Ambar N. Sengupta
We present a description of two dimensional Yang-Mills gauge theory on the plane and on compact surfaces, examining the topological, geometric and probabilistic aspects.
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To appear in Encyclopedia of Mathematical Physics, published by Elsevier in early 2006. Comments/corrections welcome. The article surveys topological aspects in gauge theories.
We show that the global aspects of Abelian and center projection of a SU(2) gauge theory on an arbitrary manifold are naturally described in terms of smooth Deligne cohomology. This is achieved through the introduction of a novel type of differential topological structure, called Cho structure. Half integral monopole charges appear naturally in this framework.
A brief review of aspects of gravity gauge theory correspondance inspired by string theory is presented.
Bethe/Gauge correspodence: a short review on an aspect of the integrability nature of supersymmetric gauge theories
本文综述了 Bethe/Gauge 对应，以 XXX 自旋链与二维 N=(2,2) 规范理论为例说明基本思想，并通过比较瞬子与涡旋配分函数探讨四维与二维理论的可积性联系。
Mathematical aspects of contemporary classical and quantum gauge theory are sketched.
Quantum theory of the gauge models in the causal approach leads to some cohomology problems. We investigate these problems in detail.
In this thesis we study some non perturbative aspects of gauge theories. We give particular emphasis on the understanding of dual confinement of magnetic monopoles by using flux tube configurations.
This thesis is about conceptual aspects of gauge theories. Gauge theories lie at the heart of modern physics: in particular, they constitute the standard model of particle physics. At its simplest, the idea of gauge is that nature is best described using a descriptively redundant language; the different descriptions are said to be related by a gauge symmetry. The over-arching question the thesis aims to answer is: how can descriptive redundancy be fruitful for physics?
We reconsider perturbative unitarity in quantum field theory and upgrade several arguments and results. The minimum assumptions that lead to the largest time equation, the cutting equations and the unitarity equation are identified. Using this knowledge and a special gauge, we give a new, simpler proof of perturbative unitarity in gauge theories and generalize it to quantum gravity, in four and higher dimensions. The special gauge interpolates between the Feynman gauge and the Coulomb gauge without double poles. When the Coulomb limit is approached, the unphysical particles drop out of the cut…
The boundary values of the time-component of the gauge potential form externally specifiable data characterizing a gauge theory. We point out some consequences such as reduced symmetries, bulk currents for manifolds with disjoint boundaries and some nuances of how the charge algebra is realized.
In this article we investigate an interpolating gauge fixing procedure in $(4l+3)$-dimensional abelian Chern-Simons theory. We show that this interpolating gauge is related to the covariant gauge in a constant anisotropic metric. We compute the corresponding propagators involved in various expressions of the linking number in various gauges. We comment on the geometric interpretations of these expressions, clarifying how to pass from one interpretation to another.
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 consider general aspects of N=2 gauge theories in three dimensions, including their multiplet structure, anomalies and non-renormalization theorems. For U(1) gauge theories, we discuss the quantum corrections to the moduli space, and their relation to ``mirror symmetries'' of 3d N=4 theories. Mirror symmetry is given an interpretation in terms of vortices. For SU(N_c) gauge groups with N_f fundamental flavors, we show that, depending on the number of flavors, there are quantum moduli spaces of vacua with various phenomena near the origin.
The confinement of quarks is one of the enduring mysteries of modern physics. There is a longstanding physics heuristic that confinement is a consequence of `unbroken center symmetry'. This article gives mathematical confirmation of this heuristic, by rigorously defining of center symmetry in lattice gauge theories and proving that a theory is confining when center symmetry is unbroken. Furthermore, a sufficient condition for unbroken center symmetry is given: It is shown that if the center of the gauge group is nontrivial, and correlations decay exponentially under arbitrary boundary conditio…
Aspects of Supergroup Gauge Theory
本文综述了超群规范理论的最新进展，从零维超矩阵模型的几何代数刻画出发，重点探讨四维超群杨-米尔斯理论的非微扰性质，包括瞬子计算、Seiberg-Witten几何、Bethe/规范对应及相交缺陷实现。
In this paper we explore some of the features of large N supersymmetric and nonsupersymmetric gauge theories using Maldacena's duality conjectures. We shall show that the resulting strong coupling behavior of the gauge theories is consistent with our qualitative expectations of these theories. Some of these consistency checks are highly nontrivial and give additional evidence for the validity of the proposed dualities.
