2026-10-01 · chapter Yang–Mills and the mass gapAppendixGuide to the literature

Guide to the literature — what to read first, where to get it, and how far this book read it

Works are divided into three kinds. A primary source is the paper that first showed a result; the exact hypotheses and statements of theorems are checked here. A review is an overall picture of results up to some point, and an entrance for deciding which primary sources to read. Textbooks and lecture notes are for acquiring the tools in order.

We attach two statuses. The first is availability: arXiv (the public repository for preprints; free), the publisher (only the journal version; often needs a library subscription; those made freely available are marked "Publisher (open)"), PDFs made public by an institute or the authors, and books. Preprints whose journal publication had not been confirmed at the time of this book are marked "not refereed".

The second is the extent to which this book read it. The "Sources and reproduction" table of each chapter writes whether a work was read in the text, only the abstract, or only the bibliographic data. We collect this over all chapters and take the maximum extent over chapters.

Here $r$ is the work, $C(r)$ is the set of chapters citing it in their source tables, and $s(c,r)$ is the extent read from the source table of chapter $c$. "Text" includes cases where only the introduction or one section was checked. For example, Osterwalder–Seiler (1978) is cited by 40 chapters, of which 25 write "bibliographic data only" (§09).

Claims of this book that rest on works marked "bibliographic data only" go through other works whose text was checked. Where they go through is in each chapter's source table.

Detour: how to read bibliographic data

First we look at the whole on one sheet. The horizontal axis is the year, the vertical the lineage. The size of a dot is the number of chapters citing the work. The fill of a dot is the extent read in this book: solid for "text", light fill for "abstract", hollow for "bibliographic data only".

The primary sources in physics cluster from 1954 to the 1970s. The work aimed at four-dimensional construction (Bałaban, Magnen–Rivasseau–Sénéor) is concentrated in the 1980s and the early 1990s. But the four-dimensional construction is not complete (6-07). Work from the side of probability theory has been increasing since around 2018.

Moving the year knob leaves dark only the works published up to that year. Moving the part knob makes dark only the works cited by the chapters of that part. Pressing a dot shows its lineage, extent read and the parts citing it below, and lets you jump to its table row (which has links to chapters).

Computed The numbers in this figure (positions, sizes and fills of dots, citations by part) were counted on this machine from the source tables of this book (Sources and reproduction). The same content is in the tables by lineage in §03–§08, readable even where the figure does not run.

If you read only one, read the official problem statement of the Clay Mathematics Institute (Jaffe–Witten). It writes, in the words of those who set it, what the problem asks for and why it is hard in four dimensions. This book reads it word by word in 0-03. It is also the work cited most often among the chapters of this book.

The core of the property the problem statement asks for is the shape of the spectrum of the Hamiltonian $H$. With the vacuum energy set to $0$, it asks that a positive gap $\Delta$ open above it.

The exact reading of this formula (uniqueness of the vacuum, relation to the decay of correlations) is in 4-11.

The second is Chatterjee's review "Yang–Mills for probabilists". It defines Yang–Mills theory on the lattice in the language of probability theory, and lists what has been shown and what is open in the form of questions (Problems 5.1 and 5.2). Many chapters of Parts 5 and 6 of this book use it as a guide. For a third, Chevyrev if you follow the probabilistic constructions in two and three dimensions, and the review by Summers for the history of axioms and construction.

Parts 1 and 2 of this book prepare, from second-year knowledge, the minimum of tools needed. This table is where to go to learn one step deeper. For groups and representations, Hall; for manifolds and connections, Carroll; for operators and spectra, Reed–Simon; for calculations in quantum field theory, Tong's lecture notes; for axioms and construction, Streater–Wightman and Glimm–Jaffe.

One example. The Baker–Campbell–Hausdorff formula used in 1-03 says that the product of matrix exponentials receives a correction that can be written with commutators alone.

This book checked it up to the third-order terms with Hall's notes. Near the identity, the product of the group is determined by the commutators of the Lie algebra alone. This formula is the reason (the group itself is not determined by the Lie algebra alone; 1-03). Many lecture notes have public versions by the authors, and the arXiv numbers are given in the table.

