The Many Faces of the Superworld

The following sections are included:

• General Acknowledgements

• List of Participants

• CONTENTS

The following sections are included:

• Basic Biographic Data

• Work on Supersymmetry; Chronology

• Likhtman

• Missed Crossroads

• Glimpses

• About This Volume

• Conclusions

Writing this was not easy, for despite the fact that Yuri passed away over five years ago, I still love him. Emotions overwhelm me, making it hard to write something logical. That's why this turned out so chaotic and fragmentary. But I am sending it anyway…

1962: Nine years before the birth of the Superworld in four dimensions. I received a passing grade on my entrance exams to the Department of Physics at Moscow State University but ran into trouble with the medical commission. I gathered my certificates of a winner of many university competitions and arranged an audience with the dean of the School of Physics. I explained to him that I wanted to become a theoretical physicist. His answer was: “No one is above the law”.

A brief history of ideas which paved the way to supersymmetry is presented.

I will refer to the “Prehistoric Era” all works relevant to supersymmetry which were done before the first publication of Wess and Zumino on this subject. Eight papers belonging to the Prehistoric Era are mentioned in Marinov's article Revealing the Path to the Superworld and/or in my Introduction. Three seminal papers of Volkov and Akulov, and Volkov and Soroka, although not explicitly cited, are of paramount importance and should be added to the list, which, thus, contains eleven publications.

Specific extensions of the algebra of Poincaré group generators by bispinor operators are considered leading to algebras which have no mirror reflection symmetry. Properties of these algebras are correlated with the symmetry properties of physical systems, in particular, with the problem of parity non-conservation in weak interactions.

The following sections are included:

• Introduction

• Superalgebras and supergroups
  • Graded linear space
  • Linear operators in K^pq
  • Grassmann algebras
  • Grassmann envelope
  • Superalgebra
  • Matrix method

• Representations and models
  • The universal superalgebra Σ
  • Representation of field multiplets
  • Superinvariants
  • The Wess-Zumino model
  • Supergauge theories

• Concluding remarks

• Acknowledgments

• References

• Reviews

• Appendix: List of publications of Yu. A. Golfand

“…die Wirklichkeit zeigt sich in unseren Erlebnissen und Forschungen nie anders wie durch ein Glas, das teils den Blick durchläßt, teils den Hineinblickenden widerspiegelt.”

We present an Sp(2n,R) duality invariant Born-Infeld U(1)²ⁿ gauge theory with scalar fields. To implement this duality we had to introduce complex gauge fields and as a result the rank of the duality group is only half as large as that of the corresponding Maxwell gauge theory with the same number of gauge fields. The latter is self-dual under Sp(4n,R), the largest allowed duality group. A special case appears for n = 1 when one can also write an SL(2,R) duality invariant Born-Infeld theory with a real gauge field. We also describe the supersymmetric version of the above construction.

A Bäcklund transformation is derived for the Yang's type (super) equations previously derived (hep-th/9811108) by M. Saveliev and the author, from the ten-dimensional super-Yang-Mills field equations in an on–shell light cone gauge. It is shown to be based upon a particular gauge transformation satisfying nonlinear conditions which ensure that the equations retain the same form. These Yang's type field equations are shown to be precisely such that they automatically provide a solution of these conditions. This Bäcklund transformation is similar to the one proposed by A. Leznov for self-dual Yang-Mills in four dimensions. In the introduction a personal recollection on the birth of supersymmetry is given.

In the examples of the N = 2 superconformal algebra in two dimensions and the affine Lie superalgebra, it is shown that representations of extended two-dimensional supersymmetry can be described in terms of representations of bosonic Lie algebras.

We study the supersymmetric quantum mechanical systems that arise from discrete light cone quantization of theories with minimal supersymmetry in various dimensions. These are distinguished by having fewer fermionic fields than the familiar Kähler and hyper-Kähler models.

