Abstract
We generalize the geometrical formulation of Wilson loops recently introduced in [1] to the description of Wilson Surfaces. For N = (2, 0) theory in six dimensions, we provide an explicit derivation of BPS Wilson Surfaces with non-trivial coupling to scalars, together with their manifestly supersymmetric version. We derive explicit conditions which allow to classify these operators in terms of the number of preserved supercharges. We also discuss kappa-symmetry and prove that BPS conditions in six dimensions arise from kappa-symmetry invariance in eleven dimensions. Finally, we discuss super-Wilson Surfaces — and higher dimensional operators — as objects charged under global p-form (super)symmetries generated by tensorial supercurrents. To this end, the construction of conserved supercurrents in supermanifolds and of the corresponding conserved charges is developed in details.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C.A. Cremonini, P.A. Grassi and S. Penati, Supersymmetric Wilson Loops via Integral Forms, JHEP 04 (2020) 161 [arXiv:2003.01729] [INSPIRE].
O.J. Ganor, Six-dimensional tensionless strings in the large N limit, Nucl. Phys. B 489 (1997) 95 [hep-th/9605201] [INSPIRE].
A.S. Cattaneo and C.A. Rossi, Wilson surfaces and higher dimensional knot invariants, Commun. Math. Phys. 256 (2005) 513 [math-ph/0210037] [INSPIRE].
S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073 [INSPIRE].
S. Gukov and E. Witten, Rigid Surface Operators, Adv. Theor. Math. Phys. 14 (2010) 87 [arXiv:0804.1561] [INSPIRE].
S. Gukov, Surface Operators, in New Dualities of Supersymmetric Gauge Theories, J. Teschner ed., pp. 223–259 (2016) [DOI] [arXiv:1412.7127] [INSPIRE].
N.R. Constable, J. Erdmenger, Z. Guralnik and I. Kirsch, Intersecting D-3 branes and holography, Phys. Rev. D 68 (2003) 106007 [hep-th/0211222] [INSPIRE].
E. Koh and S. Yamaguchi, Surface operators in the Klebanov-Witten theory, JHEP 06 (2009) 070 [arXiv:0904.1460] [INSPIRE].
S.K. Ashok, M. Billó, M. Frau, A. Lerda and S. Mahato, Surface defects from fractional branes. Part I, JHEP 07 (2020) 051 [arXiv:2005.02050] [INSPIRE].
S.K. Ashok, M. Billò, M. Frau, A. Lerda and S. Mahato, Surface defects from fractional branes. Part II, JHEP 08 (2020) 058 [arXiv:2005.03701] [INSPIRE].
E.I. Buchbinder, J. Gomis and F. Passerini, Holographic gauge theories in background fields and surface operators, JHEP 12 (2007) 101 [arXiv:0710.5170] [INSPIRE].
J.M. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].
O. Lunin, 1/2-BPS states in M-theory and defects in the dual CFTs, JHEP 10 (2007) 014 [arXiv:0704.3442] [INSPIRE].
B. Chen, W. He, J.-B. Wu and L. Zhang, M5-branes and Wilson Surfaces, JHEP 08 (2007) 067 [arXiv:0707.3978] [INSPIRE].
B. Chen, The Self-dual String Soliton in AdS4 × S7 spacetime, Eur. Phys. J. C 54 (2008) 489 [arXiv:0710.2593] [INSPIRE].
P. Agarwal, J. Kim, S. Kim and A. Sciarappa, Wilson surfaces in M5-branes, JHEP 08 (2018) 119 [arXiv:1804.09932] [INSPIRE].
A. Gustavsson, Conformal anomaly of Wilson surface observables: A Field theoretical computation, JHEP 07 (2004) 074 [hep-th/0404150] [INSPIRE].
M. Mezei, S.S. Pufu and Y. Wang, Chern-Simons theory from M5-branes and calibrated M2-branes, JHEP 08 (2019) 165 [arXiv:1812.07572] [INSPIRE].