The thermodynamic properties of a SU(3) gauge theory without quarks are calculated using a string formulation for 1.2T_c < T < 3T_c. The results are in good agreement with the lattice data. We also comment on SU(N) gauge theories.
For two decades it was believed that chiral symmetries cannot be realized in lattice field theory but this has changed now. Highlights of these new developments will be presented with emphasis on the mathematical structure of the so called ``overlap''.
At an elementary level, we present some non-perturbative aspects of non-abelian gauge theories in four dimensional space-time. Some rigorous results have been obtained in the framework of supersymmetric theories, and a very rich physics related to the strong-coupling dynamics of gauge fields has been discovered. Some of the new phenomena which appeared may play a significant role in future phenomenological models. However, quantitative computations in non-supersymmetric theories, like quantum chromodynamics, is out of reach for the moment.
The essential elements of manifestly gauge invariant exact renormalization groups are recalled. The pros and cons of the formalism are discussed and it is argued that now is the right time to try to utilize the formalism to provide new insights into the strong coupling domain of QCD.
A study of the gauged Wess-Zumino-Witten models is given focusing on the effect of topologically non-trivial configurations of gauge fields. A correlation function is expressed as an integral over a moduli space of holomorphic bundles with quasi-parabolic structure. Two actions of the fundamental group of the gauge group is defined: One on the space of gauge invariant local fields and the other on the moduli spaces. Applying these in the integral expression, we obtain a certain identity which relates correlation functions for configurations of different topologies. It gives an important inform…
It has recently been shown that generalized connections of the (A)dS space symmetry algebra provide an effective geometric and algebraic framework for all types of gauge fields in (A)dS, both for massless and partially-massless. The equations of motion are equipped with a nilpotent operator called $σ_-$ whose cohomology groups correspond to the dynamically relevant quantities like differential gauge parameters, dynamical fields, gauge invariant field equations, Bianchi identities etc. In the paper the $σ_-$-cohomology is computed for all gauge theories of this type and the field-theoretical in…
Homological aspects of topological gauge-gravity equivalence
本文利用纤维丛理论形式化了规范理论与引力间的映射，研究了内部导数约束对同调的影响，发现在 Riemann-Cartan 流形中出现了非平凡的同调。
Aspects of massive gauge fields
本文聚焦矢量场与二形式场，考察人为赋予质量的大质量规范场在取无质量极限时产生的理论问题，并揭示不同理论间的新联系。
We study aspects of gauge theory on tori which are a consequences of Morita equivalence. In particular we study the behavior of gauge theory on noncommutative tori for arbitrarily close rational values of Theta. For such values of Theta, there are Morita equivalent descriptions in terms of Yang-Mills theories on commutative tori with very different magnetic fluxes and rank. In order for the correlators of open Wilson lines to depend smoothly on Theta, the correlators of closed Wilson lines in the commutative Yang-Mills theory must satisfy strong constraints. If exactly satisfied, these constra…
For an $S_4$ space-time manifold global aspects of gauge-fixing are investigated using the relation to Topological Quantum Field Theory on the gauge group. The partition function of this TQFT is shown to compute the regularized Euler character of a suitably defined space of gauge transformations. Topological properties of the space of solutions to a covariant gauge conditon on the orbit of a particular instanton are found using the $SO(5)$ isometry group of the $S_4$ base manifold. We obtain that the Euler character of this space differs from that of an orbit in the topologically trivial secto…
In this thesis, we study the details of some fundamental cosmological problems of the gauge mediated supersymmetry breakdown and we probe the supersymmetry breaking sector by cosmological arguments. We manifest that problems like the metastable vacuum selection and the gravitino overproduction in the reheated early universe are naturally absent in the most general class of gauge mediation models without including additional ingredients or assumptions. We also find that the gravitino can generically account for the bulk dark matter of the universe. Cosmological implications of a stringy UV-comp…
We study the phase transition in gauge-Higgs unification at finite temperature. In particular, we obtain the strong first order electroweak phase transition for a simple matter content yielding the correct order of Higgs mass at zero temperature. Two stage phase transition is found for a particular matter content, which is the strong first order at each stage. We further study supersymmetric gauge models with the Scherk-Schwarz supersymmetry breaking. We again observe the first order electroweak phase transition and multi stage phase transition.
