Unification of gravity and elementary particle interactions in 26 dimensions?

The presence of an 11-cocycle in ten-dimensional theories containing an E₈×E₈ Yang-Mills theory coupled to supergravity suggests that the fundamental theory is a 26-dimensional string theory based on the Leech lattice. A theory with equal numbers of bosons and fermions is constructed by truncating a certain natural infinite-dimensional representation of the Monster sporadic group. This theory appears to be a certain combination of pieces of heterotic and N = 2 superstring.     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

Compactifications of F-theory on Calabi-Yau threefolds (II)

We continue our study of compactifications of F-theory on Calabi-Yau threefolds. We gain more insight into F-theory duals of heterotic strings and provide a recipe for building F-theory duals for arbitrary heterotic compactifications on elliptically fibered manifolds. As a byproduct we find that string/string duality in six dimensions gets mapped to fiber/base exchange in F-theory. We also construct a number of new N = 1, d = 6 examples of F-theory vacua and study transitions among them. We find that some of these transition points correspond upon further compactification to 4 dimensions to transitions through analogues of Argyres-Douglas points of N = 2 moduli. A key idea in these transitions is the notion of classifying (0,4) fivebranes of heterotic strings.     

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.     

An N = 2 dual pair and a phase transition

We carefully analyze the N = 2 dual pair of string theories in four dimensions introduced by Ferrara, Harvey, Strominger and Vafa. The analysis shows that a second discrete degree of freedom must be switched on in addition to the known “Wilson line” to achieve a non-perturbatively consistent theory. We also identify the phase transition this model undergoes into another dual pair via a process analogous to a conifold transition. This provides the first known example of a phase transition which is understood from both the type II and the heterotic string picture.     

Non critical super strings on world sheets of constant curvature

We consider correlation functions in Neveu-Schwarz string theory coupled to two dimensional gravity. The actionfor the 2D gravity consists of the string induced Liouville action and the Jackiw-Teitelboim action describing pure 2D gravity. Then gravitational dressed dimensions of vertex operators are equal to their bare conformal dimensions. There are two possible interpretations of the model. Considering the 2D dilaton and the Liouville field as additional target space coordinates one gets ad+2-dimensional critical string. In thed-dimensional non critical string picture gravitational fields retain their original meaning and ford=4 one can get a mass spectrum via consistency requirements. In both cases a GSO projection is possible.     

Euclidean D-branes and higher-dimensional gauge theory

We consider euclidean D-branes wrapping around manifolds of exceptional holonomy in dimensions seven and eight. The resulting theory on the D-brane—that is, the dimensional reduction of 10-dimensional supersymmetric Yang-Mills theory—is a cohomological field theory which describes the topology of the moduli space of instantons. The 7-dimensional theory is an N_T = 2 (or balanced) cohomological theory given by an action potential of Chern-Simons type. As a by-product of this method, we construct a related cohomological field theory which describes the monopole moduli space on a 7-manifold of G₂ holonomy.     

Variational method in hamiltonian lattice gauge theories

A general expression for the expectation value of the hamiltonian of a d + 1 dimensional lattice gauge theory as a function of the norm of the variational state (that itself has the form of a partition function of a d-dimensional lattice gauge theory) is given. Applications include U(1), SU(2), U(2) and U(N) gauge theories for large N in d = 2 + 1 dimensions. It is also demonstrated that the deconfining phase transition is of first order in every dimension above the critical one, provided it is of first or second order at the critical dimension.     

Nonstandard parafermions and string compactification

Nonstandard parafermions are built and their central charges and dimensions are calculated. We then construct new N=2 superconformal field theories by tensoring the parafermions with a free boson. We study the spectrum and modular transformations of these theories. Superstring and heterotic strings in four dimensions are then obtained by tensoring the new superconformal field theories along with some minimal models. The generations and antigenerations are studied. We give an example of the ¹²(5,7) theory which is shown to have two net generations.     

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.     

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.     

Non-perturbative superpotentials in string theory

The non-perturbative superpotential can be effectively calculated in M-theory compactification to three dimensions on a Calabi-Yau four-fold X. For certain X, the superpotential is identically zero, while for other X, a non-perturbative superpotential is generated. Using F-theory, these results carry over to certain Type IIB and heterotic string compactifications to four dimensions with N = 1 supersymmetry. In the heterotic string case, the non-perturbative superpotential can be interpreted as coming from space-time and world-sheet instantons; in many simple cases contributions come only from finitely many values of the instanton numbers.     

