Superconformal SU Research Articles

More fermionic supersymmetric Wilson loops in four dimensions

We construct supersymmetric fermionic Wilson loops along general curves in four-dimensional N=4 super Yang-Mills theory and along general planar curves in N=2 superconformal SU(N)×SU(N) quiver theory. These loops are generalizations of the Zarembo loops and are cohomologically equivalent to them. In N=4 super Yang-Mills theory, we compute their expectation values and verify the cohomological equivalence relation up to the order g4 in perturbation theory.

On AdS/CFT duality in the twisted sector of string theory on AdS5× S5/ℤ2 orbifold background

We consider type IIB string theory on an AdS5 × S5/ℤ2 orbifold background, which should be dual to 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 superconformal SU(N) × SU(N) gauge theory with two bi-fundamental hypermultiplets. The correlator of two chiral BPS operators from the twisted sector of this quiver CFT exhibits non-trivial dependence on the ’t Hooft coupling λ already in the planar limit. This dependence was recently determined using localisation and the expansion at large λ contains a subleading contribution proportional to ζ(3)λ−3/2. We address the question of how to reproduce this correction on the string theory side by starting with the ζ(3)α′3 term in the type IIB string effective action. We find a regular solution of type IIB supergravity which represents a resolution of the AdS5 × S5/ℤ2 orbifold and demonstrate that the relevant light twisted sector states may be identified as additional supergravity 2-form modes “wrapping” a finite 2-cycle in the resolution space. Reproducing the structure of the gauge theory result becomes more transparent in the large R-charge or BMN-like limit in which the resolved background takes a pp-wave form with the transverse space being a product of ℝ4 and the Eguchi-Hanson space.

N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 3 conformal superspace in four dimensions

We develop a superspace formulation for N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 3 conformal supergravity in four spacetime dimensions as a gauge theory of the superconformal group SU(2, 2|3). Upon imposing certain covariant constraints, the algebra of conformally covariant derivatives ∇A=∇a∇αi∇iα⋅\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\ abla}_A=\\left({\ abla}_a,{\ abla}_{\\alpha}^i,{\ abla}_i^{\\overset{\\cdot }{\\alpha }}\\right) $$\\end{document} is shown to be determined in terms of a single primary chiral spinor superfield, the super-Weyl spinor Wα of dimension +1/2 and its conjugate. Associated with Wα is its primary descendant Bij of dimension +2, the super-Bach tensor, which determines the equation of motion for conformal supergravity. As an application of this construction, we present two different but equivalent action principles for N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 3 conformal supergravity. We describe the model for linearised N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 3 conformal supergravity in an arbitrary conformally flat background and demonstrate that it possesses U(1) duality invariance. Additionally, upon degauging certain local symmetries, our superspace geometry is shown to reduce to the U(3) superspace constructed by Howe more than four decades ago. Further degauging proves to lead to a new superspace formalism, called SU(3) superspace, which can also be used to describe N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 3 conformal supergravity. Our conformal superspace setting opens up the possibility to formulate the dynamics of the off-shell N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 3 super Yang-Mills theory coupled to conformal supergravity.

The planar limit of the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 E-theory: numerical calculations and the large λ expansion

We study correlation functions of local operators and Wilson loop expectation values in the planar limit of a 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 superconformal SU(N) YM theory with hypermultiplets in the symmetric and antisymmetric representations of the gauge group. This so called E theory is closely related to N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM and has a holographic description in terms of a ℤ2 orientifold of AdS5 × S5. Using recent matrix model results based on supersymmetric localization we develop efficient numerical methods to calculate two- and three-point functions of certain single trace operators as well as 1/2-BPS Wilson loop expectation values as a function of the ’t Hooft coupling λ. We use our numerical results to arrive at simple analytic expressions for these correlators valid up to sixth order in the λ−1/2 strong coupling expansion. These results provide explicit field theory predictions for the α′ corrections to the supergravity approximation of type IIB string theory on the AdS5 × S5/ℤ2 orientifold.

