Mathematical Physics

We have already met the octonionic Fierz identity satisfied by spinors in 10-dimensional spacetime. This identity makes super-Yang-Mills "super" and allows the Green-Schwarz string to be kappa symmetric. But it is also the defining equation of a "higher" algebraic structure: an L-infinity algebra extending the supersymmetry algebra. We introduce this L-infinity algebra in octonionic language, and describe its cousins in various dimensions. We then survey various consequences of its existence, such as the brane bouquet of Fiorenza-Sati-Schreiber.

We explore the Z2 graded product C`10 = C`4⊗ˆC`6 (introduced by Furey) as a finite quantum algebra of the Standard Model of particle physics. The gamma matrices generating C`10 are expressed in terms of left multiplication by the imaginary octonion units and the Pauli matrices. The subgroup of Spin(10) that fixes an imaginary unit (and thus allows to write O = C⊗C 3 expressing the quark-lepton splitting) is the Pati-Salam group GP S = Spin(4) × Spin(6)/Z2 ⊂ Spin(10). If we identify the preserved imaginary unit with the C`6 pseudoscalar ω6 = γ1...γ6, ω2 6 = −1 (cf. the talk of Furey and Hughes), then Pex = 1 2 (1 − iω6) will play the role of the projector on the extended particle subspace including the right-handed (sterile) neutrino. We express the generators of C`4 and C`6 in terms of fermionic oscillators aα, a∗ α, α = 1, 2 and bj , b∗ j , j = 1, 2, 3 describing flavour and colour, respectively. The internal space observable algebra (an analog of the algebra of real functions on space-time) is then defined as the Jordan subalgebra of hermitian elements of the complexified Clifford algebra C ⊗ C`10 that commute with the weak hypercharge 1 2 Y = 1 3 P3 j=1 b ∗ j bj − 1 2 P2 α=1 a ∗ αaα. We only distinguish particles from antiparticles if they have different eigenvalues of Y . Thus the sterile neutrino and antineutrino (with Y = 0) are allowed to mix into Majorana neutrinos. Restricting C`10 to the particle subspace which consists of leptons with Y < 0 and quarks with Y > 0 allows a natural definition of the Higgs field Φ, the scalar of Quillen’s superconnection, as an element of C`1 4, the odd part of the first factor in C`10. As an application we express the ratio mH mW of the Higgs and the W-boson masses in terms of the cosine of the theoretical Weinberg angle. The talk is based on the paper arXiv:2010.15621v3 a copy of which including minor corrections is attached.

Can the 32C-dimensional algebra R(x)C(x)H(x)O offer anything new for particle physics? Indeed it can. Here we identify a sequence of complex structures within R(x)C(x)H(x)O which sets in motion a cascade of breaking symmetries: Spin(10) -> Pati-Salam -> Left-Right symmetric -> Standard model + B-L (both pre- and post-Higgs-mechanism). These complex structures derive from the octonions, then from the quaternions, then from the complex numbers. Finally, we describe a left-right symmetric Higgs system which exhibits, we believe for the first time, an explicit demonstration of quaternionic triality.

"I will begin by reviewing the unified description of pure Super Yang-Mills (SYM) Theory (consisting of just a gauge field and gaugino) in dimensions 3, 4, 6, and 10 over the four normed division algebras R, C, H, and O. Dimensionally reducing these initial theories into dimensions 3, 4, 5, 6, 7, 8, 9, 10 gives a plethora of SYM theories written over the division algebras, with a single master Lagrangian to rule them all. In particular, in D = 3 spacetime dimensions, the SYM theories with N = 1, 2, 4, and 8 supersymmetries enjoy a unified description over R, C, H, and O, respectively. In each spacetime dimension, maximally supersymmetric theories are written over the octonions. In apparently completely different developments, a popular thread in attempts to understand the quantum theory of gravity is the idea of "gravity as the square of Yang-Mills". The idea in its most basic form is that a symmetric tensor (graviton) can be built from the symmetric tensor product of two vectors (Yang-Mills fields), an idea which can be extended to obtain entire supergravity multiplets from tensor products of SYM multiplets. Having established a division-algebraic description of Super Yang-Mills theories, I will then demonstrate how tensoring these multiplets together results in supergravity theories valued over tensor products of division algebras. In D = 3, there are 4 SYM theories (N = 1, 2, 4, 8 over R, C, H, O) and so there are 4 x 4 = 16 possible supergravity theories to obtain by "squaring Yang-Mills". The global symmetries of these 16 division-algebraic SYM-squared supergravity theories are precisely those belonging to the 4 x 4 Freudenthal-Rosenfeld-Tits "magic square" of Lie algebras! Furthermore, the scalar fields in these supergravity theories describe non-linear sigma models, whose target space manifolds are division algebraic projective planes! Performing the same tensoring of SYM theories in spacetime dimensions D > 3 results in a whole "magic pyramid" of supergravities, with the magic square at the base in D = 3 and Type II supergravity at the apex in D = 10. This construction gives an explicit octonionic explanation of many of the mysterious appearances of exceptional groups within string/M-theory and supergravity."