The focus of the present thesis is on the analysis of gauge symmetry and its various impacts on gauge theories. In the first part of this thesis, we have studied different aspects of symmetries in connection with various field and string theoretic models. The manifestations of gauge symmetries and their interconnection with the diffeomorphism invariances have been discussed in the context of different generally covariant models like string, membrane, $p-$brane and metric gravity. Analysis of interacting string with proper inclusion of the stringy boundary conditions led to the noncommutative n…
Aspects of confinement within non-Abelian gauge theories
本讲座介绍讨论连续介质中QCD禁闭/退禁闭相变的工具，并在基于Curci-Ferrari模型的连续介质方法中展示其应用。
In this paper we present a study based on the use of functional techniques on the issue of insertions of massive fermionic fields in the two-dimensional massless Gauged Thirring Model. As it will be shown, the fermionic mass contributes to the Green's functions in a surprisingly simple way, leaving therefore the original nonperturbative nature of the massless results still intact in the massive theory. Also, by means of complementarity, we present a second discussion of the massive model, now at its bosonic representation.
The soliton structure of a gauge theory recently proposed to describe chiral excitations in the Fractional Quantum Hall Effect is investigated. A new type of non-linear derivative Schrödinger equation emerges as an effective description of the system that supports novel chiral solitons. We discuss the classical properties of solutions with vanishing and non-vanishing boundary conditions (dark solitons) and we explain their relation to integrable systems. The quantum analysis is also addressed in the framework of a semiclassical approximation improved by Renormalization Group arguments.
We use the Chern-Simons formulation of higher spin theories in three dimensions to study aspects of holographic W-gravity. Concepts which were useful in studies of pure bulk gravity theories, such as the Fefferman-Graham gauge and the residual gauge transformations, which induce Weyl transformations in the boundary theory and their higher spin generalizations, are reformulated in the Chern-Simons language. Flat connections that correspond to conformal and lightcone gauges in the boundary theory are considered.
We discuss the BRST cohomology and exhibit a connection between the Hodge decomposition theorem and the topological properties of a two dimensional free non-Abelian gauge theory having no interaction with matter fields. The topological nature of this theory is encoded in the vanishing of the Laplacian operator when equations of motion are exploited. We obtain two sets of topological invariants with respect to BRST and co-BRST charges on the two dimensional manifold and show that the Lagrangian density of the theory can be expressed as the sum of terms that are BRST- and co-BRST invariants.
We discuss some aspects of the topological features of a non-interacting two (1+1)-dimensional Abelian gauge theory in the framework of superfield formalism. This theory is described by a BRST invariant Lagrangian density in the Feynman gauge. We express the local and continuous symmetries, Lagrangian density, topological invariants and symmetric energy momentum tensor of this theory in the language of superfields by exploiting the nilpotent (anti-)BRST- and (anti-)co-BRST symmetries. In particular, the Lagrangian density and symmetric energy momentum tensor of this topological theory turn out…
These notes comprise the first of two articles devoted to the construction of exact solutions of noncommutative gauge theory in two spacetime dimensions. This first part deals solely with the classical theory on a noncommutative torus. Topics covered include a mathematical introduction to the geometry of the noncommutative torus, the definition, properties and symmetries of noncommutative Yang-Mills theory, and the complete solution of the classical field equations.
