Witten 1 of the e ective lagrangian for the supersymmetric eld theory with n 2 and gauge group. The theme of this book is to find the true vacuum of string theory. Seibergwitten map for noncommutative super yangmills theory. The a eckdine seiberg prepotential n 2 supersymmetric gauge theories.
Uvir mixing in noncommutative qed defined by seiberg. Solving the seiberg witten equation is equivalent to nding a twoform f ij f ija i. An introduction to the seibergwitten equations on symplectic manifolds. The di erential geometry needed to study the seibergwitten equations is the geometry of spin and. Although the same mechanism can be easily extended to the noncommutative theory, nontrivial aspects appear when one considers the seibergwitten map of. Seiberg witten geometry of mass deformed n2 superconformal ade quiver gauge theories in four dimensions is determined. In particular the use of the evolution equation is described in some detail and its relation to the cohomological approach is elucidated. By appropriately counting the solutions to the equations, one obtains smooth 4manifold invariants.
The states are described by wave functions quasiamplitudes of probability associated with wigner functions quasiprobability density. Introduction to seibergwitten theory and its stringy origin1. Cohomological methods which are applicable to gauge theories requiring the batalin. Denote by m the moduli space of solutions of the seiberg witten monopole equations up to gauge transformations. 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. Pdf string theory and noncommutative geometry semantic. Seibergwitten geometry of four dimensional n2 quiver. Magnetic bions, multiple adjoints, and seibergwitten theory. Seibergwitten geometry of mass deformed n2 superconformal ade quiver gauge theories in four dimensions is determined. Section 10 describes wittens 1988 derivation of these topological invariants using a supersymmetric topological quantum field theory.
Pure gauge su2 seibergwitten theory and modular forms. Smooth 4manifolds and the seibergwitten equations contents. The seibergwitten gauge theory serves as a simple model for the mirror symmetry of calabiyau manifolds. In this paper we derive the seibergwitten map for noncommutative super yangmills theory in wesszumino gauge. Pdf nonequivalent seibergwitten maps for noncommutative. Seibergwitten equations is that they are an abelian gauge theory with an exterior field. The first part of the book collects some preliminary notions and then gives an introduction of. Seibergwitten gauge theory matilde marcolli 1 caltech. 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. Although the same mechanism can be easily extended to the noncommutative theory, nontrivial aspects appear when one considers the seibergwitten map of that theory. Notes on seibergwitten gauge theory internet archive. Introduction in the last two years, there has been a remark. Our analysis leads us to an equivalence between ordinary gauge fields and.
Where the fire is churned, where the wind wafts, where the soma juice flows overthere the mind is born. They collect the material presented in a series of seminars. Seiberg witten theories particle physics theory group. The study of the seiberg witten equations begins in earnest in chapter 2. On a different note, i think you should avoid nicolaescus notes on seiberg witten theory. Massive vector fields can be described in a gauge invariant way with the introduction of compensating fields.
They collect the material presented in a series of seminars given by the author at the university of milano. 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. We extend earlier ideas about the appearance of noncommutative geometry in string theory with a nonzero bfield. We will discuss the case where the gauge group is u1. Lecture notes on seibergwitten invariants revised second. 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. Lerchea a cern, geneva, switzerland we give an elementary introduction to the recent solution of n 2 supersymmetric yangmills theory. Floer homology, gauge theory, and lowdimensional topology. 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. Matrixmodel description of n2 gauge theories with non. In a series of three lectures, karen gave a history of the equations of gauge theory, from the yangmills equations to the kapustinwitten equations.
They collect the material presented in a series of. 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. In fact, all the sections that refer to donaldson theory can be skipped, and this will not. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. Definition 2 is more frequently used in gauge theory, but in our presentation. The newly developed field of seiberg witten gauge theory has become a wellestablished part of the differential topology of fourmanifolds and threemanifolds. Nonequivalent seibergwitten maps for noncommutative.
The institute of the mathematical sciences of the americas. Seibergwitten geometry of four dimensional n2 quiver gauge. In the mid1990s, gaugetheoretic invariants for fourmanifolds underwent a dramatic change with the introduction of a new set of partial di. I will discuss several interesting examples and raise some questions. Lectures on the physical approach to donaldson and seiberg.
Siye wu, the geometry and physics of the seibergwitten equations, progress in mathematics. Smooth 4manifolds and the seibergwitten equations 9 1. 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. 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.
A useful discussion of the physical origins of the seibergwitten equations for mathematicians is in. The ordinary commutative theory can be gauge fixed to the socalled unitary gauge where the standard massive vector field theory is recovered. The gauge symmetry analysis provides a realization of the seibergwitten gauge theory for noncommutative fields. The study of the seibergwitten equations begins in earnest in chapter 2. 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. These notes provide an introductory exposition of the seiberg witten gauge theory. 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. Lectures on the physical approach to donaldson and. In the meantime, more advanced expositions of seibergwitten theory.
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. Realization of the noncommutative seibergwitten gauge. The seiberg witten invariants have become one of the standard tools in studying the di erential topology of fourdimensional manifolds. The a eckdineseiberg prepotential n 2 supersymmetric gauge theories. The di erential geometry needed to study the seiberg witten equations is the geometry of spin and. Nonequivalent seibergwitten maps for noncommutative massive. Lectures on gauge theory and symplectic geometry 5 2. The seibergwitten map for noncommutative yangmills theories is studied and methods for its explicit construction are discussed which are valid for any gauge group. In the unitary gauge one recovers the original formulation. On a different note, i think you should avoid nicolaescus notes on seibergwitten theory.
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. June 19, 2007 abstract we identify the spectral curve of pure gauge su2 seibergwitten theory with. These notes provide an introductory exposition of the seibergwitten gauge theory. Topological string theory on calabiyau manifolds, seiberg.
In a series of three lectures, karen gave a history of the equations of gauge theory, from the yangmills equations to the kapustin witten equations. Uvir mixing in noncommutative qed defined by seibergwitten map. Seibergwitten gauge theory matilde marcolli download. 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. 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. Introduction to seiberg witten theory and its stringy origin w. Based, in part, on lectures given at \gauge theories, applied super symmetry and quantum gravity, leuven. 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. Realization of the noncommutative seibergwitten gauge theory. 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. So far, we have developed several approaches to solving string theory, including conformal field theory, topological methods, twodimensional matrix models, and string field theory. Seiberg duality for quiver gauge theories internet archive. Good textbook or lecture notes on seibergwitten theory. 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.
This demonstrates quite explicitly that the only analytically controlled examples of con. Witten announced a new gauge theory of 4manifolds, capable of giving. What do topologists want from seibergwitten theory. Introduction to seibergwitten theory and its stringy origin w.
Seibergwitten theory and duality in n 2 supersymmetric. In fact, all the sections that refer to donaldson theory can be skipped, and this will not affect the comprehension of the remaining sections. An introduction to the seibergwitten equations on symplectic. We solve the limit shape equations derived from the gauge theory and identify the space m of vacua of the theory with the moduli. Z which assigns to every spinc structure on tx an integer swx. Pdf these notes provide an introductory exposition of the seibergwitten gauge theory. The newly developed field of seibergwitten gauge theory has become a wellestablished part of the differential topology of fourmanifolds and threemanifolds. The seibergwitten invariants have become one of the standard tools in studying the di erential topology of fourdimensional manifolds.
A good introduction to seibergwitten theory is morgans book mor. In addition to his contributions to physics, wittens work has significantly. 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. In this paper, we derive an exact expression for the seiberg witten map. Part of the texts and readings in mathematics book series volume 17.