We quantize the three-dimensional BF-model using axial gauge conditions. Exploiting the rich symm... more We quantize the three-dimensional BF-model using axial gauge conditions. Exploiting the rich symmetry-structure of the model we show that the Green-functions correspond to tree graphs and can be obtained as the unique solution of the Ward-Identities. Furthermore, we will show that the theory can be uniquely determined by symmetry considerations without the need of an action principle.
The Green functions of the Chern-Simons theory quantized in the axial gauge are shown to be calcu... more The Green functions of the Chern-Simons theory quantized in the axial gauge are shown to be calculable as the unique, exact solution of the Ward identities which express the invariance of the theory under the topological supersymmetry of Delduc, Gieres and Sorella. 1 See also [5] for more general noncovariant gauges. 2 Concerning the physical irrelevance of topological supersymmetry we disagree with the authors of Ref. [10] 3 The Lagrange multiplier fields are used for the implementation of the gauge condition.
We consider Type 0B D3-branes placed at conical singularities and analyze in detail the conifold ... more We consider Type 0B D3-branes placed at conical singularities and analyze in detail the conifold singularity. We study the non supersymmetric gauge theories on their worldvolume and their conjectured dual gravity descriptions. In the ultraviolet the solutions exhibit a logarithmic running of the gauge coupling. In the infrared we find confining solutions and IR fixed points.
We construct several new G 2 holonomy metrics that play an important role in recent studies of ge... more We construct several new G 2 holonomy metrics that play an important role in recent studies of geometrical transitions in compactifications of M-theory to four dimensions. In type IIA string theory these metrics correspond to D6 branes wrapped on the threecycle of the deformed conifold and the resolved conifold with two-form RR flux on the blown-up two-sphere, which are related by a conifold transition. We also study a G 2 metric that is related in type IIA to the line bundle over S 2 × S 2 with RR two-form flux. Our approach exploits systematically the definition of torsion-free G 2 structures in terms of three-forms which are closed and co-closed. Besides being an elegant formalism this turns out to be a practical tool to construct G 2 holonomy metrics.
We propose a method to compute the scattering angle for classical black hole scattering directly ... more We propose a method to compute the scattering angle for classical black hole scattering directly from two massive particle irreducible diagrams in a heavy-mass effective field theory approach to general relativity, without the need of subtracting iteration terms. The amplitudes in this effective theory are constructed using a recently proposed novel colour-kinematic/double copy for tree-level two-scalar, multi-graviton amplitudes, where the BCJ numerators are gauge invariant and local with respect to the massless gravitons. These tree amplitudes, together with graviton tree amplitudes, enter the construction of the requiredD-dimensional loop integrands and allow for a direct extraction of contributions relevant for classical physics. In particular the soft/heavy-mass expansions of full integrands is circumvented, and all iterating contributions can be dropped from the get go. We use this method to compute the scattering angle up to third post-Minkowskian order in four dimensions, in...
We study celestial amplitudes in (super) Yang-Mills theory using a parameterisation of the spinor... more We study celestial amplitudes in (super) Yang-Mills theory using a parameterisation of the spinor helicity variables where their overall phase is not fixed by the little group action. In this approach the spin constraint h −h = J for celestial conformal primaries emerges naturally from a new Mellin transform, and the action of conformal transformations on celestial amplitudes is derived. Applying this approach to N = 4 super Yang-Mills, we show how the appropriate definition of on-shell superspace coordinates leads naturally to a formulation of chiral celestial superamplitudes and a representation of the generators of the four-dimensional superconformal algebra on the celestial sphere, which by construction annihilate all tree-level celestial superamplitudes.
Using modern amplitude techniques we compute the leading classical and quantum corrections to the... more Using modern amplitude techniques we compute the leading classical and quantum corrections to the gravitational potential between two massive scalars induced by adding cubic terms to Einstein gravity. We then study the scattering of massless scalars, photons and gravitons off a heavy scalar in the presence of the same R 3 deformations, and determine the bending angle in the three cases from the non-analytic component of the scattering amplitude. Similarly to the Einstein-Hilbert case, we find that the classical contribution to the bending angle is universal, but unlike that case, universality is preserved also by the first quantum correction. Finally we extend our analysis to include a deformation of the form ΦR 2 , where Φ is the dilaton, which arises in the low-energy effective action of the bosonic string in addition to the R 3 term, and compute its effect on the graviton bending.
