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Surface Operators

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New Dualities of Supersymmetric Gauge Theories

Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

We give an introduction and a broad survey of surface operators in 4d gauge theories, with a particular emphasis on aspects relevant to AGT correspondence. One of the main goals is to highlight the boundary between what we know and what we don’t know about surface operators. To this end, the survey contains many open questions and suggests various directions for future research. Although this article is mostly a review, we did include a number of new results, previously unpublished.

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Notes

  1. 1.

    For example, in \({\mathcal N}=2^*\) theory with gauge group \(G=SU(N)\) this choice includes the choice of a partition of N. When G is a classical group of Cartan type B, C, or D, the choice of partition must satisfy certain conditions, as illustrated in Fig. 1. In particular, the transformation of surface operators under electric-magnetic duality becomes a rather non-trivial matter in non-abelian gauge theories.

  2. 2.

    See [5, 6] for discussion of \(\tfrac{1}{4}\)-BPS surface operators in 4d \({\mathcal N}=2\) gauge theories.

  3. 3.

    In a theory with gauge group \(G=SU(N)\) this choice is equivalent to a choice of a partition of N.

  4. 4.

    In the case of non-compact toric Calabi-Yau 3-folds mirror symmetry (often called “local mirror symmetry”) relates enumerative invariants of \(CY_3\) with complex geometry of a Riemann surface \(\Sigma \).

  5. 5.

    For simplicity, here we consider only one ramification point \(p \in C\). For the case of ramification at several points, one finds several group actions, one for each ramification point.

  6. 6.

    To be more precise, the pull-back \(p_1^*\) is left-derived and the push-forward \(p_{2*}\) is right-derived.

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Authors and Affiliations

  1. California Institute of Technology, Pasadena, CA, 91125, USA

    Sergei Gukov

  2. Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA, 91125, USA

    Sergei Gukov

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  1. Sergei Gukov
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Correspondence to Sergei Gukov .

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Editors and Affiliations

  1. String Theory Group, DESY String Theory Group, Hamburg, Hamburg, Germany

    Jörg Teschner

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Gukov, S. (2016). Surface Operators. In: Teschner, J. (eds) New Dualities of Supersymmetric Gauge Theories. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-18769-3_8

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