Self-dual Model Research Articles

Issues of duality on noncommutative manifolds: The nonequivalence between self-dual and topologically massive models

We study issues of duality and dual equivalence in noncommutative manifolds. In particular, the question of dual equivalence for the actions of the noncommutative extensions of the self-dual model in 3D space-time and the Maxwell-Chern-Simons model is investigated. We show that former model is not dual equivalent to the noncommutative extension of the Maxwell-Chern-Simons model, as widely believed, but a deformed version of it that is disclosed here. Our results are not restricted to any finite order in the Seiberg-Witten expansion involving the noncommutative parameter $\ensuremath{\theta}$.

Topological B-Model on Weighted Projective Spaces and Self-Dual Models in Four Dimensions

It was recently shown by Witten on the basis of several examples that the topological B-model whose target space is a Calabi-Yau (CY) supermanifold is equivalent to holomorphic Chern-Simons (hCS) theory on the same supermanifold. Moreover, for the supertwistor space P3|4 as target space, it has been demonstrated that hCS theory on P3|4 is equivalent to self-dual = 4 super Yang-Mills (SYM) theory in four dimensions. We consider as target spaces for the B-model the weighted projective spaces WP3|2(1,1,1,1|p,q) with two fermionic coordinates of weight p and q, respectively — which are CY supermanifolds for p+q = 4 — and discuss hCS theory on them. By using twistor techniques, we obtain certain field theories in four dimensions which are equivalent to hCS theory. These theories turn out to be self-dual truncations of = 4 SYM theory or of its twisted (topological) version.

Heating up the cold bounce

Self-dual string cosmological models provide an effective example of bouncing solutions where a phase of accelerated contraction smoothly evolves into an epoch of decelerated Friedmann–Robertson–Walker expansion dominated by the dilaton. While the transition to the expanding regime occurs at sub-Planckian curvature scales, the universe emerging after the bounce is cold, with sharply growing gauge coupling. However, since massless gauge bosons (as well as other massless fields) are super-adiabatically amplified, the energy density of the maximally amplified modes re-entering the horizon after the bounce can efficiently heat the universe. As a consequence the gauge coupling reaches a constant value, which can still be perturbative.

A New Self-Dual Embedding Method for Convex Programming

In this paper we introduce a conic optimization formulation to solve constrained convex programming, and propose a self-dual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint functions and their corresponding conjugate functions respectively. Hence they are completely symmetric. This allows for a standard primal-dual path following approach for solving the embedded problem. Moreover, there are two immediate logarithmic barrier functions for the primal and dual cones. We show that these two logarithmic barrier functions are conjugate to each other. The explicit form of the conjugate functions are in fact not required to be known in the algorithm. An advantage of the new approach is that there is no need to assume an initial feasible solution to start with. To guarantee the polynomiality of the path-following procedure, we may apply the self-concordant barrier theory of Nesterov and Nemirovski. For this purpose, as one application, we prove that the barrier functions constructed this way are indeed self-concordant when the original constraint functions are convex and quadratic. We pose as an open question to find general conditions under which the constructed barrier functions are self-concordant.

Duality in noncommutative topologically massive gauge field theory revisited

We introduce a master action in noncommutative space, out of which we obtain the action of the noncommutative Maxwell-Chern-Simons theory. Then, we look for the corresponding dual theory at both first and second orders in the noncommutative parameter. At the first order, the dual theory happens to be, precisely, the action obtained from the usual commutative Self-Dual model by generalizing the Chern-Simons term to its noncommutative version, including a cubic term. Since this resulting theory is also equivalent to the noncommutative massive Thirring model in the large fermion mass limit, we remove, as a byproduct, the obstacles arising in the generalization to noncommutative space, and to the first nontrivial order in the noncommutative parameter, of the bosonization in three dimensions. Then, performing calculations at the second order in the noncommutative parameter, we explicitly compute a new dual theory which differs from the noncommutative Self-Dual model, and further, differs also from other previous results, and involves a very simple expression in terms of ordinary fields. In addition, a remarkable feature of our results is that the dual theory is local, unlike what happens in the non-Abelian, but commutative case. We also conclude that the generalization to noncommutative space of bosonization in three dimensions is possible only when considering the first non-trivial corrections over ordinary space.

Embedding commutative and noncommutative theories in the symplectic framework

This paper is devoted to study gauge embedding of either commutative and noncommutative theories in the framework of the symplectic formalism. We illustrate our ideas in the Proca model, the irrotational fluid model and the noncommutative self-dual model. In the process of this new path of embedding, the infinitesimal gauge generators of the gauge embedded theory are easily and directly chosen. Among other advantages, this enables a greater control over the final Lagrangian and puts some light on the so called ''arbitrariness problem".

