# Higher derivative terms, toroidal compactification, and Weyl anomalies in six-dimensional (2*,* 0) theories

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## Abstract

We systematically analyze the effective action on the moduli space of (2*,* 0) superconformal field theories in six dimensions, as well as their toroidal compactification to maximally supersymmetric Yang-Mills theories in five and four dimensions. We present a streamlined approach to non-renormalization theorems that constrain this effective action. The first several orders in its derivative expansion are determined by a one-loop calculation in five-dimensional Yang-Mills theory. This fixes the leading higher-derivative operators that describe the renormalization group flow into theories residing at singular points on the moduli space of the compactified (2*,* 0) theories. This understanding allows us to compute the *a*-type Weyl anomaly for all (2*,* 0) superconformal theories. We show that it decreases along every renormalization group flow that preserves (2*,* 0) supersymmetry, thereby establishing the *a*-theorem for this class of theories. Along the way, we encounter various field-theoretic arguments for the ADE classification of (2*,* 0) theories.

## Keywords

Conformal Field Theory Extended Supersymmetry M-Theory Supersym- metric Effective Theories## Notes

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## References

- [1]S. Paban, S. Sethi and M. Stern,
*Constraints from extended supersymmetry in quantum mechanics*,*Nucl. Phys.***B 534**(1998) 137 [hep-th/9805018] [INSPIRE]. - [2]S. Paban, S. Sethi and M. Stern,
*Summing up instantons in three-dimensional Yang-Mills theories*,*Adv. Theor. Math. Phys.***3**(1999) 343 [hep-th/9808119] [INSPIRE].MathSciNetCrossRefGoogle Scholar - [3]S. Paban, S. Sethi and M. Stern,
*Supersymmetry and higher derivative terms in the effective action of Yang-Mills theories*,*JHEP***06**(1998) 012 [hep-th/9806028] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [4]S. Sethi and M. Stern,
*Supersymmetry and the Yang-Mills effective action at finite N*,*JHEP***06**(1999) 004 [hep-th/9903049] [INSPIRE]. - [5]T. Maxfield and S. Sethi,
*The Conformal Anomaly of M5-Branes*,*JHEP***06**(2012) 075 [arXiv:1204.2002] [INSPIRE]. - [6]T. Banks, W. Fischler, S.H. Shenker and L. Susskind,
*M theory as a matrix model: A Conjecture*,*Phys. Rev.***D 55**(1997) 5112 [hep-th/9610043] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [7]M. Dine and N. Seiberg,
*Comments on higher derivative operators in some SUSY field theories*,*Phys. Lett.***B 409**(1997) 239 [hep-th/9705057] [INSPIRE]. - [8]K.A. Intriligator,
*Anomaly matching and a Hopf-Wess-Zumino term in 6d, N*= (2*,*0)*field theories*,*Nucl. Phys.***B 581**(2000) 257 [hep-th/0001205] [INSPIRE]. - [9]
- [10]E. Witten,
*Some comments on string dynamics*, in*Future perspectives in string theory. Proceedings, Conference, Strings’95*, Los Angeles, U.S.A., 13–18 March 1995, pp. 501–523 (1995) [hep-th/9507121] [INSPIRE]. - [11]
- [12]E. Witten,