Non-planar corrections in orbifold/orientifold mathcal{N} = 2 superconformal theories from localization

We study non-planar corrections in two special mathcal{N} = 2 superconformal SU(N) gauge theories that are planar-equivalent to mathcal{N} = 4 SYM theory: two-nodes quiver model with equal couplings and mathcal{N} = 2 vector multiplet coupled to two hypermultiplets in rank-2 symmetric and antisymmetric representations. We focus on two observables in these theories that admit representation in terms of localization matrix model: free energy on 4-sphere and the expectation value of half-BPS circular Wilson loop. We extend the methods developed in arXiv:2207.11475 to derive a systematical expansion of non-planar corrections to these observables at strong ’t Hooft coupling constant λ. We show that the leading non- planar corrections are given by a power series in λ3/2/N2 with rational coefficients. Sending N and the coupling constant λ to infinity with λ3/2/N2 kept fixed corresponds to the familiar double scaling limit in matrix models. We find that in this limit the observables in the two models are related in a remarkably simple way: the free energies differ by the factor of 2, whereas the Wilson loop expectation values coincide. Surprisingly, these relations hold only at strong coupling, they are not valid in the weak coupling regime. We also discuss a dual string theory interpretation of the leading corrections to the free energy in the double scaling limit suggesting their relation to curvature corrections in type IIB string effective action.

Factorized class S theories and surface defects

It is known that some theories of class S are actually factorized into multiple decoupled nontrivial four-dimensional mathcal{N} = 2 theories. We propose a way of constructing examples of this phenomenon using the physics of half-BPS surface defects, and check that it works in one simple example: it correctly reproduces a known realization of two copies of mathcal{N} = 2 superconformal SU(2) QCD, describing this factorized theory as a class S theory of type A3 on a five-punctured sphere with a twist line. Separately, we also present explicit checks that the Coulomb branch of a putative factorized class S theory has the expected product structure, in two examples.

BPS Wilson loop in mathcal{N} = 2 superconformal SU(N) \u201corientifold\u201d gauge theory and weak-strong coupling interpolation

We consider the expectation value leftlangle mathcal{W}rightrangle of the circular BPS Wilson loop in mathcal{N} = 2 superconformal SU(N) gauge theory containing a vector multiplet coupled to two hypermultiplets in rank-2 symmetric and antisymmetric representations. This theory admits a regular large N expansion, is planar-equivalent to mathcal{N} = 4 SYM theory and is expected to be dual to a certain orbifold/orientifold projection of AdS5× S5 superstring theory. On the string theory side leftlangle mathcal{W}rightrangle is represented by the path integral expanded near the same AdS2 minimal surface as in the maximally supersymmetric case. Following the string theory argument in [5], we suggest that as in the mathcal{N} = 4 SYM case and in the mathcal{N} = 2 SU(N) × SU(N) superconformal quiver theory discussed in [19], the coefficient of the leading non-planar 1/N2 correction in leftlangle mathcal{W}rightrangle should have the universal λ3/2 scaling at large ’t Hooft coupling. We confirm this prediction by starting with the localization matrix model representation for leftlangle mathcal{W}rightrangle . We complement the analytic derivation of the λ3/2 scaling by a numerical high-precision resummation and extrapolation of the weak-coupling expansion using conformal mapping improved Padé analysis.

1/N expansion of circular Wilson loop in mathcal{N} = 2 superconformal SU(N) \xd7 SU(N) quiver

Localization approach to mathcal{N} = 2 superconformal SU(N) × SU(N) quiver theory leads to a non-Gaussian two-matrix model representation for the expectation value of BPS circular SU(N) Wilson loop leftlangle mathcal{W}rightrangle . We study the subleading 1/N2 term in the large N expansion of leftlangle mathcal{W}rightrangle at weak and strong coupling. We concentrate on the case of the symmetric quiver with equal gauge couplings which is equivalent to the ℤ2 orbifold of the SU(2N) mathcal{N} = 4 SYM theory. This orbifold gauge theory should be dual to type IIB superstring in AdS5 × (S5/ℤ2). We present a string theory argument suggesting that the 1/N2 term in leftlangle mathcal{W}rightrangle in the orbifold theory should have the same strong-coupling asymptotics λ3/2 as in the mathcal{N} = 4 SYM case. We support this prediction on the gauge theory side by a numerical study of the localization matrix model. We also find a relation between the 1/N2 term in the Wilson loop expectation value and the derivative of the free energy of the orbifold gauge theory on 4-sphere.

