Lectures on Fuzzy and Fuzzy Susy Physics by A. P. Balachandran

By A. P. Balachandran

Noncommutative geometry presents a strong device for regularizing quantum box theories within the kind of fuzzy physics. Fuzzy physics keeps symmetries, has no fermion-doubling challenge and represents topological beneficial properties successfully. those lecture notes offer a finished creation to the sphere. beginning with the development of fuzzy areas, utilizing the concrete examples of the bushy sphere and fuzzy advanced projective areas, the e-book strikes directly to talk about the expertise of famous person items on noncommutative R2d and at the fuzzy sphere. Scalar, spinor and gauge box theories in addition to prolonged gadgets resembling monopoles and nonlinear sigma modes are handled in enormous aspect. an in depth remedy of the regularization of supersymmetry is given utilizing the concepts of fuzzy physics.

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We briefly examine them. i. 31) to produce the free propagator. There are three such terms, two of which are planar diagrams and a non-planar diagram. 31). 32) ii. Connected Diagrams: They arise when the differentiation on external currents is applied to different factors in the product. There are 4 × 3 = 12 such terms, giving λ φ m φ¯ m C 1 =− × 4! 33) into planar and nonplanar contributions. In the planar case, the December 6, 2006 10:10 World Scientific Book - 9in x 6in book 41 Scalar Fields on the Fuzzy Sphere indices of adjacent Tˆ’s get contracted.

We consider only A-modules V whose elements are finite-dimensional vectors vi = (vi1 , · · · viK ) with vij ∈ A. The action of a ∈ A on V is then vi → avi = (avi1 , · · · , avik ). Consider the identity 1 belonging to V = M at(N + 1). Then all its elements can be got by (left- or right-) A-action. As an A-module, it is one-dimensional. It is also “generated” by 1 as an A-module. It is a “free” module as it has a basis. Generally, an A-module V is said to be free if it has a basis {ei }, ei ∈ V .

We will not need its full details here. Now replace Y m by Y m and integrate. As χ commutes with rotations, only the angular momentum 0 component of n+1 4π Tn ( 1 )2Y m ∗C December 6, 2006 10:10 World Scientific Book - 9in x 6in book 33 Star Products n+1 4π Tn ( 1 )2 Y m contributes to the integral. δ δmm Y 00 ∗C Y00 = δ 1 2 Tn (0) = 4π n+1 dΩY 1 . 107) for This component is = 0 and the value to be derived below, we get m ∗ Y m = δ δmm = dΩY mY m . 109) Hence ∗ enjoys the special identity characterizing the Weyl-Moyal product for the basis of functions in our algebra and hence for all functions.

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