Maxwell–Chern–Simons theory is free for marginally noncommutative spacetimes

We have conclusively established the duality between noncommutative Maxwell–Chern–Simons theory and self-dual model, the latter in ordinary spacetime, to the first nontrivial order in the noncommutativity parameter θμν, with θ0i=0. This shows that the former theory is free for marginally noncommutative spacetimes. A θ-generalized covariant mapping between the variables of the two models in question has been derived explicitly, that converts one model to the other, including the symplectic structure and action.

On the Behavior of the Homogeneous Self-Dual Model for Conic Convex Optimization

There is a natural norm associated with a starting point of the homogeneous self-dual (HSD) embedding model for conic convex optimization. In this norm two measures of the HSD model's behavior are precisely controlled independent of the problem instance: (i) the sizes of E-optimal solutions, and (ii) the maximum distance of E-optimal solutions to the boundary of the cone of the HSD variables. This norm is also useful in developing a stopping-rule theory for HSD-based interior-point methods such as SeDuMi. Under mild assumptions, we show that a standard stopping rule implicitly involves the sum of the sizes of the E-optimal primal and dual solutions, as well as the size of the initial primal and dual infeasibility residuals. This theory suggests possible criteria for developing starting points for the homogeneous self-dual model that might improve the resulting solution time in practice.

On Two Measures of Problem Instance Complexity and their Correlation with the Performance of SeDuMi on Second-Order Cone Problems

We evaluate the practical relevance of two measures of conic convex problem complexity as applied to second-order cone problems solved using the homogeneous self-dual (HSD) embedding model in the software SeDuMi. The first measure we evaluate is Renegar's data-based condition measure C(d), and the second measure is a combined measure of the optimal solution size and the initial infeasibility/optimality residuals denoted by S (where the solution size is measured in a norm that is naturally associated with the HSD model). We constructed a set of 144 secondorder cone test problems with widely distributed values of C(d) and S and solved these problems using SeDuMi. For each problem instance in the test set, we also computed estimates of C(d) (using Pena's method) and computed S directly. Our computational experience indicates that SeDuMi iteration counts and log(C(d)) are fairly highly correlated (sample correlation R = 0.676), whereas SeDuMi iteration counts are not quite as highly correlated with S (R = 0.600). Furthermore, the experimental evidence indicates that the average rate of convergence of SeDuMi iterations is affected by the condition number C(d) of the problem instance, a phenomenon that makes some intuitive sense yet is not directly implied by existing theory.

Gauge invariance and duality in the non-commutative plane

We show that the duality between the self-dual and Maxwell–Chern–Simons theories in 2+1 dimensions survives when the spacetime becomes non-commutative. Existence of the Seiberg–Witten map is crucial in the present analysis. It should be noted that the above models, being manifestly gauge variant and invariant, respectively, transform differently under the Seiberg–Witten map. We also discuss this duality in the Stuckelberg formalism where the self-dual model is elevated to a gauge theory. The ‘master’ Lagrangian approach has been followed throughout.

Duality between non-commutative Yang–Mills–Chern–Simons and non-Abelian self-dual models

By introducing an appropriate parent action and considering a perturbative approach, we establish, up to fourth order terms in the field and for the full range of the coupling constant, the equivalence between the non-commutative Yang–Mills–Chern–Simons theory and the non-commutative, non-Abelian self-dual model. In doing this, we consider two different approaches by using both the Moyal star-product and the Seiberg–Witten map.

Nonlinear self-duality and supergravity

The concept of self-dual supersymmetric nonlinear electrodynamics is generalized to a curved superspace of N = 1 supergravity, for both the old minimal and the new minimal versions of N = 1 supergravity. We derive the self-duality equation, which has to be satisfied by the action functional of any U(1) duality invariant model of a massless vector multiplet, and construct a family of self-dual nonlinear models. This family includes a curved superspace extension of the N = 1 super Born-Infeld action. The supercurrent and supertrace in such models are proved to be duality invariant. The most interesting and unexpected result is that the requirement of nonlinear self-duality yields nontrivial couplings of the vector multiplet to Kahler sigma models. We explicitly derive the couplings to general Kahler sigma models in the case when the matter chiral multiplets are inert under the duality rotations, and more specifically to the dilaton-axion chiral multiplet when the group of duality rotations is enhanced to SL(2,R).

On implementing a primal-dual interior-point method for conic quadratic optimization

Based on the work of the Nesterov and Todd on self-scaled cones an implementation of a primal-dual interior-point method for solving large-scale sparse conic quadratic optimization problems is presented. The main features of the implementation are it is based on a homogeneous and self-dual model, it handles rotated quadratic cones directly, it employs a Mehrotra type predictor-corrector extension and sparse linear algebra to improve the computational efficiency. Finally, the implementation exploits fixed variables which naturally occurs in many conic quadratic optimization problems. This is a novel feature for our implementation. Computational results are also presented to document that the implementation can solve very large problems robustly and efficiently.