Heisenberg groups and noncommutative fluxes

We develop a group-theoretical approach to the formulation of generalized abelian gauge theories, such as those appearing in string theory and M-theory. We explore several applications of this approach. First, we show that there is an uncertainty relation which obstructs simultaneous measurement of electric and magnetic flux when torsion fluxes are included. Next, we show how to define the Hilbert space of a self-dual field. The Hilbert space is Z₂-graded and we show that, in general, self-dual theories (including the RR fields of string theory) have fermionic sectors. We indicate how rational conformal field theories associated to the two-dimensional Gaussian model generalize to (4k + 2)-dimensional conformal field theories. When our ideas are applied to the RR fields of string theory we learn that it is impossible to measure the K-theory class of a RR field. Only the reduction modulo torsion can be measured.     

Superstring theory

Dual string theories, initially developed as phenomenological models of hadrons, now appear more promising as candidates for a unified theory of fundamental interactions. Type I superstring theory (SST I), is a ten-dimensional theory of interacting open and closed strings, with one supersymmetry, that is free from ghosts and tachyons. It requires that an SO(n) or Sp(2n) gauge group be used. A light-cone-gauge string action with space-time supersymmetry automatically incorporates the superstring restrictions and leads to the discovery of type II superstring theory (SST II). SST II is an interacting theory of closed strings only, with two D = 10 supersymmetries, that is also free from ghosts and tachyons. By taking six of the spatial dimensions to form a compact space, it becomes possible to reconcile the models with our four-dimensional perception of spacetime and to define low-energy limits in which SST I reduces to N = 4, D = 4 super Yang-Mills theory and SST II reduces to N = 8, D = 4 supergravity theory. The superstring theories can be described by a light-cone-gauge action principle based on fields that are functionals of string coordinates. With this formalism any physical quantity should be calculable. There is some evidence that, unlike any conventional field theory, the superstring theories provide perturbatively renormalizable (SST I) or finite (SST II) unifications of gravity with other interactions.     

The heterotic string is a soliton

It is shown that the Type IIA superstring compactified on K3 has a smooth string soliton with the same zero mode structure as the heterotic string compactified on a four-torus, thus providing new evidence for a conjectured exact duality between the two six-dimensional string theories. The chiral worldsheet bosons arise as zero modes of Ramond-Ramond fields of the IIA string theory and live on a signature (20,4) even, self-dual lattice. Stable, finite loops of soliton string provide the charged Ramond-Ramond states necessary for enhanced gauge symmetries at degeneration points of the K3 surface. It is also shown that Type IIB strings toroidally compactified to six dimensions have a multiplet of string solutions with Type II worldsheets.     

Kähler spinning particles

We construct the U(N) spinning particle theories, which describe particles moving on Kähler spaces. These particles have the same relation to the N = 2 string as usual spinning particles have to the NSR string. We find the restrictions on the target space of the theories coming from supersymmetry and from global anomalies. Finally, we show that the partition functions of the theories agree with what is expected from their spectra, unlike that of the N = 2 string in which there is an anomalous dependence on the proper time.     

Abelian lattice gauge theories in 3 + 1 dimensions: The linked cluster approach

We use the linked cluster expansion methods of Nickel to derive strong couping series for Z_N abelian gauge theories. These new results together with corresponding estimates using the exact linked cluster expansion algorithm are analysed and compared with previously obtained results for U(1) lattice gauge theory in 3 + 1 dimensions. We confirm the phase structure of these theories as found by other techniques. The critical value of N at which the phase structure of Z_N alters is estimated to be N_C = 4.5 ± 0.2. In each case the string tension estimates using the ELCE algorithm are found to be stable in the presence of a roughening transition.     

On the low energy spectra of the non-supersymmetric heterotic string theories

The ten dimensional string theories as well as eleven dimensional supergravity are conjectured to arise as limits of a more basic theory, traditionally dubbed M-theory. This notion is confined to the ten dimensional supersymmetric theories. String theory, however, also contains ten dimensional non-supersymmetric theories that have not been incorporated into this picture. In this note we explore the possibility of generating the low energy spectra of various non-supersymmetric heterotic string vacua from the Horava–Witten model. We argue that this can be achieved by imposing on the Horava–Witten model an invariance with respect to some extra operators which identify the orbifold fixed planes in a non-trivial way, and we demonstrate it for the E₈ and SO(16)×SO(16) heterotic string vacua in ten dimensions.     

Cavitation in holographic sQGP

We study the possibility of cavitation in the non-conformal N=^2?SU(N) theory which is a mass deformation of N=4SU(N) Yang-Mills theory. The second order transport coefficients are known from the numerical work using AdS/CFT by Buchel and collaborators. Using these and the approach of Rajagopal and Tripuraneni, we investigate the flow equations in a (1+1)-dimensional boost invariant set up. We find that the string theory model does not exhibit cavitation before phase transition is reached. We give a semi-analytic explanation of this finding.