arXiv subjects

Ben Gripaios

Publications and source records attributed to Ben Gripaios.

At least 19 recordsLinked to original sources

arXiv2026

Anomaly cancellation for two $U(1)$ factors

We show that solving the abelian part of the local anomaly cancellation conditions for a 4-d gauge theory whose gauge Lie algebra has an abelian summand with rank $K \geq 1$ is equivalent to the problem in algebraic/arithmetic geometry of finding $(K-1)$-dimensional projective linear subspaces of a cubic hypersurface over the rational numbers, where the cubic is determined by the data of the semisimple summand and the representation thereof carried by the Weyl fermions. We then use this reformulation to solve a variety of examples with rank 2. The simplest non-trivial example from physics, namely gauge Lie group $U(1)^2$ and six fermions, nevertheless has a rich (and well-studied) geometry: it corresponds to the Fano variety of lines in the Segre cubic primal threefold. This is a surface with 15 irreducible components that are planes (which correspond to lines lying in the 15 planes in the Segre cubic primal and which give rise to non-chiral fermions) and 6 components that are split del Pezzo surfaces of degree 5 (which give rise to chiral fermions). These components are all rational varieties, enabling all solutions to the anomaly cancellation conditions to be parametrized and their collective properties (e.g. their topology and asymptotic distribution) to be described.

hep-th↗

arXiv2026

TQFTs do not detect the Milnor sphere

We show that, under very general hypotheses, topological quantum field theories (TQFTs) cannot detect homotopy spheres bounding parallelisable manifolds, such as Milnor's exotic 7-dimensional sphere. The result holds for a wide variety of target categories (or $(\infty,n)$-categories) and arbitrary tangential structures. An appendix contains results on the mapping class groups of (stably-) framed manifolds that may be of independent interest.

math.AT↗

arXiv2025

Anomaly cancellation for a $U(1)$ factor

We use methods of arithmetic geometry to find solutions to the abelian local anomaly cancellation equations for a four-dimensional gauge theory whose Lie algebra has a single $\mathfrak{u}_1$ summand, assuming that a non-trivial solution exists. The resulting polynomial equations in the integer $\mathfrak{u}_1$ charges define a projective cubic hypersurface over the field of rational numbers. Generically, such a hypersurface is (by a theorem of Koll{\'a}r) unirational, making it possible to find a finitely-many-to-one parameterization of infinitely many solutions using secant and tangent constructions. As an example, for the Standard Model Lie algebra with its three generations of quarks and leptons (or even with just a single generation and two $\mathfrak{su}_3\oplus\mathfrak{su}_2$ singlet right-handed neutrinos), it follows that there are infinitely many anomaly-free possibilities for the $\mathfrak{u}_1$ hypercharges. We also discuss whether it is possible to find all solutions in this way.

hep-th↗

arXiv2025

The asymptotically-free gauge theories

We show how to classify the asymptotically-free gauge theories in four spacetime dimensions, focussing here on the case of purely fermionic matter. The classification depends on the fact (which we prove) that both the dimension and Dynkin index of irreducible representations of a simple Lie algebra are strictly increasing functions of each Dynkin label. This implies not only that the number of asymptotically-free representations of any one semisimple Lie algebra is finite, but also that they can be written down in a systematic fashion using tables for the asymptotically-free irreducible representations of simple Lie algebras, which we supply. These tables show that at most two out of a possible ten Dynkin labels can be non-zero and that no Dynkin label can exceed four. The extension to bosonic matter or supersymmetric theories is straightforward.

