Sunandan Gangopadhyay
The central theme of this thesis is to study some aspects of noncommutative quantum mechanics and noncommutative quantum field theory. We explore how noncommutative structures can emerge and study the consequences of such structures in various physical models.
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I review my results about noncommutative gauge theories and about the relation of these theories to M(atrix) theory following my lecture on ICMP 2000.
Classical and quantum aspects of noncommutative field theories are discussed. In particular, noncommutative solitons and instantons are constructed and also d=2,3 noncommutative fermion and bosonic (Wess-Zumino-Witten and Chern-Simons)theories are analyzed.
We discuss the renormalization properties of noncommutative non-gauge supersymmetric field theories.
We consider gauge theories on noncommutative euclidean space . In particular, we discuss the structure of gauge group following standard mathematical definitions and using the ideas of hep-th/0102182.
We present a pedagogical introduction into noncommutative gauge theories, their stringy origin, and non-perturbative effects, including monopole and instanton solutions
Via descent equations we derive formulas for consistent gauge anomalies in noncommutative Yang-Mills theories.
We consider the ultra-violet divergence structure of general noncommutative supersymmetric $U(N_c)$ gauge theories, and seek theories which are all-orders finite.
We give an explicit proof that the noncommutative U(N) gauge theories are one-loop renormalizable
We study the perturbative unitarity of noncommutative scalar field theories. Field theories with space-time noncommutativity do not have a unitary S-matrix. Field theories with only space noncommutativity are perturbatively unitary. This can be understood from string theory, since space noncommutative field theories describe a low energy limit of string theory in a background magnetic field. On the other hand, there is no regime in which space-time noncommutative field theory is an appropriate description of string theory. Whenever space-time noncommutative field theory becomes relevant massiv…
We establish the ultra-violet finiteness of various classes of noncommutative gauge theories.
Supersymmetric field theories on noncommutative spaces are constructed. We present two different representations of noncommutative space, but we can obtain supersymmetry algebla and supersymmetric Yang-Mills action independent of its representation. As a result, we will see that the action has a close relationship with IIB matrix model.
We discuss some aspects of noncommutative quantum field theories obtained from the Seiberg-Witten limit of string theories in the presence of an external B-field. General properties of these theories are studied as well as the phenomenological potential of noncommutative QED.
Simple argument in favour of unitarity, to all orders, of space-like noncommutative theory is given.
This is a reply to "Comment on 'Noncommutative gauge theories and Lorentz symmetry,'" Phys. Rev. D 77 (2008) 048701 by Alfredo Iorio.
First, we briefly review the description of gravity theories as gauge theories in three and four dimensions. Specifically, we recall the procedure in which the results of General Relativity in three and four dimensions are recovered in a gauge-theoretic approach. Also, the procedure is applied for the case of the Weyl gravity, too. Then, after reminding briefly the formulation of gauge theories on noncommutative spaces, we review our most recent works in which gravity models are constructed as gauge theories on noncommutative spaces.
We discuss some properties of noncommutative supersymmetric field theories which do not involve gauge fields. We concentrate on the renormalizability issue of these theories.
A BRST-cohomological analysis of Seiberg-Witten maps and results on gauge anomalies in noncommutative Yang-Mills theories with general gauge groups are reviewed.
Gauge theories are formulated on the noncommutative two-sphere. These theories have only finite number of degrees of freedom, nevertheless they exhibit both the gauge symmetry and the SU(2) "Poincaré" symmetry of the sphere. In particular, the coupling of gauge fields to chiral fermions is naturally achieved.
We construct noncommutative versions of both the minimal and the new minimal supersymmetric Grand Unified Theories. The enveloping-algebra formalism is used to carry out such constructions. The beautiful formulation of the Higgs sector of these noncommutative theories is a consequence of fact that, in the GUTs at hand, the ordinary Higgs fields can be realized as elements of the Clifford algebra Cl10(C). In the noncommutative supersymmetric GUTs we formulate, supersymmetry is linearly realized by the noncommutative fields; but it is not realized by the ordinary fields that define those noncomm…
We propose a physical interpretation of the perturbative breakdown of unitarity in time-like noncommutative field theories in terms of production of tachyonic particles. These particles may be viewed as a remnant of a continuous spectrum of undecoupled closed-string modes. In this way, we give a unified view of the string-theoretical and the field-theoretical no-go arguments against time-like noncommutative theories. We also perform a quantitative study of various locality and causality properties of noncommutative field theories at the quantum level.
We discuss some exact Seiberg--Witten-type maps for noncommutative electrodynamics. Their implications for anomalies in different (noncommutative and commutative) descriptions are also analysed.