*Five-branes and M-theory on an orbifold*,*Nucl. Phys.***B 463**(1996) 383 [hep-th/9512219] [INSPIRE]. - [13]N. Seiberg,
*Notes on theories with 16 supercharges*,*Nucl. Phys. Proc. Suppl.***67**(1998) 158 [hep-th/9705117] [INSPIRE]. - [14]
- [15]D. Gaiotto, G.W. Moore and A. Neitzke,
*Framed BPS States*,*Adv. Theor. Math. Phys.***17**(2013) 241 [arXiv:1006.0146] [INSPIRE]. - [16]G.W. Moore,
*Lecture Notes for Felix Klein Lectures*, http://www.physics.rutgers.edu/^{∼}gmoore/FelixKleinLectureNotes.pdf (2012). - [17]N. Seiberg and E. Witten,
*Comments on string dynamics in six-dimensions*,*Nucl. Phys.***B 471**(1996) 121 [hep-th/9603003] [INSPIRE]. - [18]N. Seiberg,
*Nontrivial fixed points of the renormalization group in six-dimensions*,*Phys. Lett.***B 390**(1997) 169 [hep-th/9609161] [INSPIRE]. - [19]C. Cordova, T.T. Dumitrescu and K. Intriligator,
*Deformations of Superconformal Theories*,*JHEP***11**(2016) 135 [arXiv:1602.01217] [INSPIRE]. - [20]C. Cordova, T.T. Dumitrescu and K. Intriligator,
*Multiplets of Superconformal Symmetry in Diverse Dimensions*,*JHEP***03**(2019) 163 [arXiv:1612.00809] [INSPIRE]. - [21]D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett,
*Generalized Global Symmetries*,*JHEP***02**(2015) 172 [arXiv:1412.5148] [INSPIRE]. - [22]M.R. Douglas,
*On D*= 5*super Yang-Mills theory and*(2*,*0)*theory*,*JHEP***02**(2011) 011 [arXiv:1012.2880] [INSPIRE]. - [23]N. Lambert, C. Papageorgakis and M. Schmidt-Sommerfeld,
*M5-Branes, D*4*-branes and Quantum*5*D super-Yang-Mills*,*JHEP***01**(2011) 083 [arXiv:1012.2882] [INSPIRE]. - [24]C. Papageorgakis and A.B. Royston,
*Revisiting Soliton Contributions to Perturbative Amplitudes*,*JHEP***09**(2014) 128 [arXiv:1404.0016] [INSPIRE]. - [25]
- [26]W.A. Bardeen and B. Zumino,
*Consistent and Covariant Anomalies in Gauge and Gravitational Theories*,*Nucl. Phys.***B 244**(1984) 421 [INSPIRE]. - [27]J.A. Harvey, R. Minasian and G.W. Moore,
*NonAbelian tensor multiplet anomalies*,*JHEP***09**(1998) 004 [hep-th/9808060] [INSPIRE].MathSciNetCrossRefGoogle Scholar - [28]
- [29]K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura,
*Anomaly polynomial of general*6*d SCFTs*,*PTEP***2014**(2014) 103B07 [arXiv:1408.5572] [INSPIRE]. - [30]A.A. Tseytlin and K. Zarembo,
*Magnetic interactions of D-branes and Wess-Zumino terms in superYang-Mills effective actions*,*Phys. Lett.***B 474**(2000) 95 [hep-th/9911246] [INSPIRE]. - [31]S. Deser and A. Schwimmer,
*Geometric classification of conformal anomalies in arbitrary dimensions*,*Phys. Lett.***B 309**(1993) 279 [hep-th/9302047] [INSPIRE]. - [32]A.B. Zamolodchikov,
*Irreversibility of the Flux of the Renormalization Group in a*2*D Field Theory*,*JETP Lett.***43**(1986) 730 [INSPIRE]. - [33]Z. Komargodski and A. Schwimmer,
*On Renormalization Group Flows in Four Dimensions*,*JHEP***12**(2011) 099 [arXiv:1107.3987] [INSPIRE]. - [34]Z. Komargodski,
*The Constraints of Conformal Symmetry on RG Flows*,*JHEP***07**(2012) 069 [arXiv:1112.4538] [INSPIRE]. - [35]H. Elvang, D.Z. Freedman, L.-Y. Hung, M. Kiermaier, R.C. Myers and S. Theisen,
*On renormalization group flows and the a-theorem in 6d*,*JHEP***10**(2012) 011 [arXiv:1205.3994] [INSPIRE]. - [36]F. Bastianelli, S. Frolov and A.A. Tseytlin,
