We evaluate ABJM observables at two loops, for any value of the rank N of the gauge group. We compute the color subleading contributions to the four-point scattering amplitude in ABJM at two loops. Contrary to the four dimensional case, IR divergent N-subleading contributions are proportional to leading poles in the regularization parameter. We then exploit the non-planar calculation for the amplitude to derive an expression for the two-loop Sudakov form factor at any N. In the planar limit the result coincides with the one recently obtained in literature by using Feynman diagrams and unitarity. Finally, we analyze the subleading contributions to the light-like four-cusps Wilson loop and interpret the result in terms of the non-abelian exponentiation theorem. All these perturbative results satisfy the uniform transcendentality principle, hinting at its validity in ABJM beyond the planar limit.
We report progress in computing and analyzing all tree amplitudes in ABJM theory. Inspired by the isomorphism between the orthogonal Grassmannian and the pure spinor geometries, we adopt a new gauge, called u-gauge, for evaluating the orthogonal Grassmannian integral for ABJM amplitudes. We carry out the integral explicitly for the 8-point amplitude and obtain the complete supersymmetric amplitude. The physical and spurious poles arise from the integral as expected from on-shell diagrams. We also derive a double scalar soft theorem of ABJM amplitudes and verify it for known amplitudes.
We compute the simplest one-loop planar amplitudes in Higgsed ABJM theory at a generic point of the moduli space. We explicitly check that they can be expressed in terms of integrals which are invariant under dual conformal symmetry involving masses, in a similar fashion as in N=4 SYM.
We derive an exact algebraic identity between the two-loop four-point amplitude in ABJM theory and the corresponding one-loop amplitude in N=4 SYM theory. This identity generalizes previous partial results to an exact relation valid at all orders in the IR regulator. Moreover, it allows to conjecture an exact iterative expression for the complete three dimensional amplitude in terms of the BDS ansatz for the four dimensional one, indicating that the strict relation between the two amplitudes experimented at two loops might propagate to all orders. In particular, an almost complete expression f…
We study the three-loop four-point amplitude in ABJM theory. We determine the dual conformal invariant integrals with highest number of propagators and fix their coefficients by two-particle cuts. Evaluating such a combination of integrals in dimensional regularization we provide evidence for exponentiation of the amplitude, including the finite terms. In addition we show that the three-loop amplitude can be expressed in terms of classical polylogarithms of uniform degree of transcendentality.
We study Yangian-invariant deformations of scattering amplitudes in 4d N=4 supersymmetric Yang-Mills theory and 3d N=6 Aharony-Bergman-Jafferis-Maldacena (ABJM) theory. In particular, we obtain the deformed Grassmannian integral for 4d N=4 super Yang-Mills theory, both in momentum and momentum-twistor space. For 3d ABJM theory, we initiate the study of deformed scattering amplitudes. We investigate general deformations of on-shell diagrams, and find the deformed Grassmannian integral for this theory. We furthermore introduce the algebraic R-matrix construction of deformed Yangian invariants fo…
In this paper we present the first analytic computation of the six-point two-loop amplitude of ABJM theory. We show that the two-loop amplitude consist of corrections proportional to two distinct local Yangian invariants which can be identified as the tree- and the one-loop amplitude respectively. The two-loop correction proportional to the tree-amplitude is identical to the one-loop BDS result of N=4 SYM plus an additional remainder function, while the correction proportional to the one-loop amplitude is finite. Both the remainder and the finite correction are dual conformal invariant, which …
In this paper, we introduce the momentum space amplituhedron for tree-level scattering amplitudes of ABJM theory. We demonstrate that the scattering amplitude can be identified as the canonical form on the space given by the product of positive orthogonal Grassmannian and the moment curve. The co-dimension one boundaries of this space are simply the odd-particle planar Mandelstam variables, while the even-particle counterparts are "hidden" as higher co-dimension boundaries. Remarkably, this space can be equally defined through a series of "sign flip" requirements of the projected external data…
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study and ICTP-AP, UCAS（杭州高等研究院基础物理与数学学院，ICTP-AP，中国科学院大学）；Peng Huanwu Center for Fundamental Theory（彭桓武理论物理中心）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Department of Physics, Astronomy and Mathematics, University of Hertfordshire（赫特福德大学物理、天文与数学系）