N. Drukker, M. Probst and M. Trépanier, Surface operators in the 6d N = (2, 0) theory, J. Phys. A 53 (2020) 365401 [arXiv:2003.12372] [INSPIRE].
N. Drukker, J. Gomis and S. Matsuura, Probing N = 4 SYM With Surface Operators, JHEP 10 (2008) 048 [arXiv:0805.4199] [INSPIRE].
M.-C. Tan, Nonlocal Operators and Duality in Abelian Gauge Theory on a Four-Manifold, arXiv:1312.5494 [INSPIRE].
L. Bianchi and M. Lemos, Superconformal surfaces in four dimensions, JHEP 06 (2020) 056 [arXiv:1911.05082] [INSPIRE].
R. Corrado, B. Florea and R. McNees, Correlation functions of operators and Wilson surfaces in the d = 6, (0, 2) theory in the large N limit, Phys. Rev. D 60 (1999) 085011 [hep-th/9902153] [INSPIRE].
N. Drukker, S. Giombi, A.A. Tseytlin and X. Zhou, Defect CFT in the 6d (2, 0) theory from M2 brane dynamics in AdS7 × S4 , JHEP 07 (2020) 101 [arXiv:2004.04562] [INSPIRE].
D.E. Berenstein, R. Corrado, W. Fischler and J.M. Maldacena, The Operator product expansion for Wilson loops and surfaces in the large N limit, Phys. Rev. D 59 (1999) 105023 [hep-th/9809188] [INSPIRE].
B. Chen, C.-Y. Liu and J.-B. Wu, Operator Product Expansion of Wilson surfaces from M5-branes, JHEP 01 (2008) 007 [arXiv:0711.2194] [INSPIRE].
C. Graham and E. Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence, Nucl. Phys. B 546 (1999) 52 [hep-th/9901021] [INSPIRE].
M. Henningson and K. Skenderis, Weyl anomaly for Wilson surfaces, JHEP 06 (1999) 012 [hep-th/9905163] [INSPIRE].
A. Gustavsson, On the Weyl anomaly of Wilson surfaces, JHEP 12 (2003) 059 [hep-th/0310037] [INSPIRE].
D. Young, Wilson Loops in Five-Dimensional Super-Yang-Mil ls, JHEP 02 (2012) 052 [arXiv:1112.3309] [INSPIRE].
J. Estes, D. Krym, A. O’Bannon, B. Robinson and R. Rodgers, Wilson Surface Central Charge from Holographic Entanglement Entropy, JHEP 05 (2019) 032 [arXiv:1812.00923] [INSPIRE].
K. Jensen, A. O’Bannon, B. Robinson and R. Rodgers, From the Weyl Anomaly to Entropy of Two-Dimensional Boundaries and Defects, Phys. Rev. Lett. 122 (2019) 241602 [arXiv:1812.08745] [INSPIRE].
A. Chalabi, A. O’Bannon, B. Robinson and J. Sisti, Central charges of 2d superconformal defects, JHEP 05 (2020) 095 [arXiv:2003.02857] [INSPIRE].
D. Gaiotto, Surface Operators in N = 2 4d Gauge Theories, JHEP 11 (2012) 090 [arXiv:0911.1316] [INSPIRE].
D. Gaiotto, S. Gukov and N. Seiberg, Surface Defects and Resolvents, JHEP 09 (2013) 070 [arXiv:1307.2578] [INSPIRE].
C. Cordova, D. Gaiotto and S.-H. Shao, Surface Defect Indices and 2d-4d BPS States, JHEP 12 (2017) 078 [arXiv:1703.02525] [INSPIRE].
C. Cordova, D. Gaiotto and S.-H. Shao, Surface Defects and Chiral Algebras, JHEP 05 (2017) 140 [arXiv:1704.01955] [INSPIRE].
