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The spectral action for a non-compact commutative spectral triple is computed covariantly in a gauge perturbation up to order 2 in full generality.
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V. Gayral, B. Iochum, The spectral action for Moyal planes, J. Math. Phys. 46
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E. Buzano and J. Toft, “Schatten-von Neumann properties in the Weyl calculus”, J. Funct. Anal., 259
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W. D. van Suijlekom, “Renormalization of the asymptotically expanded Yang-Mills spectral action’, Commun. Math. Phys. 312
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