From spinning conformal blocks to matrix Calogero-Sutherland models

Open Access
Regular Article - Theoretical Physics
  • 17 Downloads

Abstract

In this paper we develop further the relation between conformal four-point blocks involving external spinning fields and Calogero-Sutherland quantum mechanics with matrix-valued potentials. To this end, the analysis of [1] is extended to arbitrary dimensions and to the case of boundary two-point functions. In particular, we construct the potential for any set of external tensor fields. Some of the resulting Schrödinger equations are mapped explicitly to the known Casimir equations for 4-dimensional seed conformal blocks. Our approach furnishes solutions of Casimir equations for external fields of arbitrary spin and dimension in terms of functions on the conformal group. This allows us to reinterpret standard operations on conformal blocks in terms of group-theoretic objects. In particular, we shall discuss the relation between the construction of spinning blocks in any dimension through differential operators acting on seed blocks and the action of left/right invariant vector fields on the conformal group.

Keywords

Conformal and W Symmetry Conformal Field Theory Differential and Algebraic Geometry Field Theories in Higher Dimensions 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    V. Schomerus, E. Sobko and M. Isachenkov, Harmony of Spinning Conformal Blocks, JHEP 03 (2017) 085 [arXiv:1612.02479] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    G. Mack, Conformal Invariant Quantum Field Theory, J. Phys. Colloq. 34 (1973) 99 [INSPIRE].CrossRefGoogle Scholar
  3. [3]
    S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, Covariant expansion of the conformal four-point function, Nucl. Phys. B 49 (1972) 77 [Erratum ibid. B 53 (1973) 643] [INSPIRE].
  4. [4]
    S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].Google Scholar
  6. [6]
    G. Mack, Duality in quantum field theory, Nucl. Phys. B 118 (1977) 445 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE].
  10. [10]
    R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    L. Iliesiu, F. Kos, D. Poland, S.S. Pufu and D. Simmons-Duffin, Bootstrapping 3D Fermions with Global Symmetries, JHEP 01 (2018) 036 [arXiv:1705.03484] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    A. Dymarsky, F. Kos, P. Kravchuk, D. Poland and D. Simmons-Duffin, The 3d Stress-Tensor Bootstrap, JHEP 02 (2018) 164 [arXiv:1708.05718] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    L. Iliesiu, F. Kos, D. Poland, S.S. Pufu, D. Simmons-Duffin and R. Yacoby, Bootstrapping 3D Fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  16. [16]
    P. Kravchuk and D. Simmons-Duffin, Counting Conformal Correlators, JHEP 02 (2018) 096 [arXiv:1612.08987] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    D. Simmons-Duffin, Projectors, Shadows and Conformal Blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Radial expansion for spinning conformal blocks, JHEP 07 (2016) 057 [arXiv:1603.05552] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Projectors and seed conformal blocks for traceless mixed-symmetry tensors, JHEP 07 (2016) 018 [arXiv:1603.05551] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    A. Castedo Echeverri, E. Elkhidir, D. Karateev and M. Serone, Seed Conformal Blocks in 4D CFT, JHEP 02 (2016) 183 [arXiv:1601.05325] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    P. Kravchuk, Casimir recursion relations for general conformal blocks, JHEP 02 (2018) 011 [arXiv:1709.05347] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    J. Penedones, E. Trevisani and M. Yamazaki, Recursion Relations for Conformal Blocks, JHEP 09 (2016) 070 [arXiv:1509.00428] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight Shifting Operators and Conformal Blocks, JHEP 02 (2018) 081 [arXiv:1706.07813] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    L. Iliesiu, F. Kos, D. Poland, S.S. Pufu, D. Simmons-Duffin and R. Yacoby, Fermion-Scalar Conformal Blocks, JHEP 04 (2016) 074 [arXiv:1511.01497] [INSPIRE].ADSMathSciNetGoogle Scholar
  25. [25]
    F. Rejon-Barrera and D. Robbins, Scalar-Vector Bootstrap, JHEP 01 (2016) 139 [arXiv:1508.02676] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    M.S. Costa and T. Hansen, Conformal correlators of mixed-symmetry tensors, JHEP 02 (2015) 151 [arXiv:1411.7351] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    F. Calogero, Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971) 419 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    B. Sutherland, Exact results for a quantum many body problem in one-dimension. 2, Phys. Rev. A 5 (1972) 1372 [INSPIRE].
  29. [29]
    J. Moser, Three integrable Hamiltonian systems connnected with isospectral deformations, Adv. Math. 16 (1975) 197 [INSPIRE].ADSCrossRefMATHGoogle Scholar
  30. [30]
    G. Poschl and E. Teller, Bemerkungen zur Quantenmechanik des anharmonischen Oszillators, Z. Phys. 83 (1933) 143 [INSPIRE].ADSCrossRefMATHGoogle Scholar
  31. [31]
    O.E.M. Heckman and E.M. Opdam, Root systems and hypergeometric functions. I, Compos. Math. 64 (1987) 329.Google Scholar
  32. [32]
    G. Heckman and H. Schlicktkrull, Harmonic Analysis and Special Functions on Symmetric Spaces, Elsevier (1995).Google Scholar
  33. [33]
