# Can Algebraic Geometry Contribute to Condensed Matter Physics?

Our detour into math and physics will be concluded with this commentary.

Algebraic geometry is the branch of mathematics that used to study the roots of polynomial equations and thus connected geometry and algebra. With classical algebra turning into abstract algebra in the last century, the field of algebraic geometry took a new turn.

Modern algebraic geometry is based on more abstract techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

In the 20th century, algebraic geometry has split into several subareas.

The main stream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field.

The study of the points of an algebraic variety with coordinates in the field of the rational numbers or in a number field became arithmetic geometry (or more classically Diophantine geometry), a subfield of algebraic number theory.

The study of the real points of an algebraic variety is the subject of real algebraic geometry.

A large part of singularity theory is devoted to the singularities of algebraic varieties.

With the rise of the computers, a computational algebraic geometry area has emerged, which lies at the intersection of algebraic geometry and computer algebra. It consists essentially in developing algorithms and software for studying and finding the properties of explicitly given algebraic varieties.

Much of the development of the main stream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on “intrinsic” properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, differential and complex geometry. One key achievement of this abstract algebraic geometry is Grothendieck’s scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert’s Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. Wiles’s proof of the longstanding conjecture called Fermat’s last theorem is an example of the power of this approach.

So, the question is whether this new math can ‘modernize’ classical physics and help solve a set of unsolved problems (outside string theory) ? We, unfortunately, do not have any answer, but here are a set of papers to look into, but most are related to string theory only.

Algebraic Geometry and Physics (2000)

This article is an interdisciplinary review and an on-going progress report over the last few years made by myself and collaborators in certain fundamental subjects on two major theoretic branches in mathematics and theoretical physics: algebraic geometry and quantum physics. I shall take a practical approach, concentrating more on explicit examples rather than formal developments. Topics covered are divided in three sections: (I) Algebraic geometry on two-dimensional exactly solvable statistical lattice models and its related Hamiltonians: I will report results on the algebraic geometry of rapidity curves appeared in the chiral Potts model, and the algebraic Bethe Ansatz equation in connection with quantum inverse scattering method for the related one-dimensional Hamiltonion chain, e.g., XXZ, Hofstadter type Hamiltonian. (II) Infinite symmetry algebras arising from quantum spin chain and conformal field theory: I will explain certain progress made on Onsager algebra, the relation with the superintegrable chiral Potts quantum chain and problems on its spectrum. In conformal field theory, mathematical aspects of characters of N=2 superconformal algebra are discussed, especially on the modular invariant property connected to the theory. (III). Algebraic geometry problems on orbifolds stemming from string theory: I will report recent progress on crepant resolutions of quotient singularity of dimension greater than or equal to three. The direction of present-day research of engaging finite group representations in the geometry of orbifolds is briefly reviewed, and the mathematical aspect of various formulas on the topology of string vacuum will be discussed.

Geometric and Algebraic Topological Methods in Quantum Mechanics

In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques based on the deep interplay between algebra, differential geometry and topology. The present book aims at being a guide to advanced differential geometric and topological methods in quantum mechanics. Their main peculiarity lies in the fact that geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. Geometry is by no means the primary scope of the book, but it underlies many ideas in modern quantum physics and provides the most advanced schemes of quantization.

From Algebraic Geometry to Physics - a personal perspective

Noncommutative Geometry and Quiver algebras

We develop a new framework for noncommutative differential geometry based on double derivations. This leads to the notion of moment map and of Hamiltonian reduction in noncommutative symplectic geometry. For any smooth associative algebra B, we define its noncommutative cotangent bundle T^*B, which is a basic example of noncommutative symplectic manifold. Applying Hamiltonian reduction to noncommutative cotangent bundles gives an interesting class of associative algebras, P=P(B), that includes preprojective algebras associated with quivers. Our formalism of noncommutative Hamiltonian reduction provides the space P/[P,P] with a Lie algebra structure, analogous to the Poisson bracket on the zero fiber of the moment map. In the special case where P is the preprojective algebra associated with a quiver of non-Dynkin type, we give a complete description of the Gerstenhaber algebra structure on the Hochschild cohomology of P in terms of the Lie algebra P/[P,P].

