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Landau-Ginzburg Theories

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An Introduction to Two-Dimensional Quantum Field Theory with (0,2) Supersymmetry

Part of the book series: Lecture Notes in Physics ((LNP,volume 951))

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Abstract

In this chapter we study the simplest large class of (0,2) QFTs: the (0,2) Landau-Ginzburg theories. While they are interesting in their own right, the main goal is to introduce useful notions relevant to general (0,2) theories in the context of these simple examples.

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Notes

  1. 1.

    Recall that in our Euclidean conventions reality is defined with respect to the action of the charge conjugation operator \({\mathcal {C}}\) defined in Sect. 1.8.

  2. 2.

    We use the short-hand introduced in the first chapter: \({\overline {\boldsymbol {Q}}} \cdot {\mathcal {O}} \) stands for \({[{\overline {\boldsymbol {Q}}},{\mathcal {O}}]}\) or \({\{{\overline {\boldsymbol {Q}}},{\mathcal {O}}\}}\), depending on whether \({\mathcal {O}}\) is bosonic or fermionic.

  3. 3.

    Further discussion can be found in [129].

  4. 4.

    In what follows we will omit the distinction between \({\overline {\boldsymbol {Q}}}\)-closed operator \({\mathcal {O}}\) and its cohomology class \([{\mathcal {O}}] \in H_{{\overline {\boldsymbol {Q}}}}\), and we will drop \({\overline {\boldsymbol {Q}}}\)-exact terms in the OPEs.

  5. 5.

    We will explore these structures in greater detail in the next chapter; here we will just need some of the most basic algebraic concepts and definitions. The exercises provide some practice with these tools.

  6. 6.

    The specter of geometry appears before us. It may be useful at this point to take a look through the geometry appendix especially Sect. B.3.2. At this point we need little beyond the tensor structure, so the reader should feel free to simply skim the geometry review to get comfortable with some of the conventions.

  7. 7.

    We suppress the l label, since it does not play any role in the cohomology computation.

  8. 8.

    The latter reference, devoted to residue currents, contains many algebraic notions that should be useful to a (0,2) LG explorer.

  9. 9.

    Since the twist and BRST operator are identical to that of the full half-twisted theory, a better but longer term might be the “B/2 heterotic ring projection of the half-twisted model.”

  10. 10.

    The supercharge Q transforms as a 1-form under the twisted Lorentz symmetry, and the corresponding invariance, if it is to be maintained, must be promoted to a local one on a curved worldsheet.

  11. 11.

    If this is unclear, the reader may want to review the appendix on geometry and especially Exercise B.23.

  12. 12.

    The inevitability is easy to understand—a regularization introduces a lengthscale, and that choice is not compatible with the classical scale invariance.

  13. 13.

    The reader may want to consult the appendix on geometry to recall some notions and review our conventions for Dolbeault and De Rham cohomologies.

  14. 14.

    The author has often used this formula together with Maple’s “RootOf” command to painlessly obtain explicit expressions for correlation functions.

  15. 15.

    We will see applications of similar ideas to (0,2) non-linear sigma models and gauged linear sigma models in the next chapters. Closely related ideas also arise in four-dimensional gauge theories [58].

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  1. Department of Physics and Astronomy, James Madison University, Harrisonburg, VA, USA

    Ilarion V. Melnikov

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Melnikov, I.V. (2019). Landau-Ginzburg Theories. In: An Introduction to Two-Dimensional Quantum Field Theory with (0,2) Supersymmetry. Lecture Notes in Physics, vol 951. Springer, Cham. https://doi.org/10.1007/978-3-030-05085-6_3

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