Notes on seibergwitten gauge theory internet archive. These notes provide an introductory exposition of the seiberg witten gauge theory. Tamaritstring solitons in the m5brane worldvolume with a nambupoisson structure and seibergwitten map furuuchi kazuyuki and takimi tomohisau1 gauge field theory on minkowski space marija dimitrijevic, larisa jonke and lutz mollerrecent citations inferring the covariant exact. Following and using results of hepth0108045 we split the observer lorentz transformations into a covariant particle lorentz transformation and a remainder which gives directly the seibergwitten differential equations. The seibergwitten solution using the monodromy of periods and the holomorphicity of prepotential is similar in spirit to the bmodel calculations in mirror symmetry, while nekrasovs instanton calculations based on localization is. The institute of the mathematical sciences of the americas. The a eckdine seiberg prepotential n 2 supersymmetric gauge theories. Floer homology, gauge theory, and lowdimensional topology. Chapter 7 discusses the fundamental properties of the solutions of the seiberg witten monopole equations and shows how they can be used to construct 4manifold invariants. The newly developed field of seibergwitten gauge theory has become a wellestablished part of the differential topology of fourmanifolds and threemanifolds. Edward witten born august 26, 1951 is an american theoretical physicist and professor of mathematical physics at the institute for advanced study in princeton, new jersey. Smooth 4manifolds and the seibergwitten equations contents. Although the same mechanism can be easily extended to the noncommutative theory, nontrivial aspects appear when one considers the seibergwitten map of that theory. In the unitary gauge one recovers the original formulation.
Section 10 describes wittens 1988 derivation of these topological invariants using a supersymmetric topological quantum field theory. The newly developed field of seiberg witten gauge theory has become a wellestablished part of the differential topology of fourmanifolds and threemanifolds. Smooth 4manifolds and the seibergwitten equations 9 1. Although the same mechanism can be easily extended to the noncommutative theory, nontrivial aspects appear when one considers the seibergwitten map of. They collect the material presented in a series of seminars. The gauge symmetry analysis provides a realization of the seibergwitten gauge theory for noncommutative fields. In this paper we derive the seibergwitten map for noncommutative super yangmills theory in wesszumino gauge. Definition 2 is more frequently used in gauge theory, but in our presentation. Seibergwitten geometry of four dimensional n2 quiver. An introduction to the seibergwitten equations on symplectic. We abstract from these arguments the idea that seiberg duality arises because a configuration of gauge theory can be realized as a bound state of a collection of branes in more than one way, and we show that different brane worldvolume theories obtained this way have matching moduli spaces, the primary test of seiberg duality.
So far, we have developed several approaches to solving string theory, including conformal field theory, topological methods, twodimensional matrix models, and string field theory. Massive vector fields can be described in a gauge invariant way with the introduction of compensating fields. Solving the seiberg witten equation is equivalent to nding a twoform f ij f ija i. Denote by m the moduli space of solutions of the seiberg witten monopole equations up to gauge transformations. Realization of the noncommutative seibergwitten gauge. They collect the material presented in a series of seminars given by the author at the university of milano. We will discuss the case where the gauge group is u1. Seibergwitten gauge theory matilde marcolli 1 caltech. Lerchea a cern, geneva, switzerland we give an elementary introduction to the recent solution of n 2 supersymmetric yangmills theory. Introduction to seibergwitten theory and its stringy origin w.
Seiberg witten gauge theory matilde marcolli springer. Nonequivalent seibergwitten maps for noncommutative massive. The theme of this book is to find the true vacuum of string theory. Pdf nonequivalent seibergwitten maps for noncommutative. Introduction to seibergwitten theory and its stringy origin1. They collect the material presented in a series of. I will discuss several interesting examples and raise some questions. Lectures on the physical approach to donaldson and seiberg.
On a different note, i think you should avoid nicolaescus notes on seibergwitten theory. Pdf string theory and noncommutative geometry semantic. Seibergwitten geometry of four dimensional n2 quiver gauge. The di erential geometry needed to study the seibergwitten equations is the geometry of spin and. The ordinary commutative theory can be gauge fixed to the socalled unitary gauge where the standard massive vector field theory is recovered. Introduction in the last two years, there has been a remark. The a eckdineseiberg prepotential n 2 supersymmetric gauge theories. We solve the limit shape equations derived from the gauge theory and identify the space m of vacua of the theory with the moduli space of the genus zero holomorphic quasimaps to the moduli space of holomorphic gbundles on a possibly degenerate elliptic curve defined in. Based, in part, on lectures given at \gauge theories, applied super symmetry and quantum gravity, leuven. In fact, all the sections that refer to donaldson theory can be skipped, and this will not affect the comprehension of the remaining sections. In the meantime, more advanced expositions of seibergwitten theory. Uvir mixing in noncommutative qed defined by seibergwitten map. The first part of the book collects some preliminary notions and then gives an introduction of.