We reconsider the perturbative expansion of the Wilson loop in 2d noncommutative gauge theories, using an improved integration method. For the class of maximally crossed diagrams in the $θ\to \infty$ limit we find an intriguing formula, easily generalizable to all orders in perturbation theory.
We describe some recent progress in our understanding of Yang-Mills theories formulated on noncommutative spaces and in particular how to formulate the standard model on such spaces.
We study a class of noncommutative geometries that give rise to dimensionally reduced Yang-Mills theories. The emerging geometries describe sets of copies of an even dimensional manifold. Similarities to the D-branes in string theory are discussed.
In this thesis we study different aspects of noncommutativity in quantum mechanics, field theory and gravity. We give particular emphasis on the underlying symmetries of these theories. Deformations of usual symmetries like the external space time symmetries and the internal gauge symmetry in noncommutative theories are analyzed in detail.
In this thesis we will discuss various aspects of noncommutative geometry and compactified Little-String theories. First we will give an introduction to the use of noncommutative geometry in string theory. Thereafter we will present a proof of the connection between D-brane dynamics and noncommutative geometry. Then we will explain the concept of instantons in noncommutative gauge theories. The last chapters shift the focus to Little-String- and $(2,0)$-theories. We study compactifications of these theories on tori with twists. First we study the case of two coinciding branes in detail. Afterw…
We explicitly derive, following a Noether-like approach, the criteria for preserving Poincare invariance in noncommutative gauge theories. Using these criteria we discuss the various spacetime symmetries in such theories. It is shown that, interpreted appropriately, Poincare invariance holds. The analysis is performed in both the commutative as well as noncommutative descriptions and a compatibility between the two is also established.
Anomalies arising from nonplanar triangle diagrams of noncommutative gauge theory are studied. Local chiral gauge anomalies for both noncommutative U(1) and U(N) gauge theories with adjoint matter fields are shown to vanish. For noncommutative QED with fundamental matters, due to UV/IR mixing a finite anomaly emerges from the nonplanar contributions. It involves a generalized $\star$-product of gauge fields.
We start by reviewing the formulation of noncommutative quantum mechanics as a constrained system. Then, we address to the problem of field theories defined on a noncommutative space-time manifold. The Moyal product is introduced and the appearance of the UV/IR mechanism is exemplified. The emphasis is on finding and analyzing noncommutative quantum field theories which are renormalizable and free of nonintegrable infrared singularities. In this last connection we give a detailed discussion of the quantization of the noncommutative Wess-Zumino model as well as of its low energy behavior.
A new noncommutative model invariant with respect to U(1) gauge group is proposed. The model is free of nonintegrable infrared singularities. Its commutative classical limit describes a free scalar field. Generalization to U(N) models is also considered.
We study noncommutative deformations of Yang-Mills theories and show that these theories admit a infinite, continuous family of twisted star-gauge invariances. This family interpolates continuously between star-gauge and twisted gauge transformations. The possible physical role of these start-twisted invariances is discussed.
We study the quantum mechanical consistency of noncommutative gauge theories by perturbatively analyzing the Wilsonian quantum effective action in the matrix formulation. In the process of integrating out UV states, we find new divergences having dual UV-IR interpretations and no analogues in ordinary quantum field theories. The appearance of these new UV-IR divergences has profound consequences for the renormalizability of the theory. In particular, renormalizability fails in any nonsupersymmetric noncommutative gauge theory. In fact, we argue that renormalizability generally fails in any non…
We implement the Hamiltonian treatment of a nonAbelian noncommutative gauge theory, considering with some detail the algebraic structure of the noncommutative symmetry group. The first class constraints and Hamiltonian are obtained and their algebra derived, as well as the form of the gauge invariance they impose on the first order action.
Planarity was introduced by 't Hooft in his topological classification of diagrams in the large-N limit of U(N) gauge theories. Planarity also occurs in noncommutative field theories where amplitudes possess invariance only under cyclic permutations, a feature inherited from the parent string theory. In noncommutative gauge theories both kinds of planarity merge in a context which turns out to be particularly intriguing in the two-dimensional case where gauge invariant correlators can be explicitly computed.
In a recent paper hep-ph/0208225 it has been claimed that to one-loop order in noncommutative phi^4 scalar field theory using dimensional regularization the UV and IR divergencies decouple. We point out that this statement is incorrect.
