Skip to main content

The anisotropic chiral boson

A preprint version of the article is available at arXiv.


We construct the theory of a chiral boson with anisotropic scaling, characterized by a dynamical exponent z that takes positive odd integer values. The action reduces to that of Floreanini and Jackiw in the isotropic case (z = 1). The standard free boson with Lifshitz scaling is recovered when both chiralities are nonlocally combined. Its canonical structure and symmetries are also analyzed. As in the isotropic case, the theory is also endowed with a current algebra. Noteworthy, the standard conformal symmetry is shown to be still present, but realized in a nonlocal way. The exact form of the partition function at finite temperature is obtained from the path integral, as well as from the trace over \( \hat{u} \)(1) descendants. It is essentially given by the generating function of the number of partitions of an integer into z-th powers, being a well-known object in number theory. Thus, the asymptotic growth of the number of states at fixed energy, including subleading correc- tions, can be obtained from the appropriate extension of the renowned result of Hardy and Ramanujan.


  1. D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, The Heterotic String, Phys. Rev. Lett. 54 (1985) 502 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  2. L. Brink and M. Henneaux, Principles of string theory, Plenum, New York, U.S.A., (1988).

  3. M. Stone, Quantum Hall effect, World Scientific, Singapore, (1992).

    Book  Google Scholar 

  4. D. Tong, Lectures on the Quantum Hall Effect, 2016, arXiv:1606.06687 [INSPIRE].

  5. W. Siegel, Manifest Lorentz Invariance Sometimes Requires Nonlinearity, Nucl. Phys. B 238 (1984) 307 [INSPIRE].

    ADS  Article  Google Scholar 

  6. M. Henneaux and C. Teitelboim, Consistent quantum mechanics of chiral p forms, in 2nd Meeting on Quantum Mechanics of Fundamental Systems (CECS) Santiago, Chile, December 17–20, 1987, pp. 79–112.

  7. M. Henneaux and C. Teitelboim, Dynamics of Chiral (Selfdual) P Forms, Phys. Lett. B 206 (1988) 650 [INSPIRE].

    ADS  Article  Google Scholar 

  8. P. Pasti, D.P. Sorokin and M. Tonin, On Lorentz invariant actions for chiral p forms, Phys. Rev. D 55 (1997) 6292 [hep-th/9611100] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  9. R. Floreanini and R. Jackiw, Selfdual Fields as Charge Density Solitons, Phys. Rev. Lett. 59 (1987) 1873 [INSPIRE].

    ADS  Article  Google Scholar 

  10. M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].

  11. S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  12. S.A. Hartnoll, Horizons, holography and condensed matter, in Black holes in higher dimensions, G.T. Horowitz, ed., pp. 387–419, (2012), arXiv:1106.4324 [INSPIRE].

  13. M. Taylor, Lifshitz holography, Class. Quant. Grav. 33 (2016) 033001 [arXiv:1512.03554] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  14. E. Lifshitz, On the theory of second-order phase transitions I & II, Zh. Eksp. Teor. Fiz 11 (1941) 269.

    Google Scholar 

  15. I. Arav, S. Chapman and Y. Oz, Lifshitz Scale Anomalies, JHEP 02 (2015) 078 [arXiv:1410.5831] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  16. S. Chapman, Y. Oz and A. Raviv-Moshe, On Supersymmetric Lifshitz Field Theories, JHEP 10 (2015) 162 [arXiv:1508.03338] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  17. I. Arav, Y. Oz and A. Raviv-Moshe, Lifshitz Anomalies, Ward Identities and Split Dimensional Regularization, JHEP 03 (2017) 088 [arXiv:1612.03500] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  18. I. Arav, Y. Oz and A. Raviv-Moshe, Holomorphic Structure and Quantum Critical Points in Supersymmetric Lifshitz Field Theories, arXiv:1908.03220 [INSPIRE].

  19. J.A. Hertz, Quantum critical phenomena, Phys. Rev. B 14 (1976) 1165 [INSPIRE].

    ADS  Article  Google Scholar 

  20. S. Sachdev, Quantum Phase Transitions, Cambridge University Press, (1999).

  21. E. Bettelheim, A.G. Abanov and P. Wiegmann, Quantum Shock Waves: The case for non-linear effects in dynamics of electronic liquids, Phys. Rev. Lett. 97 (2006) 246401 [cond-mat/0606778] [INSPIRE].

  22. P. Wiegmann, Non-Linear hydrodynamics and Fractionally Quantized Solitons at Fractional Quantum Hall Edge, Phys. Rev. Lett. 108 (2012) 206810 [arXiv:1112.0810] [INSPIRE].

    ADS  Article  Google Scholar 

  23. S. Sotiriadis, Equilibration in one-dimensional quantum hydrodynamic systems, J. Phys. A 50 (2017) 424004 [arXiv:1612.00373] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  24. J. Aguilera Damia, S. Kachru, S. Raghu and G. Torroba, Two dimensional non-Fermi liquid metals: a solvable large N limit, Phys. Rev. Lett. 123 (2019) 096402 [arXiv:1905.08256] [INSPIRE].

    ADS  Article  Google Scholar 

  25. S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  26. G. Bertoldi, B.A. Burrington and A. Peet, Black Holes in asymptotically Lifshitz spacetimes with arbitrary critical exponent, Phys. Rev. D 80 (2009) 126003 [arXiv:0905.3183] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  27. G. Bertoldi, B.A. Burrington and A.W. Peet, Thermodynamics of black branes in asymptotically Lifshitz spacetimes, Phys. Rev. D 80 (2009) 126004 [arXiv:0907.4755] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  28. E. D’Hoker and P. Kraus, Holographic Metamagnetism, Quantum Criticality and Crossover Behavior, JHEP 05 (2010) 083 [arXiv:1003.1302] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  29. H.A. González, D. Tempo and R. Troncoso, Field theories with anisotropic scaling in 2D, solitons and the microscopic entropy of asymptotically Lifshitz black holes, JHEP 11 (2011) 066 [arXiv:1107.3647] [INSPIRE].

