Nonanticommutative N=1/2 supersymmetric gauge theory

Abstract

The idea behind the consideration of a noncommutative spacetime was to introduce an effective ultraviolet cutoff. However this view doesn’t hold anymore. Instead it is hoped that with the help of the dynamics of the noncommutative field theories the underlying geometry of string theory will be better understood. String theory with D–branes solved in the presence of a Neveu-Schwarz–Neveu-Schwarz background field leads to noncommutative coordinates.Noncommutativity may be introduced through the following star product which is the deformation of multiplication of functions in space–time:$f(x)*g(x)=f(x)exp(frac{1}{2}overleftarrow{partial_x}theta^{ij}overrightarrow{partial_y})g(x)$Here $theta^{ij}$ is the antisymmetric noncommutativity parameter. Another notion that we will use in this work is supersymmetry. By definition supersymmetry is a symmetry between fermions and bosons. Using the supersymmetry generators Q we can express this in the following way: Q |boson> = Q |fermion> Q |fermion> = Q |boson> N=1 supersymmetric field theories are the best candidates for a generalization of the standard model of the elementary particles. It is useful to construct supersymmetric field theories in superspace formalism. A superfield is defined as a function in superspace which is parametrized by $(x_{mu},theta_{alpha},overline{theta_{dot{alpha}}})$ μ α α θ θ 􀀅 coordinates ($mu=0,1,2,3$ and $alpha,dot{alpha}=1,2$). Here $theta_{alpha}$ and $overline{theta_{dot{alpha}}}$ are anticommutative Weyl spinors which satisfy the following anticommutation relations:${theta_{alpha},theta_{beta}}={overline{theta_{dot{alpha}}},overline{theta_{dot{beta}}}}={theta_{alpha},overline{theta_{dot{alpha}}}}=0$D–branes are hypersurfaces on which open strings can end. In a certain low energy limit the string dynamics on the world volume of the D–brane is defined by the Yang–Mills fields. Considering a D– brane in a Ramond–Ramond (graviphoton) background one finds that the superspace is deformed and the N=1 supersymmetry is broken to N=1/2 supersymmetry. Q supercharges remain as a symmetry of the superspace , while the $overline{Q}$ are broken due to their dependence on $overline{theta}$ coordinates. If functions in superspace are expressed in terms of $y, theta_{alpha}$ and $overline{theta_{dot{alpha}}}$, where $y^{mu} = x^{mu} + theta^{alpha}sigma^{mu}_{alphadot{alpha}}overline{theta^{dot{alpha}}}$ which satisfy $[y^{mu},x^{mu}]=0$, then the multiplication can be replaced by the following star product where the derivatives are taken at constant y and $theta_{alpha}$:$f(theta)*g(theta)=f(theta)exp(-frac{C^{alphabeta}}{2}frac{overleftarrow{partial}}{partialtheta^{alpha}}frac{overleftarrow{partial}}{partialtheta^{beta}})g(theta)$Here $C^{alphabeta}$, which is the symmetric deformation parameter, is defined by $C^{mu v}equiv C^{alphabeta} varepsilon_{betagamma} {sigma_{alpha}}^{mu v gamma}$ where $C^{mu v}$ a self–dual graviphoton field strength. Thus instead of coordinates which are operators, one deals with the usual superspace variables. To get a better understanding of the open string dynamics, N=1/2 supersymmetric gauge theory needs to be further investigated. Nonanticommutative supersymmetric theories are defined in Euclidean space where the fermionic coordinates are not related by complex conjugation. When the fermionic $theta$ coordinates are chosen to satisfy ${theta^{alpha},theta^{beta}}=C^{alphabeta}$ and ${overline{theta^{dot{alpha}}},overline{theta^{dot{beta}}}}=0 overline{theta}$ will no more be the complex conjugation of $theta$ . In the present work we will investigate the S–duality properties of nonanticommutative N=1/2 supersymmetric U(1) gauge theory using the parent action formalism. The notion of duality is very important as it makes the calculations easier. S– duality transformations can be obtained by exchanging original fields with their duals. It maps the states and vacua of a theory with coupling constant g to those of a theory with a coupling constant 1/g. Thus one can always benefit from perturbative calculation method. For simple theories like U(1) gauge theory S-duality property can be shown by rescaling its gauge fields. However, to study more complicated theories, such as noncommutative or nonanticommutative U(1) gauge theories, it is more convenient to use parent action formalism. By definition a parent action should give the original theory if the dual fields are eliminated using the equations of motion and vice versa. By showing the equivalence of the partition functions of the two theories we will conclude that the nonanticommutative N=1/2 supersymmetric U(1) gauge theory is invariant under S– duality.

Özet

Sicim teorisi, fonda bir Neveu-Schwarz–Neveu-Schwarz alanı varlığında çözüldüğünde bozonik koordinatların nonkomutatifliği, Ramond–Ramond alanı varlığında çözüldüğünde ise fermiyonik koordinatların nonantikomutatifliği ortaya çıkmaktadır. Nonkomutatiflik veya nonantikomutatiflik uzayın deforme edilmesiyle de elde edilebilir. Bu durumda çarpma işleminin yerini yıldız çarpımı alır. Nonkomutatif uzayda tanımlanan alan teorileri literatürde geniş ölçüde incelenmiştir. Deforme olmuş süper uzay (nonantikomutatif süperuzay) yeni yeni incelenmeye başlanmıştır. Sicim teorisi bir D–brane’in varlığında, sabit bir Ramond–Ramond alanı (gravifoton) fonunda çözüldüğünde süperuzayın deforme olduğu (nonantikomutatif hale geldiği) ve bu deformasyonu süpersimetrinin yarısını kırdığı görülmüştür. Süpersimetri üreteçleri olan Q süperyükleri korunurken, $overline{Q}$ süperyükleri $overline{theta}$ ’a bağlı olmaları nedeniyle süperuzayın bir simetrisi olmaktan çıkmaktadır. N=1 teorisinin simetrisinin yarısı kırıldığından geriye kalan simetri N=1/2 süpersimetrisi olarak adlandırılır. Uygun bir limitte buradaki D–brane yaşam alanındaki açık sicim dinamiği Yang–Mills alanlarıyla tanımlanır. Bu yüzden de Ramond–Ramond fonundaki açık sicim dinamiğinin anlaşılması N=1/2 süpersimetrik ayar teorisinin irdelenmesiyle olacaktır. Bu çalışmada nonantikomutatif N=1/2 süpersimetrik alan teorisi zayıf–kuvvetli etkileşme dualitesi (S–dualite) açısından incelenecektir. Ana eyleme ait bölüşüm fonksiyonu kullanarak, ana eylemin ürettiği teorilerin bölüşüm fonksiyonlarının denkliği gösterilecektir. S–dualitesi kuvvetli etkileşme alanlarından zayıf etkileşme alanlarına bir gönderimdir. Eğer bir teori S–dualite altında değişmez kalıyorsa zayıf etkileşme alanında yapılan hesaplamalar kuvvetli etkileşme alanındakilere dönüştürülebilir. Bu da pertürbatif hesap tekniğinden yararlanılmasına olanak sağladığı için çok önemlidir.