Quantum SU(2|1) supersymmetric \u2102N Smorodinsky-Winternitz system

We study quantum properties of SU(2|1) supersymmetric (deformed mathcal{N} = 4, d = 1 supersymmetric) extension of the superintegrable Smorodinsky-Winternitz system on a complex Euclidian space ℂN. The full set of wave functions is constructed and the energy spectrum is calculated. It is shown that SU(2|1) supersymmetry implies the bosonic and fermionic states to belong to separate energy levels, thus exhibiting the “even-odd” splitting of the spectra. The superextended hidden symmetry operators are also defined and their action on SU(2|1) multiplets of the wave functions is given. An equivalent description of the same system in terms of superconformal SU(2|1, 1) quantum mechanics is considered and a new representation of the hidden symmetry generators in terms of the SU(2|1, 1) ones is found.

N=4 super-Schwarzian derivative via nonlinear realizations

Current studies of supersymmetric extensions of the Sachdev-Ye-Kitaev model stimulate a renewed interest in super-Schwarzian derivatives. In this work, we apply the method of nonlinear realizations to the finite-dimensional superconformal group SU(1,1|2) and link its invariants to the N=4 super-Schwarzian.

Deformed mathcal{N}=8 mechanics of (8, 8, 0) multiplets

We construct new models of “curved” SU(4|1) supersymmetric mechanics based on two versions of the off-shell multiplet (8, 8, 0) which are “mirror” to each other. The worldline realizations of the supergroup SU(4|1) are treated as a deformation of flat mathcal{N}=8 , d = 1 supersymmetry. Using SU(4|1) chiral superfields, we derive invariant actions for the first-type (8, 8, 0) multiplet, which parametrizes special Kähler manifolds. Since we are not aware of a manifestly SU(4|1) covariant superfield formalism for the second-type (8, 8, 0) multiplet, we perform a general construction of SU(4|1) invariant actions for both multiplet types in terms of SU(2|1) superfields. An important class of such actions enjoys superconformal OSp(8|2) invariance. We also build off-shell actions for the SU(4|1) multiplets (6, 8, 2) and (7, 8, 1) through appropriate substitutions for the component fields in the (8, 8, 0) actions. The (6, 8, 2) actions are shown to respect superconformal SU(4|1, 1) invariance.

Superconformal SU(1, 1|n) mechanics

Recent years have seen an upsurge of interest in dynamical realizations of the superconformal group SU(1,1|2) in mechanics. Remarking that SU(1,1|2) is a particular member of a chain of supergroups SU(1,1|n) parametrized by an integer n, here we begin a systematic study of SU(1,1|n) multi-particle mechanics. A representation of the superconformal algebra su(1,1|n) is constructed on the phase space spanned by m copies of the (1,2n,2n-1) supermultiplet. We show that the dynamics is governed by two prepotentials V and F, and the Witten-Dijkgraaf-Verlinde-Verlinde equation for F shows up as a consequence of a more general fourth-order equation. All solutions to the latter in terms of root systems reveal decoupled models only. An extension of the dynamical content of the (1,2n,2n-1) supermultiplet by angular variables in a way similar to the SU(1,1|2) case is problematic.

The operator product expansion between the 16 lowest higher spin currents in the $$\mathcal{N}=4$$ N = 4 superspace

Some of the operator product expansions (OPEs) between the lowest 16 higher spin currents of spins (1, 3/2, 3/2, 3/2, 3/2, 2, 2, 2, 2, 2, 2, 5/2, 5/2, 5/2, 5/2, 3) in an extension of the large N=4 linear superconformal algebra were constructed in the N=4 superconformal coset SU(5)/SU(3) theory previously. In this paper, by rewriting the above OPEs in the N=4 superspace developed by Schoutens (and other groups), the remaining undetermined OPEs where the corresponding singular terms possess the composite fields with spins s =7/2, 4, 9/2, 5 are completely determined. Furthermore, by introducing the arbitrary coefficients in front of the composite fields in the right hand sides of the above complete 136 OPEs, reexpressing them in the N=2 superspace and using the N=2 OPEs mathematica package by Krivonos and Thielemans, the complete structures of the above OPEs with fixed coefficient functions are obtained with the help of various Jacobi identities. Then one obtains ten N=2 super OPEs between the four N=2 higher spin currents denoted by (1, 3/2, 3/2, 2), (3/2, 2, 2, 5/2), (3/2, 2, 2, 5/2) and (2, 5/2, 5/2, 3) (corresponding 136 OPEs in the component approach) in the N=4 superconformal coset SU(N+2)/SU(N) theory. Finally, one describes them as one single N=4 super OPE between the above sixteen higher spin currents in the N=4 superspace. The fusion rule for this OPE contains the next 16 higher spin currents of spins of (2, 5/2, 5/2, 5/2, 5/2, 3, 3, 3, 3, 3, 3, 7/2, 7/2, 7/2, 7/2, 4) in addition to the quadratic N=4 lowest higher spin multiplet and the large N=4 linear superconformal family of the identity operator. The various structure constants (fixed coefficient functions) appearing in the right hand side of this OPE depend on N and the level k of the bosonic spin-1 affine Kac-Moody current.