In N = 1 super Yang-Mills theory in three spacetime dimensions, with a simple gauge group G and a Chern-Simons interaction of level k, the supersymmetric index Tr(−1)^F can be computed by making a relation to a pure Chern-Simons theory or microscopically by an explicit Born-Oppenheimer calculation on a two-torus. The result shows that supersymmetry is unbroken if |k| ≥ h/2 (with h the dual Coxeter number of G) and suggests that dynamical supersymmetry breaking occurs for |k| < h/2. The theories with large |k| are massive gauge theories whose universality class is not fully described by the standard criteria.

We consider the pure supersymmetric Yang–Mills theories placed on a small 3-dimensional spatial torus with higher orthogonal and exceptional gauge groups. The problem of constructing the quantum vacuum states is reduced to a pure mathematical problem of classifying the periodic flat connections on T³. The latter problem is equivalent to the problem of classification of commuting triples of elements in a connected simply connected compact Lie group which is solved in this paper. In particular, we show that for higher orthogonal SO(N), N ≥ 7, and for all exceptional groups the moduli space of flat connections involves several distinct connected components. The total number of vacuum states is given in all cases by the dual Coxeter number of the group which agrees with the result obtained earlier with the instanton technique. We also solve the problem of classification of periodic flat connections on the torus of arbitrary dimension.

An overview of supersymmetry and its different applications is presented. We motivate supersymmetry in particle physics. We then explain how supersymmetry helps us analyze field theories exactly, and what dynamical lessons these solutions teach us. Finally, we describe how supersymmetry is used to derive exact results in string theory. These results have led to a revolution in our understanding of the theory.

I briefly review some of the new insights into the dynamics of supersymmetric gauge theories in different dimensions that were obtained in the last few years by realizing them as low energy theories on branes.

Abelian theories in three dimensions can have linearly confining phases as a result of monopole-instantons, as shown, for SU(2) Yang-Mills theory broken to its abelian subgroup, by Polyakov. In this article the generalization of this phase for supersymmetric abelian theories is identified, using a dual description. Topologically stable BPS-saturated and unsaturated particle and string soli tons play essential roles. A plasma of monopoles of charge 1 and −1 (along with their antimonopoles) are required for a stable confining vacuum. Yang-Mills theory broken to U(1) lacks this phase because its monopoles all have the same charge, which causes a runaway instability. A few observations regarding possible analogous confining phases of string theory are presented.

Generalising ideas of¹⁶, we address the problem of constructing Brane Box Models of what we call the Z-D Type from a new point of view, so as to establish the complete correspondence between these brane setups and orbifold singularities of the non-Abelian G generated by Z_k and D_d under certain group-theoretic constraints to which we refer as the BBM conditions. Moreover, we present a new class of quiver theories of the ordinary dihedral group d_k as well as the ordinary exceptional E_6,7,8 which have non-chiral matter content and discuss issues related to brane setups thereof.

This is a brief introductory review of the AdS/CFT correspondence and of the ideas that led to its formulation. Emphasis is placed on dualities between conformal large N gauge theories in 4 dimensions and string backgrounds of the form AdS₅ × X₅ where X₅ is an Einstein manifold. The simplest case is when X₅ is a 5-sphere and the dual gauge theory is the supersymmetric SU(N) Yang-Mills theory. We also discuss the case when X₅ is a coset space T^1,1 = (SU(2) × SU(2))/U(1) which is dual to a certain superconformal field theory with gauge group SU(N) × SU(N).