*Conformal anomaly of*(2*,*0)*tensor multiplet in six-dimensions and AdS/CFT correspondence*,*JHEP***02**(2000) 013 [hep-th/0001041] [INSPIRE]. - [37]C. Beem, L. Rastelli and B.C. van Rees,
*W symmetry in six dimensions*,*JHEP***05**(2015) 017 [arXiv:1404.1079] [INSPIRE]. - [38]M. Henningson and K. Skenderis,
*The Holographic Weyl anomaly*,*JHEP***07**(1998) 023 [hep-th/9806087] [INSPIRE]. - [39]A.A. Tseytlin,
*R*^{4}*terms in*11*dimensions and conformal anomaly of*(2*,*0)*theory*,*Nucl. Phys.***B 584**(2000) 233 [hep-th/0005072] [INSPIRE]. - [40]M. Beccaria, G. Macorini and A.A. Tseytlin,
*Supergravity one-loop corrections on AdS*_{7}*and AdS*_{3}*, higher spins and AdS/CFT*,*Nucl. Phys.***B 892**(2015) 211 [arXiv:1412.0489] [INSPIRE]. - [41]C. Vafa,
*Geometric origin of Montonen-Olive duality*,*Adv. Theor. Math. Phys.***1**(1998) 158 [hep-th/9707131] [INSPIRE]. - [42]Y. Tachikawa,
*On S-duality of*5*d super Yang-Mills on S*^{1},*JHEP***11**(2011) 123 [arXiv:1110.0531] [INSPIRE]. - [43]Y. Tachikawa,
*On the*6*d origin of discrete additional data of*4*d gauge theories*,*JHEP***05**(2014) 020 [arXiv:1309.0697] [INSPIRE]. - [44]D. Anselmi, D.Z. Freedman, M.T. Grisaru and A.A. Johansen,
*Nonperturbative formulas for central functions of supersymmetric gauge theories*,*Nucl. Phys.***B 526**(1998) 543 [hep-th/9708042] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [45]C. Cordova, T.T. Dumitrescu and K. Intriligator,
*Anomalies, renormalization group flows and the a-theorem in six-dimensional*(1*,*0)*theories*,*JHEP***10**(2016) 080 [arXiv:1506.03807] [INSPIRE]. - [46]E. Witten,
*Conformal Field Theory In Four And Six Dimensions*, in*Topology, geometry and quantum field theory. Proceedings, Symposium in the honour of the 60th birthday of Graeme Segal*, Oxford, U.K., 24–29 June 2002, pp. 405–419 (2007) [arXiv:0712.0157] [INSPIRE]. - [47]H. Elvang and T.M. Olson,
*RG flows in d dimensions, the dilaton effective action and the a-theorem*,*JHEP***03**(2013) 034 [arXiv:1209.3424] [INSPIRE]. - [48]A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi,
*Causality, analyticity and an IR obstruction to UV completion*,*JHEP***10**(2006) 014 [hep-th/0602178] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [49]
- [50]M.A. Luty, J. Polchinski and R. Rattazzi,
*The a-theorem and the Asymptotics of*4*D Quantum Field Theory*,*JHEP***01**(2013) 152 [arXiv:1204.5221] [INSPIRE]. - [51]Y. Wang and X. Yin,
*Constraining Higher Derivative Supergravity with Scattering Amplitudes*,*Phys. Rev.***D 92**(2015) 041701 [arXiv:1502.03810] [INSPIRE]. - [52]Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin,
*Higher derivative couplings in theories with sixteen supersymmetries*,*Phys. Rev.***D 92**(2015) 125017 [arXiv:1503.02077] [INSPIRE]. - [53]A. Kapustin,
*Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality*,*Phys. Rev.***D 74**(2006) 025005 [hep-th/0501015] [INSPIRE]. - [54]C.W. Bernard, N.H. Christ, A.H. Guth and E.J. Weinberg,