A novel understanding of scattering amplitudes in terms of on-shell diagrams and positive Grassmannian has been recently established for four dimensional Yang-Mills theories and three dimensional Chern-Simons theories of ABJM type. We give a detailed construction of the positroid stratification of orthogonal Grassmannian relevant for ABJM amplitudes. On-shell diagrams are classified by pairing of external particles. We introduce a combinatorial aid called `OG tableaux' and map each equivalence class of on-shell diagrams to a unique tableau. The on-shell diagrams related to each other through B…
We explicitly show that the Bern-Carrasco-Johansson color-kinematic duality holds at tree level through at least eight points in Aharony-Bergman-Jafferis-Maldacena theory with gauge group SU(N) x SU(N). At six points we give the explicit form of numerators in terms of amplitudes, displaying the generalized gauge freedom that leads to amplitude relations. However, at eight points no amplitude relations follow from the duality, so the diagram numerators are fixed unique functions of partial amplitudes. We provide the explicit amplitude-numerator decomposition and the numerator relations for eigh…
For N=6 superconformal Chern-Simons-matter theories in three dimensions, by a direct superspace Feynman diagram approach, we compute the two-loop four-point scattering amplitude with external chiral matter fields. We find that the result is in perfect agreement with the two-loop result for a light-like four-polygon Wilson loop. This is a nontrivial evidence of the scattering amplitudes/Wilson loop duality in three dimensions. Moreover, both the IR divergent and the finite parts of our two-loop result agree with a BDS-like ansatz for all-loop amplitudes where the scaling function is given in te…
For three dimensional N=6 superconformal field theories we compute one-loop scattering amplitudes for any number of external particles. We focus on a particular subsector of N=2 invariant superamplitudes for which the ordinary perturbative evaluation becomes very easy. The result we obtain is in general non-vanishing. For six external particles our findings are sufficient for determining the complete expression of the N=6 superamplitude at this order. We discuss the symmetries of the result and its anomalous variation under superconformal generators.
A remarkable connection between perturbative scattering amplitudes of four-dimensional planar SYM, and the stratification of the positive grassmannian, was revealed in the seminal work of Arkani-Hamed et. al. Similar extension for three-dimensional ABJM theory was proposed. Here we establish a direct connection between planar scattering amplitudes of ABJM theory, and singularities there of, to the stratification of the positive orthogonal grassmannian. In particular, scattering processes are constructed through on-shell diagrams, which are simply iterative gluing of the fundamental four-point …
We propose a recursion relation for tree-level scattering amplitudes in three-dimensional Chern-Simons-matter theories. The recursion relation involves a complex deformation of momenta which generalizes the BCFW-deformation used in higher dimensions. Using background field methods, we show that all tree-level superamplitudes of the ABJM theory vanish for large deformations, establishing the validity of the recursion formula. Furthermore, we use the recursion relation to compute six-point and eight-point component amplitudes and match them with independent computations based on Feynman diagrams…
In this paper, we present the two-loop correction to scattering amplitudes in three-dimensional $\mathcal{N}=6$ Chern-Simons matter theory. We use eight-point case as our main example, but the method generalizes to all multiplicities. The integrand is completely fixed by dual conformal symmetry, maximal cuts, constraints from soft-collinear behavior and from vanishing of odd-multiplicity amplitudes. After performing integrations with Higgs regularizations, the integrated results demonstrate that the infrared divergence is again identical to that of ${\cal N}=4$ super Yang-Mills. After subtract…
We use unitarity techniques to compute the two-loop non-planar corrections to the Sudakov form factor and the four-point amplitude in ABJM theory. We start by reconstructing non-planar integrals from two-particle cuts in three dimensions. This causes ambiguities, due to the one-loop four-point amplitude being subleading in dimensional regularization. We provide a prescription to circumvent them and show that it leads to the correct results, as checked against the recent Feynman diagram computation. For the amplitude we point out an alternative basis of integrals, including a non-planar double-…
We compute the two-loop Sudakov form factor in three-dimensional N=6 superconformal Chern-Simons theory, using generalised unitarity. As an intermediate step, we derive the non-planar part of the one-loop four-point amplitude in terms of box integrals. Our result for the Sudakov form factor is given by a single non-planar tensor integral with uniform degree of transcendentality, and is in agreement with the known infrared divergences of two-loop amplitudes in ABJM theory. We also discuss a number of interesting properties satisfied by related three-dimensional integral functions.
