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Slavnov Determinants, Yang–Mills Structure Constants, and Discrete KP

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Symmetries, Integrable Systems and Representations

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 40))

Abstract

Using Slavnov’s scalar product of a Bethe eigenstate and a generic state in closed XXZ spin-\(\frac{1}{2}\) chains, with possibly twisted boundary conditions, we obtain determinant expressions for tree-level structure constants in 1-loop conformally-invariant sectors in various planar (super) Yang-Mills theories. When certain rapidity variables are allowed to be free rather than satisfy Bethe equations, these determinants become discrete KP τ-functions.

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Notes

  1. 1.

    Superstrings with tree-level interactions only, and no spacetime loops.

  2. 2.

    The limit in which the number of colours N c →∞, the gauge coupling g YM →0, while the ’t Hooft coupling \(\lambda= g^{2}_{\mathit{YM}} N_{c}\) remains finite.

  3. 3.

    Further highlights of integrability in modern quantum field theory and in string theory include (1) Classical integrable hierarchies in matrix models of non-critical strings, from the late 1980’s [10], (2) Finite gap solutions in Seiberg-Witten theory of low-energy SYM2 in the mid 1990’s [11–14], (3) Integrability in QCD scattering amplitudes in the mid 1990’s [8, 15–17], (4) Free fermion methods in works of Nekrasov, Okounkov, Nakatsu, Takasaki and others on Seiberg-Witten theory, in the 2000’s [18, 19], (5) Integrable spin chains in works of Nekrasov, Shatashvili and others on SYM2, in the 2000’s [20], (6) Integrable structures, particularly the Yangian, that appear in recent studies of SYM4 scattering amplitudes [21, 22]. There are many more.

  4. 4.

    In this work, we restrict our attention to this class of local composite operators. In particular, we do not consider descendants or operators with non-zero spin, for which the 2-point and 3-point functions are different.

  5. 5.

    Three operators \(\mathcal{O}_{i}\), of length L i , i∈{1,2,3}, are non-extremal if l ij =L i +L j −L k >0.

  6. 6.

    The SYM4 expression of [29] is a special case of the general expression obtained here.

  7. 7.

    There are definitely more gauge theories that are conformally-invariant at 1-loop or more, with SU(2) sectors that map to states in spin-\(\frac{1}{2}\) chains. Here we consider only samples of theories with different supersymmetries and operator content.

  8. 8.

    XXX spin-\(\frac{1}{2}\) chains are XXZ spin-\(\frac{1}{2}\) chains with an anisotropy parameter Δ=1.

  9. 9.

    The fact that the structure constants in these two types of theories should be handled differently was pointed out to us by C. Ahn and R. Nepomechie.

  10. 10.

    In [29], S[L,N 1,N 2] was denoted by S[L,{N}].

  11. 11.

    Minahan and Zarembo obtained their remarkable result in the context of the complete scalar sector of SYM4. The relevant spin chain in that case is SO(6) symmetric. Here we focus our attention on the restriction of their result to the SU(2) scalar subsector.

  12. 12.

    We are interested in local single-trace composite operators that consist of many fundamental fields. These fields are interacting. In a weakly-interacting quantum field theory, one can consistently choose to ignore all interactions beyond a chosen order in perturbation theory. In the planar theory under consideration, perturbation theory can be arranged according to the number of loops in Feynman diagrams computed. In a 1-loop approximation, one keeps only 1-loop diagrams.

  13. 13.

    We use β in two different ways. 1. To indicate the deformation parameter in \(\mathrm{SYM}_{4}^{\beta}\) theories, and 2. To indicate that a certain state is a Bethe eigenstate of the spin-chain Hamiltonian. There should be no confusion with 1, in which β is a parameter but never a subscript, while in 2 it is always a subscript.

  14. 14.

    To simplify the notation, we use N 1, N 2 and N 3=N 1−N 2, instead of the corresponding notation used in [42, 43]. These variables match the corresponding ones in Sect. 5.

  15. 15.

    The weights of the six-vertex model (52) and the entries of the XXZ R-matrix (12) are identical. This is the origin of the connection between the two models. We have chosen to write down these functions twice for clarity and to emphasize this fact.

  16. 16.

    For visual clarity, we have allowed for a gap between the B-lines and the C-lines in Fig. 5. There is also a gap between the N 3-th and (N 3+1)-th vertical lines, where N 3=3 in the example shown, that indicates separate portions of the lattice that will be relevant shortly. The reader should ignore this at this stage.

  17. 17.

    The following result does not require that any set of rapidities satisfy Bethe equations.

  18. 18.

    The conclusion that, in order to obtain a determinant formula, one of the single-trace operators should be BPS-like, was obtained in discussions with C. Ahn and R. Nepomechie.

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Acknowledgements

O.F. thanks C. Ahn, N. Gromov, G. Korchemsky, I. Kostov, R. Nepomechie, D. Serban, P. Vieira and K. Zarembo for discussions on the topic of this work and the Inst. H. Poincare for hospitality where it started. Both authors thank the Australian Research Council for financial support, and the anonymous referee for remarks that helped us improve the text.

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  1. Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria, 3010, Australia

    Omar Foda & Michael Wheeler

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  1. Omar Foda
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Correspondence to Omar Foda .

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  1. Inst. Camille Jordan, UMR5028 CNRS, Université Claude Bernard Lyon 1, Villeurbanne Cedex, 69622, France

    Kenji Iohara

  2. Inst. de Mathém. de Jus., UMR 7586 CNRS, Université Pierre et Marie Curie, Place Jussieu, Case 247 4, Paris, 75252, France

    Sophie Morier-Genoud

  3. Inst. Camille Jordan, UMR5028 CNRS, Université Claude Bernard Lyon I, Villeurbanne Cedex, 69622, France

    Bertrand Rémy

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Dedicated to Professor M. Jimbo on his 60th birthday.

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Foda, O., Wheeler, M. (2013). Slavnov Determinants, Yang–Mills Structure Constants, and Discrete KP. In: Iohara, K., Morier-Genoud, S., Rémy, B. (eds) Symmetries, Integrable Systems and Representations. Springer Proceedings in Mathematics & Statistics, vol 40. Springer, London. https://doi.org/10.1007/978-1-4471-4863-0_5

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