Why can algebraic geometry be applied into theoretical physics?

It is to be said at the outset that I do not have much familiarity with physics beyond what is in a semi-popular book; say, the Feynman Lectures Vol 1 and 2.

As I progressed in math graduate school specializing in number theory and algebraic geometry, it was astounding to discover a certain class of researchers who were doing very serious and nontrivial cutting-edge stuff connecting algebraic geometry and mathematical physics.

At the risk of appearing like a completely naive person, I must confess that I am completely baffled at how can this be possible. From one perspective, algebraic geometry is at its basics about studying solutions of algebraic equations which also have certain geometric aspects. Certainly way of looking at is not enough. How to philosophically explain this connection?

To start with, we can look at many different possible spacetimes, and it turns out that looking at certain algebraic varieties (or more generally, symplectic manifolds) is very fruitful. There are many reasons for this, but I’ll mention just one: it turns out that the number of algebraic curves in these varieties is something which appears in physics.

Here is an attempt at explaining one piece of this (warning: I only understand the mathematical side, so what I say about physics may be completely wrong. If so, someone who actually knows physics please correct me. Also, note that I do not know how to explain this in a way that is not at least somewhat handwavy, though I will try to not say anything false.)

In classical mechanics, there is the Lagrangian formulation. What this says is that the path of an object must (at least locally) minimize a quantity called the “action” of the path. The calculus of variations lets us prove that this is equivalent to Newton’s laws.

Now when you go to quantum mechanics (and in particular quantum field theory, which is where this is really useful), there is the Feynman path integral formulation: A particle may be treated as taking every path it could possibly take. However, most of these paths “cancel out” , and the only ones we actually end up seeing are the ones that are critical points of the action. What this means is that we can evaluate certain things in quantum field theory by integrating over all paths, as most of the paths will cancel out. (This “cancelling out” is something that can happen in quantum mechanics; to give an analogue, think of wave interference.)

Now if you we start working with strings, then as a string moves, we get a 2d- surface instead of a path, called the worldsheet of a string. As with the path integral, we now get integrals over these 2d-surfaces. It now turns out that the surfaces which don’t cancel must be pseudoholomorphic curve, and when our spacetime is an algebraic variety, this pseudoholomorphic curve corresponds to an algebraic curve.

So counts of algebraic curves in complex varieties are something which appear in string theory.

Now of course there is no rigorous formulation of infinite-dimensional integration in mathematics. Therefore, many of the results obtainable via the Feynman path integral are not obtainable via traditional techniques, which is why mathematicians are interested in this relation.

Why study algebraic geometry? (a very informative comment with many links to get started on algebraic geometry)

In my personal case, I started as a theoretical physicists but switched completely to pure mathematics because of algebraic geometry, and I also began by self-learning. It is a very deep subject with connections to almost everything else, once one has learned enough to realize that. It is also a very demanding field because of the tremendous background one has to master, in commutative and homological algebra for example, before being able to get to the most modern and interesting results. The effort nevertheless pays off! In fact, the route through commutative algebra actually paves the way not only to algebraic geometry but to algebraic number theory and arithmetic geometry. I had a strong background in differential geometry so I arrived at algebraic geometry through complex (Khler) geometry, and ended up fascinated by even the most abstract incarnations of it.

“Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate…” - David Mumford.

Topological quantum field theory

Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory.

In condensed matter physics, topological quantum field theories are the low energy effective theories of topologically ordered states, such as fractional quantum Hall states, string-net condensed states, and other strongly correlated quantum liquid states.

Regarding the original question, we have not seen much application in condensed matter physics, but may be because we have not looked hard enough.

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**GAGA Paper** of Jean-Pierre Serre connects algebraic geometry and analytic geometry.

Foundations for the many relations between the two theories were put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory. The major paper consolidating the theory was Gometrie Algbrique et Gomtrie Analytique Serre (1956) by Serre, now usually referred to as GAGA. It proves general results that relate classes of algebraic varieties, regular morphisms and sheaves with classes of analytic spaces, holomorphic mappings and sheaves. It reduces all of these to the comparison of categories of sheaves.

Nowadays the phrase GAGA-style result is used for any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, to a well-defined subcategory of analytic geometry objects and holomorphic mappings.