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Coulomb Branches of 3-Dimensional Gauge Theories and Related Structures

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Book cover Geometric Representation Theory and Gauge Theory

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2248))

Abstract

These are (somewhat informal) lecture notes for the CIME summer school “Geometric Representation Theory and Gauge Theory” in June 2018. In these notes we review the constructions and results of Braverman et al. (Adv Theor Math Phys 22(5):1017–1147, 2018; Adv Theor Math Phys 23(1):75–166, 2019; Adv Theor Math Phys 23(2):253–344, 2019) where a mathematical definition of Coulomb branches of 3d N = 4 quantum gauge theories (of cotangent type) is given, and also present a framework for studying some further mathematical structures (e.g. categories of line operators in the corresponding topologically twisted theories) related to these theories.

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Notes

  1. 1.

    The main ideas are due to T. Braden, A. Licata, N. Proudfoot and B. Webster.

  2. 2.

    (T ∗T ∨)∕W is actually the Coulomb branch of the corresponding classical field theory and the fact that the above birational isomorphism is not in general biregular means that “in the quantum theory the Coulomb branch acquires quantum corrections”.

  3. 3.

    The reader is also advised to consult the papers [39, 41, 42] by Nakajima. In particular, the papers [13,14,15] are based on the ideas developed earlier in [39]. Also [41] contains a lot of interesting open problems in the subject most of which will not be addressed in these notes.

  4. 4.

    In fact, this is already a simplification: non-algebraic holomorphic symplectic manifolds should also arise in this way, but we are not going to discuss such theories.

  5. 5.

    By Hamiltonian action we mean a symplectic action with fixed moment map.

  6. 6.

    The nature of this additional structure will become more clear in Sect. 1.7.6.

  7. 7.

    Because we plunge ourselves into world of derived algebraic geometry here, it doesn’t make sense to talk about either quasi-coherent sheaves or D-modules as an abelian category: only the corresponding derived category makes sense.

  8. 8.

    The reader should be warned that although we have a natural functor \(\operatorname {IndCoh}_{\mathcal W}(\mathcal Z)\to \operatorname {IndCoh}(\mathcal Z)\), this functor is not fully faithful, so \(\operatorname {IndCoh}_{\mathcal W}(\mathcal Z)\) is not a full subcategory of \(\operatorname {IndCoh}(\mathcal Z)\).

  9. 9.

    Here we see that \(\operatorname {Maps}(\mathcal D_{dR},\mathcal Y)\) should be defined with some extra care. Namely, if we just used the naive definition then the equivalence \(\operatorname {Maps}(\mathcal D_{dR},\mathcal Y)\simeq \mathcal Y\) would imply that \(\mathcal D_{dR}=\operatorname {pt}\) which is far from being the case.

  10. 10.

    Here we want to stress once again that all fibered products must be understood in the dg-sense!

  11. 11.

    Again, the reader should keep in mind Sect. 1.7.7.

  12. 12.

    It is known that this is only an approximate conjecture. The correct conjecture (due to A. Arinkin) requires a (rather tricky) modification of the notion category over \(\operatorname {LocSys}_G(\mathcal D^*)\) (which again has to do with the difference between QCoh and IndCoh).

  13. 13.

    Such a tensor product does make sense as long as we live in the world of dg-categories.

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Acknowledgements

We are greatly indebted to our coauthor H. Nakajima who taught us everything we know about Coulomb branches of 3d N = 4 gauge theories and to the organizers of the CIME summer school “Geometric Representation Theory and Gauge Theory” in June 2018, for which these notes were written. In addition we would like to thank T. Dimofte, D. Gaiotto, J. Hilburn and P. Yoo for their patient explanations of various things (in particular, as was mentioned above the main idea of Sect. 1.7 is due to them). Also, we are very grateful to R. Bezrukavnikov, K. Costello, D. Gaitsgory and S. Raskin for their help with many questions which arose during the preparation of these notes. The research of M.F. was supported by the grant RSF 19-11-00056.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto and Perimeter Institute of Theoretical Physics, Waterloo, ON, Canada

    Alexander Braverman

  2. Skolkovo Institute of Science and Technology, Moskva, Russia

    Alexander Braverman

  3. National Research University Higher School of Economics, Russian Federation, Department of Mathematics, Moscow, Russia

    Michael Finkelberg

  4. Skolkovo Institute of Science and Technology, Institute for Information Transmission Problems, Moskva, Russia

    Michael Finkelberg

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  1. Alexander Braverman
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Correspondence to Alexander Braverman .

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Editors and Affiliations

  1. Department of Mathematics, Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy

    Ugo Bruzzo

  2. Department of Mathematics, Università di Bologna, Bologna, Italy

    Antonella Grassi

  3. Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, University of Tokyo, Kashiwa-shi, Chiba, Japan

    Francesco Sala

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Braverman, A., Finkelberg, M. (2019). Coulomb Branches of 3-Dimensional Gauge Theories and Related Structures. In: Bruzzo, U., Grassi, A., Sala, F. (eds) Geometric Representation Theory and Gauge Theory. Lecture Notes in Mathematics(), vol 2248. Springer, Cham. https://doi.org/10.1007/978-3-030-26856-5_1

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