Mathematical Gauge Theory
In Low and Higher Dimensions
(after Donaldson and Floer)
In Low and Higher Dimensions
(after Donaldson and Floer)
Table of Contents:
1. A Brief History of Mathematical Gauge Theory
2. Recommended References for Learning the Subject
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1. A Brief History of Mathematical Gauge Theory
Manifolds are central objects of study in geometry and topology. These are topological spaces that resemble Euclidean spaces locally but may differ significantly on a global scale. Understanding the geometry and topology of manifolds is challenging, with many unknowns and open questions. Topology in low dimensions, particularly in three and four dimensions, has unique properties with numerous theorems and counterexamples that don’t extend to higher dimensions. Manifolds in these dimensions are also closely linked to mathematical physics, and many advancements in understanding three- and four-dimensional manifolds have emerged from mathematical physics, in particular, relations to String theory, through the works of Edward Witten (for an introduction to him, you might find this short video helpful).
In 1982, Simon Donaldson introduced new methods to study low-dimensional manifolds using "invariants" based on moduli spaces of solutions to partial differential equations defined on these manifolds. What it means is that in order to study a manifold (say 3- or 4-dimensional) one would study solutions to certain (elliptic) partial differential equations defined on these manifolds. These equations, known as the anti-self-duality or instanton equations, are special cases of the Yang-Mills equations. Donaldson's work transformed the study of smooth four-manifolds, answering many questions while raising new ones (for an introduction to him, you might find this short video helpful).
A decade later, in 1994, Nathan Seiberg and Edward Witten developed new gauge-theoretic equations, now known as the Seiberg-Witten equations, that achieved similar results to Donaldson’s theory but with greater simplicity. This made it possible to prove new theorems that were previously out of reach.
In 1982, Witten proposed defining homology groups using a Morse function on a manifold, which led Andreas Floer in 1988 to develop a type of Morse homology on infinite-dimensional spaces. For example, when the infinite-dimensional space is the loop space of a symplectic manifold, this homology theory—Hamiltonian Floer Homology—has rich applications in understanding Hamiltonian dynamics, including presenting a proof of Arnold’s conjecture. When applied to the space of SU(2) connections on a three-manifold, Floer defined a homology theory, now known as instanton Floer homology, that is deeply connected to the Donaldson invariants of four-manifolds. This theory was later extended by Kronheimer and Mrowka using Seiberg-Witten equations (there is a vide of Floer explaining the idea of Floer homology here; however, the quality is low, and hopefully one can recover the good quality with a use of AI).
Heegaard Floer homology, defined by Ozsvath and Szabo, and Khovanov homology are additional invariants for 3-manifolds and knots in 3-manifolds. Heegaard Floer homology relates closely to another Floer theory, called Lagrangian Floer homology, while Khovanov homology, though algebraic, is expected to have symplectic and gauge-theoretic interpretations. Donaldson suggested categorifying Floer theories in 1992 to produce categorical invariants of manifolds. Building on this, Kenji Fukaya developed Fukaya category, where objects are Lagrangians, and morphisms are Floer complexes. This category connects to algebraic geometry and forms part of mirror symmetry, where the derived category of coherent sheaves on a Calabi-Yau manifold aligns with the Fukaya category of its mirror.
In three- and four-dimensional gauge theory, challenges often arise from non-compact moduli spaces of solutions. Compactness typically requires understanding and controlling sources of non-compactness, which can be quite difficult. While gauge theory has yielded significant insights in low dimensions, its application to higher-dimensional manifolds often requires specific geometric structures, such as special holonomy metrics.
Manifolds with special holonomy allow similar gauge theories and potential invariants. Proposed by Donaldson-Thomas-Segal, this program aims to define numerical invariants for 8-dimensional Spin(7)-manifolds, Floer homology groups for 7-dimensional G2-manifolds, and categorical invariants for (real) 6-dimensional Calabi-Yau 3-folds. Progress has been made, but challenges remain, especially regarding non-compactness issues. In the Calabi-Yau case, Donaldson-Thomas invariants can be approached through algebraic geometry, avoiding some of the analytic difficulties. Another related method involves calibrated submanifolds, which are minimal submanifolds that, as conjectured, relate to gauge-theoretic invariants, similar to Taubes’s conjecture connecting Seiberg-Witten and Gromov invariants of symplectic four-manifolds.