The non-renormalization of the 3-point functions 〈trX^k₁ trX^k₂ trX^k₃) of chiral primary operators in super-Yang-Mills theory is one of the most striking facts to emerge from the AdS/CFT correspondence. A two–fold puzzle appears in the extremal case, e.g. k₁ = k₂ + k₃. First, the supergravity calculation involves analytic continuation in the k_i variables to define the product of a vanishing bulk coupling and an infinite integral over AdS. Second, extremal correlators are uniquely sensitive to mixing of the single–trace operators trX^k with protected multi-trace operators in the same representation of SU(4). We show that the calculation of extremal correlators from supergravity is subject to the same subtley of regularization known for 2-point functions, and we present a careful method which justifies the analytic continuation and shows that supergravity fields couple to single traces without admixture. We also study extremal n-point functions of chiral primary operators, and argue that type IIB supergravity requires that their space-time form is a product of n − 1 two–point functions (as in the free–field approximation) multiplied by a non-renormalized coefficient. This non–renormalization property of extremal n–point functions is a new prediction of the AdS/CFT correspondence. As a by-product of this work we obtain the cubic couplings tϕϕ and sϕϕ of fields in the dilaton and 5-sphere graviton towers of type IIB supergravity on AdS₅ × S₅.

I briefly review the outcomes of two very different old questions – where global SUSY improves on ordinary QFT – when they are in turn posed in the local, SUGRA, context. The first concerns the unexpectedly powerful role of the “Dirac square root” graded algebra relation E = Q², originally found in D=4 SUSY by Gol'fand and Likhtman, in proving positive energy theorems both in SUGRA and in ordinary gravity theories, where E and Q have very different – gauge generator – definitions. The second seeks SUGRA counterparts of the boson/fermion loop ultraviolet cancellations in SUSY models. Here the result is negative: local supersymmetry cannot overcome the infinities associated with the dimensional gravitational κ² even, as recently shown, in maximal D=11 SUGRA.

The following sections are included:

• Introduction

• The 1978 Picture

• References

• Subsequent Developments

• The Supercurrent and the Adler-Bardeen Theorem

The T-dual formulation of Type I superstring theory, sometimes called Type I′ theory, has a number of interesting features. Here we review some of them including the role of D0-branes and D8-branes in controlling possible gauge symmetry enhancement.

After a brief historical survey that emphasizes the role of the algebra obeyed by the Dirac operator, we examine an algebraic Dirac operator associated with the Lie algebras and Lie algebra cosets. For symmetric cosets, its “massless” solutions display non-relativistic supersymmetry, and can be identified with the massless degrees of freedom of some supersymmetric theories: N = 1 supergravity in eleven dimensions (M-theory), type IIB string theory in ten and four dimensions, and in four dimensions, N = 8 supergravity, N = 4 super-Yang-Mills, and the N = 1 Wess-Zumino multiplet. By generalizing this Dirac operator to the affine case, we generate superconformal algebras associated with cosets g/h, where h contains the space little group. Only for eleven-dimensional supergravity h is simple. This suggests, albeit in a non-relativistic setting, that these may be the limit of theories with underlying two-dimensional superconformal structure.

We review and elaborate on some aspects of Born-Infeld action and its supersymmetric generalizations in connection with string theory.

We study instanton-corrected renormalization group flow in the two-dimensional sigma models and four dimensional gauge theory. In two dimensions we do that by replacing the non-linear supersymmetric CP^{N − 1} model by the gauged linear sigma model which is in the same universality class. We compare the moduli spaces of the instantons in the non-linear model and that of BPS field configurations in the linear model. We reduce the problem of matching of the parameters of the two systems to the intersection theory on the compact moduli space of the latter model. In four dimensions we find that the analogue of the linear sigma model is the gauge theory on the non-commutative space-time. Its BPS moduli space is the space of torsion free sheaves. Both cases (2d and 4d) are unified by the notion of the freckled instantons. We also put an end to the discussion of the nature of the superpotentials W ~ σ log σ in 2d and 4d and discover the surprising disconnectnessness of the effective target space.

We recall the obstacles which seemed, long ago, to prevent one from viewing supersymmetry as a possible fundamental symmetry of Nature. Is spontaneous supersymmetry breaking possible? Where is the spin- Goldstone fermion of supersymmetry, if not a neutrino? Which bosons and fermions could be related? Can one define conserved baryon and lepton numbers in such theories, although they systematically involve self-conjugate Majorana fermions? If we have to postulate the existence of new bosons carrying B and L (the new spin-0 squarks and sleptons), can we prevent them from mediating new unwanted interactions?