In this paper we study the combinatorics associated with the positive orthogonal Grassmannian OG_k and its connection to ABJM scattering amplitudes. We present a canonical embedding of OG_k into the Grassmannian Gr(k,2k), from which we deduce the canonical volume form that is invariant under equivalence moves. Remarkably the canonical forms of all reducible graphs can be converted into irreducible ones with products of dLog forms. Unlike N=4 super Yang-Mills, here the Jacobian plays a crucial role to ensure the dLog form of the reduced representation. Furthermore, we identify the functional ma…
We show that by studying the flat spacetime limit of the Mellin amplitude associated with the four-point correlation function of scalar operators in the stress tensor multiplet of ABJM theory, one can produce the momentum expansion of the M-theory four-graviton S-matrix elements. Using CFT data previously obtained from the supersymmetric localization method, we carry out this procedure explicitly to the second nontrivial order in the momentum expansion, and recover precisely the known $R^4$ contribution to the scattering amplitude of super-gravitons in M-theory in eleven dimensions.
In previous investigations, it was found that four-sided polygonal light-like Wilson loops in ABJM theory calculated to two-loop order have the same form as the corresponding Wilson loop in N = 4 SYM at one-loop order. Here we study light-like polygonal Wilson loops with n cusps in planar three-dimensional Chern-Simons and ABJM theory to two loops. Remarkably, the result in ABJM theory precisely agrees with the corresponding Wilson loop in N = 4 SYM at one-loop order for arbitrary n. In particular, anomalous conformal Ward identites allow for a so-called remainder function of conformal cross r…
We revisit the computation of the two-loop light-like tetragonal Wilson loop for three dimensional pure Chern-Simons and N=6 Chern-Simons-matter theory, within dimensional regularization with dimensional reduction scheme. Our examination shows that, contrary to prior belief, the result respects maximal transcendentality as is the case of the four-point scattering amplitude of the theory. Remarkably, the corrected result matches exactly the scattering amplitude both in the divergent and in the finite parts, constants included.
In this paper we analyse the ABJM theory in ${\cal N}=1$ superspace. Firstly we study the linear and non-linear BRST transformations for the ABJM theory. Then we derive the finite field dependent version of these BRST (FFBRST) transformations. Further we show that such FFBRST transformations relate the generating functional in linear gauge to the generating functional in the non-linear gauge of ABJM theory.
There is a number of indications that scattering amplitudes in the Aharony-Bergman-Jafferis-Maldacena theory might have a dual description in terms of a holonomy of a supergauge connection on a null polygonal contour in a way analogous to the four-dimensional maximally supersymmetric Yang-Mills theory. However, so far its explicit implementations evaded a successful completion. The difficulty is intimately tied to the lack of the T-self-duality of the sigma model on the string side of the gauge/string correspondence. Unscathed by the last misfortune, we initiate with this study an application …
Scattering amplitudes in 4d $\mathcal{N}=4$ super Yang-Mills theory (SYM) can be described by Grassmannian contour integrals whose form depends on whether the external data is encoded in momentum space, twistor space, or momentum twistor space. After a pedagogical review, we present a new, streamlined proof of the equivalence of the three integral formulations. A similar strategy allows us to derive a new Grassmannian integral for 3d $\mathcal{N}=6$ ABJM theory amplitudes in momentum twistor space: it is a contour integral in an orthogonal Grassmannian with the novel property that the internal…
Supersymmetry allows a $D^4 R^4$ interaction in M-theory, but such an interaction is inconsistent with string theory dualities and so is known to be absent. We provide a novel proof of the absence of the $D^4 R^4$ M-theory interaction by calculating 4-point scattering amplitudes of 11d supergravitons from ABJM theory. This calculation extends a previous calculation performed to the order corresponding to the $R^4$ interaction. The new ingredient in this extension is the interpretation of the fourth derivative of the mass deformed $S^3$ partition function of ABJM theory, which can be determined…
We calculate fermionic response in domain wall backgrounds of four-dimensional gauged supergravity interpolating between distinct stable AdS vacua. The backgrounds, found by Bobev et al., are holographically dual to zero-temperature states of ABJM theory at finite density for monopole charge and are similar to zero-temperature limits of holographic superconductors, but with a symmetry-breaking source as well. The condensed scalar mixes charged and neutral fields dual to composite fermionic operators in the Dirac equations. Both gapped and gapless bands of stable quasiparticles are found.