In this PhD thesis, we first look at some manifestations of CP violation, such as electric dipole moments (EDMs) of elementary particles, both in the weak sector in the presence of neutrino masses (absent in the SM) as well as in the strong sector. Then, in a second study, we build an effective field theory for gauge bosons. We generalize the Euler-Heisenberg Lagrangian for the study of photon interactions, considered as the archetype of an effective field theory; to gauge bosons of arbitrary Lie algebras, several Grand Unification groups and some mixed symmetries.
In this thesis, we present two aspects of higher-spin gauge field theories: dualities and interactions. We first consider dualities of the free theories at the level of the action. Then, external "electric" and "magnetic" sources are introduced a la Dirac, which leads to a quantization condition on the product of conserved "electric" and "magnetic" higher-spin charges. In the second part, higher-spin interactions are studied in the field-antifield (or BRST) formalism. Theorems are proved about local interactions of sets of exotic spin-two fields, as well as of sets of symmetric spin-three fiel…
We study almost-forward scattering in the context of usual and non-commutative QED. We study the semi-classical behaviour of particles undergoing this scattering process in the two theories, and show that the shock wave picture, effective in QED fails for NCQED. Further, we show that whereas in QED, there are no leading logarithmic contributions to the amplitude upto sixth order, uncancelled logarithms appear in NCQED.
A large class of non-minimal gauge mediation models, such as (semi-)direct gauge mediation, predict a hierarchy between the masses of the supersymmetric standard model gauginos and those of scalar particles. We perform a comprehensive study of these non-minimal gauge mediation models, including mass calculations in semi-direct gauge mediation, to illustrate these features, and discuss the phenomenology of the models. We point out that the cosmological gravitino problem places stringent constraints on mass splittings, when the Bino is the NLSP. However, the GUT relation of the gaugino masses is…
The gauge transformation properties of the Skyrme--Chern-Simons (SCS) densities is studied. Two types of SCS actions are identified, Type_{I} in which the gauge group is smaller than the largest possible one, and Type_{II} which are gauged with the largest allowed gauge group. Type_{I} SCS feature only one power of the gauge connection and no curvature, while Type_{II} feature both the gauge connection and the curvature. The Abelian Type_{I} SCS turn out to be explicitly gauge invariant while non-Abelian Type_{I} and all Type_{II} SCS are gauge invariant only up to a total divergence term, and…
In these lecture notes we will try to give an introduction to the use of the mathematics of fibre bundles in the understanding of some global aspects of gauge theories, such as monopoles and instantons. They are primarily aimed at beginning PhD students. First, we will briefly review the concept of a fibre bundle and define the notion of a connection and its curvature on a principal bundle. Then we will introduce some ideas from topology such as homotopy, topological degree and characteristic classes. Finally, we will apply these notions to the bundle setup corresponding to monopoles and insta…
We give an introductory review of gauge/gravity duality, and associated ideas of holography, emphasising the conceptual aspects. The opening Sections gather the ingredients, viz. anti-de Sitter spacetime, conformal field theory and string theory, that we need for presenting, in Section 5, the central and original example: Maldacena's AdS/CFT correspondence. Sections 6 and 7 develop the ideas of this example, also in applications to condensed matter systems, QCD, and hydrodynamics. Sections 8 and 9 discuss the possible extensions of holographic ideas to de Sitter spacetime and to black holes. S…
New aspects of gauge-gravity relation
