We generalise the construction of fuzzy CP N in a manner that allows us to access all noncommutative equivariant complex vector bundles over this space. We give a simplified construction of polarization tensors on S 2 that generalizes to complex projective space, identify Laplacians and natural noncommutative covariant derivative operators that map between the modules that describe noncommutative sections. In the process we find a natural generalization of the Schwinger-Jordan construction to su(n) and identify composite oscillators that obey a Heisenberg algebra on an appropriate Fock space.
We present a universal Dirac operator for noncommutative spin and spin c bundles over fuzzy complex projective spaces. We give an explicit construction of these bundles, which are described in terms of finite dimensional matrices, calculate the spectrum and explicitly exhibit the Dirac eigenspinors. To our knowledge the spin c spectrum for CP n with n ≥ 3 is new. *
We study the three-loop Euler-Heisenberg Lagrangian in spinor quantum electrodynamics in 1+1 dimensions. In this first part we calculate the one-fermion-loop contribution, applying both standard Feynman diagrams and the worldline formalism which leads to two different representations in terms of fourfold Schwinger-parameter integrals. Unlike the diagram calculation, the worldline approach allows one to combine the planar and the non-planar contributions to the Lagrangian. Our main interest is in the asymptotic behaviour of the weak-field expansion coefficients of this Lagrangian, for which a nonperturbative prediction has been obtained in previous work using worldline instantons and Borel analysis. We develop algorithms for the calculation of the weak-field expansions coefficients that, in principle, allow their calculation to arbitrary order. Here for the nonplanar contribution we make essential use of the polynomial invariants of the dihedral group D 4 in Schwinger parameter space to keep the expressions manageable. As expected on general grounds, the coefficients are of the form r 1 + r 2 ζ 3 with rational numbers r 1 , r 2 . We compute the first two coefficients analytically, and four more by numerical integration. arXiv:1812.08380v1 [hep-th]
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