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Towards Precision Holography for Latitude Wilson Loops

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Book cover Perturbative and Non-perturbative Approaches to String Sigma-Models in AdS/CFT

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Abstract

In recent years a new wealth of exact results has become available for path-integrals in supersymmetric QFTs on curved manifolds by means of supersymmetric localization.

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Notes

  1. 1.

    See below (5.2) for Wilson loops.

  2. 2.

    Here we consider the spacetime in Euclidean signature, for which the path-ordered exponential is not a pure complex phase and there is no unitarity bound \(\langle \mathcal {W} \rangle \le 1\), see comments in [14]. We recommend reading the review in [15], which also deals with loop operators in ABJM theory.

  3. 3.

    By this counting over the full set of 16 super-Poincaré \(\mathcal {Q}\)’s and 16 superconformal \(\mathcal {S}\)’s generators, an operator that breaks all the \(\mathcal {Q}\)’s while preserving at least one \(\mathcal {S}\) is still called supersymmetric.

  4. 4.

    In gauge theory this is equivalent to the statement that gauge and scalar propagators coincide in Feynman gauge and cancel order by order in perturbation theory. At strong coupling, it is easy to check that the dual classical worldsheet has zero regularized area [19, 20] (see 3.100 in this thesis), but the vanishing of the subleading corrections is less transparent. The one-loop order is is rather subtle because one needs an ad hoc prescription to subtract divergences [20, 21].

  5. 5.

    It can be computed with the matrix integral technology in [35] and references therein.

  6. 6.

    However there exists a claim of disagreement in the subleading order of correlators of latitude loops at strong coupling [43].

  7. 7.

    A derivation and other practical expansions are in [53, 54].

  8. 8.

    These are diagrams that “stretch” across the circular loop without carrying interaction vertices.

  9. 9.

    The proof is not completed at the same level of rigour of [2] for the circle: one still needs to compute the one-loop determinants around the localization locus and effectively prove that the Wilson loops localizes in YM\(_2\) on \(S^2\).

  10. 10.

    About the topological contribution of the measure, its relevance in canceling the divergences occurring in evaluating quantum corrections to the string partition function has been first discussed in [19] after the observations of [67, 68]. We use this general argument below, see discussion around (5.79).

  11. 11.

    In presence of zero modes of the classical solution, a possible dependence of the path-integral measure on the classical solution comes from the integration over collective coordinates associated to them, see arguments in [18].

  12. 12.

    There exist other solutions with more wrapping in \(S^5\), but they are not supersymmetric [65].

  13. 13.

    See also [71] for an analysis of the contribution to the string partition function due to (broken) zero modes of the solution in [18].

  14. 14.

    The reason is that the initial angular coordinates (5.24) in the background (5.25) would not yield a bosonic quadratic Lagrangian in the standard form for the kinetic terms of the eight physical fields.

  15. 15.

    In the language of [73], this shift corresponds to a different choice of orthonormal vectors that are orthogonal to the string surface.

  16. 16.

    In the free Lagrangian \(\mathcal {L}_F^{(2)}\) the spinor field \(\Psi \) couples only to the classical background (5.25), which lies in the timeslice \(t=0\) and has Euclidean signature (5.33). This may cause some issues with the fact that the Green-Schwarz action and the Majorana condition are only defined for a worldsheet of Lorentzian signature. Notice that the analytic continuation of the AdS time t does not affect the signature of the classical solution. Here we simply think of doing the expansion for imaginary worldsheet time \(\tau \) and only at the end Wick-rotate back to Euclidean signature (5.25).

  17. 17.

    A non-trivial matrix structure is also encountered in the fermionic sector of the circular Wilson loop [20], but the absence of a background geometry in \(S^5\) leads to a simpler gamma structure. It comprises only three gamma combinations (\(\Gamma _0,\Gamma _4,\Gamma _{04}\)), whose algebra allows their identification with the three Pauli matrices without the need of labelling the subspaces.

  18. 18.

    The same property is showed by (5.26) in [20].

  19. 19.

    In this reference a regularization slightly different from [75, 76] was adopted.

  20. 20.

    This is expected from an analysis of the Seeley-DeWitt coefficients in (3.104), with logarithmic divergencies given by (3.111) instead of (3.113) because in (5.78) we are not including the effect of the non-trivial Jacobian explained below (3.111).

  21. 21.

    This is convenient because of the different form for the special modes (5.68)–(5.70) together with the relabeling discussed above.

  22. 22.

    This can be proved analytically since the summand behaves as \(\mu \, e^{-\mu \ell } \ell ^{-2}\) for large \(\ell \). Removing the cutoff makes the sum vanish.

  23. 23.

    This is also due to the volume part of the Euler number \(\chi _v(\theta _0)\) being independent of \(\sigma _0\) up to \(\epsilon \) corrections, see (5.79).

  24. 24.

    The SU(2|2) quantum numbers of the fields are summarized in (4.26) of [66]. For vanishing \(\theta _0\) it was known that they fill representations of the symmetry group \(OSp(4^*|4)\) of the half-supersymmetric circular Wilson loop [78].

  25. 25.

    We are referring to formulas (4.27) and (4.28) in [66].

  26. 26.

    See Eq. (5.50) in [66].

  27. 27.

    This resembles the quest for an “integrability-preserving” regularization scheme, different from the most natural one suggested by worldsheet field theory considerations, in the worldsheet calculations of light-like cusps in \(\mathcal {N}=4\) SYM [80] and ABJM theory [81].

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    Edoardo Vescovi

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Vescovi, E. (2017). Towards Precision Holography for Latitude Wilson Loops. In: Perturbative and Non-perturbative Approaches to String Sigma-Models in AdS/CFT. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-63420-3_5

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