The chiral de Rham complex and quantum non-linear sigma models

Abstract We study quantum corrections to hypersurfaces of dimension d + 1 > 2 embedded in generic higher-dimensional spacetimes. Manifest covariance is maintained throughout the analysis and our methods are valid for arbitrary co-dimension and arbitrary bulk metric. A variety of theories which are prominent in the modern amplitude literature arise as special limits: the scalar sector of Dirac-Born-Infeld theories and their multi-field variants, as well as generic non-linear sigma models and extensions thereof. Our explicit one-loop results unite the leading corrections of all such models under a single umbrella. In contrast to naive computations which generate effective actions that appear to violate the non-linear symmetries of their classical counterparts, our efficient methods maintain manifest covariance at all stages and make the symmetry properties of the quantum action clear. We provide an explicit comparison between our compact construction and other approaches and demonstrate the ultimate physical equivalence between the superficially different results.

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Loop calculations in quantum-mechanical non-linear sigma models

Nuclear Physics B ◽

10.1016/0550-3213(95)00241-j ◽

1995 ◽

Vol 446 (1-2) ◽

pp. 211-222 ◽

Cited By ~ 43

Author(s):

Jan de Boer ◽

Bas Peeters ◽

Kostas Skenderis ◽

Peter van Nieuwenhuizen

Keyword(s):

Sigma Models ◽

Quantum Mechanical ◽

Non Linear ◽

Linear Sigma Models

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Integral calculus on quantum exterior algebras

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887814500261 ◽

2014 ◽

Vol 11 (04) ◽

pp. 1450026 ◽

Cited By ~ 2

Author(s):

Serkan Karaçuha ◽

Christian Lomp

Keyword(s):

Differential Calculus ◽

Polynomial Algebra ◽

Exterior Algebra ◽

Integral Calculus ◽

De Rham Complex ◽

Integral Forms ◽

Rham Complex ◽

De Rham ◽

Quantum Polynomial

Hom-connections and associated integral forms have been introduced and studied by Brzeziński as an adjoint version of the usual notion of a connection in non-commutative geometry. Given a flat hom-connection on a differential calculus (Ω, d) over an algebra A yields the integral complex which for various algebras has been shown to be isomorphic to the non-commutative de Rham complex (in the sense of Brzeziński et al. [Non-commutative integral forms and twisted multi-derivations, J. Noncommut. Geom.4 (2010) 281–312]). In this paper we shed further light on the question when the integral and the de Rham complex are isomorphic for an algebra A with a flat Hom-connection. We specialize our study to the case where an n-dimensional differential calculus can be constructed on a quantum exterior algebra over an A-bimodule. Criteria are given for free bimodules with diagonal or upper-triangular bimodule structure. Our results are illustrated for a differential calculus on a multivariate quantum polynomial algebra and for a differential calculus on Manin's quantum n-space.

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