Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On actions for (entangling) surfaces and DCFTs


The dynamics of surfaces and interfaces describe many physical systems, including fluid membranes, entanglement entropy and the coupling of defects to quantum field theories. Based on the formulation of submanifold calculus developed by Carter, we introduce a new variational principle for (entangling) surfaces. This principle captures all diffeomorphism constraints on surface/interface actions and their associated spacetime stress tensor. The different couplings to the geometric tensors appearing in the surface action are interpreted in terms of response coefficients within elasticity theory. An example of a surface action with edges at the two-derivative level is studied, including both the parity-even and parity-odd sectors. Its conformally invariant counterpart restricts the type of conformal anomalies that can appear in two-dimensional submanifolds with boundaries. Analogously to hydrodynamics, it is shown that classification methods can be used to constrain the stress tensor of (entangling) surfaces at a given order in derivatives. This analysis reveals a purely geometric parity-odd contribution to the Young modulus of a thin elastic membrane. Extending this novel variational principle to BCFTs and DCFTs in curved spacetimes allows to obtain the Ward identities for diffeomorphism and Weyl transformations. In this context, we provide a formal derivation of the contact terms in the stress tensor and of the displacement operator for a broad class of actions.

A preprint version of the article is available at ArXiv.


  1. [1]

    J. Guven and P. Vázquez-Montejo, The Geometry of Fluid Membranes: Variational Principles, Symmetries and Conservation Laws, Springer International Publishing, Cham (2018), pp. 167-219.

  2. [2]

    D.M. McAvity and H. Osborn, Energy momentum tensor in conformal field theories near a boundary, Nucl. Phys. B 406 (1993) 655 [hep-th/9302068] [INSPIRE].

  3. [3]

    K. Jensen and A. O’Bannon, Constraint on Defect and Boundary Renormalization Group Flows, Phys. Rev. Lett. 116 (2016) 091601 [arXiv:1509.02160] [INSPIRE].

  4. [4]

    M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].

  5. [5]

    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].

  6. [6]

    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].

  7. [7]

    B. Carter, Brane dynamics for treatment of cosmic strings and vortons, in Recent developments in gravitation and mathematical physics. Proceedings, 2nd Mexican School, Tlaxcala, Mexico, December 1-7, 1996 (1997) [hep-th/9705172] [INSPIRE].

  8. [8]

    R. Emparan, T. Harmark, V. Niarchos and N.A. Obers, World-Volume Effective Theory for Higher-Dimensional Black Holes, Phys. Rev. Lett. 102 (2009) 191301 [arXiv:0902.0427] [INSPIRE].

  9. [9]

    R. Emparan, T. Harmark, V. Niarchos and N.A. Obers, Essentials of Blackfold Dynamics, JHEP 03 (2010) 063 [arXiv:0910.1601] [INSPIRE].

  10. [10]

    J. Armas, How Fluids Bend: the Elastic Expansion for Higher-Dimensional Black Holes, JHEP 09 (2013) 073 [arXiv:1304.7773] [INSPIRE].

  11. [11]

    J. Armas, J. Bhattacharya and N. Kundu, Surface transport in plasma-balls, JHEP 06 (2016) 015 [arXiv:1512.08514] [INSPIRE].

  12. [12]

    C.V. Johnson, D-brane primer, in Strings, branes and gravity. Proceedings, Theoretical Advanced Study Institute, TASI’99, Boulder, U.S.A., May 31-June 25, 1999, pp. 129-350, DOI: [hep-th/0007170] [INSPIRE].

  13. [13]

    B. Carter, Outer curvature and conformal geometry of an imbedding, J. Geom. Phys. 8 (1992) 53 [INSPIRE].

  14. [14]

    B. Carter, Perturbation dynamics for membranes and strings governed by Dirac Goto Nambu action in curved space, Phys. Rev. D 48 (1993) 4835 [INSPIRE].

  15. [15]

    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].

  16. [16]

    J. Camps, Generalized entropy and higher derivative Gravity, JHEP 03 (2014) 070 [arXiv:1310.6659] [INSPIRE].

  17. [17]

    X. Dong, Holographic Entanglement Entropy for General Higher Derivative Gravity, JHEP 01 (2014) 044 [arXiv:1310.5713] [INSPIRE].

  18. [18]

    X. Dong and A. Lewkowycz, Entropy, Extremality, Euclidean Variations and the Equations of Motion, JHEP 01 (2018) 081 [arXiv:1705.08453] [INSPIRE].

  19. [19]

    J.L. Cardy, Conformal Invariance and Surface Critical Behavior, Nucl. Phys. B 240 (1984) 514 [INSPIRE].

  20. [20]

    J.L. Cardy and D.C. Lewellen, Bulk and boundary operators in conformal field theory, Phys. Lett. B 259 (1991) 274 [INSPIRE].

  21. [21]

    D.M. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys. B 455 (1995) 522 [cond-mat/9505127] [INSPIRE].

  22. [22]

    P. Liendo, L. Rastelli and B.C. van Rees, The Bootstrap Program for Boundary CFT d, JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].