This is a very active field. There are many open problems to work on. In the following, I will introduce some references to start studying this subject.
2. Recommended References for Learning the Subject
We begin with low-dimensional topology, starting with knot theory, which is an accessible and engaging entry point for those new to topology. Knots can be thought of as closed, knotted curves inside a three-dimensional space (say R^3). While they may appear simple or unimportant, the mathematical theory behind knots is quite intricate and unexpectedly rich. Knots are not only beautiful and interesting in their own right, but they also play a crucial role in the study of 3-manifold topology and, by extension, 4-manifolds. While some aspects of knot theory are relatively accessible, others require more advanced tools such as gauge theory, Floer theory, or even algebraic geometry.
For an introductory reference with minimal prerequisites, I recommend The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots by Colin Adams. It is a well-written text that covers fundamental concepts in knot theory and introduces important topics with ease.
1. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots by Colin Adams
If you prefer video lectures, you can watch Jake Rasmussen’s course on knot theory.
1.1 Knot Theory by Jake Rasmussen, YouTube link
To build geometric and topological intuition, I also suggest Intuitive Topology by Victor Prasolov. Like The Knot Book, it contains numerous problems with solutions, but it's beneficial to try solving them independently before checking the answers. This book focuses on continuous transformations like knot deformations, flows, and fixed points, which are essential in low-dimensional topology.
2. Intuitive Topology by Victor V. Prasolov
As you gain familiarity with knots and surfaces, it's time to move toward a more systematic study of manifolds. A grate starting point for studying manifolds is John Milnor's Topology from the Differentiable Viewpoint.
3. Topology from the Differentiable Viewpoint by John Milnor
If you enjoy Milnor’s book, I recommend watching his recorded lectures from 1965.
3.1 Differential Topology Lectures by John Milnor, YouTube Link
However, Milnor's book does not discuss the concept of transversality, which is a critical concept in differential topology. As Blaine Lawson once stated, "Transversality is the key which opens many doors". To fill this gap, I suggest reading the first two chapters of Differential Topology by Guillemin and Pollack, which will cover the basics. Chapter 4, which delves into differential forms and integration, may not go into enough depth, so additional resources could be useful.
4. Differential Topology by Victor Guillemin and Alan Pollack
Next, it's important to deepen your understanding of homology and cohomology theories. For a continuation of Differential Topology, I recommend From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes by Madsen and Tornehave. It covers the essentials, but make sure to brush up on your multivariable calculus first.
5. From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes by Ib Madsen and Jørgen Tornehave
Algebraic topology emerges from the study of differential forms, though the heavy notation can obscure the deeper insights. With patience, the elegance of these algebraic structures becomes clear. A good reference for algebraic topology is William Massey's A Basic Course in Algebraic Topology.
6. A Basic Course in Algebraic Topology by William Massey
This book covers important topics such as compact surface classification, fundamental groups, homotopy types, and covering spaces. Rather than reading it all in one go, I recommend taking a more flexible approach—perhaps reading it alongside other texts and revisiting sections that are particularly challenging (a more popular reference might be Hatcher's book; however, for my taste, Massey's book is more direct and straighforward).
For a more geometric focus, especially in Riemannian geometry, Milnor’s Morse Theory (Part II, pages 43–67) is an excellent starting point. You can skip Part I if you're specifically interested in Riemannian geometry.
7. Morse Theory by John Milnor
If you like to spend more time on the foundations of Riemannian geometry I suggest the following book. The first 6 chapters would be sufficient.
7.1. Riemannian Geometry and Geometric Analysis by Jurgen Jost
At this stage, you should be comfortable with the basics of manifolds. To explore low-dimensional manifolds—particularly 3- and 4-manifolds—I suggest watching Simon Donaldson’s talk on the topic for a broad overview (up to 2010).
