The ultra-violet question in maximally supersymmetric field theories

Abstract

We discuss various approaches to the problem of determining which supersymmetric invariants are permitted as counterterms in maximally supersymmetric super Yang--Mills and supergravity theories in various dimensions. We review the superspace non-renormalisation theorems based on conventional, light-cone, harmonic and certain non-Lorentz covariant superspaces, and we write down explicitly the relevant invariants. While the first two types of superspace admit the possibility of one-half BPS counterterms, of the form $F^4$ and $R^4$ respectively, the last two do not. This suggests that UV divergences begin with one-quarter BPS counterterms, i.e. $d^2{F}^4$ and $d^4{R}^4$, and this is supported by an entirely different approach based on algebraic renormalisation. The algebraic formalism is discussed for non-renormalisable theories and it is shown how the allowable supersymmetric counterterms can be determined via cohomological methods. These results are in agreement with all the explicit computations that have been carried out to date. In particular, they suggest that maximal supergravity is likely to diverge at four loops in D=5 and at five loops in D=4, unless other infinity suppression mechanisms not involving supersymmetry or gauge invariance are at work.

arXiv comment56 pages, 1 figure, uses youngtab.sty. Contribution to the proceedings of the W.E. Heraeus Workhop "Quantum Gravity: Challenges and Perspectives", Bad Honnef, 14-16 April 2008. References and clarifying comments added

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Cited by in corpus (56)

• The Ultraviolet Behavior of N=8 Supergravity at Four Loops
• E7(7) constraints on counterterms in N=8 supergravity
• Ultraviolet Properties of N = 8 Supergravity at Five Loops
• The Complete Four-Loop Four-Point Amplitude in N=4 Super-Yang-Mills Theory
• Basics of Generalized Unitarity
• The vanishing volume of D=4 superspace
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• Stringy KLT relations, global symmetries, and E_7(7) violation
• The three-loop form factor in N=4 super Yang-Mills
• The Tree Formula for MHV Graviton Amplitudes
• On duality symmetries of supergravity invariants
• Solution to the Ward Identities for Superamplitudes
• Tropical Amplitudes
• A note on the UV behaviour of maximally supersymmetric Yang-Mills theories
• Non-renormalization conditions for four-gluon scattering in supersymmetric string and field theory
• On the Structure of Supersymmetric Sums in Multi-Loop Unitarity Cuts
• Anomalies and divergences in N=4 supergravity
• Invariants and divergences in half-maximal supergravity theories
• One-loop divergences in 6D, N=(1,0) SYM theory
• Minimal unitary representations from supersymmetry
• N=8 Supergravity on the Light Cone
• The two D6R4 type invariants and their higher order generalisation
• Integral invariants in maximally supersymmetric Yang-Mills theories
• Supergraph analysis of the one-loop divergences in $6D$, $\mathcal{N}=(1,0)$ and $\mathcal{N}=(1,1)$ gauge theories
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• SUSY Ward identities, Superamplitudes, and Counterterms
• Supersymmetric field theory with benign ghosts
• The Ultraviolet Finiteness of N=8 Supergravity
• One-loop four-graviton amplitudes in N=4 supergravity models
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• Ultraviolet behavior of 6D supersymmetric Yang-Mills theories and harmonic superspace
• Loop amplitudes in maximal supergravity with manifest supersymmetry
• Spinor frame formalism for amplitudes and constrained superamplitudes of 10D SYM and 11D supergravity
• Four-derivative couplings and BPS dyons in heterotic CHL orbifolds
• Ultraviolet divergences in non-renormalizable supersymmetric theories
• On the two-loop divergences of the 2-point hypermultiplet supergraphs for $6D$, $\mathcal{N}=(1,1)$ SYM theory
• Chiral supergravity actions and superforms
• Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds
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• Off-Shell Scalar Supermultiplet in the Unfolded Dynamics Approach
• Off-shell invariants of linearized $4D,\mathcal{N}=2$ supergravity in the harmonic approach
• Low-energy $6D$, $\mathcal{N}=(1,1)$ SYM effective action beyond the leading approximation
• On the two-loop divergences in 6D, $\mathcal{N}=(1,1)$ SYM theory
• String Theory, Unification and Quantum Gravity
• Maximally Supersymmetric Yang-Mills in five dimensions in light-cone superspace
• On two-loop divergences of effective action in $6D$, $\mathcal{N}=(1,1)$ SYM theory
• Deformation of d=4, N> 4 Supergravities Breaks Nonlinear Local Supersymmetry
• Higher-dimensional invariants in $6D$ super Yang-Mills theory
• One-loop divergences in the six-dimensional $\mathcal{N}=(1,0)$ hypermultiplet self-coupling model
• Off- and on-shell harmonic superspaces for 6D SYM theories