In particular the use of the evolution equation is described in some detail and its relation to the cohomological approach is elucidated. Seiberg witten geometry of mass deformed n2 superconformal ade quiver gauge theories in four dimensions is determined. June 19, 2007 abstract we identify the spectral curve of pure gauge su2 seibergwitten theory with. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. In theoretical physics, seibergwitten theory is a theory that determines an exact lowenergy effective action for massless degrees of freedom of a supersymmetric gauge theorynamely the metric of the moduli space of vacua. Where the fire is churned, where the wind wafts, where the soma juice flows overthere the mind is born. Topological string theory on calabiyau manifolds, seiberg. Lecture notes on seibergwitten invariants revised second. This makes contact with mathematical results in the representation theory of algebras. The seiberg witten map for noncommutative yangmills theories is studied and methods for its explicit construction are discussed which are valid for any gauge group. Introduction to seiberg witten theory and its stringy origin w. The seibergwitten invariants have become one of the standard tools in studying the di erential topology of fourdimensional manifolds. Nonequivalent seibergwitten maps for noncommutative. Z which assigns to every spinc structure on tx an integer swx.
Lectures on gauge theory and symplectic geometry 5 2. Realization of the noncommutative seibergwitten gauge theory. Witten announced a new gauge theory of 4manifolds, capable of giving. In a series of three lectures, karen gave a history of the equations of gauge theory, from the yangmills equations to the kapustinwitten equations. Good textbook or lecture notes on seibergwitten theory. Pure gauge su2 seibergwitten theory and modular forms. We identify a limit in which the entire string dynamics is described by a minimally coupled supersymmetric gauge theory on a noncommutative space, and discuss the corrections away from this limit. Magnetic bions, multiple adjoints, and seibergwitten theory. The seibergwitten map for noncommutative gauge theories. The seibergwitten map for noncommutative yangmills theories is studied and methods for its explicit construction are discussed which are valid for any gauge group. Part of the texts and readings in mathematics book series volume 17. In fact, all the sections that refer to donaldson theory can be skipped, and this will not.
Seibergwitten theory and duality in n 2 supersymmetric. Seibergwitten geometry of mass deformed n2 superconformal ade quiver gauge theories in four dimensions is determined. These notes provide an introductory exposition of the seibergwitten gauge theory. In addition to his contributions to physics, wittens work has significantly. Seiberg witten theories particle physics theory group. The states are described by wave functions quasiamplitudes of probability associated with wigner functions quasiprobability density. Matrixmodel description of n2 gauge theories with non.
A good introduction to seibergwitten theory is morgans book mor. Seibergwitten equations is that they are an abelian gauge theory with an exterior field. We extend earlier ideas about the appearance of noncommutative geometry in string theory with a nonzero bfield. The di erential geometry needed to study the seiberg witten equations is the geometry of spin and. Seiberg witten gauge theory and complex surfaces kyoungseog lee most part of this talk will be a survey about seiberg witten gauge theory and how it can be understood for complex smooth projective surfaces. Siye wu, the geometry and physics of the seibergwitten equations, progress in mathematics. We also present the general structure of the homogeneous solutions of the defining. Equations of gauge theory karen uhlenbeck notes by laura fredrickson these notes are based on a series of lectures professor karen uhlenbeck gave in 2012 at temple university in philadelphia. This demonstrates quite explicitly that the only analytically controlled examples of con. Lectures on the physical approach to donaldson and. Pdf these notes provide an introductory exposition of the seibergwitten gauge theory.
What do topologists want from seibergwitten theory. As we show here, only a particular class of its solutions. The seibergwitten gauge theory serves as a simple model for the mirror symmetry of calabiyau manifolds. In their seminal work, seiberg and witten derived the wilsonian lowenergy e ective action of the n 2 su2 gauge theory by encoding the problem in a twodimensional holomorphic curve. Seibergwitten map for noncommutative super yangmills theory. Uvir mixing in noncommutative qed defined by seiberg.
Witten 1 of the e ective lagrangian for the supersymmetric eld theory with n 2 and gauge group. The seiberg witten invariants have become one of the standard tools in studying the di erential topology of fourdimensional manifolds. Chapter 7 discusses the fundamental properties of the solutions of the seibergwitten monopole equations and shows how they can be used to construct 4manifold invariants. Through all these parts of the book, seibergwitten gauge theory is considered as a completely selfcontained subject and no a priori knowledge of donaldson theory is assumed. The study of the seibergwitten equations begins in earnest in chapter 2. A useful discussion of the physical origins of the seibergwitten equations for mathematicians is in.
In the mid1990s, gaugetheoretic invariants for fourmanifolds underwent a dramatic change with the introduction of a new set of partial di. Seibergwitten gauge theory and complex surfaces kyoungseog lee most part of this talk will be a survey about seibergwitten gauge theory and how it can be understood for complex smooth projective surfaces. The seibergwitten curves and differentials for the n 2 gauge theories considered in this paper were previously obtained using mtheory methods, following the approach of ref. Although this gauging mechanism can be extended to noncommutative spaces in a straightforward way, non trivial aspects show up when we consider the seiberg witten map. By appropriately counting the solutions to the equations, one obtains smooth 4manifold invariants. Seibergwitten gauge theory matilde marcolli download. Seiberg duality for quiver gauge theories internet archive. The study of the seiberg witten equations begins in earnest in chapter 2. Although this gauging mechanism can be extended to noncommutative spaces in a straightforward way, non trivial aspects show up when we consider the seibergwitten map. Through all these parts of the book, seiberg witten gauge theory is considered as a completely selfcontained subject and no a priori knowledge of donaldson theory is assumed. An introduction to the seibergwitten equations on symplectic manifolds. In this paper, we derive an exact expression for the seiberg witten map.
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