Yang–Mills theory appeared in the 1954 paper of Yang and Mills, in the form of building a field from the local symmetry of isospin. In the 1960s the rules of gauge fixing and ghosts (Faddeev–Popov) were set in order, and in 1973 asymptotic freedom was found: the discovery of the sign that makes the coupling weaker at short distances.

In this book's notation (formula (8) of A-01), the one-loop running has the following form.

Since $b_0\gt0$, raising the energy $\mu$ makes $g$ smaller. Known This is a one-loop perturbative calculation, not a theorem about the four-dimensional theory. Physics This sign is the ground for expecting Yang–Mills theory to make sense at short distances. The account of its content ($11/3=4-1/3$) is in 4-06.

Many in this lineage are "abstract" or "bibliographic data only" in the chapters of this book. This is because they are in old journals with no public version; formulas and numbers are checked against reviews whose text was checked (PDG, Tong, Chatterjee).

The table also includes three that come after the skeleton of the 1970s. Gribov (1978) showed that gauge fixing is not uniquely determined. Uhlenbeck (1982) is a mathematical paper in geometric analysis showing that singularities of classical Yang–Mills fields can be removed. Baulieu–Zwanziger (1999) is the viewpoint that sees stochastic quantisation as a five-dimensional field theory.

In 1974 Wilson put forward the formulation that replaces spacetime by a lattice and puts group elements on the links. With it Yang–Mills theory becomes an integral over finitely many variables, and can be measured by computer. In this book's notation the action has the following form (5-01).

The sum is over all unit squares (plaquettes) $P$, and $U_P$ is the product of the links around it. Osterwalder–Seiler (1978) gave the mathematical foundation of this formulation (reflection positivity and the transfer matrix). Among the chapters of this book, it is cited most often, together with Jaffe–Witten and Chatterjee.

On the numerical side, starting with the Monte Carlo calculations of Creutz (1980), Lucini–Teper (2001) and Athenodorou–Teper (2020, 2021) have measured glueball masses and their ratios to the string tension. Physics These are values supported by numerical computation, not theorems. The correction coming from fluctuations of the string (Lüscher 1981) has the following form (7-03).

In four dimensions ($d=4$) the last term is $-\pi/(12R)$.

The mass gap problem consists of two stages, "existence" and "gap". The works approaching it from the mathematical side fall into three groups: the axioms; the four-dimensional ultraviolet; and probability theory in low dimensions and on the lattice.

Axioms: returning from probability to quantum mechanics

The Osterwalder–Schrader axioms are conditions for rebuilding a Hilbert space and a Hamiltonian from Euclidean (imaginary-time) correlation functions. At the centre is reflection positivity, the condition that the expectation of the product of an observable reflected in time, $\Theta F$, and the original $F$ is not negative (4-10).

If this holds, a positive inner product can be made, and translation in time appears as $e^{-sH}$ ($H\ge0$). $H$ in formula (2) comes from here.

The four-dimensional ultraviolet: Bałaban and MRS

The work that came closest to the four-dimensional continuum limit is Bałaban's series (a three-dimensional paper of 1985 and four-dimensional papers of 1987–1989; ultraviolet stability of lattice Yang–Mills by the renormalisation group) and the construction with an infrared cutoff by Magnen–Rivasseau–Sénéor (1993). Neither reaches a stage that this book can read as "complete" in the sense of the problem statement. What has been shown and what remains is read in 6-07.

Two and three dimensions, and probability theory on the lattice

Known In two dimensions the Yang–Mills measure has been constructed (Driver 1989, Lévy 2001 and others). In the 2020s, the road that makes the measure as the equilibrium of a stochastic dynamics (stochastic quantisation) is advancing in two and three dimensions. On the lattice, exponential decay of correlations at strong coupling follows from the cluster expansion of Osterwalder–Seiler (1978), and Shen–Zhu–Zhu showed Poincaré-type inequalities and exponential decay by methods of stochastic analysis in an explicit range of $\beta$ Known. The reach and where it stops in four dimensions are in 6-10 and 9-01. Unrefereed preprints are marked "not refereed" in the table.

On the physics side, there is a view that reads confinement and glueball masses of Yang–Mills theory in the geometry of gravity in one more dimension. The starting points are the correspondence proposed by Maldacena (submitted 1997) and Witten (1998), who used it for a model of confinement. Csáki–Ooguri–Oz–Terning derived ratios of glueball masses from this model as solutions of an eigenvalue problem.

