We show that any solitonic representation of a conformal (diffeomorphism covariant) net on S 1 has positive energy and construct an uncountable family of mutually inequivalent solitonic representations of any conformal net, using nonsmooth diffeomorphisms. On the loop group nets, we show that these representations induce representations of the subgroup of loops compactly supported in $${S^1{\setminus} \{-1\}}$$ S 1 \ { - 1 } that do not extend to the whole loop group. In the case of the U (1)-current net, we extend the diffeomorphism covariance to the Sobolev diffeomorphisms $${\mathcal{D}^s(S^1), s > 2}$$ D s ( S 1 ) , s > 2 , and show that the positive-energy vacuum representations of $${{\rm Diff}_{+}(S^1)}$$ Diff + ( S 1 ) with integer central charges extend to $${\mathcal{D}^s(S^1)}$$ D s ( S 1 ) . The solitonic representations constructed above for the $${\mathrm{U}(1)}$$ U ( 1 ) -current net and for Virasoro nets with integral central charge are continuously covariant with respect to the stabilizer subgroup of $${{\rm Diff}_{+}(S^1)}$$ Diff + ( S 1 ) of $${-1}$$ - 1 of the circle.

中文翻译：

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共形网络中的孤子和非光滑微分同胚

我们证明了 S 1 上的共形（微分同胚协变）网络的任何孤子表示都具有正能量，并使用非光滑微分同胚构造任何共形网络的互不等价孤子表示的不可数族。在循环组网络上，我们表明这些表示诱导了 $${S^1{\setminus} \{-1\}}$$ S 1 \ { - 1 } 中紧密支持的循环子组的表示扩展到整个循环组。在 U (1)-current 网络的情况下，我们将微分同胚协方差扩展到 Sobolev 微分同胚 $${\mathcal{D}^s(S^1), s > 2}$$ D s ( S 1 ) , s > 2 ，并表明具有整数中心电荷的 $${{\rm Diff}_{+}(S^1)}$$ Diff + ( S 1 ) 的正能量真空表示扩展到 $${ \mathcal{D}^s(S^1)}$$ D s ( S 1 ) 。