Geometric approach to a massivepform duality

Massive theories of Abelian p forms are quantized in a generalized path representation that leads to a description of the phase space in terms of a pair of dual nonlocal operators analogous to the Wilson loop and the 't Hooft disorder operators. Special attention is devoted to the study of the duality between the topologically massive and self-dual models in $2+1$ dimensions. It is shown that these models share a geometric representation in which just one nonlocal operator suffices to describe the observables.

Space of signed points and the self-dual model

We study a generalization of the group of loops based on sets of signed points, instead of paths or loops. This geometrical setting incorporates the kinematical constraints of the Sigma Model, inasmuch as the the group of loops does with the Bianchi identities of Yang-Mills theories. We employ an Abelian version of this construction to quantize the Self-Dual Model, which allows us to relate this theory with that of a massless scalar field obeying non-trivial boundary conditions.

Duality in non-linear B∧F models: equivalence between self-dual and topologically massive Born–Infeld B∧F models

We study the dual equivalence between the non-linear generalization of the self-dual (NSDB∧F) and the topologically massive B∧F models with particular emphasis on the non-linear electrodynamics proposed by Born and Infeld. This is done through a dynamical gauge embedding of the non-linear self-dual model yielding to a gauge invariant and dynamically equivalent theory. We clearly show that non-polinomial NSDB∧F models can be map, through a properly defined duality transformation into TMB∧F actions. The general result obtained is then particularized for a number of examples, including the Born–Infeld-BF (BIBF) model that has experienced a revival in the recent literature.

Erratum: Equivalence of the self-dual model and Maxwell-Chern-Simons theory on arbitrary manifolds [Phys. Rev. D59, 065019 (1999)

Using a group-invariant version of the Faddeev-Popov method we explicitly obtain the partition functions of the Self-Dual Model and Maxwell-Chern-Simons theory. We show that their ratio coincides with the partition function of abelian Chern-Simons theory to within a phase factor depending on the geometrical properties of the manifold.

Criticality in self-dual sine-Gordon models

We discuss the nature of criticality in the β 2=2 πN self-dual extension of the sine-Gordon model. This field theory is related to the two-dimensional classical XY model with a N-fold degenerate symmetry-breaking field. We briefly overview the already studied cases N=2,4 and analyze in detail the case N=3 where a single phase transition in the three-state Potts universality class is expected to occur. The Z 3 infrared critical properties of the β 2=6 π self-dual sine-Gordon model are derived using two non-perturbative approaches. On one hand, we map the model onto an integrable deformation of the Z 4 parafermion theory. The latter is known to flow to a massless Z 3 infrared fixed point. Another route is based on the connection with a chirally asymmetric, su(2) 4⊗su(2) 1 Wess–Zumino–Novikov–Witten model with anisotropic current–current interaction, where we explore the existence of a decoupling (Toulouse) point.

Topologically massive gauge theories from first order theories in arbitrary dimensions

We thereby prove that a large class of topologically massive theories of the Cremmer–Scherk–Kalb–Ramond-type in any d dimensions corresponds to gauge non-invariant first-order theories that can be interpreted as self-dual models.

A dynamic large-update primal‐dual interior-point method for linear optimization

Primal-dual interior-point methods (IPMs) have shown their power in solving large classes of optimization problems. However, at present there is still a gap between the practical behavior of these algorithms and their theoretical worst-case complexity results, with respect to the strategies of updating the duality gap parameter in the algorithm. The so-called small-update IPMs enjoy the best known theoretical worst-case iteration bound but work very poorly in practice, while the so-called large-update IPMs have superior practical performance but with relatively weaker theoretical results. In this article, by restricting us to linear optimization (LO), we first exploit some interesting properties of a self-regular proximity function, proposed recently by the authors of this work and Roos, and use it to define a neighborhood of the central path. These simple but interesting properties of the proximity function indicate that, when the current iterate is in a large neighborhood of the central path, then the large-update IPM emerges to be the only natural choice. Then, we apply these results to design a specific self-regularity based IPM. Among others, we show that this self-regularity based IPM can also predict precisely the change of the duality gap as the standard IPM does. Therefore, we can directly apply the modified IPM to the simplified self-dual homogeneous model for LO. This provides a remedy for an implementation issue of the new self-regular IPMs. A dynamic large-update IPM in large neighborhood is proposed. Different from traditional large-update IPMs, the new dynamic IPM always takes large-updates and does not utilize any inner iteration to get centered. An worst-case iteration bound of the algorithm is established.