S.K. Ashok et al., Surface operators, chiral rings and localization in \( \mathcal{N} \) = 2 gauge theories, JHEP 11 (2017) 137 [arXiv:1707.08922] [INSPIRE].
S.K. Ashok et al., Surface operators, dual quivers and contours, Eur. Phys. J. C 79 (2019) 278 [arXiv:1807.06316] [INSPIRE].
P. Liendo, L. Rastelli and B.C. van Rees, The Bootstrap Program for Boundary CFTd, JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].
D. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014) 100 [arXiv:1310.5078] [INSPIRE].
F. Gliozzi, P. Liendo, M. Meineri and A. Rago, Boundary and Interface CFTs from the Conformal Bootstrap, JHEP 05 (2015) 036 [arXiv:1502.07217] [INSPIRE].
F. Gliozzi, Truncatable bootstrap equations in algebraic form and critical surface exponents, JHEP 10 (2016) 037 [arXiv:1605.04175] [INSPIRE].
P. Liendo and C. Meneghelli, Bootstrap equations for \( \mathcal{N} \) = 4 SYM with defects, JHEP 01 (2017) 122 [arXiv:1608.05126] [INSPIRE].
A. Bissi, T. Hansen and A. Söderberg, Analytic Bootstrap for Boundary CFT, JHEP 01 (2019) 010 [arXiv:1808.08155] [INSPIRE].
A. Kaviraj and M.F. Paulos, The Functional Bootstrap for Boundary CFT, JHEP 04 (2020) 135 [arXiv:1812.04034] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
C. Córdova, T.T. Dumitrescu and K. Intriligator, Exploring 2-Group Global Symmetries, JHEP 02 (2019) 184 [arXiv:1802.04790] [INSPIRE].
N. Seiberg, Field Theories With a Vector Global Symmetry, SciPost Phys. 8 (2020) 050 [arXiv:1909.10544] [INSPIRE].
M. Pretko, Subdimensional Particle Structure of Higher Rank U(1) Spin Liquids, Phys. Rev. B 95 (2017) 115139 [arXiv:1604.05329] [INSPIRE].
M. Pretko, The Fracton Gauge Principle, Phys. Rev. B 98 (2018) 115134 [arXiv:1807.11479] [INSPIRE].
A. Gromov, Towards classification of Fracton phases: the multipole algebra, Phys. Rev. X 9 (2019) 031035 [arXiv:1812.05104] [INSPIRE].
N. Seiberg and S.-H. Shao, Exotic Symmetries, Duality, and Fractons in 2 + 1-Dimensional Quantum Field Theory, arXiv:2003.10466 [INSPIRE].
N. Seiberg and S.-H. Shao, Exotic U(1) Symmetries, Duality, and Fractons in 3 + 1-Dimensional Quantum Field Theory, SciPost Phys. 9 (2020) 046 [arXiv:2004.00015] [INSPIRE].
N. Seiberg and S.-H. Shao, Exotic ℤN Symmetries, Duality, and Fractons in 3 + 1-Dimensional Quantum Field Theory, arXiv:2004.06115 [INSPIRE].
H. Ouyang, J.-B. Wu and J.-j. Zhang, BPS Wilson loops in Minkowski spacetime and Euclidean space, Eur. Phys. J. C 75 (2015) 606 [arXiv:1504.06929] [INSPIRE].
N. Berkovits, Towards covariant quantization of the supermembrane, JHEP 09 (2002) 051 [hep-th/0201151] [INSPIRE].
I. Chepelev, NonAbelian Wilson surfaces, JHEP 02 (2002) 013 [hep-th/0111018] [INSPIRE].
C. Hofman, NonAbelian 2 forms, hep-th/0207017 [INSPIRE].
J.C. Baez and J. Huerta, An Invitation to Higher Gauge Theory, Gen. Rel. Grav. 43 (2011) 2335 [arXiv:1003.4485] [INSPIRE].