We calculate conformal anomalies in noncommutative gauge theories by using the path integral method (Fujikawa's method). Along with the axial anomalies and chiral gauge anomalies, conformal anomalies take the form of the straightforward Moyal deformation in the corresponding conformal anomalies in ordinary gauge theories. However, the Moyal star product leads to the difference in the coefficient of the conformal anomalies between noncommutative gauge theories and ordinary gauge theories. The $β$ (Callan-Symanzik) functions which are evaluated from the coefficient of the conformal anomalies coi…
We elaborate on the dynamics of noncommutative two-dimensional gauge field theories. We consider U(N) gauge theories with fermions in either the fundamental or the adjoint representation. Noncommutativity leads to a rather non-trivial dependence on theta (the noncommutativity parameter) and to a rich dynamics. In particular the mass spectrum of the noncommutative U(1) theory with adjoint matter is similar to that of ordinary (commutative) two-dimensional large-N SU(N) gauge theory with adjoint matter. The noncommutative version of the 't Hooft model receives a non-trivial contribution to the v…
The relation of the noncommutative self-dual Chern-Simons (NCSDCS) system to the noncommutative generalizations of Toda and of affine Toda field theories is investigated more deeply. This paper continues the programme initiated in $JHEP {\bf 10} (2005) 071$, where it was presented how it is possible to define Toda field theories through second order differential equation systems starting from the NCSDCS system. Here we show that using the connection of the NCSDCS to the noncommutative chiral model, exact solutions of the Toda field theories can be also constructed by means of the noncommutativ…
We present a formalism to explicitly construct non-Abelian gauge theories on noncommutative spaces (induced via a star product with aconstant Poisson tensor) from a consistency relation. This results in an expansion of the gauge parameter, the noncommutative gauge potential and fields in the fundamental representation, in powers of a parameter of the noncommutativity. This allows the explicit construction of actions for these gauge theories.
T-duality of gauge theories on a noncommutative $T^d$ can be extended to include fields with twisted boundary conditions. The resulting T-dual theories contain novel nonlocal fields. These fields represent dipoles of constant magnitude. Several unique properties of field theories on noncommutative spaces have simpler counterparts in the dipole-theories.
We prove by explicit calculation that Feynman graphs in noncommutative Yang-Mills theory made of repeated insertions into itself of arbitrarily many one-loop ghost propagator corrections are renormalizable by local counterterms. This provides a strong support for the renormalizability conjecture of that model.
The perturbative approach to quantum field theory using retarded functions is extended to noncommutative theories. Unitarity as well as quantized equations of motion are studied and seen to cause problems in the case of space-time noncommutativity. A modified theory is suggested that is unitary and preserves the classical equations of motion on the quantum level.
We study chiral symmetries of fermionic non commutative dipole theories. By using Fujikawa's approach we obtain explicit expressions of the anomalies for Dirac and chiral fermions in 2 and 4 dimensions.
We develop a formalism to realize algebras defined by relations on function spaces. For this porpose we construct the Weyl-ordered star-product and present a method how to calculate star-products with the help of commuting vector fields. Concepts developed in noncommutative differential geometry will be applied to this type of algebras and we construct actions for noncommutative field theories. Derivations of star-products makes it further possible to extend noncommutative gauge theory in the Seiberg-Witten formalism with covariant derivatives. In the commutative limit these theories are becom…
We show that a solution of the type IIB supergravity representing D3-branes in the presence of a 2-form background has 16 supersymmetries by explicitly constructing the transformation parameter of unbroken supersymmetry. This solution is dual to a noncommutative Yang-Mills theory in a certain limit.
In this brief report, the microcausility of quantum field theory on spin-induced noncom- mutative spacetime is discussed. It is found that for spacelike seperation the microcausality is not obeyed by the theory generally. It means that Lorentz covariance can not guaran- tee microcausality in quantum field thoery. We also give some comments about quantum field thoeries on such noncommutative spacetime and the relations between noncommutative spacetime and causality.
Free noncommutative fields constitute a natural and interesting example of constrained theories with higher derivatives. The quantization methods involving constraints in the higher derivative formalism can be nicely applied to these systems. We study real and complex free noncommutative scalar fields where momenta have an infinite number of terms. We show that these expressions can be summed in a closed way and lead to a set of Dirac brackets which matches the usual corresponding brackets of the commutative case.
The possibility of noncommutative topological gravity arising in the same manner as Yang-Mills theory is explored. We use the Seiberg-Witten map to construct such a theory based on a SL(2,C) complex connection, from which the Euler characteristic and the signature invariant are obtained. This gives us a way towards the description of noncommutative gravitational instantons as well as noncommutative local gravitational anomalies.