We present a systematic procedure to compute complete, analytic form factors of gauge-invariant o... more We present a systematic procedure to compute complete, analytic form factors of gauge-invariant operators at loop level in pure Yang-Mills. We consider applications to operators of the form Tr F n where F is the gluon field strength. Our approach is based on an extension to form factors of the dimensional reconstruction technique, in conjunction with the six-dimensional spinor-helicity formalism and generalised unitarity. For form factors this technique requires the introduction of additional scalar operators, for which we provide a systematic prescription. We also discuss a generalisation of dimensional reconstruction to any number of loops, both for amplitudes and form factors. Several novel results for one-loop minimal and non-minimal form factors of Tr F n with n > 2 are presented. Finally, we describe the Mathematica package SpinorHelicity6D, which is tailored to handle six-dimensional quantities written in the spinorhelicity formalism.
Form factors of the stress-tensor multiplet operator in $$ \mathcal{N}=4 $$ N = 4 supersymmetric ... more Form factors of the stress-tensor multiplet operator in $$ \mathcal{N}=4 $$ N = 4 supersymmetric Yang-Mills reveal surprisingly simple structures similar to those appearing in scattering amplitudes. In this paper we show that, as for the case of amplitudes, they also enjoy dual conformal symmetry. We compute the dual conformal anomaly at one loop for an arbitrary number of particles and generic helicities, which matches the expression of the dual conformal anomaly of amplitudes, and perform explicit checks for MHV and NMHV one-loop form factors. In the NMHV case the realisation of dual conformal symmetry requires a delicate cancellation of offending terms arising from three-mass triangles, which we explicitly check in the case of the four-point NMHV form factor.
It is known that the Yangian of P SU (2, 2|4) is a symmetry of the tree-level S-matrix of N = 4 s... more It is known that the Yangian of P SU (2, 2|4) is a symmetry of the tree-level S-matrix of N = 4 super Yang-Mills. On the other hand, the complete one-loop dilatation operator in the same theory commutes with the level-one Yangian generators only up to certain boundary terms found by Dolan, Nappi and Witten. Using a result by Zwiebel, we show how the Yangian symmetry of the tree-level S-matrix of N = 4 super Yang-Mills implies precisely the Yangian invariance, up to boundary terms, of the one-loop dilatation operator.
We derive a compact expression for the three-point MHV form factors of half-BPS operators in N = ... more We derive a compact expression for the three-point MHV form factors of half-BPS operators in N = 4 super Yang-Mills at two loops. The main tools of our calculation are generalised unitarity applied at the form factor level, and the compact expressions for supersymmetric tree-level form factors and amplitudes entering the cuts. We confirm that infrared divergences exponentiate as expected, and that collinear factorisation is entirely captured by an ABDK/BDS ansatz. Next, we construct the two-loop remainder function obtained by subtracting this ansatz from the full two-loop form factor and compute it numerically. Using symbology, combined with various physical constraints and symmetries, we find a unique solution for its symbol. With this input we construct a remarkably compact analytic expression for the remainder function, which contains only classical polylogarithms, and compare it to our numerical results. Furthermore, we make the surprising observation that our remainder is equal to the maximally transcendental piece of a closely related two-loop amplitude in QCD.
We show how to calculate the one-loop scattering amplitude with all gluons of negative helicity i... more We show how to calculate the one-loop scattering amplitude with all gluons of negative helicity in non-supersymmetric Yang-Mills theory using MHV diagrams. We argue that the amplitude with all positive helicity gluons arises from a Jacobian which occurs when one performs a Bäcklund-type holomorphic change of variables in the lightcone Yang-Mills Lagrangian. This also results in contributions to scattering amplitudes from violations of the equivalence theorem. Furthermore, we discuss how the one-loop amplitudes with a single positive or negative helicity gluon arise in this formalism. Perturbation theory in the new variables leads to a hybrid of MHV diagrams and lightcone Yang-Mills theory.
