We examine the three-dimensional N = 6 superconformal Chern--Simons theory with U(N) X U(N) gauge symmetry, which was recently constructed by Aharony, Bergman, Jafferis, and Maldacena (ABJM). Using a formulation with manifest SU(4) R-symmetry and no auxiliary fields, we verify in complete detail both the Poincare supersymmetry and the conformal supersymmetry of the action. Together, these imply the complete OSp(6|4) superconformal symmetry of the theory. The potential, which is sixth order in scalar fields, is recast as a sum of squares.
The ABJM theory refers to superconformal Chern-Simons-matter theory with product gauge group U(L)xU(R) and level +k, -k, respectively. The theory is a candidate for worldvolume dynamics of M2-branes sitting at C(4)/Z(k)k. By utilizing monopole operators, we prove that ABJM theory gets enhanced N=8 supersymmetry and SO(8) R-symmetry at Chern-Simons levels k=1,2. We first show that the ABJM Lagrangian can be written in a manifestly SO(8) invariant form up to certain extra terms. We then show that upon integrating out Chern-Simons gauge fields these extra terms vanish precisely at levels k=1,2. U…
We analyze the quantum ABJM theory on ${\cal N} = 1$ superspace in different gauges. We study the Batalin-Vilkovisky (BV) formulation for this model. By developing field/antifield dependent BRST transformation we establish connection between the two different solutions of the quantum master equation within the BV formulation.
We construct the classical action of the Aharony-Bergman-Jafferis-Maldacena (ABJM) model in the N=3, d=3 harmonic superspace. In such a formulation three out of six supersymmetries are realized off shell while the other three mix the superfields and close on shell. The superfield action involves two hypermultiplet superfields in the bifundamental representation of the gauge group and two Chern-Simons gauge superfields corresponding to the left and right gauge groups. The N=3 superconformal invariance allows only for a minimal gauge interaction of the hypermultiplets. Amazingly, the correct sex…
We extend the hermitian three-algebra formulation of ABJM theory to include $U(1)$ factors. With attention payed to extra $U(1)$ factors, we refine the classification of $\N=6$ ABJM theories. We argue that essentially the only allowed gauge groups are $SU(N)\times SU(N)$, $U(N)\times U(M)$ and $Sp(N)\times U(1)$ and that we have only one independent Chern-Simons level in all these cases. Our argument is based on integrality of the $U(1)$ Chern-Simons levels and supersymmetry. A relation between monopole operators and Wilson lines in Chern-Simons theory suggests certain gauge representations of…
Studying the supersymmetry enhancement mechanism of Aharony, Bergman, Jafferis and Maldacena, we find a simple condition on the gauge group generators for the matter fields. We analyze all possible compact Lie groups and their representations. The only allowed gauge groups leading to the manifest N=6 supersymmetry are, up to discrete quotients, SU(n) x U(1), Sp(n) x U(1), SU(n) x SU(n), and SU(n) x SU(m) x U(1) with possibly additional U(1)'s. Matter representations are restricted to be the (bi)fundamentals. As a byproduct we obtain another proof of the complete classification of the three alg…
We consider general supersymmetric Wilson loops in ABJM model, i.e. Chern-Simons-matter theory in 2+1 dimensions with N=6 supersymmetry. They are so-called Zarembo-type: the Wilson loops of our interest have generic contours in spacetime, but the scalar field coupling is arranged accordingly so that there are unbroken supersymmetries. Based on the supermatrix formulation of Wilson loops by Drukker and Trancanelli, we construct explicitly 1/6-BPS Wilson loops and check that their expectation value is protected using perturbation up to two loops. We also study the dual string configuration in Ad…
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 construct a new NS5-brane solution in AdS_4 x CP^3 with S^2 x S^3 topology. This solution belongs to the general class of non-Einstein N_{11} metrics to which T^{1,1} belongs, and carries a non-vanishing D0-brane charge. In eleven dimensions it gives rise to a squashed S^2 x S^3 M5-brane giant graviton which is now of the N_{10} type. The energies of both solutions satisfy the BPS bound E=kQ/2, indicating supersymmetric configurations, where Q is interpreted as D0-brane charge for the NS5-brane and as angular momentum for the dynamically stable M5-brane giant graviton. The ground state is d…
We give a brief account of the recently proposed N=3 superfield formulation of the N=6, 3D superconformal theory of Aharony et al (ABJM) describing a low-energy limit of the system of multiple M2-branes on the AdS_4 x S^7/Z_k background. This formulation is given in harmonic N=3 superspace and reveals a number of surprising new features. In particular, the sextic scalar potential of ABJM arises at the on-shell component level as the result of eliminating appropriate auxiliary fields, while there is no any explicit superpotential at the off-shell superfield level.