Minimal unitary representation of [formula omitted] and its [formula omitted] deformations and [formula omitted], [formula omitted] superconformal models

Quantization of the geometric quasiconformal realizations of noncompact groups and supergroups leads directly to their minimal unitary representations (minreps). Using quasiconformal methods massless unitary supermultiplets of superconformal groups SU(2,2|N) and OSp(8⁎|2n) in four and six dimensions were constructed as minreps and their U(1) and SU(2) deformations, respectively. In this paper we extend these results to SU(2) deformations of the minrep of N=4 superconformal algebra D(2,1;λ) in one dimension. We find that SU(2) deformations can be achieved using n pair of bosons and m pairs of fermions simultaneously. The generators of deformed minimal representations of D(2,1;λ) commute with the generators of a dual superalgebra OSp(2n⁎|2m) realized in terms of these bosons and fermions. We show that there exists a precise mapping between symmetry generators of N=4 superconformal models in harmonic superspace studied recently and minimal unitary supermultiplets of D(2,1;λ) deformed by a pair of bosons. This can be understood as a particular case of a general mapping between the spectra of quantum mechanical quaternionic Kähler sigma models with eight super symmetries and minreps of their isometry groups that descends from the precise mapping established between the 4d, N=2 sigma models coupled to supergravity and minreps of their isometry groups.

Superembedding methods for 4DN=1SCFTs

We extend SO(4,2) covariant lightcone embedding methods of four-dimensional CFTs to N=1 superconformal field theory (SCFT). Manifest superconformal SU(2,2|1) invariance is achieved by realizing 4D superconformal space as a surface embedded in the projective superspace spanned by certain complex chiral supermatrices. Because SU(2,2|1) acts linearly on the ambient space, the constraints on correlators implied by superconformal Ward identities are automatically solved in this formalism. Applications include new, compact expressions for correlation functions containing one anti-chiral superfield and arbitrary chiral superfield insertions, and manifestly invariant expressions for the superconformal cross-ratios that parametrize the four-point function of two chiral and two anti-chiral fields. Superconformal expressions for the leading singularities in the OPE of chiral and anti-chiral operators are also given. Because of covariance, our expressions are valid in any superconformally flat background, e.g., AdS_4 or R times S^3.

A Clifford Cl(5, C) Unified Gauge Field Theory of Conformal Gravity, Maxwell and U(4) × U(4) Yang-Mills in 4D

A Clifford Cl(5, C) Unified Gauge Field Theory of Conformal Gravity, Maxwell and U(4) × U(4) Yang-Mills in 4D is rigorously presented extending our results in prior work. The \({Cl(5, C) = Cl(4, C) \oplus Cl(4, C)}\) algebraic structure, behind the 4D (complexified) Conformal Gravity-Maxwell and U(4) × U(4) Yang-Mills unification program advanced in this work, is such that it encodes the direct group product U(2, 2) × U(4) × U(4) = [SU(2, 2)] spacetime × [U(1) × U(4) × U(4)] internal and which does not violate the Coleman-Mandula theorem because the spacetime symmetries (conformal group SU(2, 2) in the absence of a mass gap, Poincare group when there is mass gap) do not mix with the internal symmetries. Similar considerations apply to the supersymmetric case when the symmetry group structure is given by the direct product of the superconformal group (in the absence of a mass gap) with an internal symmetry group so that the Haag-Lopuszanski-Sohnius theorem is not violated. A generalization of the de Sitter and Anti de Sitter gravitational theories based on the gauging of the Cl(4, 1, R), Cl(3, 2, R) algebras follows. We conclude with a few remarks about the complex extensions of the Metric Affine theories of Gravity (MAG) based on GL(4, C) × s C 4, the realizations of twistors and the \({\mathcal{N}}\) = 1 superconformal su(2, 2|1) algebra purely in terms of Clifford algebras and their plausible role in Witten’s formulation of perturbative \({\mathcal{N}}\) = 4 super Yang- Mills theory in terms of twistor-string variables.