  23. [23]

    L. Bianchi, M. Meineri, R.C. Myers and M. Smolkin, Rényi entropy and conformal defects, JHEP 07 (2016) 076 [arXiv:1511.06713] [INSPIRE].

  24. [24]

    P. Liendo and C. Meneghelli, Bootstrap equations for \( \mathcal{N} \) = 4 SYM with defects, JHEP 01 (2017) 122 [arXiv:1608.05126] [INSPIRE].

  25. [25]

    L. Rastelli and X. Zhou, The Mellin Formalism for Boundary CFT d, JHEP 10 (2017) 146 [arXiv:1705.05362] [INSPIRE].

  26. [26]

    A. Karch and L. Randall, Open and closed string interpretation of SUSY CFT’s on branes with boundaries, JHEP 06 (2001) 063 [hep-th/0105132] [INSPIRE].

  27. [27]

    O. DeWolfe, D.Z. Freedman and H. Ooguri, Holography and defect conformal field theories, Phys. Rev. D 66 (2002) 025009 [hep-th/0111135] [INSPIRE].

  28. [28]

    J. Erdmenger, Z. Guralnik and I. Kirsch, Four-dimensional superconformal theories with interacting boundaries or defects, Phys. Rev. D 66 (2002) 025020 [hep-th/0203020] [INSPIRE].

  29. [29]

    O. Aharony, O. DeWolfe, D.Z. Freedman and A. Karch, Defect conformal field theory and locally localized gravity, JHEP 07 (2003) 030 [hep-th/0303249] [INSPIRE].

  30. [30]

    J. Erdmenger, M. Flory and M.-N. Newrzella, Bending branes for DCFT in two dimensions, JHEP 01 (2015) 058 [arXiv:1410.7811] [INSPIRE].

  31. [31]

    J. Erdmenger, M. Flory, C. Hoyos, M.-N. Newrzella and J.M.S. Wu, Entanglement Entropy in a Holographic Kondo Model, Fortsch. Phys. 64 (2016) 109 [arXiv:1511.03666] [INSPIRE].

  32. [32]

    M. de Leeuw, C. Kristjansen and K. Zarembo, One-point Functions in Defect CFT and Integrability, JHEP 08 (2015) 098 [arXiv:1506.06958] [INSPIRE].

  33. [33]

    I. Buhl-Mortensen, M. de Leeuw, A.C. Ipsen, C. Kristjansen and M. Wilhelm, One-loop one-point functions in gauge-gravity dualities with defects, Phys. Rev. Lett. 117 (2016) 231603 [arXiv:1606.01886] [INSPIRE].

  34. [34]

    M. de Leeuw, A.C. Ipsen, C. Kristjansen, K.E. Vardinghus and M. Wilhelm, Two-point functions in AdS/dCFT and the boundary conformal bootstrap equations, JHEP 08 (2017) 020 [arXiv:1705.03898] [INSPIRE].

  35. [35]

    P. Canham, The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell, J. Theor. Biol. 26 (1970) 61.

  36. [36]

    W. Helfrich, Elastic Properties of Lipid Bilayers — Theory and Possible Experiments, Z. Naturforsch. C 28 (1973) 693.

  37. [37]

    U. Seifert, Configurations of fluid membranes and vesicles, Adv. Phys. 46 (1997) 13.

  38. [38]

    Z.C. Tu and Z.C. Ou-Yang, A geometric theory on the elasticity of bio-membranes, J. Phys. A 37 (2004) 11407.

  39. [39]

    L. Landau and E.M. Lifshitz, Theory of elasticity, Course of Theoretical Physics 7 (1959) 134.

  40. [40]

    J. Guven, Perturbations of a topological defect as a theory of coupled scalar fields in curved space, Phys. Rev. D 48 (1993) 5562 [gr-qc/9304033] [INSPIRE].

  41. [41]

    R. Capovilla and J. Guven, Geometry of deformations of relativistic membranes, Phys. Rev. D 51 (1995) 6736 [gr-qc/9411060] [INSPIRE].

  42. [42]

    J. Guven, Membrane geometry with auxiliary variables and quadratic constraints, J. Phys. A 37 (2004) L313 [math-ph/0404064] [INSPIRE].

  43. [43]

    J. Guven and P. Vazquez-Montejo, Metric variations become a surface, Phys. Lett. A 377 (2013) 1507 [arXiv:1211.7154] [INSPIRE].

  44. [44]

    G. Arreaga, R. Capovilla and J. Guven, Noether currents for bosonic branes, Annals Phys. 279 (2000) 126 [hep-th/0002088] [INSPIRE].

  45. [45]

    M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy, Lect. Notes Phys. 931 (2017) pp.1 [arXiv:1609.01287] [INSPIRE].

  46. [46]

    O.-Y. Zhong-can and W. Helfrich, Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders, Phys. Rev. A 39 (1989) 5280.

  47. [47]

    C. Charmousis and R. Zegers, Matching conditions for a brane of arbitrary codimension, JHEP 08 (2005) 075 [hep-th/0502170] [INSPIRE].