The Lagrangian frame-like formulation of free higher spin symmetric bosonic AdS(d) fields is given within a manifestly sp(2) invariant framework. It is designed to deal with infinite multiplets of fields appearing as gauge connections of the higher spin algebras.
We propose a new formulation of the $D=3$ type II superstring which is manifestly invariant under both target-space $N=2$ supersymmetry and worldsheet $N=(1,1)$ super reparametrizations. This gives rise to a set of twistor (commuting spinor) variables, which provide a solution to the two Virasoro constraints. The worldsheet supergravity fields are shown to play the rôle of auxiliary fields.
We write explicit Lagrangian and supersymmetry transformation rules using the component fields in the ${\cal N}=2,3$ GT theories. In the component field expansion, the manifestation of an additional ${\cal N}=1$ supersymmetry is verified in the ${\cal N}=3$ GT theory. We find maximal supersymmetry preserving mass-deformation of the GT theories and their classical supersymmetric discrete vacua. Some interesting aspects of the set of discrete vacua are discussed in comparison with the ABJM case.
In $\mathcal N \geq 2$ superconformal Chern-Simons-matter theories we construct the infinite family of Bogomol'nyi-Prasad-Sommerfield (BPS) Wilson loops featured by constant parametric couplings to scalar and fermion matter, including both line Wilson loops in Minkowski spacetime and circle Wilson loops in Euclidean space. We find that the connection of the most general BPS Wilson loop cannot be decomposed in terms of double-node connections. Moreover, if the quiver contains triangles, it cannot be interpreted as a supermatrix inside a superalgebra. However, for particular choices of the param…
A superspace formulation of type II superstring background with manifest T-duality symmetry is presented. This manifestly T-dual formulation is constructed in a space spanned by two sets of nondegenerate super-Poincare algebra. Supertorsion constraints are obtained from consistency of the kappa-symmetric Virasoro constraints. All superconnections and vielbein fields are solved in terms of a prepotential which is one of the vielbein components. AdS5xS5 background is explained in this formulation.
We derive a manifestly duality-invariant formulation of the action principle for linearized gravity on anti de Sitter background. The analysis is based on the two-potential formalism, obtained upon resolution of the constraints in the Hamiltonian formulation. We discuss the relevance of our result in the context of holography.
We propose a new formulation of the heterotic $D=10$ Green-Schwarz superstring whose worldsheet is a superspace with two even and eight odd coordinates. The action is manifestly invariant under both target-space supersymmetry and a worldsheet reparametrisation supergroup. It contains only a finite set of auxiliary fields. The key ingredient are the commuting spinor (twistor) variables, which naturally arise as worldsheet superpartners of the target space Grassmann coordinates. These spinors parametrise the sphere $S^8$ regarded as a coset space of the $D=10$ Lorentz group. The sphere is associ…
Born-Infeld theory is formulated using an infinite set of gauge fields, along the lines of McClain, Wu and Yu. In this formulation electromagnetic duality is generated by a fully local functional. The resulting consistency problems are analyzed and the formulation is shown to be consistent.
