Superconformal symmetry and higher-derivative Lagrangians

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Superconformal symmetry and higher-derivative Lagrangians. Antoine Van Proeyen KU Leuven. Mc Gill university, workshop in honour of Marc Grisaru April 19, 2013. My experience with Marc. First encounter: on boat to Cargèse 1979 Only one paper together .
Superconformal symmetry and higher-derivative LagrangiansAntoine Van ProeyenKU LeuvenMc Gill university, workshop in honour of Marc GrisaruApril 19, 2013My experience with Marc
  • First encounter: on boat to Cargèse 1979
  • Only one paper together
  • but many long discussions, e.g. while walking along the sea in Trieste on massive gravitino superfieldand very nice moments togetherInterest in higher-derivative terms
  • appear as ®0terms in effective action of string theory
  • corrections to black hole entropy
  • higher order to AdS/CFT correspondence
  • counterterms for UV divergences of quantum loops
  • Plan
  • What we know about general sugra/susy theories
  • The superconformal method
  • Higher derivative sugra actions and sugra loop results
  • Dirac-Born-Infeld− Volkov-Akulovand deformation of supersymmetry (example of an all order higher-derivative susy action)
  • Conclusions
  • 1. What we know about general sugra/susy theoriesStrathdee,1987vector multiplets +multiplets up to spin 1/2vector multipletstensor multipletThe map: dimensions and # of supersymmetriesHigher-derivative actions: no systematic knowledge
  • without higher derivatives: ungauged and gauged supergravities.
  • Insight thanks to embedding tensor formalism
  • Various constructions of higher derivative terms
  • e.g. susy Dirac-Born-Infeld: Deser, Puzalowski, 1980; Cecotti, Ferrara; Tseytlin; Bagger, Galperin; Roček; Kuzenko, Theisen; Ivanov, Krivonos, Ketov; Bellucci,
  • but no systematic construction, or classification of what are the possibilities; (certainly not in supergravity)
  • 2. The superconformal method from one thousand and one lecturesconformal scalar action (contains Weyl fields)local conformal symmetryGauge fix dilatations and special conformal transformationsPoincaré gravity actionlocal ¡symmetrySuperconformal constructionThe idea of superconformal methods
  • Difference susy- sugra: the concept of multiplets is clear in susy, they are mixed in supergravity
  • Superfields are an easy conceptual tool for rigid susy
  • (Super)gravity can be obtained by starting with (super)conformal symmetry and gauge fixing.
  • With matter:Before gauge fixing: everything looks like in rigid supersymmetry + covariantizations
  • Gauge fix dilatations, special conformal transformations, local R-symmetry and special supersymmetryPoincaré supergravity actionThe strategy :superconformal construction of N=1 supergravitychiral multiplet + Weyl multiplet superconformal actionS. Ferrara, M. Kaku, P.K. Townsend. P. van Nieuwenhuizen, 1977-78More explanationsSuperconformal construction of N=4 supergravityDe Roo, 1985Weyl multiplet+6 gauge compensating multiplets (on-shell)superconformal actionno flexibility in field equations !! gauge-fixing Weyl symmetry, local SU(4), local U(1) , S-supersymmetry and K-conformal boostspure N=4 Cremmer-Scherk-Ferrara supergravity3. Higher derivative supergravity actions and supergravity loop results
  • many miraculous cancellations have been found
  • is there an underlying hidden symmetry ?
  • If there are divergences: supersymmetric counterterms should exist (or supersymmetry anomalies)
  • We do not know enough to be sure whether (or which type of) invariants do exist.
  • Higher derivative sugra actions
  • for low N there is a superconformal tensor calculus: allows to construct counterterms and even anomalies → superconformal symmetry allows divergences.
  • for N=4: superconformal symmetry can be defined, but no construction of counterterms (or anomalies)
  • difficulty: on-shell susy algebra: to change action (add counterterms), one has to change simultaneously the transformations.
  • How do we get anomalies for low N from tensor calculus ?de Wit,Grisaru 1985Anomalies from tensor calculusfor low N ? (here N=1)de Wit,Grisaru 1985N=1: local superconformal anomalies satisfying Wess-Zumino consistency condition can be constructed using superconformal tensor calculusCompensator superfield transforms as forHow for N=4 ?
  • No tensor calculus; no auxiliary fields; only on-shell construction: no flexibility in field equations
  • How to establish the existence/non-existence of the consistent order by order deformation of N=4 on shell superspace ?