We exploit a recently found connection between special triple-cut diagrams and tree-level recursi... more We exploit a recently found connection between special triple-cut diagrams and tree-level recursive diagrams to derive a general formula capturing the multi-particle factorisation of arbitrary one-loop amplitudes in the ABJM theory. This formula contains certain anomalous contributions which are reminiscent of the so-called non-factorising contributions appearing in the factorisation of one-loop amplitudes in four-dimensional gauge theory. In the second part of the paper we derive a recursion relation for the supercoefficients of one-loop amplitudes in ABJM theory. By applying this recursion relation, any one-loop supercoefficient can be reduced to special triple-cut diagrams involving at least one four-point tree amplitude. In turn, this implies that any one-loop supercoefficient can be derived from tree-level recursive diagrams.
We propose a new, twistor string theory inspired formalism to calculate loop amplitudes in N = 4 ... more We propose a new, twistor string theory inspired formalism to calculate loop amplitudes in N = 4 super Yang-Mills theory. In this approach, maximal helicity violating (MHV) tree amplitudes of N = 4 super Yang-Mills are used as vertices, using an offshell prescription introduced by Cachazo, Svrcek and Witten, and combined into effective diagrams that incorporate large numbers of conventional Feynman diagrams. As an example, we apply this formalism to the particular class of supersymmetric MHV one-loop scattering amplitudes with an arbitrary number of external legs in N = 4 super Yang-Mills. Remarkably, our approach naturally leads to a representation of the amplitudes as dispersion integrals, which we evaluate exactly. This yields a new, simplified form for the MHV amplitudes, which is equivalent to the expressions obtained previously by Bern, Dixon, Dunbar and Kosower using the cut-constructibility approach.
We use the recently proposed supergravity approach to large N gauge theories to calculate ordinar... more We use the recently proposed supergravity approach to large N gauge theories to calculate ordinary and spatial Wilson loops of gauge theories in various dimensions. In this framework we observe an area law for spatial Wilson loops in four and five dimensional supersymmetric Yang-Mills at finite temperature. This can be interpreted as the area law of ordinary Wilson loops in three and four dimensional non-supersymmetric gauge theories at zero temperature which indicates confinement in these theories. Furthermore, we show that super Yang Mills theories with 16 supersymmetries at finite temperature do not admit phase transitions between the weakly coupled super Yang Mills and supergravity regimes. This result is derived by analyzing the entropy and specific heat of those systems as well as by computing ordinary Wilson loops at finite temperature. The calculation of the entropy was carried out in all different regimes and indicates that there is no first order phase transition in these systems. For the same theories at zero temperature we also compute the dependence of the quark anti-quark potential on the separating distance.
We propose a connected prescription formula in twistor space for all tree-level form factors of t... more We propose a connected prescription formula in twistor space for all tree-level form factors of the stress tensor multiplet operator in N = 4 super Yang-Mills, which is a generalisation of the expression of Roiban, Spradlin and Volovich for superamplitudes. By introducing link variables, we show that our formula is identical to the recently proposed four-dimensional scattering equations for form factors. Similarly to the case of amplitudes, the link representation of form factors is shown to be directly related to BCFW recursion relations, and is considerably more tractable than the scattering equations. We also discuss how our results are related to a recent Grassmannian formulation of form factors, and comment on a possible derivation of our formula from ambitwistor strings.
We apply MHV diagrams to the derivation of the one-loop dilatation operator of N = 4 super Yang-M... more We apply MHV diagrams to the derivation of the one-loop dilatation operator of N = 4 super Yang-Mills in the SO(6) sector. We find that in this approach the calculation reduces to the evaluation of a single MHV diagram in dimensional regularisation. This provides the first application of MHV diagrams to an off-shell quantity. We also discuss other applications of the method and future directions. §
We apply MHV diagrams to the derivation of the one-loop dilatation operator of N=4 super Yang-Mil... more We apply MHV diagrams to the derivation of the one-loop dilatation operator of N=4 super Yang-Mills in the SO(6) sector. We find that in this approach the calculation reduces to the evaluation of a single MHV diagram in dimensional regularisation. This provides the first application of MHV diagrams to an off-shell quantity. We also discuss other applications of the method and future directions.
