Quantum vortex strings: A review

The quantum worldsheet dynamics of vortex strings contains information about the 4d non-Abelian gauge theory in which the string lives. Here I tell this story. The string worldsheet theory is typically some variant of the CP^N-1 sigma-model, describing the orientation of the string in a U(N) gauge group. Qualitative parallels between 2d sigma-models and 4d non-Abelian gauge theories have been known since the 1970s. The vortex string provides a quantitative link between the two. In 4d theories with N=2 supersymmetry, the exact BPS spectrum of the worldsheet coincides with the bulk spectrum in 4d. Moreover, by tuning parameters, the CP^N-1 sigma-model can be coaxed to flow to an interacting conformal fixed point which is related to the 4d Argyres-Douglas fixed point. For theories with N=1 supersymmetry, the worldsheet theory suffers dynamical supersymmetry breaking and, more interestingly, supersymmetry restoration, in a way which captures the physics of Seiberg’s quantum deformed moduli space.     

Model building on asymmetric Z3 orbifolds: Non-supersymmetric models

Four-dimensional string models arising in the asymmetric Z₃ orbifold compactifications of the heterotic string are studied. A mechanism for supersymmetry breaking that gives rise to chiral models in four dimensions is presented, and some typical models are discussed. A formalism for calculating one-loop partition functions in Z₃ models is developed. One partition function is constructed that may correspond to a non-supersymmetric, tachyon-free theory, with a vanishing cosmological constant as a consequence of Atkin-Lehner symmetry. The negative result of a search for the model corresponding to this partition function is reported.     

Strings and D=4

It is argued that the unique infrared behaviour of non-abelian gauge couplings in four dimensions could be responsible, in closed super- (or heterotic) string theory, for dynamical supersymmetry breaking and for compactification to D=4 with string-scale compactified radii.     

Supersymmetry breaking,open strings and M-theory

We study supersymmetry breaking by Scherk-Schwarz compactifications in type I string theory. While in the gravitational sector all mass splittings are proportional to a (large) compactification radius, supersymmetry remains unbroken for the massless excitations of D-branes orthogonal to the large dimension. In this sector, supersymmetry breaking can then be mediated by gravitational interactions alone, that are expected to be suppressed by powers of the Planck mass. The mechanism is non-perturbative from the heterotic viewpoint and requires a compactification radius at intermediate energies of order 10¹²−10¹⁴ GeV This can also explain the value of Newton's constant if the string scale is close to the unification scale, of order 10¹⁶ GeV.     

Grand unification with local supersymmetry

We study general conditions for obtaining spontaneous breaking of local supersymmetry in N = 1 supergravity coupled to supersymmetric matter. We consider in particular the coupling of N = 1 supergravity to grand unified theories like SU(5) and study the conditions which must be met in order to obtain a realistic model. Specific models are built in which local supersymmetry is broken at a scale √M_Wm_p ～ 10¹⁰ GeV. This breaking of supersymmetry is only detected at low energies through soft terms breaking explicitly the global supersymmetry. These soft terms (scalar masses, gaugino masses and trilinear scalar couplings) are renormalized at low energies according to the renormalization group. The (mass)² of the Higgs doublet evolve towards negative values at low energies giving rise to SU(2) × U(1) breaking as a radiative effect of local supersymmetry breaking. We finally point out the possible relevance of non-renormalizable superpotentials for the problem of fermion masses.     

A heterotic N=2 string with space–time supersymmetry

We reconsider the issue of embedding space–time fermions into the four-dimensional N=2 worldsheet supersymmetric string. A new heterotic theory is constructed, taking the right-movers from the N=4 topological extension of the conventional N=2 string but a c=0 conformal field theory supporting target-space supersymmetry for the left-moving sector. The global bosonic symmetry of the full formalism proves to be U(1,1), just as in the usual N=2 string. Quantization reveals a spectrum of only two physical states, one boson and one fermion, which fall in a multiplet of (1,0) supersymmetry.     

