Class lecture notes at a beginning graduate level on the mathematical background needed to understand classical gauge theory. Covers group actions, fiber bundles, principal bundles, connections, gauge transformations, parallel transport, curvature, covariant derivatives, pseudo-riemannian manifolds, lagrangians, clifford algebras, spin bundles, and the Dirac operator. Requires an elementary knowledge of groups, manifolds, lie groups and algebras, and mutlilinear algebra.
Gravitation theory is formulated as gauge theory on natural bundles with spontaneous symmetry breaking where gauge symmetries are general covariant transformations, gauge fields are general linear connections, and Higgs fields are pseudo-Riemannian metrics.
In these lectures I present a review of non-perturbative instanton effects in quantum theories, with a focus on large N gauge theories and matrix models. I first consider the structure of these effects in the case of ordinary differential equations, which provide a model for more complicated theories, and I introduce in a pedagogical way some technology from resurgent analysis, like trans-series and the resurgent version of the Stokes phenomenon. After reviewing instanton effects in quantum mechanics and quantum field theory, I address general aspects of large N instantons and then present a d…
These are notes for four lectures on higher structures in M-theory as presented at workshops at the Erwin Schroedinger Institute and Tohoku University. The first lecture gives an overview of systems of multiple M5-branes and introduces the relevant mathematical structures underlying a local description of higher gauge theory. In the second lecture, we develop the corresponding global picture. A construction of non-abelian superconformal gauge theories in six dimensions using twistor spaces is discussed in the third lecture. The last lecture deals with the problem of higher quantization and its…
A gauge system is a classical field theory where among the fields there are connections in a principal G-bundle over the space-time manifold and the classical action is either invariant or transforms appropriately with respect to the action of the gauge group. The lectures are focused on the path integral quantization of such systems. Here two main examples of gauge systems are Yang-Mills and Chern-Simons.
This is a redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. The lectures may be viewed online at https://goo.gl/3DJdOr. Please send typos or corrections to strominger@physics.harvard.edu.
Lecture notes for the course "Batalin-Vilkovisky formalism and applications in topological quantum field theory" given at the University of Notre Dame in the Fall 2016 for a mathematical audience. In these lectures we give a slow introduction to the perturbative path integral for gauge theories in Batalin-Vilkovisky formalism and the associated mathematical concepts.
This is a pedagogical review on the integrability-based approach to the three-point function in N=4 supersymmetric Yang-Mills theory. We first discuss the computation of the structure constant at weak coupling and show that the result can be recast as a sum over partitions of the rapidities of the magnons. We then introduce a non-perturbative framework, called the "hexagon approach", and explain how one can use the symmetries (i.e. superconformal and gauge symmetries) and integrability to determine the structure constants. This article is based on the lectures given in Les Houches Summer Schoo…
These Lectures summarize the relevant material on existent applications of jet manifold techniques to classical and quantum field theory. The following topics are included: 1. Fibre bundles, 2. Jet manifolds, 3. Connections, 4. Lagrangian field theory, 5. Gauge theory of principal connections, 6. Higher order jets, 7. Infinite order jets, 8. The variational bicomplex, 9. Geometry of simple graded manifolds, 10. Jets of ghosts and antifields.
These notes contain a survey of some aspects of the theory of differential modules and complexes as well as of their generalization, that is, the theory of $N$-differential modules and $N$-complexes. Several applications and examples coming from physics are discussed. The commun feature of these physical applications is that they deal with the theory of constrained or gauge systems. In particular different aspects of the BRS methods are explained and a detailed account of the $N$-complexes arising in the theory of higher spin gauge fields is given.
These lecture notes in the De Rham-Hodge theory are designed for a 1-semester undergraduate course (in mathematics, physics, engineering, chemistry or biology). This landmark theory of the 20th Century mathematics gives a rigorous foundation to modern field and gauge theories in physics, engineering and physiology. The only necessary background for comprehensive reading of these notes is Green's theorem from multivariable calculus.