We quantize the three-dimensional BF-model using axial gauge conditions. Exploiting the rich symm... more We quantize the three-dimensional BF-model using axial gauge conditions. Exploiting the rich symmetry-structure of the model we show that the Green-functions correspond to tree graphs and can be obtained as the unique solution of the Ward-Identities. Furthermore, we will show that the theory can be uniquely determined by symmetry considerations without the need of an action principle.
The Green functions of the Chern-Simons theory quantized in the axial gauge are shown to be calcu... more The Green functions of the Chern-Simons theory quantized in the axial gauge are shown to be calculable as the unique, exact solution of the Ward identities which express the invariance of the theory under the topological supersymmetry of Delduc, Gieres and Sorella. 1 See also [5] for more general noncovariant gauges. 2 Concerning the physical irrelevance of topological supersymmetry we disagree with the authors of Ref. [10] 3 The Lagrange multiplier fields are used for the implementation of the gauge condition.
We consider Type 0B D3-branes placed at conical singularities and analyze in detail the conifold ... more We consider Type 0B D3-branes placed at conical singularities and analyze in detail the conifold singularity. We study the non supersymmetric gauge theories on their worldvolume and their conjectured dual gravity descriptions. In the ultraviolet the solutions exhibit a logarithmic running of the gauge coupling. In the infrared we find confining solutions and IR fixed points.
We construct several new G 2 holonomy metrics that play an important role in recent studies of ge... more We construct several new G 2 holonomy metrics that play an important role in recent studies of geometrical transitions in compactifications of M-theory to four dimensions. In type IIA string theory these metrics correspond to D6 branes wrapped on the threecycle of the deformed conifold and the resolved conifold with two-form RR flux on the blown-up two-sphere, which are related by a conifold transition. We also study a G 2 metric that is related in type IIA to the line bundle over S 2 × S 2 with RR two-form flux. Our approach exploits systematically the definition of torsion-free G 2 structures in terms of three-forms which are closed and co-closed. Besides being an elegant formalism this turns out to be a practical tool to construct G 2 holonomy metrics.
We propose a method to compute the scattering angle for classical black hole scattering directly ... more We propose a method to compute the scattering angle for classical black hole scattering directly from two massive particle irreducible diagrams in a heavy-mass effective field theory approach to general relativity, without the need of subtracting iteration terms. The amplitudes in this effective theory are constructed using a recently proposed novel colour-kinematic/double copy for tree-level two-scalar, multi-graviton amplitudes, where the BCJ numerators are gauge invariant and local with respect to the massless gravitons. These tree amplitudes, together with graviton tree amplitudes, enter the construction of the requiredD-dimensional loop integrands and allow for a direct extraction of contributions relevant for classical physics. In particular the soft/heavy-mass expansions of full integrands is circumvented, and all iterating contributions can be dropped from the get go. We use this method to compute the scattering angle up to third post-Minkowskian order in four dimensions, in...
We study celestial amplitudes in (super) Yang-Mills theory using a parameterisation of the spinor... more We study celestial amplitudes in (super) Yang-Mills theory using a parameterisation of the spinor helicity variables where their overall phase is not fixed by the little group action. In this approach the spin constraint h −h = J for celestial conformal primaries emerges naturally from a new Mellin transform, and the action of conformal transformations on celestial amplitudes is derived. Applying this approach to N = 4 super Yang-Mills, we show how the appropriate definition of on-shell superspace coordinates leads naturally to a formulation of chiral celestial superamplitudes and a representation of the generators of the four-dimensional superconformal algebra on the celestial sphere, which by construction annihilate all tree-level celestial superamplitudes.
Using modern amplitude techniques we compute the leading classical and quantum corrections to the... more Using modern amplitude techniques we compute the leading classical and quantum corrections to the gravitational potential between two massive scalars induced by adding cubic terms to Einstein gravity. We then study the scattering of massless scalars, photons and gravitons off a heavy scalar in the presence of the same R 3 deformations, and determine the bending angle in the three cases from the non-analytic component of the scattering amplitude. Similarly to the Einstein-Hilbert case, we find that the classical contribution to the bending angle is universal, but unlike that case, universality is preserved also by the first quantum correction. Finally we extend our analysis to include a deformation of the form ΦR 2 , where Φ is the dilaton, which arises in the low-energy effective action of the bosonic string in addition to the R 3 term, and compute its effect on the graviton bending.