We derive a manifestly duality-symmetric formulation of the action principle for conformal gravity linearized around Minkowski space-time. The analysis is performed in the Hamiltonian formulation, the fourth-order character of the equations of motion requiring the formal treatment of the three-dimensional metric perturbation and the extrinsic curvature as independent dynamical variables. The constraints are solved in terms of two symmetric potentials that are interpreted as a dual three-dimensional metric and a dual extrinsic curvature. The action principle can be written in terms of these fou…
A manifestly covariant formulation of quantum field Maslov complex-WKB theory (semiclassical field theory) is investigated for the case of scalar field. The main object of the theory is "semiclassical bundle". Its base is the set of all classical states, fibers are Hilbert spaces of quantum states in the external field. Semiclassical Maslov states may be viewed as points or surfaces on the semiclassical bundle. Semiclassical analogs of QFT axioms are formulated. A relationship between covariant semiclassical field theory and Hamiltonian formulation is discussed. The constructions of axiomatic …
We give a Lagrangian and display all local symmetries for N=4 topological string by Berkovits, Vafa and Siegel, the latter previously known in the superconformal gauge. Leading to a small N=4 superconformal algebra and exhibiting the manifest Lorentz invariance the model is proposed to be a framework for restoring the manifest Lorentz invariance in N=2 string scattering amplitudes.
We consider the open superstring ending on a D-brane in the presence of a constant NS-NS B field, using the Green-Schwarz formalism. Quantizing in the light-cone gauge, we find that the anti-commutation relations for the fermionic variables of superspace remain unmodified. We also derive the unbroken supersymmetry algebra living on the D-brane. This establishes how the Moyal product is extended in a superspace formulation of non-commutative field theories. The superfield formulation of non-commutative supersymmetric field theories is briefly considered.
We discuss the extended BRST and anti-BRST symmetry (including shift symmetry) in the Batalin-Vilkovisky (BV) formulation for two and three form gauge theories. Further we develop the superspace formulation for the BV actions for these theories. We show that the extended BRST invariant BV action for these theories can be written manifestly covariant manner in a superspace with one Grassmann coordinate. On the hand a superspace with two Grassmann coordinates are required for a manifestly covariant formulation of the extended BRST and extended anti-BRST invariant BV actions for higher form gauge…
We give a brief review of Bagger-Lambert-Gustavsson (BLG) model, with emphasis on its version invariant under the volume preserving diffeomorphisms (SDiff3) symmetry. We describe the on-shell superfield formulation of this SDiff3 BLG model in standard N=8, d=3 superspace, as well as its superfield action in the pure spinor N=8 superspace. We also briefly address the Aharony-Bergman-Jafferis-Maldacena (ABJM/ABJ) model invariant under $SU(M)_{k}\times SU(N')_{-k}$ gauge symmetry, and discuss the possible form of their N=6 and, for the case of Chern-Simons level k=1,2, N=8 superfield equations.
These notes provide a brief introduction to the ABJM theory, the level k U(N) x U(N) superconformal Chern-Simons matter theory which has been conjectured to describe N coincident M2-branes. We discuss its dual formulation in terms of M-theory on AdS_4 x S^7/Z_k and review some of the evidence in favor of the conjecture. We end with a brief discussion of the important role played by the monopole operators.
We propose a new formulation of the $D=10$ Brink-Schwarz superparticle which is manifestly invariant under both the target-space super-Poincaré group and the world-line local $N=8$ superconformal group. This twistor-like construction naturally involves the sphere $S^8$ as a coset space of the $D=10$ Lorentz group. The action contains only a finite set of auxiliary fields, but they appear in unusual trilinear combinations. The origin of the on-shell $D=10$ fermionic $κ$ symmetry of the standard Brink-Schwarz formulation is explained. The coupling to a $D=10$ super-Maxwell background requires a …
The self-dual superstring has been described previously in a Neveu-Schwarz-Ramond formulation with local N=2 or 4 world-sheet supersymmetry. We present a Green-Schwarz-type formulation, with manifest spacetime supersymmetry.
A manifestly Lorentz invariant effective action for Yang-Mills theory depending only on commuting fields is constructed. This action posesses a bosonic symmetry, which plays a role analogous to the BRST symmetry in the standard formalism.