  • Conjecture: if it does not exist: explanation of finiteness (if Bern et al do not find N=4, D=4 is divergent at higher loops)
  • Until invariant counterterms are constructed (conformal?) we have no reason to expect UV divergences
  • Two points of view1. Legitimate counterterms are not available yet2. Legitimate counterterms are not available, period???S. Ferrara, R. Kallosh, AVP, 1209.0418N=4 conjecture“We are trying to prove ourselves wrong as quickly as possible, because only in that way can we find progress.” (Feynman)If the UV finiteness will persist in higher loops, one would like to view this as an opportunity to test some new ideas about gravity.E.g. : is superconformal symmetry more fundamental ?Repeat: Classical N=4 is obtained from gauge fixing a superconformal invariant action: The mass MPlappears in the gauge-fixing procedureBergshoeff, de Roo, de Wit, van Holten and AVP, 1981; de Roo, 1984
  • Analogy:
  • Mass parameters MW and MZof the massive vector mesons are not present in the gauge invariant action of the standard model.
  • Show up when the gauge symmetry is spontaneously broken.
  • In unitary gauge they give an impression of being fundamental.
  • In renormalizable gauge (where UV properties analyzed) : absent
  • S. Ferrara, R. Kallosh, AVP, 1209.04184.Dirac-Born-Infeld− Volkov-Akulovand deformation of supersymmetryon the search of deformations of N=4 theories, we find all-order invariant actions in rigid susy with extra supersymmetries (Volkov-Akulov (VA) – type)E. Bergshoeff, F. Coomans, C. Shahbazi, R. Kallosh, AVP, arXiv:1303.5662Bottom-up approach: start deformationsgauge field (D-2) on-shell dof; fermion = #spinor comp / 2vector multiplets +multiplets up to spin 1/2vector multipletstensor multipletThe map: dimensions and # of supersymmetriesBottom-up approach: start deformationsD=10: MW; D=6 SW; D=4 M; D=3 M; D=2 MWgauge field (D-2) on-shell dof; fermion = #spinor comp / 2extra (trivial) fermionic shift symmetrynormalization for later useBottom-up deformation
  • answer unique up to field redefinitions
  • also transformation rules deformed. As well ² as ´ parameter transformations can be defined
  • e.g. Only identities used that are valid in D=10,6,4,3, e.g. cyclic (Fierz) identitylooks hopeless to continue to all orderscontinues from work in D=6 : E. Bergshoeff, M. Rakowski and E. Sezgin, 1987Dp-brane actionStart with –symmetric Dp brane actionDp brane: IIB theory m=0,..., 9 and ¹=0,..., p=2n+1space-time coördinates Xm; µ is doublet of MW spinors; F¹ºAbelian field strength Same applies for D=6 (2,0) (also called iib): m=0,...,5 Symmetries:rigid susy doublet ²1; ²2local  symmetry doublet (effectively only half (reducible) symmetry)world volume gct Also D=4 N=2, m=0,...,3 (BH solutions)After gauge fixingworldvolume gct »m’ and  symmetry gauge-fixed: only one fermion, ¸, remains
  •  ²1 and ²2 supersymmetries preserved
  • suitable combinations are called ² and ³ : act as
  • ordinary susy: deformed at order ®
  • VA susy
  • Complete DBI-VA model for the p=9 caseExpanding the all-order result, one re-obtains indeed the result that was obtained in the bottom-up calculation to order ®2This proves that our all-order result is indeed the full deformation that we were looking for !vector multiplets +multiplets up to spin 1/2vector multipletstensor multipletThe map: dimensions and # of supersymmetriesD9D7D5D3vector multiplets +multiplets up to spin 1/2vector multipletstensor multipletThe map: dimensions and # of supersymmetriesD9D7V5D5D3V35. Conclusions
  • Superconformal symmetry has been used as a tool for constructing classical actions of supergravity.
  • Uses many concepts of superspace, reformulated as multiplets transforming under superconformal group.
  • We do not have a systematic knowledge of higher-derivative supergravity actions.
  • Can (broken) superconformal symmetry be an extra quantum symmetry?
  • The non-existence of (broken) superconformal-invariant counterterms and anomalies in N=4, D=4 could in that case explain ‘miraculous’ vanishing results.
  • We are studying new constructions of higher-derivative actions using gauge-fixed brane actions. For now: rigid supersymmetry.
  • More work to do ...
  • Import in superconformal
  • or there is still always
  • many happy returns !
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