Twisted sigma-model solitons on the quantum projective line
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On the configuration space of projections in a noncommutative algebra, and for an automorphism of the algebra, we use a twisted Hochschild cocycle for an action functional and a twisted cyclic cocycle for a topological term. The latter is Hochschild-cohomologous to the former and positivity in twisted Hochschild cohomology results into a lower bound for the action functional. While the equations for the critical points are rather involved, the use of the positivity and the bound by the topological term lead to self-duality equations (thus yielding twisted noncommutative sigma-model solitons, or instantons). We present explicit nontrivial solutions on the quantum projective line.
KeywordsNoncommutative sigma-models Harmonic maps Self-duality equations Complex and holomorphic structures Twisted cyclic and Hochschild cocycles Twisted Hochschild positivity Quantum projective line Projections and line bundles
Mathematics Subject Classification58B32 32L05 55R91 58E20
Part of this work was done during a visit at the Tohoku Forum for Creativity in Sendai. I am grateful to Yoshi Maeda for the kind invitation and to him, to Ursula Carow–Watamura and to Satoshi Watamura for the great hospitality in Sendai. Francesco D’Andrea read a first version of the paper and made useful suggestions.
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