bunke, Schick and collaborators have given a functorial description using algebraic topology methods, which is valid for pairs in which both bundles are commutative spaces. In complex algebraic geometry, T- duality is effected by the Fourier-Mukai transform; in that context duals to singular toric fibrations can be constructed (e.g. [DP],[BBP]). Mathai, Rosenberg and collaborators have constructed T-dual pairs using C ∗ -algebra methods, and arrived at the remarkable discovery that in certain situations one side of the duality must be a family of **noncommutative** **tori**.

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In **noncommutative** geometry, the metric information of a **noncommutative** space is en- coded in the data of a spectral triple (A, H, D), where D plays the role of the Dirac operator acting on the Hilbert space of spinors. Ideas of spectral geometry can then be used to define suitable notions such as volume, scalar curvature, and Ricci curvature. In particular, one can construct the Ricci curvature from the asymptotic expansion of the heat trace Tr(e −tD 2 ). In Chapter 2, we will compute the Ricci curvature of a curved **noncommutative** three torus. The computation is done for both conformal and a non-conformal perturbation of the flat metric. By applying Connes’ pseudodi ff erential calculus for the **noncommutative** **tori**, we explicitly compute the second density of the heat trace expansion for the perturbed Lapla- cians on both functions and 1−forms. On the other hand, in **noncommutative** geometry one also wants to get a good notion of an action functional which depends only on the spectrum of D, called spectral action functional. It is known that such a functional can be expressed as Tr(f(D)) for some function f . In chapter 3, we show that the von Neumann entropy, average energy, and negative free energy of the Gibbs state of the second quantized Dirac operator d Γ D has a spectral action functional interpretation of the original Dirac operator D. To be able to carry on the computations, we have to incorporate the chemical poten- tial µ. All those spectral action coe ffi cients can be given in terms of the modified Bessel functions.

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assume that a group G acts on K by automorphism and maps F to itself. Noether’s rationality problem asks under what conditions the extension K G /F G is also rational. The origin of the problem goes back to some problems in constructive Galois theory (see [29] and [17]). The special case of multiplicative G-fields (here, K(L) for some lattice L with a multiplicative ac- tion of G) for a finite group G has received particular attention. One of the main reasons of this attention is the connection of multiplicative invariants with algebraic **tori**.

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Perhaps, any researcher had observed the wrong pub- lications, statements, presentations, where the mistake could be revealed with axioms **TORI**, without to drill through the deduction, nor dig deep into the experimental set-up. Usually, there is no need to declare the publication as “wrong”, it is sufficient to indicate, which basic principle should be revised on the base of the new result. I suggest only one example: “Medium with such a wonderful prop- erties is very interesting. It should allow to make not only new efficient laser, but also the perpetual motion ma- chine of Second kind.” Usually, the authors are not ready to negate the fundamental principles, and consider such a diplomatic construction as kind of accusation, and some- times found mistakes in their deductions or experiments.

this fixed torus, also minimizes energy locally with respect to variations of the period lattice. The key idea is that varying the torus among all flat **tori** with Euclidean metric is equivalent to fixing the torus but varying the metric on the torus among all constant coefficient metrics. This manoeuvre allows us to formulate the energy variation with respect to the torus in terms of the stress tensor of the field. We compute both the first and second variation formulae to give two criteria, involving the stress tensor, for the field to be a critical point of energy with respect to variations of the lattice (the first variation) and then, further, a local minimum of energy with respect to such variations (the second variation). The first criterion is that the stress tensor should be L 2 orthogonal to the 1

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dimension have got a lot of attention lately because of their connections to various parts of mathematics, such as commutative algebra, **noncommutative** algebraic geometry and representation theory. The problem of constructing explicit NCRs is difficult in general and one only knows some scattered examples. In particular, NCRs for non-normal rings have not been considered much in the literature, mostly only in examples where R has Krull-dimension ≤ 2 (geometrically: R is the coordinate ring of a collection of points, curves or surfaces).

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Describing the nucleons as **tori**, we calculate in [8] the potential energy and the interaction force between the nucleon couples. This approach is based on recently de- rived for the first time analytical expressions describing the electrostatic interaction between two charged spheres in the most general case [10]. According to the above mentioned formulae one can determine the electrostatic interaction between spheres at distances much less than their radii.

There has been a great deal of recent interest in **noncommutative** (NC) quantum field theories, stimulated by a connection with string theory and M -theory; see for example Refs. [1]–[19]. The theories have, moreover, novel properties which make them worthy of attention in their own right; for example NC quantum electrodynamics exhibits both asymptotic freedom and charge quantisation.