hep-th↗

arXiv2025

More varieties of 4-d gauge theories: product representations

Recently, we used methods of arithmetic geometry to study the anomaly-free irreducible representations of an arbitrary gauge Lie algebra. Here we generalize to the case of products of irreducible representations, where it is again possible to give a complete description. A key result is that the projective variety corresponding to $m$-fold product representations of the Lie algebra $\mathfrak{su}_n$ is a rational variety for every $m$ and $n$. We study the simplest case of $\mathfrak{su}_3$ (corresponding to the strong interaction) in detail. We also describe the implications of a number-theoretic conjecture of Manin (and related theorems) for the number of chiral representations of bounded size $B$ (measured roughly by the Dynkin labels) compared to non-chiral ones, giving a precise meaning to the sense in which the former (which are those most relevant for phenomenology) are rare compared to the latter. As examples, we show that, for both irreducible representations of $\mathfrak{su}_5$ and once-reducible product representations of $\mathfrak{su}_3$ that are non-anomalous, the number of chiral representations is asymptotically between $B (\log B)^5$ and $B^\frac{4}{3}$, while the number of non-chiral representations is asymptotically $B^2$. Despite this rarity of chiral, anomaly-free, product representations, we show that there are examples relevant for phenomenology, including one that gives an asymptotically-free gauge theory with Lie algebra $\mathfrak{su}_7$.

hep-th↗

arXiv2024

Varieties of four-dimensional gauge theories

We use algebraic geometry to study the anomaly-free representations of an arbitrary gauge Lie algebra for 4-dimensional spacetime fermions. For irreducible representations, the problem reduces to studying the Lie algebras $\mathfrak{su}_n$ for $n\geq 3$. We show that there exist equivalence classes of such representations that are in bijection with the rational points on a projective variety that are dense in a region of the underlying real variety diffeomorphic to $\mathbb{R}^{n-3}$. It follows that the chiral ones overwhelm the non-chiral ones for $n \geq 5$. We present an efficient algorithm to find explicit anomaly-free irreducible representations and discuss the generalization to reducible representations.

hep-th↗

arXiv2024

Two-dimensional gauge anomalies and $p$-adic numbers

We show how methods of number theory can be used to study anomalies in gauge quantum field theories in spacetime dimension two. To wit, the anomaly cancellation conditions for the abelian part of the local anomaly admit solutions if and only if they admit solutions in the reals and in the $p$-adics for every prime $p$ and we use this to build an algorithm to find all solutions.

hep-th↗

arXiv2024

Smooth generalized symmetries of quantum field theories

Dynamical quantum field theories (QFTs), such as those in which spacetimes are equipped with a metric and/or a field in the form of a smooth map to a target manifold, can be formulated axiomatically using the language of $\infty$-categories. According to a geometric version of the cobordism hypothesis, such QFTs collectively assemble themselves into objects in an $\infty$-topos of smooth spaces. We show how this allows one to define and study generalized global symmetries of such QFTs. The symmetries are themselves smooth, so the `higher-form' symmetry groups can be endowed with, e.g., a Lie group structure. Among the more surprising general implications for physics are, firstly, that QFTs in spacetime dimension $d$, considered collectively, can have $d$-form symmetries, going beyond the known $(d-1)$-form symmetries of individual QFTs and, secondly, that a global symmetry of a QFT can be anomalous even before we try to gauge it, due to a failure to respect either smoothness (in that a symmetry of an individual QFT does not smoothly extend to QFTs collectively) or locality (in that a symmetry of an unextended QFT does not extend to an extended one). Smoothness anomalies are shown to occur even in 2-state systems in quantum mechanics (here formulated axiomatically by equipping $d=1$ spacetimes with a metric, an orientation, and perhaps some unitarity structure). Locality anomalies are shown to occur even for invertible QFTs defined on $d=1$ spacetimes equipped with an orientation and a smooth map to a target manifold. These correspond in physics to topological actions for a particle moving on the target and the relation to an earlier classification of such actions using invariant differential cohomology is elucidated.