本文探讨了四维$SO(4)$纯Yang-Mills理论与引力的关系，将理论泛函积分改写为引力度规和Riemann张量，并借此推导出平均Wilson loop产生的线性势。
We discuss some aspects of cohomological properties of a two-dimensional free Abelian gauge theory in the framework of BRST formalism. We derive the conserved and nilpotent BRST- and co-BRST charges and express the Hodge decomposition theorem in terms of these charges and a conserved bosonic charge corresponding to the Laplacian operator. It is because of the topological nature of free U(1) gauge theory that the Laplacian operator goes to zero when equations of motion are exploited. We derive two sets of topological invariants which are related to each-other by a certain kind of duality transf…
Recently we provided a microscopic derivation of the exact supergravity profile for the twisted scalar field emitted by systems of fractional D3-branes at a Z2 orbifold singularity. In this contribution we focus on a set-up supporting an N = 2 SYM theory with SU(2) gauge group and Nf=4. We take into account the tower of D-instanton corrections to the source terms for the twisted scalar and find that its profile can be expressed in terms of the chiral ring elements of the gauge theory. We show how the twisted scalar, which at the perturbative level represents the gravity counterpart of the gaug…
We investigate new aspects related to the abelian gauge-Higgs model with the addition of the Carroll-Field-Jackiw term. We focus on one-loop quantum corrections to the photon and Higgs sectors due to spontaneous breaking of gauge symmetry and show that new finite and definite Lorentz-breaking terms are induced. Specifically in the gauge sector, a CPT-even aether term is induced. Besides, aspects of the one-loop renormalization of the background vector dependent terms are discussed.
The renormalization of N=1 Super Yang-Mills theory is analysed in the Wess-Zumino gauge, employing the Landau condition. An all orders proof of the renormalizability of the theory is given by means of the Algebraic Renormalization procedure. Only three renormalization constants are needed, which can be identified with the coupling constant, gauge field and gluino renormalization. The non-renormalization theorem of the gluon-ghost-antighost vertex in the Landau gauge is shown to remain valid in N=1 Super Yang-Mills. Moreover, due to the non-linear realization of the supersymmetry in the Wess-Zu…
Using a synthesis of the functional integral and operator approaches we discuss the fermion-boson mapping and the role played by the Bose field algebra in the Hilbert space of two-dimensional gauge and anomalous gauge field theories with massive fermions. In the $QED_2$ with quartic self-interaction among massive fermions, the use of an auxiliary vector field introduces a redundant Bose field algebra that should not be considered as an element of the intrinsic algebraic structure defining the model. In the anomalous chiral $QED_2$ with massive fermions the effect of the chiral anomaly leads to…
A supersymmetric non-Abelian self-dual gauge theory with the explicit introduction of Kaluza-Klein modes is proposed to give a classical description of multiple M5-branes on $R^5 \times S^1$. The gauge symmetry is parametrized by Lie-algebra valued 1-forms with the redundancy of a 0-form, and the supersymmetry transformations without gauge-fixing are given. We study BPS configurations involving KK modes, including M-waves and M2-branes with non-trivial distributions around the circle. Finally, this supersymmetric gauge theory of two-forms can be equipped with more general non-Abelian gerbes in…
Two aspects of the color charge in Coulomb gauge continuum Yang-Mills theory are discussed. The first aspect is the existence of a conserved and vanishing total charge exhibited within the first order functional formalism. The second aspect is the closure of the set of Slavnov-Taylor identities in the second order functional formalism, such that the exact solution for temporal Green's functions is in principle possible and thereby preserving the color charge.