8. Invariants of Manifolds and Classification Problems (Talk by Simon Donaldson), YouTube Link.
For a more in-depth study of 3-manifolds, Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant by Nikolai Saveliev is a great place to start. It covers knots, 3- and 4-manifolds, and provides a solid introduction to the Casson invariant, a key concept related to the Euler characteristic of instanton Floer homology.
9. Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant by Nikolai Saveliev
For those interested in gauge theory and its applications to low-dimensional topology, Invariants of Homology 3-Spheres by Saveliev is a comprehensive resource.
10. Invariants of Homology 3-Spheres by Nikolai Saveliev
Finally, to study 3- and 4-manifolds using analytic tools, you’ll need to understand moduli spaces of solutions to partial differential equations defined on these manifolds (or their associated bundles). A basic introduction to these ideas, including regularity, can be found in the following lecture notes:
11- Introduction to Partial Differential Equations by Thomas Walpuski
Slightly more advanced equation is the Poisson equation, which can be thought of as a non-homogenous (generalized) version of Laplace's equation. A good place to read about this equation is the 2nd chapter of Donaldson's notes,
A more complicated version of this equation, usually called Kazdan-Warner equation, appears in the study of prescribed scalar curvature problems (and many other places). One good reference to study this equation is the following paper by Kazdan and Warner
13- Curvature functions for compact 2-manifolds by Jerry Kazdan and Frank Warner
Or if you want to also learn more about partial differential equations and their geometric applications you can read the following notes.
13.1- Applications of Partial Differential Equations To Problems in Geometry by Jerry Kazdan
Similar equations appear in mathematical gauge theory but these equations usually are written on a bundle over the manifold. A good, but lengthy, reference to learn about bundles and connection formalism is Walpuski's lecture notes (the first 76 pages would suffice).
14- Differential Geometry III Gauge Theory by Thomas Walpuski
A shorter reference to learn about gauge theory formalism is the following lecture note by Andriy Haydys (first 25 pages).
14.1- Introduction to Gauge Theory by Andriy Haydys
After understanding the mathematical framework of gauge theory one can write down the Vortex equations, which is a 2-dimensional gauge theory. Clifford Taubes studied these equations on R^2, during his PhD, and solved them.
15- Arbitrary N-vortex solutions to the first order Vortex equations by Clifford Taubes
It's good to spend more time with these equations, since gauge theory in dimension 2 is much easier than gauge theory over 3- and 4-manifolds. There are some features which are clear here, but are quite complicated in higher dimensions. One can study the vortex equations over any closed Riemann surface. Garcia-Prada solves the vortex equations by realizing them as a moment map of some Hamiltonian action.
These equations can in fact be written over any Kahler manifold, which has been studied by Bradlow. Moreover, these equations can be transformed into the Kazdan-Warner equation which we mentioned before.
17- Vortices in Holomorphic Line Bundles over Closed Kahler Manifolds Steven B. Bradlow
The vortices can be thought of as minima of the so-called 2D Yang-Mills-Higgs functional. A 3D version results in monopoles. One very important tool used to study monopoles is the gluing technique (which appears in studying of many other PDEs). A very good reference is the following book by Jaffe and Taubes.
18- Vortices and Monopoles by Arthur Jaffe and Clifford Taubes
A shorter version of this proof can be found in the following paper of Lorenzo Foscolo (Proposition 5.8)
18.1- A Gluing Construction for Periodic Monopoles by Lorenzo Foscolo
Atiyah and Hitchin studied the moduli spaces of monopoles on R^3, and showed these spaces are hyperkahler (and much more).
19- The Geometry and Dynamics of Magnetic Monopoles by Michael Atiyah and Nigel Hitchin
Smooth monopoles on closed 3-manifolds are somewhat trivial. However, the moduli spaces of singular monopoles are still quite complicated and interesting. In the following paper, I show the existence of singular monopoles on rational homology 3-spheres.
20- Singular Monopoles on Closed 3-Manifolds by Saman Habibi Esfahani
Magic happens in dimension four! The 4-dimensional version, so called Donaldson theory, is a strong tool to study 4-manifolds. Using this approach Donaldson proved the diagonalization theorem. This theorem states that a definite intersection form of a compact, oriented, simply connected, smooth 4-manifold is diagonalisable. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the integers.
