Equivalence of quantum field theories related by the θ-exact Seiberg-Witten map

The equivalence of the noncommutative U(N) quantum field theories related by the θ-exact Seiberg-Witten maps is, in this paper, proven to all orders in the perturbation theory with respect to the coupling constant. We show that this holds for super Yang-Mills theories with N=0, 1, 2, 4 supersymmetry. A direct check of this equivalence relation is performed by computing the one-loop quantum corrections to the quadratic part of the effective action in the noncommutative U(1) gauge theory with N=0, 1, 2, 4 supersymmetry.

Branes and supersymmetric quantum field theories

Since the discovery of D-branes as non-perturbative, dynamic objects in string theory, various configurations of branes in type IIA/B string theory and M-theory have been considered to study their low-energy dynamics described by supersymmetric quantum field theories.

One example of such a construction is based on the description of Seiberg-Witten curves of four-dimensional N = 2 supersymmetric gauge theories as branes in type IIA string theory and M-theory. This enables us to study the gauge theories in strongly-coupled regimes. Spectral networks are another tool for utilizing branes to study non-perturbative regimes of two- and four-dimensional supersymmetric theories. Using spectral networks of a Seiberg-Witten theory we can find its BPS spectrum, which is protected from quantum corrections by supersymmetry, and also the BPS spectrum of a related two-dimensional N = (2,2) theory whose (twisted) superpotential is determined by the Seiberg-Witten curve. When we don’t know the perturbative description of such a theory, its spectrum obtained via spectral networks is a useful piece of information. In this thesis we illustrate these ideas with examples of the use of Seiberg-Witten curves and spectral networks to understand various two- and four-dimensional supersymmetric theories.

First, we examine how the geometry of a Seiberg-Witten curve serves as a useful tool for identifying various limits of the parameters of the Seiberg-Witten theory, including Argyres-Seiberg duality and Argyres-Douglas fixed points. Next, we consider the low-energy limit of a two-dimensional N = (2, 2) supersymmetric theory from an M-theory brane configuration whose (twisted) superpotential is determined by the geometry of the branes. We show that, when the two-dimensional theory flows to its infra-red fixed point, particular cases realize Kazama-Suzuki coset models. We also study the BPS spectrum of an Argyres-Douglas type superconformal field theory on the Coulomb branch by using its spectral networks. We provide strong evidence of the equivalence of superconformal field theories from different string-theoretic constructions by comparing their BPS spectra.

Quaterninic Kähler manifolds, constrained instantons, and the magic square

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

Noncommutativity and Some of Its Present-Day Developments

This masters thesis explores some of the most recent developments in noncommutative quantum field theory. This old theme, first suggested by Heisenberg in the late 1940s, has had a renaissance during the last decade due to the firmly held belief that space-time becomes noncommutative at small distances and also due to the discovery that string theory in a background field gives rise to noncommutative field theory as an effective low energy limit. This has led to interesting attempts to create a noncommutative standard model, a noncommutative minimal supersymmetric standard model, noncommutative gravity theories etc. This thesis reviews themes and problems like those of UV/IR mixing, charge quantization, how to deal with the non-commutative symmetries, how to solve the Seiberg-Witten map, its connection to fluid mechanics and the problem of constructing general coordinate transformations to obtain a theory of noncommutative gravity. An emphasis has been put on presenting both the group theoretical results and the string theoretical ones, so that a comparison of the two can be made.

Noncommutative Quantum Field Theories and UV/IR Mixing

Our present-day understanding of fundamental constituents of matter and their interactions is based on the Standard Model of particle physics, which relies on quantum gauge field theories. On the other hand, the large scale dynamical behaviour of spacetime is understood via the general theory of relativity of Einstein. The merging of these two complementary aspects of nature, quantum and gravity, is one of the greatest goals of modern fundamental physics, the achievement of which would help us understand the short-distance structure of spacetime, thus shedding light on the events in the singular states of general relativity, such as black holes and the Big Bang, where our current models of nature break down. The formulation of quantum field theories in noncommutative spacetime is an attempt to realize the idea of nonlocality at short distances, which our present understanding of these different aspects of Nature suggests, and consequently to find testable hints of the underlying quantum behaviour of spacetime. The formulation of noncommutative theories encounters various unprecedented problems, which derive from their peculiar inherent nonlocality. Arguably the most serious of these is the so-called UV/IR mixing, which makes the derivation of observable predictions especially hard by causing new tedious divergencies, to which our previous well-developed renormalization methods for quantum field theories do not apply. In the thesis I review the basic mathematical concepts of noncommutative spacetime, different formulations of quantum field theories in the context, and the theoretical understanding of UV/IR mixing. In particular, I put forward new results to be published, which show that also the theory of quantum electrodynamics in noncommutative spacetime defined via Seiberg-Witten map suffers from UV/IR mixing. Finally, I review some of the most promising ways to overcome the problem. The final solution remains a challenge for the future.