We propose a new formulation of the D=11 supermembrane theory that involves commuting spinors (twistor--like variables) and exhibits a manifest $n$--extended world volume supersymmetry $(1\leq n\leq 8)$. This supersymmetry replaces $n$ components of the usual $κ$--symmetry. We show that this formulation is classically equivalent to the standard one.
We derive the complete supergravity description of the N=2 scalar potential which realizes a generic flux-compactification on a Calabi-Yau manifold (generalized geometry). The effective potential V_{eff}=V_{(\partial_Z V=0)}$, obtained by integrating out the massive axionic fields of the special quaternionic manifold, is manifestly mirror symmetric, i.e. invariant with respect to {\rm Sp}(2 h_2+2)\times {\rm Sp}(2 h_1+2) and their exchange, being h_1, h_2 the complex dimensions of the underlying special geometries. {\Scr V}_{eff} has a manifestly N=1 form in terms of a mirror symmetric superpo…
We present the (0,4) superspace version of Witten's sigma model construction for ADHM instantons. We use the harmonic superspace formalism, which exploits the three complex structures common to both (0,4) supersymmetry and self-dual Yang-Mills theory. A novel feature of the superspace formulation is the manifest interplay between the ADHM construction and its twistor counterpart.
We consider a space-time invariant duality symmetric action for a free Maxwell field and an $SL(2,{\bf R})\times SO(6,22)$ invariant effective action describing a low-energy bosonic sector of the heterotic string compactified on a six-dimensional torus. The manifest Lorentz and general coordinate invariant formulation of the models is achieved by coupling dual gauge fields to an auxiliary vector field from an axionic sector of the theory.
Previous studies of the type IIB superstring in an ${AdS_5 \times S^5}$ background are based on a description of the superspace geometry as the quotient space $PSU(2,2|4)/SO(4,1) \times SO(5)$. This paper develops an alternative approach in which the Grassmann coordinates provide a nonlinear realization of $PSU(2,2|4)$ based on the quotient space $PSU(2,2|4)/SU(2,2) \times SU(4)$, and the bosonic coordinates are described as a submanifold of $SU(2,2) \times SU(4)$. This formulation keeps all bosonic symmetries manifest, and it provides the complete dependence on the Grassmann coordinates in te…
The frame-like covariant Lagrangian formulation of bosonic and fermionic mixed-symmetry type higher spin massless fields propagating on the AdS(d) background is proposed. Higher spin fields are described in terms of gauge p-forms which carry tangent indices representing certain traceless tensor or gamma transversal spinor-tensor representations of the AdS(d) algebra o(d-1,2) (or o(d,1) for bosonic fields in dS(d)). Manifestly gauge invariant Abelian higher spin field strengths are introduced for the general case. We describe the general framework and demonstrate how it works for the mixed-symm…
Tree-level amplitudes in N=4 SYM can be decomposed into partial or color-ordered amplitudes. Identities relating various partial amplitudes have been known since the 80's. They are Kleiss-Kuijf (KK) identities. In 2008, Bern, Carrasco and Johansson (BCJ) introduced a new set of identities which reduce the number of independent partial amplitudes to (n-3)!. In recent years, several formulations for partial amplitudes have been discovered and shown to be equivalent to each other. These can be thought of as simple dualities in the sense that different formulations make manifest different properti…
Unfolded equations of motion for symmetric massive bosonic fields of any spin in Minkowski and (A)dS spaces are presented. Manifestly gauge invariant action for a spin $s \ge 2$ massive field in any dimension is constructed in terms of gauge invariant curvatures.
Demonstrating the split octonion formalism for unified fields of dyons (electromagnetic fields) and gravito-dyons (gravito-Heavisidian fields of linear gravity), relevant field equations are derived in compact, simpler and manifestly covariant forms. It has been shown that this unified model reproduces the dynamics of structure of fields associated with individual charges (masses) in the absence of others.