*Instanton Parameters for Arbitrary Gauge Groups*,*Phys. Rev.***D 16**(1977) 2967 [INSPIRE]. - [55]M. Movshev and A. Schwarz,
*Supersymmetric Deformations of Maximally Supersymmetric Gauge Theories*,*JHEP***09**(2012) 136 [arXiv:0910.0620] [INSPIRE]. - [56]G. Bossard, P.S. Howe, U. Lindström, K.S. Stelle and L. Wulff,
*Integral invariants in maximally supersymmetric Yang-Mills theories*,*JHEP***05**(2011) 021 [arXiv:1012.3142] [INSPIRE]. - [57]C.-M. Chang, Y.-H. Lin, Y. Wang and X. Yin,
*Deformations with Maximal Supersymmetries Part 1: On-shell Formulation*, arXiv:1403.0545 [INSPIRE]. - [58]J. Polchinski,
*String theory. Vol. 2: Superstring theory and beyond*, Cambridge University Press (2007) [INSPIRE]. - [59]E.A. Bergshoeff, A. Bilal, M. de Roo and A. Sevrin,
*Supersymmetric nonAbelian Born-Infeld revisited*,*JHEP***07**(2001) 029 [hep-th/0105274] [INSPIRE]. - [60]Z. Bern, J.J. Carrasco, L.J. Dixon, M.R. Douglas, M. von Hippel and H. Johansson,
*D*= 5*maximally supersymmetric Yang-Mills theory diverges at six loops*,*Phys. Rev.***D 87**(2013) 025018 [arXiv:1210.7709] [INSPIRE]. - [61]
- [62]S. Deser, A. Gomberoff, M. Henneaux and C. Teitelboim,
*Duality, selfduality, sources and charge quantization in Abelian N form theories*,*Phys. Lett.***B 400**(1997) 80 [hep-th/9702184] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [63]K. Intriligator,
*6d, N*= (1*,*0)*Coulomb branch anomaly matching*,*JHEP***10**(2014) 162 [arXiv:1408.6745] [INSPIRE]. - [64]M. Henningson,
*Self-dual strings in six dimensions: Anomalies, the ADE-classification and the world-sheet WZW-model*,*Commun. Math. Phys.***257**(2005) 291 [hep-th/0405056] [INSPIRE]. - [65]N. Seiberg and W. Taylor,
*Charge Lattices and Consistency of 6D Supergravity*,*JHEP***06**(2011) 001 [arXiv:1103.0019] [INSPIRE]. - [66]Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin,
*Interpolating the Coulomb Phase of Little String Theory*,*JHEP***12**(2015) 022 [arXiv:1502.01751] [INSPIRE]. - [67]
- [68]A. Kapustin and N. Seiberg,
*Coupling a QFT to a TQFT and Duality*,*JHEP***04**(2014) 001 [arXiv:1401.0740] [INSPIRE]. - [69]O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3
*d dualities from*4*d dualities*,*JHEP***07**(2013) 149 [arXiv:1305.3924] [INSPIRE]. - [70]O. Aharony, S.S. Razamat, N. Seiberg and B. Willett,
*3d dualities from*4*d dualities for orthogonal groups*,*JHEP***08**(2013) 099 [arXiv:1307.0511] [INSPIRE]. - [71]D. Gaiotto, G.W. Moore and Y. Tachikawa,
*On*6*d N*= (2*,*0)*theory compactified on a Riemann surface with finite area*,*PTEP***2013**(2013) 013B03 [arXiv:1110.2657] [INSPIRE]. - [72]C. Cheung and D. O’Connell,
*Amplitudes and Spinor-Helicity in Six Dimensions*,*JHEP***07**(2009) 075 [arXiv:0902.0981] [INSPIRE]. - [73]T. Dennen, Y.-t. Huang and W. Siegel,
*Supertwistor space for*6*D maximal super Yang-Mills*,*JHEP***04**(2010) 127 [arXiv:0910.2688] [INSPIRE]. - [74]R.H. Boels and D. O’Connell,
*Simple superamplitudes in higher dimensions*,*JHEP***06**(2012) 163 [arXiv:1201.2653] [INSPIRE]. - [75]M. Movshev,
*Deformation of maximally supersymmetric Yang-Mills theory in dimensions 10. An Algebraic approach*, hep-th/0601010 [INSPIRE].