hep-th↗

arXiv2022

Anomalies of non-Abelian finite groups via cobordism

We use cobordism theory to analyse anomalies of finite non-abelian symmetries in 4 spacetime dimensions. By applying the method of `anomaly interplay', which uses functoriality of cobordism and naturality of the $η$-invariant to relate anomalies in a group of interest to anomalies in other (finite or compact Lie) groups, we derive the anomaly for every representation in many examples motivated by flavour physics, including $S_3$, $A_4$, $Q_8$, and $\mathrm{SL}(2,\mathbb{F}_3)$. In the case of finite abelian groups, it is well known that anomalies can be `truncated' in a way that has no effect on low-energy physics, by means of a group extension. We extend this idea to non-abelian symmetries. We show, for example, that a system with $A_4$ symmetry can be rendered anomaly-free, with only one-third as many fermions as naïvely required, by passing to a larger symmetry. As another example, we find that a well-known model of quark and lepton masses utilising the $\mathrm{SL}(2,\mathbb{F}_3)$ symmetry is anomalous, but that the anomaly can be cancelled by enlarging the symmetry to a $\mathbb{Z}/3$ extension of $\mathrm{SL}(2,\mathbb{F}_3)$.

hep-th↗

arXiv2022

Generalized symmetries of topological field theories

We study generalized symmetries in a simplified arena in which the usual quantum field theories of physics are replaced with topological field theories and the smooth structure with which the symmetry groups of physics are usually endowed is forgotten. Doing so allows many questions of physical interest to be answered using the tools of homotopy theory. We study both global and gauge symmetries, as well as `t Hooft anomalies, which we show fall into one of two classes. Our approach also allows some insight into earlier work on symmetries (generalized or not) of topological field theories.

hep-th↗

arXiv2022

Floccinaucinihilipilification: Semisimple extensions of the Standard Model gauge algebra

We show how one may classify all semisimple algebras containing the $\mathfrak{su}(3)\oplus \mathfrak{su}(2) \oplus \mathfrak{u}(1)$ symmetry of the Standard Model and acting on some given matter sector, enabling theories beyond the Standard Model with unification (partial or total) of symmetries (gauge or global) to be catalogued. With just a single generation of Standard Model fermions plus a singlet neutrino, the only {gauge} symmetries correspond to the well-known algebras $\mathfrak{su}(5),\mathfrak{so}(10),$ and $\mathfrak{su}(4)\oplus \mathfrak{su}(2) \oplus \mathfrak{su}(2)$, but with two or more generations a limited number of exotic symmetries mixing flavour, colour, and electroweak degrees of freedom become possible. We provide a complete catalogue in the case of 3 generations or fewer and outline how our method generalizes to cases with additional matter.

hep-th↗

arXiv2022

Lorentz and permutation invariants of particles III: constraining non-standard sources of parity violation

Comparisons of the positive and negative halves of the distributions of parity-odd event variables in particle-physics experimental data can provide sensitivity to sources of non-standard parity violation. Such techniques benefit from lacking first-order dependence on simulations or theoretical models, but have hitherto lacked systematic means of enumerating all discoverable signals. To address that issue this paper seeks to construct sets of parity-odd event variables which may be proved to be able to reveal the existence of any Lorentz-invariant source of non-standard parity violation which could be visible in data consisting of groups of real non space-like four-momenta exhibiting certain permutation symmetries.

hep-ph↗

arXiv2022

Differential cohomology and topological actions in physics

We use differential cohomology to systematically construct a large class of topological actions in physics, including Chern-Simons terms, Wess-Zumino-Novikov-Witten terms, and theta terms (continuous or discrete). We introduce a notion of invariant differential cohomology and use it to describe theories with global symmetries and we use equivariant differential cohomology to describe theories with gauge symmetries. There is a natural map from equivariant to invariant differential cohomology whose failure to surject detects 't Hooft anomalies, i.e. global symmetries which cannot be gauged. We describe a number of simple examples from quantum mechanics, such as a rigid body or an electric charge coupled to a magnetic monopole. We also describe examples of sigma models, such as those describing non-abelian bosonization in two dimensions, for which we offer an intrinsically bosonic description of the mod-2-valued 't Hooft anomaly that is traditionally seen by passing to the dual theory of Majorana fermions. Along the way, we describe a smooth structure on equivariant differential cohomology and prove various exactness and splitting properties that help with the characterization of both the equivariant and invariant theories.