A new gauge invariant formulation of the relativistic scalar field interacting with Chern-Simons gauge fields is considered. This formulation is consistent with the gauge fixed formulation. Furthermore we find that canonical (Noether) Poincaré generators are not gauge invariant even on the constraints surface and do not satisfy the (classical) Poincaré algebra. It is the improved generators, constructed from the symmetric energy-momentum tensor, which are (manifestly) gauge invariant and obey the classical Poincaré algebra.
Light cone gauge manifestly supersymmetric formulations of type IIB 10-dimensional supergravity in $AdS_5 \times S^5$ background and related boundary conformal field theory representations are developed. A precise correspondence between the bulk fields of IIB supergravity and the boundary operators is established. The formulations are given entirely in terms of light cone scalar superfields, allowing us to treat all component fields on an equal footing.
The manifestly gauge invariant formulation for free symmetric partially massless fields in $(A)dS_d$ is given in terms of gauge connections and linearized curvatures that take values in the irreducible representations of $(o(d-1,2)) o(d,1)$ described by two-row Young tableaux, in which the lengths of the first and second row are, respectively, associated with spin and depth of partial masslessness.
We show that the Lorentz covariant formulation of N=2 string in a curved space reveals an explicit hyper--Kähler structure. Apart from the metric, the superconformal currents couple to a background two--form. By superconformal symmetry the latter is constrained to be holomorphic and covariantly constant and allows one to construct three complex structures obeying a (pseudo)quaternion algebra.
A dual formulation of the S Matrix for N=4 SYM has recently been presented, where all leading singularities of n-particle N^{k-2}MHV amplitudes are given as an integral over the Grassmannian G(k,n), with cyclic symmetry, parity and superconformal invariance manifest. In this short note we show that the dual superconformal invariance of this object is also manifest. The geometry naturally suggests a partial integration and simple change of variable to an integral over G(k-2,n). This change of variable precisely corresponds to the mapping between usual momentum variables and the "momentum twisto…
We propose the bosonic part of an action that defines M-theory. It possesses manifest SO(1, 10) symmetry and constructed based on the Lorentzian 3-algebra associated with U(N) Lie algebra. From our action, we derive the bosonic sector of BFSS matrix theory and IIB matrix model in the naive large N limit by taking appropriate vacua. We also discuss an interaction with fermions.
We study a three-dimensional symmetric Chern-Simons field theory with a general covariance and it turns out that the original Chern-Simons theory is just a gauge fixed action of the symmetric Chern-Simons theory whose constraint algebra belongs to fully first class constraint system. The Abelian Chern-Simons theory with matter coupling is studied for the construction of anyon operators without any ordering ambiguity with the help of this symmetric Chern-Simons action. Finally we shall discuss some connections between the present symmetric formulation of Chern-Simons theory and the Stükelberg m…
The scalar fields of supersymmetric models are coordinates of a geometric space. We propose a formulation of supersymmetry that is covariant with respect to reparametrizations of this target space. Employing chiral multiplets as an example, we introduce modified supersymmetry variations and redefined auxiliary fields that transform covariantly under reparametrizations. The resulting action and transformation laws are manifestly covariant and highlight the geometric structure of the supersymmetric theory. The covariant methods are developed first for general theories (not necessarily supersymme…
This paper presents a 6d world-volume action that describes the dynamics of the M theory five-brane in a flat 11d space-time background. The world-volume action has global 11d super-Poincare invariance, as well as 6d general coordinate invariance and kappa symmetry, which are realized as local symmetries. The paper mostly considers a formulation in which general coordinate invariance is not manifest in one direction. However, it also describes briefly an alternative formulation, due to Pasti, Sorokin, and Tonin, in which general coordinate invariance is manifest. The latter approach requires a…
We present a formulation of a matrix model which manifestly possesses the general coordinate invariance when we identify the large $N$ matrices with differential operators. In order to build a matrix model which has the local Lorentz invariance, we investigate how the $so(9,1)$ Lorentz symmetry and the $u(N)$ gauge symmetry are mixed together. We first analyze the bosonic part of the model, and we find that the Einstein gravity is reproduced in the classical low-energy limit. And we present a proposal to build a matrix model which has ${\cal N}=2$ SUSY and reduces to the type IIB supergravity …
The generalized metric is a T-duality covariant symmetric matrix constructed from the metric and two-form gauge field and arises in generalized geometry. We view it here as a metric on the doubled spacetime and use it to give a simple formulation with manifest T-duality of the double field theory that describes the massless sector of closed strings. The gauge transformations are written in terms of a generalized Lie derivative whose commutator algebra is defined by a double field theory extension of the Courant bracket.