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dichotomous branchiae are arranged on segments 2 and 3, with short stems and short tips. Anterior branchiae (on segment 2) are always longer than posterior ones and are nearer to the middorsum (Fig. 2c, e, Fig. 3b). On segments 2, 3 and 4 lateral lobes are present, not much developed but distinctly visible. Ten distinct ventral shields are visible on thorax (Fig. 2a) with comparable dimension and shape, except for the first one (on the second segment) thinner than the others (Fig. 3a). Noto- chaetae start from segment 4 and are present on 13 thoracic chaetigers. Prominent uncinigerous **tori** are present on thorax from segment 5, moderately long and arranged in single row (Fig. 3a). Uncini in double rows start on segment 11 (Fig. 2a) to segment 20 (4th abdom- inal segment). On the rest of abdomen uncini are again arranged in a single row (Fig. 2a, g). Uncini are small and avicular and appear to have six rows of teeth above the main fang (MF). Dental formula sensu Day (1967) MF:4–5:ca6:ca8:∞:∞:∞.

class of maximal k-split **tori** containing a maximal (θ, k)-split torus. (By Proposition 1.4.2, such a θ exists in SL (n, k) .) Let T be the representative torus of this conjugacy class. It can be shown that one can associate each θ -stable maximal k -split torus with an element of W H k (T ). To obtain subproblem (4), one can use the following general

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It is well known that for a left Noetherian, connected graded k-algebra, A, Tors A is a well-behaved Serre subcategory. In the commutative case, this of course covers the class of all finitely generated connected graded k-algebras. However, as noncom- mutative rings are generally less well behaved than their commutative counterparts, we note that even the **noncommutative** polynomial algebra khx, yi is no longer left Noetherian (see, e.g. Goodearl and Warfield (2004, Exercise 1, p. 8)). The following proposition allows us to consider non-Noetherian rings.

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These results are deeply related to the **noncommutative** geometrization program for CFT recently developed in [73, 17, 16, 14, 15], cf. also [82, 83] for related work. Actu- ally, this program was one of the initial motivations for the present article. We obtain here for the first time examples for which this **noncommutative** geometrization program can be carried out for all sectors of the corresponding conformal nets. This means that all irreducible sectors can be separated by JLO cocycles associated to spectral triples naturally arising from supersymmetry. The central new idea which allows us to over- come certain technical difficulties found e.g. in [15, 16] is the one to consider spectral triples associated to degenerate representations. As a consequence we can relax the re- quirement of superconformal symmetry by considering superconformal tensor product extensions, the superconformal tensor product considered in Section 5. By the results in the latter section these examples include quite a large family of completely rational conformal nets beyond loop group conformal nets. Another consequence of the results of this paper is that the **noncommutative** geometric description of the representation theory of these conformal nets is directly related to the K-theoretic description recently given in [12, 13].

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Whereas the momenta and coordinates in (2.3) are Hermitian operators acting on a Hilbert space with standard inner product, this is no longer true for the variables associated to the deformed algebra (2.1) as they become in general non-Hermitian with regard to these inner products. Thus a quantum mechanical or quantum field theoretical models on these spaces will, in general, not be Hermitian in that space. However, it is by now well accepted that one may consider complex PT symmetric non-Hermitian systems as self-consistent descriptions of physical systems [29, 30]. Guided by these results one may try to identify this symmetry for the **noncommutative** space relations (2.3). In [31] the authors argue that this would not be possible and one is therefore forced to take the **noncommutative** constants to be complex. We reason here that this is incorrect and even the standard **noncommutative** space relations are in fact symmetric under many different versions of PT -symmetry. All one requires to formulate a consistent quantum description is an antilinear involutory map [32] that leaves the relations (2.3) invariant. We identify here the several possibilities:

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2. Mesh-of-**tori** interconnection topology. This discussion leads us to the following question. As- suming that the focal plane I/O like approach is used to provide an effective data transfer to a rectangularly arranged group of processing units, what network topology should be used to connect them. Here, the experi- ences from development of parallel supercomputers could be useful. Historically, a number of topologies have been proposed, and the more interesting of them were: (1) hypercube – scaled up to 64000+ processor in the Connection Machine CM-1, (2) mesh – scaled up to 4000 processors in the Intel Paragon, (3) processor array – scaled up to 16000+ processor in the MassPar computer, (4) rings of rings – scaled up to 1000+ processors in the Kendall Square KSR-1 machines, and (5) torus – scaled up to 2048 units in the Cray T3D.

[X, P ] = i 1 + ˇ τ P 2 , X = (1 + ˇ τ p 2 )x, P = p. (3.2) Here ˇ τ := τ /(mω ) has the dimension of an inverse squared momentum with τ being dimensionless. We also provided in (3.2) a representation for the **noncommutative** variables in terms of the standard canonical variables x, p satisfying [x, p] = i . The ground state energy is conveniently shifted to allow for a factorization of the energy. The Hamiltonian in (3.1) in terms of x, p diﬀers from the one treated recently in [29] as we take a diﬀerent representation for X and P , which we believe to be incorrect in [29] even up to O (τ ). The so-called Dyson map η, whose adjoint action relates the non-Hermitian Hamiltonian in (3.1) to its isospectral Hermitian counterpart h, is easily found to be η = (1 + ˇ τ p 2 ) −1/2 . With the help of this expression we evaluate

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**Tori** are small bony growths routinely found in adult patients. They are usually asymptomatic and often incidentally found during routine clinical examinations. They vary in size, shape, and number, but are often bilateral and sometimes can attain very large sizes. **Tori** are classified according to their shapes as flat **tori** which are smooth and convex in shape, spindle-shaped **tori** usually present in the mid-palatine region, lobular **tori** which are lobulated masses arising from a single base, and nodular **tori** which appear as multiple protuberances with individual bases. 3,5