Supersymmetrical string theories

A supersymmetrical light-cone-gauge string action is presented. It provides a basis for understanding the previously- studied supersymmetrical dual string theory as well as two new closed-string theories that have extended supersymmetry in ten dimensions, corresponding to N=8 supersymmetry in four dimensions.     

Phenomenology and conformal field theory or can string theory predict the weak mixing angle?

We show that the weak mixing angle θ_w is the same for continuously connected classical vacua of the heterotic string which have chiral fermions in their massless spectra. We also show that the world-sheet quantum field theory for any classical vacuum with spacetime supersymmetry possesses an N = 2 superconformal invariance.     

Effective lagrangian for supersymmetric theories with intermediate breaking scale

We investigate effective lagrangians in supersymmetric models broken spontaneously at an intermediate energy (～ μ ? M_GUT). It is shown to all orders in perturbation theory that the low-energy interactions of light particles are described by an effective lagrangian with explicit but soft supersymmetry breaking of order μ² / M_GUT.     

Cohomological Yang–Mills theories on Kähler 3-folds

We study topological gauge theories with N_c=(2,0) supersymmetry based on stable bundles on general Kähler 3-folds. In order to have a theory that is well defined and well behaved, we consider a model based on an extension of the usual holomorphic bundle by including a holomorphic 3-form. The correlation functions of the model describe complex 3-dimensional generalizations of Donaldson–Witten type invariants. We show that the path integral can be written as a sum of contributions from stable bundles and a complex 3-dimensional version of Seiberg–Witten monopoles. We study certain deformations of the theory, which allow us to consider the situation of reducible connections. We shortly discuss situations of reduced holonomy. After dimensional reduction to a Kähler 2-fold, the theory reduces to Vafa–Witten theory. On a Calabi–Yau 3-fold, the supersymmetry is enhanced to N_c=(2,2). This model may be used to describe classical limits of certain compactifications of (matrix) string theory.     

Spontaneous supersymmetry breaking in supersymmetric string theories

A generalization of the dimensional reduction of supersymmetric string theories is introduced which leads to spontaneous breaking of supersymmetry. This supersymmetry breaking has non-trivial consequences for the quantization and dynamics of the theory. The lowest quantum correction to the cosmological constant is calculated and found to be unacceptably large.     

Use of compensators to break supersymmetry in an N = 1 supergravity theory

We consider an N = 1 supergravity theory with multiple compensators and show that supersymmetry is broken by a solution to the equation of motion of a compensator. When a chiral scalar superfield is coupled to supergravity, we discuss various aspects of supersymmetry breaking and show that the super-Higgs-Kibble effect is operative. Possible applications of this mechanism of supersymmetry breaking in model building and extended supergravity theories are indicated.     

Low energy aspects of superstring theories

The effective theory obtained as low energy limit of theE ₈×E ₈ superstring is analyzed from the phenomenological point of view. The supersymmetry breaking manifests itself in the low energy sector only after the radiative corrections involving the goldstino field have been taken into account. A universal mass for the scalar fields, generally smaller thanm _3/2, is generated. An intermediate scale can be obtained through the radiative breaking of part of the gauge group. As a consequence, the additional particles present in the spectrum get a heavy mass, and effective interactions are induced, which make possible the breaking of the electroweak group and the radiative generation of gaugino masses.     

Anti-de Sitter BPS black holes in N=2 gauged supergravity

Electrically charged solutions breaking half of the supersymmetry in Anti-de Sitter four dimensional N=2 supergravity coupled to vector supermultiplets are constructed. These static black holes live in an asymptotic AdS₄ space time. The Killing spinor, i.e., the spinor for supersymmetry variation is explicitly constructed for these solutions.     