21- Self-dual connections and the topology of smooth 4-manifolds by Simon Donaldson
This is highly in contrast with topological 4-manifolds. Michael Freedman had previously shown that any unimodular symmetric bilinear form is realized as the intersection form of some closed, oriented four-manifolds. This results in very Exotic phenomena in dimension four.
The main reference to learn about Donaldson's theory is the following book of Donaldson and Kronheimer; however, if you decide to read it, know that it is not a very easy read.
22- The Geometry of Four-manifolds by Simon Donaldson and Peter Kronheimer
Andrew Floer used flat connections and instantons to define a homological invariant of 3-manifolds, which categorifies the Casson invariant. There are different references to learn about the instanton Floer theory. One can read the original paper of Andrew Floer on the subject.
23- An instanton-invariant for 3-manifolds by Andreas Floer
An easier introduction to the subject is the following paper by Daniel Ruberman.
23.1- An Introduction to Floer Homology by Daniel Ruberman
A more detailed book about the subject is the following book of Simon Donaldson (which again is not a very easy read).
23.2- Floer Homology Groups in Yang-Mills Theory by Simon Donaldson
Moreover, if you like to hear about this theory, rather than reading the papers, you can watch the following good mini lectures on the subject by Simon Donaldson.
23.3- Introduction to Floer Homology By Simon Donaldson: Part 1, Part 2, Part 3
I am not aware of any other reference on the subject. Let me know if you know one.
As we mentioned above, the Seiberg-Witten theory changed the face of the field, since they are easier to work with and essentially contain the same amount of information about 4-manifolds. In fact, the vortex equations can be thought of as dimensional reduction of the Seiberg-Witten equations. Haydys notes (which we mentioned before too) is a good place to start studying about them (along with an introduction to Chern-Weil theory, Dirac operators...).
24- Introduction to gauge theory Andriy Haydys
To get a deeper understanding of these equations you can also read the following book by John Morgan.
Another good book on the subject, which is probably too long to read cover to cover, and can be used as a reference, is the following book by Dietmar Salamon.
24.2- Spin Geometry and Seiberg-Witten Invariants by Dietmar Salamon
And the following book which is on the monopole Floer homology (the Floer theory which comes out of Seiberg-Witten equations).
24.3- Monopoles and Three-Manifolds by Peter Kronheimer and Tomasz Mrowka
At this point, it is probably a good idea to learn more about the symplectic side of the story. Symplectic topology is a very interesting topic, which is also related to low dimensional manifold. Andreas Floer discovered a novel way to study symplectic manifolds which resulted in proving Arnold's conjecture. The best place to read about this is Audin-Damian's book. This book is not only the best place to start studying about Hamiltonian-Floer theory but also Morse homology and very first introduction to symplectic topology,
25- Morse Theory and Floer Homology by Michèle Audin and Mihai Damian
One can also take a look at Dietmar Salamon's note, it is shorter and not as detailed.
25.1- Lectures on Floer homology by Dietmar Salamon (up to page 38, the end of the section 3.4)
There is a more general version of this Floer theory, called Lagrangian Floer homology. This is a quite technical topic and unfortunately there is no good, detailed, elementary reference here. An obvious reference is the Floer's original paper; however, Floer's papers are not always very easy to go through. There are also many FOOO papers on the subject; however, definitely not elementary. A good note to understand the basics is the following note (Let me know if you are aware of other good references on the subject (which are not hand-written)).
26- A Quick View of Lagrangian Floer Homology by Andres Pedroza
An obvious next step is to learn the basics of the Fukaya category. A short good note is the following paper by Denis Auroux.
27- A Beginner's Introduction to Fukaya Categories by Denis Auroux
There are of course more detailed technical references on Fukaya categories, by FOOO and Seidel.
One can combine the symplectic picture and holomorphic curves with the theory of vortex equations which we saw earlier, to get some generalized vortex equations, called symplectic vortex equations. These equations turn out to be quite interesting and appear in many different problems in gauge theory and symplectic topology.
One can similarly generalize the Seiberg-Witten equations, by extending the Dirac operator to sections of hyperkahler bundles where the fibers are not flat. This has been done by Taubes.