In 2008, Aharony, Bergman, Jafferis, and Maldacena (ABJM) discovered a three-dimensional Chern-Simons theory with N = 6 supersymmetry and conjectured that in a certain limit, this theory is dual to type IIA string theory on AdS4xCP3. Since then, a great deal of evidence has been accumulated which suggests that the ABJM theory is integrable in the planar limit. Integrability is a very useful property that allows many physical observables, such as anomalous dimensions and scattering amplitudes, to be computed efficiently. In the first half of this thesis, we will explain how to use integrabilty …
We introduce ABJM quantum field theory in the noncommutative spacetime by using the component formalism and show that it is N=6 supersymmetric. For the U(1)_κ x U(1)_{-κ} case, we compute all one-loop 1PI two and three point functions in the Landau gauge and show that they are UV finite and have well-defined commutative limits theta^{μν} -> 0, corresponding exactly to the 1PI functions of the ordinary ABJM field theory. This result also holds for all one-loop functions which are UV finite by power counting. It seems that the quantum noncommutative ABJM field theory is free from the noncommutat…
We study remarkable connections between twistor-string formulas for tree amplitudes in ${\cal N}=4$ SYM and ${\cal N}=6$ ABJM, and the corresponding momentum amplituhedron in the kinematic space of $D=4$ and $D=3$, respectively. Based on the Veronese map to positive Grassmannians, we define a twistor-string map from $G_{+}(2,n)$ to a $(2n{-}4)$-dimensional subspace of the 4d kinematic space where the momentum amplituhedron of SYM lives. We provide strong evidence that the twistor-string map is a diffeomorphism from $G_+(2,n)$ to the interior of momentum amplituhedron; the canonical form of the…
We apply the conjecture of arXiv:2111.06903 for gravitational building blocks to the effective supergravity description of M-theory on S$^7/\mathbb{Z}_k$. Utilizing known localization results for the holographically dual ABJM theory, we determine a complete tower of higher derivative corrections to the AdS$_4$ supergravity and a further set of quantum corrections. This uniquely fixes the gravitational block, leading to holographic predictions for a number of exact ABJM observables, excluding only constant and non-perturbative corrections in the gauge group rank $N$. The predicted S$^3$ partiti…
We study light-like polygonal Wilson loops in three-dimensional Chern-Simons and ABJM theory to two-loop order. For both theories we demonstrate that the one-loop contribution to these correlators cancels. For pure Chern-Simons, we find that specific UV divergences arise from diagrams involving two cusps, implying the loss of finiteness and topological invariance at two-loop order. Studying those UV divergences we derive anomalous conformal Ward identities for n-cusped Wilson loops which restrict the finite part of the latter to conformally invariant functions. We also compute the four-cusp Wi…
In this paper, using the localization technique we analyze the large N limit of the mass deformed Aharony-Bergman-Jafferis-Maldacena (ABJM) theory on the three sphere with a finite mass parameter and finite Chern-Simons levels. We find two different solutions of the saddle point equations in the large N limit. With these solutions we compute the free energy and find that there is a first order phase transition. Our results may predict a phase transition in the dual gravity theory.
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 r…
We establish a connection between tree-level superamplitudes in ABJM theory and leading singularities associated to special three-particle cuts of one-loop superamplitudes where one of the tree amplitudes entering the cut is a four-point amplitude. Using these relations, we show that certain intriguing similarities between one-loop and tree-level superamplitudes observed recently become completely manifest. This connection is reminiscent of a similar relation in the maximally supersymmetric gauge theory in four dimensions, where the sum of two-mass hard and one-mass box coefficients of a one-l…
To study thermal aspects of the ABJM theory in the strongly coupled regime, we carry out the CP3 invariant dimensional reduction of the type IIA supergravity down to four dimensions. We then investigate zero and finite temperature responses of the operators which are dual to the AdS scalar and vector fields. Two scalar operators are shown to have finite-temperature condensations by coupling of constant source term. The currents dual to the massless and massive gauge fields are not induced by coupling of constant boundary vector potential, which implies that the phase described by black brane b…
Yau Mathematical Sciences Center (YMSC), Tsinghua University（清华大学丘成桐数学科学中心）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京国际数学研究中心）；Peng Huanwu Center for Fundamental Theory（彭桓武理论物理中心）；Department of Mathematical Sciences, Tsinghua University（清华大学数学科学系）；Jefferson Physical Laboratory, Harvard University（哈佛大学杰斐逊物理实验室）
Using an effective vertex method we explicitly derive the two-loop dilatation generator of ABJM theory in its SU(2)xSU(2) sector, including all non-planar corrections. Subsequently, we apply this generator to a series of finite length operators as well as to two different types of BMN operators. As in N=4 SYM, at the planar level the finite length operators are found to exhibit a degeneracy between certain pairs of operators with opposite parity - a degeneracy which can be attributed to the existence of an extra conserved charge and thus to the integrability of the planar theory. When non-plan…
We study non-analytic behavior in the static charge susceptibility in finite density states of the ABJM theory using its holographic dual. Emphasis is placed on a particular state characterized by vanishing entropy density at zero temperature, and Fermi surface-like singularities in various fermionic correlation functions. The susceptibility exhibits branch points in the complex momentum plane, with a real part quantitatively very similar to the location of the Fermi surface singularities.