Dual electrodynamics and corresponding Maxwell's equations (in the presence of monopole only) are revisited from the symmetry of duality and gauge invariance. Accordingly, the manifestly covariant, dual symmetric and gauge invariant two potential theory of generalized electromagnetic fields of dyons has been developed consistently from U(1)\times U(1) gauge symmetry. Corresponding field equations and equation of motion are derived from Lagrangian formulation adopted for U(1)\times U(1) gauge symmetry for the justification of two four potentials of dyons.
We present the superspace formulation of the local RG equation, a framework for the study of supersymmetric RG flows in which the constraints of holomorphy and R-symmetry are manifest. We derive the consistency conditions associated with super-Weyl symmetry off-criticality and initiate the study of their implications. As examples, we derive an expression for the a-function, and present an analog of the a-maximization equation, which is valid off-criticality. We also apply this machinery to the study of conformal manifolds and give a simple proof that the metric on such manifolds is Kahler.
We discuss the extended BRST and anti-BRST symmetry (including shift symmetry) in the Batalin-Vilkovisky (BV) formulation for the Chern-Simons (CS) theories in $(2+1)$ spacetime dimensions. Further we develop the superspace description of BV formulation for such theories. Interestingly, the extended BRST invariant CS theories can be described in superspace in covariant manner with the help of one more (Grassmann) coordinate. However, a superspace with two Grassmann coordinates are required for a manifestly covariant formulation of the extended BRST and extended anti-BRST invariant actions for …
Some features of Einstein gravity are most easily understood from string theory but are not manifest at the level of the usual Lagrangian formulation. One example is the factorization of gravity amplitudes into gauge theory amplitudes. Based on the recently constructed `double field theory' and a geometrical frame-like formalism developed by Siegel, we provide a framework of perturbative Einstein gravity coupled to a 2-form and a dilaton in which, as a consequence of T-duality, the Feynman rules factorize to all orders in perturbation theory. We thereby establish the precise relation between t…
In this paper we consider the previously proposed generalised space-time and investigate the structure of the field theory upon which it is based. In particular, we derive a SO(D,D) formulation of the bosonic string as a non-linear realisation at lowest levels of the semi-direct product of E11 with the first fundamental representation l1. We give a Hamiltonian formulation of this theory and carry out its quantisation. We argue that the choice of representation of the quantum theory breaks the manifest SO(D,D) symmetry but that the symmetry is manifest in a non-commutative field theory. We disc…
Generalized complex geometry is a new mathematical framework that is useful for describing the target space of N=(2,2) nonlinear sigma-models. The most direct relation is obtained at the N=(1,1) level when the sigma model is formulated with an additional auxiliary spinorial field. We revive a formulation in terms of N=(2,2) semi-(anti)chiral multiplets where such auxiliary fields are naturally present. The underlying generalized complex structures are shown to commute (unlike the corresponding ordinary complex structures) and describe a Generalized Kahler geometry. The metric, B-field and gene…
In the doubled field theory approach to string theory, the T-duality group is promoted to a manifest symmetry at the expense of replacing ordinary Riemannian geometry with generalised geometry on a doubled space. The local symmetries are then given by a generalised Lie derivative and its associated algebra. This paper constructs an analogous structure for M-theory. A crucial by-product of this is the derivation of the physical section condition for M-theory formulated in an extended space.