F-theoretic vs microscopic description of a conformal $ \mathcal{N} = 2 $ SYM theory

The F-theory background of four D7 branes in a type I′ orientifold was conjectured to be described by the Seiberg-Witten curve for the superconformal SU(2) gauge theory with four flavors. This relation was explained by considering in this background a probe D3 brane, which supports this theory with SU(2) realized as Sp(1). Here we explicitly compute the non-perturbative corrections to the D7/D3 system in type I′ due to D-instantons. This computation provides both the quartic effective action on the D7 branes and the quadratic effective action on the D3 brane; the latter agrees with the F-theoretic prediction. The action obtained in this way is related to the one derived from the usual instanton calculus a la Nekrasov (or from its AGT realization in terms of Liouville conformal blocks) by means of a non-perturbative redefinition of the coupling constant. We also point out an intriguing relation between the four-dimensional theory on the probe D3 brane with SO(8) flavor symmetry and the eight-dimensional dynamics on the D7 branes. On the latter, SO(8) represents a gauge group and the flavor masses correspond to the vacuum expectation values of an adjoint scalar field m: what we find is that the exact effective coupling in four dimensions is obtained from its perturbative part by taking into account in its mass dependence the full quantum dynamics of the field m in eight dimensions.

Proving the AGT relation for N f = 0, 1, 2 antifundamentals

Using recursive relations satisfied by Nekrasov partition functions and by irregular conformal blocks we prove the AGT correspondence in the case of N=2 superconformal SU(2) quiver gauge theories with N_f = 0,1,2 antifundamental hypermultiplets

Remark about scaling limit of ABJ theory

We generalize the suggestion presented in arXiv:0806.3498 that the 3d N=8 superconformal SU(N) Chern-Simons-matter theory of Lorentzian Bagger-Lambert-Gustavson type (L-BLG) can be obtained through the scaling limit from N=6 superconformal U(N)xU(N) Chern-Simons-matter theory of Aharony, Bergman, Jafferis and Maldacena (ABJM) to the case when we study the scaling limit of N=6 superconformal U(M)xU(N) Chern-Simons-matter theory of Aharony, Bergman and Jafferis (ABJ). We show that if we extend the ABJ theory in the same way as in arXiv:0811.1540 we can define a consistent limit that leads to SU(N) L-BLG theory together with U(M-N) Chern-Simons theory of level k.

Supersymmetric field theory in two-time physics

We construct N=1 supersymmetry in 4+2 dimensions compatible with the theoretical framework of 2T physics field theory and its gauge symmetries. The fields are arranged into 4+2-dimensional chiral and vector supermultiplets, and their interactions are uniquely fixed by supersymmetry (SUSY) and 2T-physics gauge symmetries. Many 3+1 spacetimes emerge from 4+2 by gauge fixing. Gauge degrees of freedom are eliminated as one comes down from 4+2 to 3+1 dimensions without any remnants of Kaluza-Klein modes. In a special gauge, the remaining physical degrees of freedom, and their interactions, coincide with ordinary N=1 supersymmetric field theory in 3+1 dimensions. In this gauge, SUSY in 4+2 is interpreted as superconformal symmetry SU(2,2|1) in 3+1 dimensions. Furthermore, the underlying 4+2 structure imposes some interesting restrictions on the emergent 3+1 SUSY field theory, which could be considered as part of the predictions of 2T-physics. One of these is the absence of the troublesome renormalizable CP violating F*F terms. This is good for curing the strong CP violation problem of QCD. An additional feature is that the superpotential is required to have no dimensionful parameters. To induce phase transitions, such as SUSY or electroweak symmetry breaking, a coupling to the dilaton is needed. This suggests amore » common origin of phase transitions that is driven by the vacuum value of the dilaton and needs to be understood in a cosmological scenario as part of a unified theory that includes the coupling of supergravity to matter. Another interesting aspect of the proposed theory is the possibility to utilize the inherent 2T gauge symmetry to explore dual versions of the N=1 theory in 3+1 dimensions, such as the minimal supersymmetric standard model (MSSM) and its duals. This is expected to reveal nonperturbative aspects of ordinary 1T field theory.« less