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Submanifold Dirac operators with torsion. (English) Zbl 1210.53003

The submanifold Dirac operator has been studied for this decade, which is closely related to Frenet-Serret and generalized Weierstrass relations. Following the algebraic construction, the author introduces another submanifold Dirac operator which has a gauge field associated with U(1)-gauge field as torsion in the sense of the Frenet-Serret relation, which also has data of immersion of the surface in \(\mathbb E^4\). The authors provide the theorems which are connected with the generalized Weierstrass relation. For a space curve in three dimensional Euclidean space \(\mathbb E^3\), Takagi and Tanzawa found a submanifold Schrödinger operator with a gauge field, \[ S:=-(\partial_s-\sqrt{-1}a)^2-\frac{1}{4}k^2,\tag{1} \] where \(k\) is a curvature of the curve. Using the algebraic construction, the author generalizes the Takagi-Tanzawa Schrödinger operator (1) to the submanifold Dirac operator over a surface immersed in \(\mathbb E^4\).

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
51N20 Euclidean analytic geometry
81T20 Quantum field theory on curved space or space-time backgrounds
53A05 Surfaces in Euclidean and related spaces
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