Nonlinear supersymmetry without the GSO projection and unstable D9-brane

Orientable open string theories containing both bosons and fermions without the GSO projection are expected to have the 10-dimensional N=2 space–time supersymmetry in a spontaneously broken phase. We study the low-energy theorem for the nonlinearly realized N=2 supersymmetry using the effective action for an unstable D9-brane. It is explicitly confirmed that the 4-fermion open string amplitudes without the GSO projection obey the low-energy theorem derived from the nonlinear N=2 supersymmetry. An intimate connection between the existence of the hidden supersymmetry and the open–open string (s–t) duality is pointed out.     

RG-invariant sum rule in a generalization of anomaly-mediated SUSY-breaking models

We study a generalization of anomaly-mediated supersymmetry-breaking (AMSB) scenarios, under the assumption that the effects of the high-scale theory do not completely decouple and that D-term type contributions can therefore be present. We investigate the effect of such possible D-term additional contributions to soft scalar masses by requiring that, for non-vanishing, renormalizable Yukawa couplings Y^ijk, the sum of squared soft supersymmetry breaking mass parameters, M²_ijk≡m_i²+m_j²+m_k², is RG-invariant, in the sense that it becomes independent of the specific ultraviolet boundary conditions as it occurs in the AMSB models. This type of models can avoid the problem of tachyonic solutions for the slepton mass spectrum present in AMSB scenarios. We implement the electroweak symmetry breaking condition and explore the sparticle spectrum associated with this framework. To show the possible diversity of the sparticle spectrum, we consider two examples, one in which the D-terms induce a common soft supersymmetry breaking mass term for all sfermion masses, and another one in which a light stop can be present in the spectrum.     

A grand unified theory based onSU(n, 1)-minimal supergravity

We discuss a grand unified theory in the framework ofSU(n, 1) minimal supergravity with the Planck mass as the only input mass scale.M _W ≈m _3/2 is fixed by radiative corrections to be naturally ?M _P1. Due to the particular form of explicit soft supersymmetry breaking a light singlet can be used to obtain naturally light Higgs doublets and for a new mechanism for radiativeSU (2)×U(1) breaking. The low energy particle spectrum is very restricted withm _3/2≈10⁴ GeV.     

Supersymmetrical dual string theory

It is proved that the physical states of the open-string sector of the ten-dimensional string theory form supersymmetry multiplets. The proof is achieved by first constructing a new formulation of the spectrum generating algebra, and then forming the supersymmetry operator (as well as associated operators) and demonstrating that it transforms as a spinor under Lorentz transformations and has the correct anticommutation relations. The results can be interpreted either in terms of a covariant formulation or in terms of a light-cone gauge formulation. In the former case certain formulas are not completely proved, although they are in the latter. When interpreted in four dimensions (by dimensional reduction) the string theory provides an interacting theory of an infinite number of massive representations of N = 4 supersymmetry involving particles of arbitrarily high spin.     

Radiative decays of quarkonia into Goldstone fermions

We consider the decay of³ S ₁ quarkonia into a photon and two Goldstone fermions in a model realizing supersymmetry in a nonlinear way. We derive a lower bound of 3.5 GeV for the scale of spontaneous supersymmetry breaking from Υ-decays. We show that the decay of a³ S ₁ quarkonium into two Goldstone fermions is forbidden in the order we consider.     

Orbifold models and modular transformation

The orbifold models of the heterotic string are constructed on the quotient spaces of generalized tori by translational and rotational discrete symmetries. In order to obtain the consistent orbifold models, the conditions of the modular invariance are derived from a one-loop vacuum amplitude. Z₃ orbifold models satisfying such conditions are searched systematically. It is shown that there are infinite possible models with N = 2 supersymmetry. Among these models, two examples having E₆ and E₇ gauge groups are discussed. The orbifold models with N = 1 supersymmetry are also discussed in detail. It is shown that there are only five consistent models in the class of these models based on E₈ ⊗ E′₈ heterotic string in which the extra six-dimensional torus and the E₈ ⊗ E′₈ maximal torus are modded out by the rotational and the translational Z₃ symmetries respectively.