Physics These are correspondences and models at the level of physics, not mathematical theorems. Nor have they been shown in a form that applies to pure four-dimensional Yang–Mills theory itself. What the model assumes, and what happens in the limit back to four dimensions, are read in 7-08, 7-09 and 7-10.

The source tables of the 110 chapters of this book (excluding this one) have 1035 rows in total. The arXiv numbers appearing in them are of 206 kinds, and bibliographic data in the form "volume (year) page" of about 602 kinds (the latter may pick up sequences of numbers inside formulas, so it is approximate). Computed

For each work, we counted the number of chapters citing it and the extent read in each chapter. Table 1 is the top 10 in order of the number of citing chapters. "Undetermined" counts chapters whose source-table row had no extent written, or where matching of words could not decide.

Table 1 Works cited often by the chapters of this book (breakdown of the extent in each chapter's source table)

Osterwalder–Seiler is cited by 40 chapters, but many chapters go through the abstract and reviews whose text was checked (Chatterjee §4 and others). For Lucini–Teper, 12 of 20 chapters write that they checked as far as the tables of the text.

We also counted by part. Among the source-table rows, for those containing bibliographic data of primary sources, we count the extent in that row. The fraction of text is the following ratio.

Computed The numbers in this figure were counted on this machine from the source tables of this book. Table 2 below has the same numbers.

Table 2 Extent of source-table rows containing primary sources, by part

The extent was decided by the words in the source tables ("text checked", "abstract", "bibliographic data only" and so on), so it is affected by variations in wording. Table 1 reads with priority the words inside the separators (the full-width slash and semicolon) in which the name of the work appears.

Differences between versions. The arXiv version and the journal version may differ in page and theorem numbers. The chapters of this book write in their source tables which version was checked.

Publication status. The "journal-ref" in the arXiv catalogue is a field filled in by the authors, and volume and pages are sometimes empty. For example, for Chevyrev–Shen the catalogue gives only "Comm. Pure Appl. Math. (2026)", and volume and pages are not confirmed in this book. For the preprints of 2025 and 2026 (Cao–Nissim–Sheffield, and the two of 2026), publication had not been confirmed at the time of this book. Unrefereed results are not treated with the same weight as refereed results.

Finding old journals. Many papers of the 1970s and 80s exist only in journal versions. Bibliographic data can be checked in INSPIRE (the catalogue of high-energy physics literature) or Crossref. Some old volumes of Comm. Math. Phys. have public versions on Project Euclid (for example Driver 1989, CMP 123). Otherwise, going through a university library is the sure way.

Bibliographic data checked in primary sources for chapters' citations

The tables of §03–§08 are the 52 works that matter across chapters. Separately, among the works supporting claims of one or two chapters, those whose bibliographic data or text were rechecked in primary catalogues (Crossref, INSPIRE, DML-CZ, Math-Net.Ru, publishers' pages) are listed in Table 3. "Level of checking" is how far each work was checked. They are not included in the counts of Figure 1, Table 1 or Table 2.

Table 3 Bibliographic data for individual chapters' sources, with the level of checking confirmed

The two papers by Zwanziger are different papers. The 1981 paper is gauge fixing within stochastic quantisation (stochastic gauge fixing), and the 1990 paper, from the same year as Parrinello–Jona-Lasinio, is gauge fixing by a weighted average over orbits. What Part 10 of this book calls "the scheme of Parrinello–Jona-Lasinio and Zwanziger" is the latter pair of 1990, and the source table of 10-02 lists both. For Cao–Nissim–Sheffield in the table of §07, 5-11 checked arXiv:2509.04688 as far as Definition 1.4 and Remark 1.5.

Chapter authors and overall authors. The PDG review as a whole is written as Navas et al., and when a chapter within it (for example the QCD chapter) is cited, it is written with the names of the chapter's authors. The difference in how 0-03 and 4-06 write it is this distinction, and both point to the same edition (2024).

The meaning of the labels is in A-04, the verification procedure in A-05, and the papers of this site in the list of papers.

What this chapter can say

Sources and reproduction

Next chapter to read: A-04 Lean and machine checking — what "Lean" means among the labels of this book.

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