We then recall how we obtained the three basic ingredients of the Supersymmetric Standard Model: 1) the SU(3) × SU(2) × U(1) gauge superfields; 2) the chiral quark and lepton superfields; 3) the two doublet Higgs superfields responsible for the electroweak breaking, and the generation of quark and lepton masses. The original continuous “R-invariance” of this model was soon abandoned in favor of its discrete version, R-parity, so that the gravitino, and gluinos, can acquire masses – gluinos getting their masses from supergravity, or radiative corrections.

R-parity forbids unwanted squark and slepton exchanges, and guarantees the stability of the “lightest supersymmetric particle”. It is present only since we restricted the initial superpotential to be an even function of quark and lepton superfields (so as to allow for B and L conservation laws), as made apparent by the formula re-expressing R-parity as (−1)^2S (−1)^(3B+L). Whether it turns out to be absolutely conserved, or not, R-parity plays an essential role in the phenomenology of supersymmetric theories, and the experimental searches for the new sparticles.

We discuss the possibility that the electro-weak and strong interactions arise as the low energy effective description of branes in M-theory. As a step towards constructing such a model we show how one can naturally obtain SU(N₁) × SU(N₂) × U(1) gauge theories from branes, including matter in the bi-fundamental representation of SU(N₁) × SU(N₂) which are fractionally charged under U(1).

We discuss phenomenology of extra time dimensions in a scenario where the standard model particles are localized in “our” time, whereas gravity can propagate in all time dimensions. For an odd number of extra times, at small distances, the (real part of the) Newtonian potential is “screened” by tachyonic Kaluza-Klein gravitons. In general, the gravitational self-energy of objects acquires an imaginary part. This complexity may either be interpreted as an amplitude for the disappearance into “nothing,” leading to the causality and probability violation in low-energy processes, or as an artifact of the fictitious decay into the unphysical negative energy tachyons. The former case would put the most severe phenomenological restrictions on such theories. In the latter case the size of extra times may be within the reach of the proposed gravitational experiments. In such a case these experiments should observe that the strength of the Newtonian gravity diminishes at short distances.

We review the theory of higher spin gauge fields in 2 + 1 and 3 + 1 dimensional anti-de Sitter space and present some new results on the structure of higher spin currents and explicit solutions of the massless equations. A previously obtained d = 3 integrating flow is generalized to d = 4 and is shown to give rise to a perturbative solution of the d = 4 nonlinear higher spin equations. A particular attention is paid to the relationship between the star-product origin of the higher spin symmetries, AdS geometry and the concept of space-time locality.

The following sections are included:

• Introduction

• Should 4=10 be read in English or in Hebrew?

• String thermodynamics in D-brane backgrounds

• Phases in Gravity
  • Bulk Phase Diagram
  • World-Volume Phase Diagram

• Concluding remark

• References

In theories with TeV string scale and sub-millimeter extra dimensions the attractive picture of logarithmic gauge coupling unification at 10¹⁶ GeV is seemingly destroyed. In this paper we argue to the contrary that logarithmic unification can occur in such theories. The rationale for unification is no longer that a gauge symmetry is restored at short distances, but rather that a geometric symmetry is restored at large distances in the bulk away from our 3-brane. The apparent ‘running’ of the gauge couplings to energies far above the string scale actually arises from the logarithmic variation of classical fields in (sets of) two large transverse dimensions. We present a number of N = 2 and N = 1 supersymmetric D-brane constructions illustrating this picture for unification.

The following sections are included:

• Rigid N = 1 Supersymmetry and Supergravity in x-Space

• Rigid and Local Superspace

• Extended Supersymmetries

The following sections are included:

• Epilogue (Editor's Concluding Remarks)