    E.M. Opdam, Part I: Lectures on Dunkl Operators, in Lecture Notes on Dunkl Operators for Real and Complex Reflection Groups, The Mathematical Society of Japan, Tokyo, Japan (2000), pp. 2-62.Google Scholar
  34. [34]
    I. Cherednik, Double Affine Hecke Algebras, Cambridge University Press (2005).Google Scholar
  35. [35]
    M.A. Olshanetsky and A.M. Perelomov, Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Invent. Math. 37 (1976) 93.ADSMathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    M.A. Olshanetsky and A.M. Perelomov, Classical integrable finite dimensional systems related to Lie algebras, Phys. Rept. 71 (1981) 313 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    M.A. Olshanetsky and A.M. Perelomov, Quantum Integrable Systems Related to Lie Algebras, Phys. Rept. 94 (1983) 313 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    P. Etingof and A. Kirillov, On a unified representatiuon theoretical approach to special functions, Funk. Anal. Prilozh. 28 (1994) 91.Google Scholar
  39. [39]
    P.I. Etingof, I.B. Frenkel and A.A. Kirillov Jr., Spherical functions on affine Lie groups, Duke Math. J. 80 (1995) 79 [hep-th/9407047] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    P.I. Etingof, Quantum integrable systems and representations of Lie algebras, J. Math. Phys. 36 (1995) 2636 [hep-th/9311132] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    M. Isachenkov and V. Schomerus, Superintegrability of d-dimensional Conformal Blocks, Phys. Rev. Lett. 117 (2016) 071602 [arXiv:1602.01858] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    M. Isachenkov and V. Schomerus, Integrability of Conformal Blocks I: Calogero-Sutherland Scattering Theory, arXiv:1711.06609 [DESY-17-178] [INSPIRE].
  43. [43]
    L. Feher and B.G. Pusztai, Derivations of the trigonometric BC n Sutherland model by quantum Hamiltonian reduction, Rev. Math. Phys. 22 (2010) 699 [arXiv:0909.5208] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    L. Feher and B.G. Pusztai, Hamiltonian reductions of free particles under polar actions of compact Lie groups, Theor. Math. Phys. 155 (2008) 646 [arXiv:0705.1998] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    P. Liendo, L. Rastelli and B.C. van Rees, The Bootstrap Program for Boundary CFT d, JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].
  46. [46]
    A. Castedo Echeverri, E. Elkhidir, D. Karateev and M. Serone, Deconstructing Conformal Blocks in 4D CFT, JHEP 08 (2015) 101 [arXiv:1505.03750] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    V.K. Dobrev, G. Mack, V.B. Petkova, S.G. Petrova and I.T. Todorov, Harmonic Analysis on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory, Lect. Notes Phys. 63 (1977) 1 [INSPIRE].ADSCrossRefMATHGoogle Scholar
  48. [48]
    M. Isachenkov and V. Schomerus, Integrablity of Conformal Blocks II: Algebraic Structures, work in progress.Google Scholar
  49. [49]
    N. Reshetikhin, Degenerate integrability of quantum spin Calogero-Moser systems, Lett. Math. Phys. 107 (2017) 187 [arXiv:1510.00492] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  50. [50]
    N. Reshetikhin, Degenerately Integrable Systems, J. Math. Sci. 213 (2016) 769 [arXiv:1509.00730] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    A.L. Fitzpatrick, J. Kaplan, Z.U. Khandker, D. Li, D. Poland and D. Simmons-Duffin, Covariant Approaches to Superconformal Blocks, JHEP 08 (2014) 129 [arXiv:1402.1167] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    M. Cornagliotto, M. Lemos and V. Schomerus, Long Multiplet Bootstrap, JHEP 10 (2017) 119 [arXiv:1702.05101] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  53. [53]
    A. Gadde, Conformal constraints on defects, arXiv:1602.06354 [INSPIRE].
  54. [54]
    M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].ADSMathSciNetGoogle Scholar
  55. [55]
    P. Liendo and C. Meneghelli, Bootstrap equations for \( \mathcal{N} \) = 4 SYM with defects, JHEP 01 (2017) 122 [arXiv:1608.05126] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  56. [56]
    M. Fukuda, N. Kobayashi and T. Nishioka, Operator product expansion for conformal defects, JHEP 01 (2018) 013 [arXiv:1710.11165] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    C.P. Herzog and K.-W. Huang, Boundary Conformal Field Theory and a Boundary Central Charge, JHEP 10 (2017) 189 [arXiv:1707.06224] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  58. [58]
    V. Schomerus, Harmony of Defects, talk given at Boundary and Defect Conformal Field Theory: Open Problems and Applications, Chicheley Hall, September 2017.Google Scholar
  59. [59]
    I. Balitsky, V. Kazakov and E. Sobko, Two-point correlator of twist-2 light-ray operators in N = 4 SYM in BFKL approximation,arXiv:1310.3752 [INSPIRE].
  60. [60]
    I. Balitsky, V. Kazakov and E. Sobko, Structure constant of twist-2 light-ray operators in the Regge limit, Phys. Rev. D 93 (2016) 061701 [arXiv:1506.02038] [INSPIRE].ADSMathSciNetGoogle Scholar
  61. [61]
    I. Balitsky, V. Kazakov and E. Sobko, Three-point correlator of twist-2 light-ray operators in N = 4 SYM in BFKL approximation, arXiv:1511.03625 [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.DESY Hamburg, Theory GroupHamburgGermany
  2. 2.Nordita, Stockholm University and KTH Royal Institute of TechnologyStockholmSweden

Personalised recommendations