We write down an $\mathcal{N}=1$ supersymmetric extension for non-Abelian gauge theories in (1+3) dimensions with a Lorentz- and CPT-violating term of the Carroll-Field-Jackiw type. By including effects of the background (supersymmetric) fermion bilinears that accompany Lorentz-symmetry violation in a Carroll-Field-Jackiw scenario, we investigate both the gauge boson and gaugino dispersion relations in order to compute their respective masses in terms of the background structures. Such results open up a potential path towards a possible mechanism for gaugino-gauge boson conversion, an analogue…
I analyze the IR fixed point structure of Landau gauge QCD. Precisely the fixed point with a strong kinematic singularity of the quark-gluon vertex that proved crucial for the recently proposed confinement mechanism in the quenched approximation is absent in dynamical QCD. Therefore, the IR singularities do not induce asymptotic quark confinement but the long-range interaction is screened by unquenching loops at scales of the order of the quark mass. This provides the prerequisite for a microscopic description of deconfinement and string breaking. The fixed points determine the qualitative for…
We present and study a 4d Chern-Simons (CS) model whose gauge symmetry is encoded in a balanced Lie group crossed module. Using the derived formal set-up recently found, the model can be formulated in a way that in many respects closely parallels that of the familiar 3d CS one. In spite of these formal resemblance, the gauge invariance properties of the 4d CS model differ considerably. The 4d CS action is fully gauge invariant if the underlying base 4fold has no boundary. When it does, the action is gauge variant, the gauge variation being a boundary term. If certain boundary conditions are im…
The introduction and quantization of a center-of-mass coordinate is demonstrated for the one-soliton sector of nonlinear field theories in (1+1) dimensions. The present approach strongly emphazises the gauge and BRST-symmetry aspects of collective coordinate quantization. A gauge is presented which is independent of any approximation scheme and which allows to interpret the new degree of freedom as the {\em quantized} center of mass coordinate of a soliton. Lorentz invariance is used from the beginning to introduce fluctuations of the collective coordinate in the {\em rest frame} of the {\em m…
Phenomenological consequences of the infrared singular, instantaneous part of the gluon propagator in Coulomb gauge are investigated. The corresponding quark Dyson-Schwinger equation is solved, neglecting retardation and transverse gluons and regulating the resulting infrared singularities. While the quark propagator vanishes as the infrared regulator goes to zero, the frequency integral over the quark propagator stays finite and well-defined. Solutions of the homogeneous Bethe-Salpeter equation for the pseudoscalar and vector mesons as well as for scalar and axial-vector diquarks are obtained…
N=2 supersymmetric Yang-Mills theories for all classical gauge groups, that is, for SU(N), SO(N), and Sp(N) is considered. The formal expression for almost all models accepted by the asymptotic freedom are obtained. The equations which define the Seiberg-Witten curve are proposed. In some cases they are solved. It is shown that for all considered the 1-instanton corrections which follows from these equations agree with the direct computations. Also they agree with the computations based on Seiberg-Witten curves which come from the M-theory consideration. It is shown that for a large class of m…
Absolute confinement of its color charges is a natural property of gauge theories such as quantum chromodynamics. On the one hand, it can be attributed to the existence of color-magnetic monopoles, a topological feature of the theory, but one can also maintain that all non-Abelian gauge theories confine. It is illustrated how ``confinement'' works in the SU(2) sector of the Standard Model, and why for example the electron and its neutrino can be viewed as SU(2)-hadronic bound states rather than a gauge doublet. The mechanism called `Abelian projection' then puts the Abelian sector of any gauge…
In this Letter we consider SU(2) Yang-Mills theory analysed in Cho-Faddeev-Niemi variables which remains invariant under local gauge transformations. The BRST symmetries of this theory is generalized by making the infinitesimal parameter finite and field-dependent. Further, we show that under appropriate choices of finite and field-dependent parameter, the gauge-fixing and ghost terms corresponding to Landau as well as maximal Abelian gauge for such Cho-Faddeev-Niemi decomposed theory appear naturally within functional integral through Jacobian calculation.
The metric algebroid proposed by Vaisman (the Vaisman algebroid) governs the gauge symmetry algebra generated by the C-bracket in double field theory (DFT). We show that the Vaisman algebroid is obtained by an analogue of the Drinfel'd double of Lie algebroids. Based on a geometric realization of doubled space-time as a para-Hermitian manifold, we examine exterior algebras and a para-Dolbeault cohomology on DFT and discuss the structure of the Drinfel'd double behind the DFT gauge symmetry. Similar to the Courant algebroid in the generalized geometry, Lagrangian subbundles $(L,\tilde{L})$ in a…
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