  30. S.A. Hartnoll, D.M. Ramirez and J.E. Santos, Emergent scale invariance of disordered horizons, JHEP 09 (2015) 160 [arXiv:1504.03324] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  31. J. Matulich and R. Troncoso, Asymptotically Lifshitz wormholes and black holes for Lovelock gravity in vacuum, JHEP 10 (2011) 118 [arXiv:1107.5568] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  32. E. Ayon-Beato, A. Garbarz, G. Giribet and M. Hassaine, Lifshitz Black Hole in Three Dimensions, Phys. Rev. D 80 (2009) 104029 [arXiv:0909.1347] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  33. A. Pérez, D. Tempo and R. Troncoso, Boundary conditions for General Relativity on AdS3 and the KdV hierarchy, JHEP 06 (2016) 103 [arXiv:1605.04490] [INSPIRE].

  34. O. Fuentealba et al., Integrable systems with BMS3 Poisson structure and the dynamics of locally flat spacetimes, JHEP 01 (2018) 148 [arXiv:1711.02646] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  35. D. Melnikov, F. Novaes, A. Pérez and R. Troncoso, Lifshitz Scaling, Microstate Counting from Number Theory and Black Hole Entropy, JHEP 06 (2019) 054 [arXiv:1808.04034] [INSPIRE].

  36. H.A. González, J. Matulich, M. Pino and R. Troncoso, Revisiting the asymptotic dynamics of General Relativity on AdS3, JHEP 12 (2018) 115 [arXiv:1809.02749] [INSPIRE].

  37. D. Grumiller and W. Merbis, Near horizon dynamics of three dimensional black holes, arXiv:1906.10694 [INSPIRE].

  38. E. Ojeda and A. Pérez, Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV/mKdV hierarchies, JHEP 08 (2019) 079 [arXiv:1906.11226] [INSPIRE].

  39. J.L. Cardy, Critical exponents of the chiral Potts model from conformal field theory, Nucl. Phys. B 389 (1993) 577 [hep-th/9210002] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  40. A.V. Chubukov, S. Sachdev and T. Senthil, Quantum phase transitions in frustrated quantum antiferromagnets, Nucl. Phys. B 426 (1994) 601 [Erratum ibid. B 438 (1995) 649] [INSPIRE].

  41. K. Yang, Ferromagnetic transition in one-dimensional itinerant electron systems, Phys. Rev. Lett. 93 (2004) 066401.

    ADS  Article  Google Scholar 

  42. R.N. Caldeira Costa and M. Taylor, Holography for chiral scale-invariant models, JHEP 02 (2011) 082 [arXiv:1010.4800] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  43. D.M. Hofman and A. Strominger, Chiral Scale and Conformal Invariance in 2D Quantum Field Theory, Phys. Rev. Lett. 107 (2011) 161601 [arXiv:1107.2917] [INSPIRE].

    ADS  Article  Google Scholar 

  44. G.H. Hardy and S. Ramanujan, Asymptotic formulaæ in combinatory analysis, Proc. Lond. Math. Soc. 2 (1918) 75.

    Article  Google Scholar 

  45. L.D. Faddeev and R. Jackiw, Hamiltonian Reduction of Unconstrained and Constrained Systems, Phys. Rev. Lett. 60 (1988) 1692 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  46. P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer Science & Business Media, (2012).

  47. P. Senjanovic, Path Integral Quantization of Field Theories with Second Class Constraints, Annals Phys. 100 (1976) 227 [Erratum ibid. 209 (1991) 248] [INSPIRE].

  48. L.D. Faddeev and A.A. Slavnov, Gauge fields. Introduction to quantum theory, Front. Phys. 50 (1980) 1 [INSPIRE].

  49. M. Henneaux and C. Teitelboim, Quantization of gauge systems, Princeton University Press, Princeton, U.S.A., (1992).

    Book  Google Scholar 

  50. J. Cotler and K. Jensen, A theory of reparameterizations for AdS3 gravity, JHEP 02 (2019) 079 [arXiv:1808.03263] [INSPIRE].

    ADS  Article  Google Scholar 

  51. A. Gafni, Power partitions, J. Number Theory 163 (2016) 19 [arXiv:1506.06124].

    MathSciNet  Article  Google Scholar 

  52. E.M. Wright, Asymptotic partition formulae. III. Partitions into k-th powers, Acta Math. 63 (1934) 143.

  53. R.C. Vaughan, Squares: additive questions and partitions, Int. J. Number Theory 11 (2015) 1367.

    MathSciNet  Article  Google Scholar 

  54. G. Tenenbaum, J. Wu and Y.-L. Li, Power partitions and saddle-point method, J. Number Theory 204 (2019) 435 [arXiv:1901.02234].

    MathSciNet  Article  Google Scholar 

  55. F. Luca and D. Ralaivaosaona, An explicit bound for the number of partitions into roots, J. Number Theory 169 (2016) 250.

    MathSciNet  Article  Google Scholar 

  56. Y.-L. Li and Y.-G. Chen, On the r-th root partition function, II, J. Number Theory 188 (2018) 392.

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Miguel Pino.

Additional information

ArXiv ePrint: 1909.02699

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fuentealba, O., González, H.A., Pino, M. et al. The anisotropic chiral boson. J. High Energ. Phys. 2019, 123 (2019).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI:


  • Conformal and W Symmetry
  • Space-Time Symmetries
  • Field Theories in Lower Dimensions