I consider three-point functions of twist-one operators in ABJM at weak coupling. I compute the structure constant of correlators involving one twist-one un-protected operator and two protected ones for a few finite values of the spin, up to two-loop order. As an application I enforce a limit on the gauge group ranks, in which I relate the structure constant for three chiral primary operators to the expectation value of a supersymmetric Wilson loop. Such a relation is then used to perform a successful five-loop test on the matrix model conjectured to describe the supersymmetric Wilson loop.
The partition function of the ABJM theory receives non-perturbative corrections due to instanton effects. We study these non-perturbative corrections, including bound states of worldsheet instantons and membrane instantons, in the Fermi-gas approach. We require that the total non-perturbative correction should be always finite for arbitrary Chern-Simons level. This finiteness is realized quite non-trivially because each bound state contribution naively diverges at some levels. The poles of each contribution should be canceled out in total. We use this pole cancellation mechanism to find unknow…
We add massive fundamental matter to the ABJM model by adding D6-branes wrapped asymptotically over RP3. We find two types of solutions at finite temperature, one that enters the black hole and one that ends before the black hole. We analyze the behavior of the free energy as a function of temperature, and find that the system exhibits a phase transition between the two types of solutions, similar to what happens in the D3-D7 system. We also analyze the meson spectrum in the model and find several massive scalar modes, again, quite like the D3-D7 system. We end with a calculation of the conduc…
We analyze orbifolds with discrete torsion of the ABJM theory by a finite subgroup $Γ$ of $SU(2)\times SU(2)$ . Discrete torsion is implemented by twisting the crossed product algebra resulting after orbifolding. It is shown that, in general, the order $m$ of the cocycle we chose to twist the algebra by enters in a non trivial way in the moduli space. To be precise, the M-theory fiber is multiplied by a factor of $m$ in addition to the other effects that were found before in the literature. Therefore we got a $\mathbb{Z}_{\frac{k|Γ|}{m}}$ action on the fiber. We present a general analysis on h…
We consider the U(1) charged sector of ABJM theory at finite temperature, which corresponds to the Reissner-Nordstrom AdS black hole in the dual type IIA supergravity description. Including back-reaction to the bulk geometry, we show that phase transitions occur to a broken phase where SU(4) R-symmetry of the field theory is broken spontaneously by the condensation of dimension one or two operators. We show both numerically and analytically that the relevant critical exponents for the dimension one operator agree precisely with those of mean field theory in the strongly coupled regime of the l…
We use supertwistor space to construct scattering amplitudes of maximal superconformal theories in three and six dimensions. In both cases, the constraints of superconformal invariance and rationality imply that the three-point amplitude vanishes on-shell, which constrains the four-point amplitude to have vanishing residues in all channels. In three dimensions, we find a unique solution for the four-point amplitude and demonstrate that it agrees with the component result in the BLG theory. This suggests that BLG is the unique three-dimensional theory with classical OSp(8|4) symmetry that admit…
We compute the null cusp anomalous dimension of ABJM theory at strong coupling up to two-loop order. This is done by evaluating corrections to the corresponding superstring partition function, weighted by the $AdS_4\times \mathbb{CP}^3$ action in AdS light-cone gauge. We compare our result, where we use an anomalous shift in the $AdS_4$ radius, with the cusp anomaly of $\mathcal{N}=4$ SYM, and extract the two-loop contribution to the non-trivial integrable coupling $h(λ)$ of ABJM theory. It coincides with the strong coupling expansion of the exact expression for $ h(λ)$ recently conjectured by…
We compute the four-point amplitude in 3d N=8 SYM at two loops, by solving the three dimensional scalar doublebox in dimensional regularization. We compare it to the same result in the ABJM theory, to which maximal SYM should flow in the infrared at strong coupling. After proper rescalings, we find that the two amplitudes coincide in the Regge limit.
