Duality in noncommutative gauge theories

Abstract

It is largely believed that spacetime structure differs from the usual definiton by going to the high energies or equivalently at small distances. This belief has been justified by using different approaches in different fields of the physics. However, noncommutativity of space coordinates arise in a natural way in string theory and appear as a consequence of the nonlocal property of the string. Noncommutative spaces emerge as a consequence of the quantization of worldsheet theory of the string, which is ending on a D-brane, in presence of a background magnetic field. This configuration leads to the noncommutativity of the open string coordinates at the end where they are attached to the brane and hence the worldvolume of the D-brane also becomes noncommutative. The effective physics on the D-branes in presence of a background field can be described either by a commutative gauge theory or by a noncommutative one. Seiberg-Witten proved that these two different descriptions arise from the same field theory with different regularizations. Since the physics does not depend on the regularization, theories obtained with different regularizations can be related to each other. Hence they have enabled to construct a map between these two descriptions by a field redefinition. This is a perturbative expansion of the noncommutative fields in terms of the ordinary fields with respect to the noncommutativity parameter. By using the Seiberg-Witten map one can pass from a gauge theory defined in terms of the noncommutative fields to a gauge theory in terms of the ordinary fields. Duality appears in several different contexts of physics. Dual theories provide two different but equivalent descriptions of the same model in the diverse interaction regimes by using in general, different fields. The relation between the fields is in general not known explicitly and in the most of the cases it contains nonlinear terms. Thus, knowing the explicit relation between the fields allows perturbative calculations in the variables of the original theory both in the strong and weak coupling regimes. In this work we deal with noncommutative U(1) gauge theory and its S duality. When the initial theory has a noncommutativity between the space coordinates, dual of that results in a time/space noncommutative gauge theory. These types of time/space noncommutative theories correspond to the certain configurations of the string theory. When there is a noncommutativity between time and space it is not obvious how to apply canonical quantization methods. In the ordinary case time variable is the evolution parameter of system however it is not clear what one means by the noncommutativity of time. Hence, Hamiltonian formulation of dual theory should be clarified. Parent action method is an appropriate tool to study dual theories. Parent action is constructed from the initial theory by a Legendre transformation with respect to the initial field and contains the dual field as a Lagrange multiplier. Starting from the parent action one obtains the dual theory by taking the variation of the parent action with respect to the initial field. On the other hand if one performs the variation with respect to the dual field obtains the initial theory. Hence it becomes possible to study the dual Hamiltonian bypassing the space/time noncommutative dual Lagrangian.This Hamilton formulation playes important roles in the D3-brane worldvolume theories. Secondly we investigate the electric-magnetic duality relations in the noncommutative U(1) gauge theory. Electric-magnetic duality exchanges the electric degrees of freedom of theory with the magnetic degrees of freedom. It also exchanges the electric charge quanta with the managnetic charge quanta.Electric charge quanta at the same time related to coupling constant of theory. Such a transformation, if it can be constructed, will map the strongly coupled electric degrees of freedom of theory to weakly coupled magnetic degrees of freedom of it. Hence different phases of the gauge theories can be investigated. We provide the electricmagnetic duality transformations for both the Lagrange and Hamilton densities. A well known property of the ordinary gauge theories is verified for the noncommutative U(1) gauge theory: Duality maps the Lagrangian to itself up to an overall minus sign and keeps intact the Hamiltonian of U(1) gauge theory. However, electric-magnetic duality transformation in configuration space is shown to be defined by a reversed one with respect to that of in phase space.

Özet

Komütatif olmayan (noncommutative) uzaylar farklı çerçevelerde karşımıza çıksa da bu tip kuramların gittikçe artan bir şekilde çalışılması, bunların sicim kuramı ile olan yakın ilişkilerinin kurulması ile olmuştur. Komütatif olmayan uzaylar üzerinde tanımlanan alan kuramları sicim kuramının ayrışma (decoupling) limitinde ortaya çıkmaktadırlar ve bu tip kuramlar komütatif benzerlerinden farklı özellikler göstermeleri nedeniyle ilgi çekmektedirler. Bu şekilde komütatif olmayan ayar kuramlarından elde edilecek sonuçların sicim kurumlarının çeşitli özelliklerinin anlaşılmasında önemli olması beklenmektedir. Dual kuramlar bir fiziksel modelin iki farklı, fakat eşdeğer formülasyonunu tanımlarlar. S dualite olarak adlandırılan kuvvetli ve zayıf etkileşimli modeller arasındaki ilişki, birinden diğerine gidilerek bir fiziksel modelin farklı etkileşme bölgelerindeki özelliklerinin çalışılmasına imkân verir. Bu çalışmada biz komütatif olmayan U(1) ayar kuramında S dualiteyi inceleyeceğiz. Öncelikle komütatif olmayan U(1) kuramının duali için hamilton fonksiyonunun nasıl kurulabileceğini inceleyeceğiz. Dual kuramda uzay ve zaman koordinatları arasıda komütatiflik özelliğinin bulunmaması nedeniyle bilinen kuantizasyon yöntemleri ile bunun nasıl yapılacağı açık değildir. Ana (parent) eylem bu iş için uygun bir araç olmaktadır. Ana eylemde uygun değişkenlere göre hareket denklemleri çözülür ve bu çözümler ana eylemde yerine konulursa orijinal kuram veya onun duali elde edilebilmektedir. Ana eylemin bu tanımlaması dual Lagrange fonksiyonu kullanılmaksızın hamilton fonksiyonunun elde edilebilmesine imkân verir. Yöntem öncelikle normal (komütatif) durum için geliştirilecek daha sonra elde edilen sonuçlar komütatif olmayan duruma genelleştirilecektir. İkinci olarak komütatif olmayan U(1) ayar kuramı ve onun duali için hem Lagrange hem de Hamilton yoğunluklarında elektrik-manyetik dualite dönüşümlerinin nasıl tanımlanacağını göstereceğiz.