We construct noncommutative gauge theories based on the notion of the Weyl bundle, which appears in Fedosov's construction of deformation quantization on an arbitrary symplectic manifold. These correspond to D-brane worldvolume theories in non-constant B-field and curved backgrounds in string theory. All such theories are embedded into a "universal" gauge theory of the Weyl bundle. This shows that the combination of a background field and a noncommutative field strength has universal meaning as a field strength of the Weyl bundle. We also show that the gauge equivalence relation is a part of s…
The energy-momentum tensor of Matrix Theory is derived by computing disk amplitudes with one closed string and an arbitrary number of open strings and by taking the DKPS limit. We clarify its relation to the energy-momentum tensor of the noncommutative gauge theory derived in our previous paper.
We construct a modification of the Poisson bracket which is suitable for a canonical analysis of space-time noncommutative field theories. We show that this bracket satisfies the Jacobi identities and generates equations of motion. In this modified canonical formalism one can define the notion of the first-class constraints, demonstrate that they generate gauge symmetries, and derive an explicit form of these symmetry transformations.
Field theory and gauge theory on noncommutative spaces have been established as their own areas of research in recent years. The hope prevails that a noncommutative gauge theory will deliver testable experimental predictions and will thus be a serious candidate for an extension of the Standard Model. This note contains the results for expanded gauge theory actions on a noncommutative space with constant theta, up to second order, together with a discussion of the ambiguities of the expanded theory and how they affect the action.
Open Wilson line operators and generalized star product have been studied extensively in noncommutative gauge theories. We show that they also show up in noncommutative scalar field theories as universal structures. We first point out that dipole picture of noncommutative geometry provides an intuitive argument for robustness of the open Wilson lines and generalized star products therein. We calculate one-loop effective action of noncommutative scalar field theory with cubic self-interaction and show explicitly that the generalized star products arise in the nonplanar part. It is shown that, a…
The extension of the noncommutative u*(N) Lie algebra to noncommutative orthogonal and symplectic Lie algebras is studied. Using an anti-automorphism of the star-matrix algebra, we show that the u*(N) can consistently be restricted to o*(N) and usp*(N) algebras that have new mathematical structures. We give explicit fundamental matrix representations of these algebras, through which the formulation for the corresponding noncommutative gauge field theories are obtained. In addition, we present a D-brane configuration with an orientifold which realizes geometrically our algebraic construction, t…
In this talk I briefly review recent developments in quantum field theories on a noncommutative Euclidean space, with Heisenberg-like commutation relations between coordinates. I will be concentrated on new physics learned from this simplest class of non-local field theories, which has applications to both string theory and condensed matter systems, and possibly to particle phenomenology.
Generalizations of GL(n) abelian Toda and $\widetilde{GL}(n)$ abelian affine Toda field theories to the noncommutative plane are constructed. Our proposal relies on the noncommutative extension of a zero-curvature condition satisfied by algebra-valued gauge potentials dependent on the fields. This condition can be expressed as noncommutative Leznov-Saveliev equations which make possible to define the noncommutative generalizations as systems of second order differential equations, with an infinite chain of conserved currents. The actions corresponding to these field theories are also provided.…
Some equivalence classes in symmetric group lead to an interesting class of noncommutive and associative algebras. From these algebras we construct noncommutative Frobenius algebras. Based on the correspondence between Frobenius algebras and two-dimensional topological field theories, the noncommutative Frobenius algebras can be interpreted as topological open string theories. It is observed that the centre of the algebras are related to closed string theories via open-closed duality. Area-dependent two-dimensional field theories are also studied.
We study the perturbative approach to the Wilsonian integration of noncommutative gauge theories in the matrix representation. We begin by motivating the study of noncommutative gauge theories and reviewing the matrix formulation. We then systematically develop the perturbative treatment of UV states and calculate both the leading and next to leading order one- and two-loop corrections to the quantum effective action. Throughout, we discuss how our formalism clarifies problems associated with UV-IR mixing, a particular emphasis being placed on the dipole structure imposed by noncommutative gau…
We argue that Yang-Mills theory on noncommutative torus, expressed in the Fourrier modes, is described by a gauge theory in a usual commutative space, the gauge group being a generalization of the area-preserving diffeomorphisms to the noncommutative case. In this way, performing the loop calculation in this gauge theory in the continuum limit we show that this theory is {\it one loop renormalizable}, and discuss the UV and IR limits. The moduli space of the vacua of the noncommutative super Yang-Mills theories in (2+1) dimensions is discussed.
The recently found non-critical open string theories is reviewed. These open strings, noncommutative open string theories (NCOS), arise as consistent quantum theories describing the low energy theory of D-branes in a background electric B-field in the critical limit. Focusing on the D3-brane case, we construct the most general (3+1) NCOS, which is described by four parameters. We study S and T -dualities of these theories and argue the existence of a U-duality group.
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