((1+1)-dimensional ${\cal N}=1$ super-symmetric field theory and (3+1)-dimensional ${\cal N}=2$ super-symmetric gauge theory are discussed in a, more or less, unified way, designed to identify the quantum BPS states in both systems. Euclidean 4-dimensional gauge theory with ${\cal N}=2$ twisted super-symmetry is also analized. ${\bf C}^\infty$-topological invariants are identified as certain n-point correlation functions in this QFT framework. The twist of the effective dual Abelian gauge theory is briefly described, both from mathematical and physical viewpoints. The physical nature of the to…
These notes grew out of a lecture course on mathematical methods of classical physics for students of mathematics and mathematical physics at the master's level. Also, physicists with a strong interest in mathematics may find this text useful as a resource complementary to existing textbooks on classical physics. Topics include Lagrangian Mechanics, Hamiltonian Mechanics, Hamilton-Jacobi Theory, as well as Classical Field Theory formulated in the language of jet bundles. The latter topic also covers important examples of field theories such as sigma models, gauge theory, and Einstein's theory …
These third-year lecture notes are designed for a 1-semester course in topological quantum field theory (TQFT). Assumed background in mathematics and physics are only standard second-year subjects: multivariable calculus, introduction to quantum mechanics and basic electromagnetism. Keywords: quantum mechanics/field theory, path integral, Hodge decomposition, Chern-Simons and Yang-Mills gauge theories, conformal field theory
These lecture notes in Lie Groups are designed for a 1--semester third year or graduate course in mathematics, physics, engineering, chemistry or biology. This landmark theory of the 20th Century mathematics and physics gives a rigorous foundation to modern dynamics, as well as field and gauge theories in physics, engineering and biomechanics. We give both physical and medical examples of Lie groups. The only necessary background for comprehensive reading of these notes are advanced calculus and linear algebra.
In these lectures I talk about simplifications and universalities found in scattering amplitudes for gauge and gravity theories. In contrast to Ward identities, which are understood to arise from familiar symmetries of the classical action, these structures are currently only understood in terms of graphical organizational principles, such as the gauge-theoretic color-kinematics duality and the gravitational double-copy structure, for local representations of multi-loop S-matrix elements. These graphical principles make manifest new relationships in and between gauge and gravity scattering amp…
The goal of these lectures is to exhibit the framing anomaly in the Batalin-Vilkovisky formulation of perturbative Chern-Simons theory. Concretely, we show that the partition function fails to satisfy the Quantum Master Equation, and show that this can be remedied at the cost of introducing a framing. Along the way, we discuss principal bundles and connections, the Chern-Simons action functional, and perturbative quantization of gauge theories in the Batalin-Vilkovisky formalism.
In these lecture notes, we survey the landscape of Calabi-Yau threefolds, and the use of machine learning to explore it. We begin with the compact portion of the landscape, focusing in particular on complete intersection Calabi-Yau varieties (CICYs) and elliptic fibrations. Non-compact Calabi-Yau manifolds are manifest in Type II superstring theories, they arise as representation varieties of quivers, used to describe gauge theories in the bulk familiar four dimensions. Finally, given the huge amount of Calabi-Yau data, whether and how machine learning can be applied to algebraic geometry and …
Lectures on resolutions à la Kronheimer of orbifold singularities, McKay quivers for Gauge Theories on D3 branes, and the issue of Ricci flat metrics on the resolved three-folds
This series of lectures is planned as a generalization of author's large (more than fifteen years) experience of work in the theoretical physics. The modern theoretical physics is based on the group-theoretical approach which generates the formalism of the principal fibre bundles and the instanton approach. The latter is based on the Pontrjagin's degree of map theorem and this theorem is the original ``bridge'' between homology and cohomology theories. The author plans to devote his two first lectures to fibre bundle theory: this is the foundation on which the modern physics rests -- the theor…
This set of lecture notes first gives an introduction to the geometry of principal bundles. Next, it demonstrates how they can be used to formalize the concept of gauge theories, arising in physics. A basic familiarity is assumed with the differential geometry of manifolds and classical field theories of general relativity and electromagnetism.
These notes provide an introduction to recent work by Kevin Costello in which integrable lattice models of classical statistical mechanics in two dimensions are understood in terms of quantum gauge theory in four dimensions. This construction will be compared to the more familiar relationship between quantum knot invariants in three dimensions and Chern-Simons gauge theory. (Based on a Whittaker Colloquium at the University of Edinburgh and a lecture at Strings 2016 in Beijing.)
These notes accompany an introductory lecture course on the twistor approach to supersymmetric gauge theories aimed at early-stage PhD students. It was held by the author at the University of Cambridge during the Michaelmas term in 2009. The lectures assume a working knowledge of differential geometry and quantum field theory. No prior knowledge of twistor theory is required.