29- Nonlinear Generalizations of a 3-Manifold’s Dirac Operator by Cliff Taubes
An interesting case appears when the fibers are modeled on the moduli space of monopoles on R3. One would expect it would be possible to define a Floer-theoretic invariant of 3-manifolds using these non-linear harmonic spinors, called the
operators. You can read about these operators in the 3rd chapter of my PhD thesis.
30- Monopoles, Singularities and Hyperkähler Geometry (Chapter 3) by Saman Habibi Esfahani
Fueter sections can be used to define a Fukaya category for hyperkahler manifolds. The following paper of Doan and Rezchikov is a good reference on the subject.
31- Holomorphic Floer Theory and the Fueter Equation by Aleksander Doan and Semon Rezchikov
As mentioned earlier, one can extend the gauge theory methods to higher-dimensional manifolds, especially manifolds with special holonomy groups. There are some good references on the subject. The standard one is the following book by Dominic Joyce.
31*-An essential difficulty in working with Fueter sections is the non-compactness problem which I address here: Monopole Fueter Floer homology: Saman H Esfahani. and later joint with Yang Li: Fueter sections and Z2-harmonic 1-forms, as a first step towards defining a new 3-manifold invariant using Fueter sections (which is expected to be ralated to Rozansky-Wittein invariant) and also motivated by Donaldson-Segal conjecture.
32- Compact Manifolds with Special Holonomy by Dominic Joyce
There is also a shorter version.
32.1- Lectures on Calabi-Yau and Special Lagrangian Geometry by Dominic Joyce
There is also the following lecture note by Robert Bryant which contains interesting approaches to the study of G2-manifolds.
32.2- Some remarks on G2-structures by Robert Bryant
The original references for gauge theory on manifolds with special holonomy groups are the papers of Donaldson-Thomas and Donaldson-Segal.
33- Gauge Theory in Higher Dimensions by Donaldson and Thomas
34- Gauge Theory in Higher Dimensions II by Donaldson and Segal
For understanding the Donaldson-Thomas program to study manifolds with special holonomy and how it is related to generalized Seiberg-Witten equations you can read the introduction of the well-written thesis of Aleksander Doan
35- Monopoles and Fueter sections on three-manifolds (introduction) by Aleksander Doan
For more information on instantons in higher dimensions a good reference is the PhD thesis of Thomas Walpuski.
36- Gauge theory on G2–manifolds by Thomas Walpuski
For more information on monopoles in higher dimensions a good reference is the PhD thesis of Goncalo Oliveira.
37- Monopoles in Higher Dimensions by Goncalo Oliveira
To understand the role of calibrated submanifolds in higher-dimensional gauge theory, especially relevant to the non-compactness problems, a good reference is the following paper of Tian.
38- Gauge Theory and Calibrated Geometry, I by Gang Tian
(This paper is numbered 1; however, I am not aware of any sequel to this paper.)
Another way of studying the manifolds with special holonomy is by counting the calibrated submanifolds inside them. In the case of Special Lagrangians in Calabi-Yau 3-folds one can look at
39- On Counting Special Lagrangian Homology 3-Spheres by Dominic Joyce
Or in case of associative 3-folds in G2 manifolds,
40- Conjectures on Counting Associative 3-Folds in G_2-Manifolds by Dominic Joyce
It is expected that one would be able to define an invariant of G2-manifolds by somehow considering the solutions to the gauge-theoretic equations and calibrated submanifolds at the same time. The following is an example of this approach.
Another approach, closely related to other things mentioned above, to G2 manifolds is the point of view of co-associative fibrations, and in particular the “adiabatic limit”, introduced in the following paper by Simon Donaldson.
42- Adiabatic Limits of Co-associative Kovalev-Lefschetz Fibrations by Simon Donaldson
And developed further by Donaldson and Scaduto.
43- Associative Submanifolds and Gradient Cycles by Simon Donaldson and Chris Scaduto
Donaldson and Scaduto raise an important conjecture about calibrated submanifolds in CY3/G2 manifolds with K3/hyperkahler fibration. With Yang Li we addressed this conjecture:
44- On the Donaldson-Scaduto conjecture, Saman Habibi Esfahani, Yang Li