TeV scale implications of non commutative space time in laboratory frame with polarized beams

We analyze e(+)e(-) -> gamma gamma, e(-)gamma -> e(-)gamma and gamma gamma -> e(+)e(-) processes within the Seiberg-Witten expanded noncommutative scenario using polarized beams. With unpolarized beams the leading order effects of non commutativity starts from second order in non commutative(NC) parameter i.e. O(Theta(2)), while with polarized beams these corrections appear at first order (O(Theta')) in cross section. The corrections in Compton case can probe the magnetic component(Theta(B)) while in Pair production and Pair annihilation probe the electric component((Theta) over right arrow (E)) of NC parameter. We include the effects of earth rotation in our analysis. This study is done by investigating the effects of non commutativity on different time averaged cross section observables. The results which also depends on the position of the collider, can provide clear and distinct signatures of the model testable at the International Linear Collider(ILC).

Anomalous triple gauge boson couplings in e(-)e(+) -> gamma gamma for noncommutative standard model

We investigate e(+)e(-) -> gamma gamma process within the Seiberg-Witten expanded noncommutative standard model (NCSM) scenario in the presence of anomalous triple gauge boson couplings. This study is done with and without initial beam polarization and we restrict ourselves to leading order effects of noncommutativity i.e. O(Theta). The noncommutative (NC) corrections are sensitive to the electric component ((Theta) over bar (E)) of NC parameter. We include the effects of Earth's rotation in our analysis. This study is done by investigating the effects of noncommutativity on different time averaged cross section observables. We have also defined forward backward asymmetries which will be exclusively sensitive to anomalous couplings. We have looked into the sensitivity of these couplings at future experiments at the International Linear Collider (ILC). This analysis is done under realistic ILC conditions with the center of mass energy (cm.) root s = 800 GeV and integrated luminosity L = 500 fb(-1). The scale of noncommutativity is assumed to be Lambda = 1 TeV. The limits on anomalous couplings of the order 10(-1) from forward backward asymmetries while much stringent limits of the order 10(-2) from total cross section are obtained if no signal beyond SM is seen. (C) 2012 Elsevier B.V. All rights reserved.

Inner structure of Spin(c)(4) gauge potential on 4-dimensional manifolds

The decomposition of Spin(c)(4) gauge potential in terms of the Dirac 4-spinor is investigated, where an important characterizing equation Delta A(mu) = -lambda A(mu) has been discovered. Here, lambda is the vacuum expectation value of the spinor field, lambda = parallel to Phi parallel to(2), and A(mu) the twisting U(1) potential. It is found that when), takes constant values, the characterizing equation becomes an eigenvalue problem of the Laplacian operator. It provides a revenue to determine the modulus of the spinor field by using the Laplacian spectral theory. The above study could be useful in determining the spinor field and twisting potential in the Seiberg-Witten equations. Moreover, topological characteristic numbers of instantons in the self-dual sub-space are also discussed.

Decomposition of Noncommutative U(1) Gauge Potential

We investigate the decomposition of noncommutative gauge potential (A) over cap (i), and find that it has inner structure, namely, (A) over cap (i) can he decomposed in two parts, (b) over cap (i) and (a) over cap (i), where (b) over cap (i) satisfies gauge transformations while (a) over cap (i) satisfies adjoint transformations, so close the Seiberg-Witten mapping of noncommutative, U(1) gauge potential. By, means of Seiberg-Witten mapping, we construct a mapping of unit vector field between noncommutative space and ordinary space, and find the noncommutative U(1) gauge potential and its gauge field tensor can be expressed in terms of the unit vector field. When the unit vector field has no singularity point, noncommutative gauge potential and gauge field tensor will equal ordinary gauge potential and gauge field tensor

O gás de elétrons degenerado não-comutativo

A dinâmica quântica de sistemas de partícula única não-relativística envolvendo coordenadas não-comutativas, usualmente chamada mecânica quântica não-comutativa, tem sido objeto de numerosas investigações. Neste trabalho estendemos esses estudos para o caso de sistema de muitas partículas. Usamos como protótipo o modelo do gás de elé- trons degenerado cuja dinâmica é bem conhecida no limite comutativo. Nosso objetivo principal aqui é entender qualitativamente, em vez de quantitativamente, as principais modi cações induzidas pela presença de coordenadas não-comutativas. Primeiro veremos que a não-comutatividade modi ca a energia de correlação de troca enquanto preserva a neutralidade elétrica do modelo. Logo, através da teoria de perturbações independente do tempo juntamente com o mapa de Seiberg-Witten mostramos que o potencial de ioniza ção é modi cado pela não-comutatividade e, também, que o parâmetro não-comutativo atua como uma temperatura de referência. Sendo assim, a não-comutatividade levanta a degenerescência do gás de elétrons a temperatura zero.

Duality between non-commutative Yang-Mills-Chem-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-ChernSimons 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. (C) 2003 Elsevier B.V. B.V. All rights reserved.

Super-Maxwell actions with manifest duality

Superstring field theory was recently used to derive a four-dimensional Maxwell action with manifest duality. This action is related to the McClain-Wu-Yu Hamiltonian and can be locally coupled to electric and magnetic sources. In this letter, the manifestly dual Maxwell action is supersymmetrized using N = 1 and N = 2 superspace. The N = 2 version may be useful for studying Seiberg-Witten duality. © 1997 Elsevier Science B.V.

Kallen-Lehmann representation of noncommutative quantum electrodynamics

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

N=2 supersymmetric QCD and elliptic potentials

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Quasimodular instanton partition function and the elliptic solution of Korteweg-de Vries equations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)