  48. [48]

    P. Fonda, V. Jejjala and A. Veliz-Osorio, On the Shape of Things: From holography to elastica, Annals Phys. 385 (2017) 358 [arXiv:1611.03462] [INSPIRE].

  49. [49]

    J. Armas, J. Gath, V. Niarchos, N.A. Obers and A.V. Pedersen, Forced Fluid Dynamics from Blackfolds in General Supergravity Backgrounds, JHEP 10 (2016) 154 [arXiv:1606.09644] [INSPIRE].

  50. [50]

    R.A. Battye and B. Carter, Gravitational perturbations of relativistic membranes and strings, Phys. Lett. B 357 (1995) 29 [hep-ph/9508300] [INSPIRE].

  51. [51]

    R. Capovilla and J. Guven, Large deformations of relativistic membranes: A Generalization of the Raychaudhuri equations, Phys. Rev. D 52 (1995) 1072 [gr-qc/9411061] [INSPIRE].

  52. [52]

    B. Carter, Amalgamated Codazzi-Raychaudhuri identity for foliation, Contemp. Math. 203 (1997) 207 [hep-th/9705083] [INSPIRE].

  53. [53]

    R.A. Porto, The effective field theorist’s approach to gravitational dynamics, Phys. Rept. 633 (2016) 1 [arXiv:1601.04914] [INSPIRE].

  54. [54]

    Y. Aminov, The Geometry of Submanifolds, Taylor & Francis (2001).

  55. [55]

    A. Papapetrou, Spinning test particles in general relativity. 1., Proc. Roy. Soc. Lond. A 209 (1951) 248 [INSPIRE].

  56. [56]

    J. Armas and T. Harmark, Constraints on the effective fluid theory of stationary branes, JHEP 10 (2014) 063 [arXiv:1406.7813] [INSPIRE].

  57. [57]

    L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries, Phys. Rev. D 94 (2016) 084046 [arXiv:1609.00207] [INSPIRE].

  58. [58]

    M. Vasilic and M. Vojinovic, Classical spinning branes in curved backgrounds, JHEP 07 (2007) 028 [arXiv:0707.3395] [INSPIRE].

  59. [59]

    J. Armas, (Non)-Dissipative Hydrodynamics on Embedded Surfaces, JHEP 09 (2014) 047 [arXiv:1312.0597] [INSPIRE].

  60. [60]

    J. Armas, J. Camps, T. Harmark and N.A. Obers, The Young Modulus of Black Strings and the Fine Structure of Blackfolds, JHEP 02 (2012) 110 [arXiv:1110.4835] [INSPIRE].

  61. [61]

    A. Schwimmer and S. Theisen, Entanglement Entropy, Trace Anomalies and Holography, Nucl. Phys. B 801 (2008) 1 [arXiv:0802.1017] [INSPIRE].

  62. [62]

    J. Armas, J. Bhattacharya, A. Jain and N. Kundu, On the surface of superfluids, JHEP 06 (2017) 090 [arXiv:1612.08088] [INSPIRE].

  63. [63]

    M. Cvitan, P. Dominis Prester, S. Pallua, I. Smolić and T. Štemberga, Parity-odd surface anomalies and correlation functions on conical defects, arXiv:1503.06196 [INSPIRE].

  64. [64]

    A. Castro, S. Detournay, N. Iqbal and E. Perlmutter, Holographic entanglement entropy and gravitational anomalies, JHEP 07 (2014) 114 [arXiv:1405.2792] [INSPIRE].

  65. [65]

    T. Ali, S. Shajidul Haque and J. Murugan, Holographic Entanglement Entropy for Gravitational Anomaly in Four Dimensions, arXiv:1611.03415 [INSPIRE].

  66. [66]

    S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B 309 (1993) 279 [hep-th/9302047] [INSPIRE].

  67. [67]

    C.R. Graham and E. Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence, Nucl. Phys. B 546 (1999) 52 [hep-th/9901021] [INSPIRE].

  68. [68]

    M. Taylor and W. Woodhead, Renormalized entanglement entropy, JHEP 08 (2016) 165 [arXiv:1604.06808] [INSPIRE].

  69. [69]

    T. Azeyanagi, R. Loganayagam and G.S. Ng, Holographic Entanglement for Chern-Simons Terms, JHEP 02 (2017) 001 [arXiv:1507.02298] [INSPIRE].

  70. [70]

    E. Caceres, R. Mohan and P.H. Nguyen, On holographic entanglement entropy of Horndeski black holes, JHEP 10 (2017) 145 [arXiv:1707.06322] [INSPIRE].

Download references

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.

Author information

Correspondence to Jay Armas.

Additional information

ArXiv ePrint: 1709.06766

Rights and permissions

This article is published under an open access license. Please check the 'Copyright Information' section for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Armas, J., Tarrío, J. On actions for (entangling) surfaces and DCFTs. J. High Energ. Phys. 2018, 100 (2018).

Download citation


  • Brane Dynamics in Gauge Theories
  • D-branes
  • p-branes