hep-th↗

arXiv2022

Gauge anomalies of finite groups

We show how the theory of characters can be used to analyse an anomaly corresponding to chiral fermions carrying an arbitrary representation of a gauge group that is finite, but otherwise arbitrary. By way of example, we do this for some groups of relevance for the study of quark and lepton masses and mixings.

hep-th↗

arXiv2022

Inverse Higgs phenomena as duals of holonomic constraints

The inverse Higgs phenomenon, which plays an important rôle in physical systems with Goldstone bosons (such as the phonons in a crystal) involves nonholonomic mechanical constraints. By formulating field theories with symmetries and constraints in a general way using the language of differential geometry, we show that many examples of constraints in inverse Higgs phenomena fall into a special class, which we call coholonomic constraints, that are dual (in the sense of category theory) to holonomic constraints. Just as for holonomic constraints, systems with coholonomic constraints are equivalent to unconstrained systems (whose degrees of freedom are known as essential Goldstone bosons), making it easier to study their consistency and dynamics. The remaining examples of inverse Higgs phenomena in the literature require the dual of a slight generalisation of a holonomic constraint, which we call (co)meronomic. Our formalism simplifies and clarifies the many ad hoc assumptions and constructions present in the literature. In particular, it identifies which are necessary and which are merely convenient. It also opens the way to studying much more general dynamical examples, including systems which have no well-defined notion of a target space.

hep-th↗

arXiv2020

Effective field theory analysis of composite higgsino-like and wino-like thermal relic dark matter

We study the effective field theory (including operators up to dimension five) of models in which dark matter is composite, consisting of either an electroweak doublet Dirac fermion (`higgsino-like dark matter') or an electroweak triplet Majorana fermion (`wino-like dark matter'). Some of the dimension-five operators in the former case cause mass splittings between the neutralino and chargino states, leading to a depleted rate of coannihilations and viable thermal relic dark matter with masses of the order of tens to hundreds of GeV rather than the usual pure higgsino thermal relic mass of 1 TeV. No such effects are found in the latter case (where the usual thermal relic mass is 3 TeV). Other operators, present for both wino- and higgsino-like dark matter, correspond to inelastic electromagnetic dipole moment interactions and annihilation through these can lead to viable models with dark matter masses up by an order of magnitude compared to the usual values.

hep-ph↗

arXiv2020

Lorentz and permutation invariants of particles II

Two theorems of Weyl tell us that the algebra of Lorentz- (and parity-) invariant polynomials in the momenta of $n$ particles are generated by the dot products and that the redundancies which arise when $n$ exceeds the spacetime dimension $d$ are generated by the $(d+1)$-minors of the $n \times n$ matrix of dot products. Here, we use the Cohen-Macaulay structure of the invariant algebra to provide a more direct characterisation in terms of a Hironaka decomposition. Among the benefits of this approach is that it can be generalized straightforwardly to cases where a permutation group acts on the particles, such as when some of the particles are identical. In the first non-trivial case, $n=d+1$, we give a homogeneous system of parameters that is valid for the action of an arbitrary permutation symmetry and make a conjecture for the full Hironaka decomposition in the case without permutation symmetry. An appendix gives formulæ for the computation of the relevant Hilbert series for $d \leq 4$.

hep-th↗

arXiv2020

Lectures: From quantum mechanics to the Standard Model

The goal of these lectures is to introduce readers with a basic knowledge of undergraduate physics (specifically non-relativistic quantum mechanics, special relativity, and electromagnetism) to the `current theory of everything': the Standard Model of particle of physics. By the end of the course, readers should be able to make predictions for simple processes at the Large Hadron Collider, such as decay rates of the Higgs boson. Some discussion of the ongoing search for physics beyond the Standard Model is also included. Based on lectures given at the Universities of Cambridge (UK) and Canterbury (New Zealand).

hep-ph↗