We present a systematic procedure to compute complete, analytic form factors of gauge-invariant o... more We present a systematic procedure to compute complete, analytic form factors of gauge-invariant operators at loop level in pure Yang-Mills. We consider applications to operators of the form Tr F n where F is the gluon field strength. Our approach is based on an extension to form factors of the dimensional reconstruction technique, in conjunction with the six-dimensional spinor-helicity formalism and generalised unitarity. For form factors this technique requires the introduction of additional scalar operators, for which we provide a systematic prescription. We also discuss a generalisation of dimensional reconstruction to any number of loops, both for amplitudes and form factors. Several novel results for one-loop minimal and non-minimal form factors of Tr F n with n > 2 are presented. Finally, we describe the Mathematica package SpinorHelicity6D, which is tailored to handle six-dimensional quantities written in the spinorhelicity formalism.
Form factors of the stress-tensor multiplet operator in $$ \mathcal{N}=4 $$ N = 4 supersymmetric ... more Form factors of the stress-tensor multiplet operator in $$ \mathcal{N}=4 $$ N = 4 supersymmetric Yang-Mills reveal surprisingly simple structures similar to those appearing in scattering amplitudes. In this paper we show that, as for the case of amplitudes, they also enjoy dual conformal symmetry. We compute the dual conformal anomaly at one loop for an arbitrary number of particles and generic helicities, which matches the expression of the dual conformal anomaly of amplitudes, and perform explicit checks for MHV and NMHV one-loop form factors. In the NMHV case the realisation of dual conformal symmetry requires a delicate cancellation of offending terms arising from three-mass triangles, which we explicitly check in the case of the four-point NMHV form factor.
It is known that the Yangian of P SU (2, 2|4) is a symmetry of the tree-level S-matrix of N = 4 s... more It is known that the Yangian of P SU (2, 2|4) is a symmetry of the tree-level S-matrix of N = 4 super Yang-Mills. On the other hand, the complete one-loop dilatation operator in the same theory commutes with the level-one Yangian generators only up to certain boundary terms found by Dolan, Nappi and Witten. Using a result by Zwiebel, we show how the Yangian symmetry of the tree-level S-matrix of N = 4 super Yang-Mills implies precisely the Yangian invariance, up to boundary terms, of the one-loop dilatation operator.
We derive a compact expression for the three-point MHV form factors of half-BPS operators in N = ... more We derive a compact expression for the three-point MHV form factors of half-BPS operators in N = 4 super Yang-Mills at two loops. The main tools of our calculation are generalised unitarity applied at the form factor level, and the compact expressions for supersymmetric tree-level form factors and amplitudes entering the cuts. We confirm that infrared divergences exponentiate as expected, and that collinear factorisation is entirely captured by an ABDK/BDS ansatz. Next, we construct the two-loop remainder function obtained by subtracting this ansatz from the full two-loop form factor and compute it numerically. Using symbology, combined with various physical constraints and symmetries, we find a unique solution for its symbol. With this input we construct a remarkably compact analytic expression for the remainder function, which contains only classical polylogarithms, and compare it to our numerical results. Furthermore, we make the surprising observation that our remainder is equal to the maximally transcendental piece of a closely related two-loop amplitude in QCD.
We show how to calculate the one-loop scattering amplitude with all gluons of negative helicity i... more We show how to calculate the one-loop scattering amplitude with all gluons of negative helicity in non-supersymmetric Yang-Mills theory using MHV diagrams. We argue that the amplitude with all positive helicity gluons arises from a Jacobian which occurs when one performs a Bäcklund-type holomorphic change of variables in the lightcone Yang-Mills Lagrangian. This also results in contributions to scattering amplitudes from violations of the equivalence theorem. Furthermore, we discuss how the one-loop amplitudes with a single positive or negative helicity gluon arise in this formalism. Perturbation theory in the new variables leads to a hybrid of MHV diagrams and lightcone Yang-Mills theory.
