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arXiv 2201.09468hep-lathep-th
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带$θ$项的自对偶U（1）格点场论

Self-dual U(1) lattice field theory with a $θ$-term

Mariia Anosova, Christof Gattringer, Tin Sulejmanpasic

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摘要：

研究了具有修正的Villain作用量的U（1）规范理论.这样的理论可以自然地耦合到电和磁物质，并显示出精确的电磁对偶性。在没有$θ$ -项的最简单的公式中，这样的理论是超局部的。我们将讨论扩展到具有$θ$ -项的U（1）规范理论，使得$θ$周期性对于自由理论是精确的，并且表明，施加电磁对偶性导致定域而非超定域格点作用量，这让人想起4d中保持轴对称费米子的Lüscher构造。我们讨论了耦合到电和磁物质以及双子。对于双子物质，电磁对偶性和$θ$ -角位移$2π$一起产生了SL $(2,\mathbb Z)$对偶变换群，就像在连续统中一样。最后，我们说明如何SL $(2,\mathbb Z)$对偶可用于探索理论在有限的$θ$没有一个标志的问题。

英文摘要：

We study U(1) gauge theories with a modified Villain action. Such theories can naturally be coupled to electric and magnetic matter, and display exact electric-magnetic duality. In their simplest formulation without a $θ$-term, such theories are ultra-local. We extend the discussion to U(1) gauge theories with $θ$-terms, such that $θ$ periodicity is exact for a free theory, and show that imposing electric-magnetic duality results in a local, but not ultra-local lattice action, which is reminiscent of the Lüscher construction of axial-symmetry preserving fermions in 4d. We discuss the coupling to electric and magnetic matter as well as to dyons. For dyonic matter the electric-magnetic duality and shifts of the $θ$-angle by $2π$ together generate an SL$(2,\mathbb Z)$ duality group of transformations, just like in the continuum. We finally illustrate how the SL$(2,\mathbb Z)$ duality may be used to explore theories at finite $θ$ without a sign problem.

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