P.-M. Ho and Y. Matsuo, Note on non-Abelian two-form gauge fields, JHEP 09 (2012) 075 [arXiv:1206.5643] [INSPIRE].
H. Kim and C. Sämann, Adjusted Paral lel Transport for Higher Gauge Theories, J. Phys. A 52 (2020) 445206 [arXiv:1911.06390] [INSPIRE].
C.A. Cremonini, P.A. Grassi and S. Penati, Superfractons, in preparation.
A. Belopolsky, Picture changing operators in supergeometry and superstring theory, hep-th/9706033 [INSPIRE].
E. Witten, Notes On Supermanifolds and Integration, arXiv:1209.2199 [INSPIRE].
R. Catenacci, P.A. Grassi and S. Noja, Superstring Field Theory, Superforms and Supergeometry, J. Geom. Phys. 148 (2020) 103559 [arXiv:1807.09563] [INSPIRE].
L. Castellani, R. Catenacci and P.A. Grassi, Supergravity Actions with Integral Forms, Nucl. Phys. B 889 (2014) 419 [arXiv:1409.0192] [INSPIRE].
L. Castellani, R. Catenacci and P.A. Grassi, Integral representations on supermanifolds: super Hodge duals, PCOs and Liouvil le forms, Lett. Math. Phys. 107 (2017) 167 [arXiv:1603.01092] [INSPIRE].
R. Catenacci, P.A. Grassi and S. Noja, A∞-Algebra from Supermanifolds, Annales Henri Poincaré 20 (2019) 4163 [arXiv:1901.00818] [INSPIRE].
C.A. Cremonini and P.A. Grassi, Pictures from Super Chern-Simons Theory, JHEP 03 (2020) 043 [arXiv:1907.07152] [INSPIRE].
C.A. Cremonini and P.A. Grassi, Super Chern-Simons theory: Batalin-Vilkovisky formalism and A∞ algebras, Phys. Rev. D 102 (2020) 025009 [arXiv:1912.10807] [INSPIRE].
P. Deligne et al., Quantum fields and strings: A course for mathematicians. Vol. 1, 2. AMS (1999) [INSPIRE].
P.S. Howe, G. Sierra and P.K. Townsend, Supersymmetry in Six-Dimensions, Nucl. Phys. B 221 (1983) 331 [INSPIRE].
E. Bergshoeff, E. Sezgin and A. Van Proeyen, Superconformal Tensor Calculus and Matter Couplings in Six-dimensions, Nucl. Phys. B 264 (1986) 653 [Erratum ibid. 598 (2001) 667] [INSPIRE].
P. Claus, R. Kallosh and A. Van Proeyen, M five-brane and superconformal (0, 2) tensor multiplet in six-dimensions, Nucl. Phys. B 518 (1998) 117 [hep-th/9711161] [INSPIRE].
S. Ferrara and E. Sokatchev, Representations of (1, 0) and (2, 0) superconformal algebras in six-dimensions: Massless and short superfields, Lett. Math. Phys. 51 (2000) 55 [hep-th/0001178] [INSPIRE].
H. Ooguri, J. Rahmfeld, H. Robins and J. Tannenhauser, Holography in superspace, JHEP 07 (2000) 045 [hep-th/0007104] [INSPIRE].
A.D. Polyanin, V.F. Zaitsev and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London (2002).
L. Castellani, R. Catenacci and P.A. Grassi, Hodge Dualities on Supermanifolds, Nucl. Phys. B 899 (2015) 570 [arXiv:1507.01421] [INSPIRE].
R. Bott and L.W. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics (Book 82), Springer, 1st edition (1995).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2006.08633v2
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.
About this article
Cite this article
Cremonini, C.A., Grassi, P.A. & Penati, S. Surface operators in superspace. J. High Energ. Phys. 2020, 50 (2020). https://doi.org/10.1007/JHEP11(2020)050
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2020)050