We exploit a recently found connection between special triple-cut diagrams and tree-level recursi... more We exploit a recently found connection between special triple-cut diagrams and tree-level recursive diagrams to derive a general formula capturing the multi-particle factorisation of arbitrary one-loop amplitudes in the ABJM theory. This formula contains certain anomalous contributions which are reminiscent of the so-called non-factorising contributions appearing in the factorisation of one-loop amplitudes in four-dimensional gauge theory. In the second part of the paper we derive a recursion relation for the supercoefficients of one-loop amplitudes in ABJM theory. By applying this recursion relation, any one-loop supercoefficient can be reduced to special triple-cut diagrams involving at least one four-point tree amplitude. In turn, this implies that any one-loop supercoefficient can be derived from tree-level recursive diagrams.
We propose a new, twistor string theory inspired formalism to calculate loop amplitudes in N = 4 ... more We propose a new, twistor string theory inspired formalism to calculate loop amplitudes in N = 4 super Yang-Mills theory. In this approach, maximal helicity violating (MHV) tree amplitudes of N = 4 super Yang-Mills are used as vertices, using an offshell prescription introduced by Cachazo, Svrcek and Witten, and combined into effective diagrams that incorporate large numbers of conventional Feynman diagrams. As an example, we apply this formalism to the particular class of supersymmetric MHV one-loop scattering amplitudes with an arbitrary number of external legs in N = 4 super Yang-Mills. Remarkably, our approach naturally leads to a representation of the amplitudes as dispersion integrals, which we evaluate exactly. This yields a new, simplified form for the MHV amplitudes, which is equivalent to the expressions obtained previously by Bern, Dixon, Dunbar and Kosower using the cut-constructibility approach.
We use the recently proposed supergravity approach to large N gauge theories to calculate ordinar... more We use the recently proposed supergravity approach to large N gauge theories to calculate ordinary and spatial Wilson loops of gauge theories in various dimensions. In this framework we observe an area law for spatial Wilson loops in four and five dimensional supersymmetric Yang-Mills at finite temperature. This can be interpreted as the area law of ordinary Wilson loops in three and four dimensional non-supersymmetric gauge theories at zero temperature which indicates confinement in these theories. Furthermore, we show that super Yang Mills theories with 16 supersymmetries at finite temperature do not admit phase transitions between the weakly coupled super Yang Mills and supergravity regimes. This result is derived by analyzing the entropy and specific heat of those systems as well as by computing ordinary Wilson loops at finite temperature. The calculation of the entropy was carried out in all different regimes and indicates that there is no first order phase transition in these systems. For the same theories at zero temperature we also compute the dependence of the quark anti-quark potential on the separating distance.
We propose a connected prescription formula in twistor space for all tree-level form factors of t... more We propose a connected prescription formula in twistor space for all tree-level form factors of the stress tensor multiplet operator in N = 4 super Yang-Mills, which is a generalisation of the expression of Roiban, Spradlin and Volovich for superamplitudes. By introducing link variables, we show that our formula is identical to the recently proposed four-dimensional scattering equations for form factors. Similarly to the case of amplitudes, the link representation of form factors is shown to be directly related to BCFW recursion relations, and is considerably more tractable than the scattering equations. We also discuss how our results are related to a recent Grassmannian formulation of form factors, and comment on a possible derivation of our formula from ambitwistor strings.
We apply MHV diagrams to the derivation of the one-loop dilatation operator of N = 4 super Yang-M... more We apply MHV diagrams to the derivation of the one-loop dilatation operator of N = 4 super Yang-Mills in the SO(6) sector. We find that in this approach the calculation reduces to the evaluation of a single MHV diagram in dimensional regularisation. This provides the first application of MHV diagrams to an off-shell quantity. We also discuss other applications of the method and future directions. §
We apply MHV diagrams to the derivation of the one-loop dilatation operator of N=4 super Yang-Mil... more We apply MHV diagrams to the derivation of the one-loop dilatation operator of N=4 super Yang-Mills in the SO(6) sector. We find that in this approach the calculation reduces to the evaluation of a single MHV diagram in dimensional regularisation. This provides the first application of MHV diagrams to an off-shell quantity. We also discuss other applications of the method and future directions.
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Papers by A. Brandhuber