Human contact patterns are highly heterogeneous in terms of the both the number and nature of interactions. To incorporate these heterogeneities into infectious disease models one naturally represents a population as a weighted network. While there is a large literature on the spread of diseases on networks, most techniques are highly computational in nature. In this talk I will talk about an analytical framework for modeling infectious diseases as a percolation process on weighted networks based on probability generating functions. 

In the context of vaccination, human contact heterogeneities become a resource: Vaccinating individuals with greater total exposure leads to a greater reduction in disease spread. We have proposed that exposure notification apps, such as COVID Alert, can be leveraged to improve vaccine uptake among high exposure individuals, thereby optimizing our limited COVID-19 vaccine supply. We demonstrate the efficiency of this proposal using our weighted percolation theory framework.

Can gravity, in certain regards, be the `product' of two gauge theories, such as those appearing in the Standard Model? I will begin by reviewing the Bern—Carrasco—Johansson colour—kinematics duality conjecture, which implies that one can write the scattering amplitudes of Einstein-Hilbert gravity (coupled to a Kalb-Ramond 2-form and dilaton scalar) as the double copy of Yang—Mills amplitudes. Although the colour—kinematics duality, and therefore the double copy, was quickly established at the tree level, it remains a longstanding open problem at the loop level, despite highly non-trivial explicit examples. I will then describe how one can take this `gravity = gauge x gauge' amplitude paradigm `off-shell’ as a product of spacetime fields: the Yang-Mills BRST-Lagrangian itself double copies into perturbatively quantised Einstein-Hilbert gravity coupled to a Kalb-Ramond 2-form and dilaton, establishing the validity of the double copy to all orders, tree and loop. I will end by briefly discussing the homotopy algebras underpinning this result and the inclusion of supersymmetry, which reveals fascinating octonionic structures (some very well-known, others completely new) that will be the subject of Mia Hughes's talk in the following week.

"Non-equilibrium statistical mechanics has seen some impressive developments in the last three decades, thank to the pioneering works of Evans, Cohen, Morris and Searles on the violation of the second law, soon followed by the ground-breaking formulation of the Fluctuation Theorem by Gallavotti and Cohen for entropy fluctuation in the early nineties. Their work was by vast literature, both theoretical and experimental The extension of these results to the quantum setting has turned out to be surprisingly challenging and it is still an undergoing effort. Kurchan’s seminal work (2000) showed the measurement role has to be taken in account, leading to the introduction of the so called two-time measurement statistics (also known as full counting statistics). However in this context, the lack of a trajectory notion leads to both conceptual and technical problems, or phenomena with no classical counterpart, as underlined by some of our recent results. In this talk I will review some of the key concept involved in the Fluctuation Theorem and its extensions to the quantum setting; I will present some recent results exploring the role played by ultraviolet regularity conditions (joint work with R.Raquépas and T.Benoist )."

"I discuss applications of a hidden $U_q(\mathfrak{sl}_2)$-symmetry in CFT with central charge $c \leq 1$ (focusing on the generic, semisimple case, with $c$ irrational). This symmetry provides a systematic method for solving Belavin-Polyakov-Zamolodchikov PDE systems, and in particular for explicit calculation of the asymptotics and monodromy properties of the solutions. Using a quantum Schur-Weyl duality, one can understand solution spaces of such PDE systems in a detailed way. The solutions, in turn, are useful both for CFT questions and for rigorous understanding of the connections of 2D CFT with critical models of statistical physics."

The Boltzmann--Gibbs entropy is a functional on the space of probability measures. One characterization of the Boltzmann--Gibbs entropy is given by the Shannon--Khinchin axioms, which consist of continuity, maximality, expandability and extensivity. The extensivity is expressed in terms of the linear combinations of conditional probabilities. Replacing the coefficients in the linear combinations with a power function provides a characterization of the Tsallis entropy. I talk about the impossibility to replace the coefficients with a non-power function.

" In this talk, I posit two concepts from the economics literature as hypotheses for the observed data on women in academia. This talk includes time for discussion about how these concepts can inform our approach to mentoring junior women. "

"Geometrically, a gauge theory consists of a spinor bundle describing the matter fields, associated to some principal bundle whose gauge group rules the internal symmetries of the system. The gauge fields are the local expressions of a principal connection inducing a covariant derivative which settles the dynamics of the matter fields. Principal connections can be seen as parallel displacements on the fibres of the principal bundle along curves on the base manifold. In this talk I shortly present a generalisation of gauge fields given by direct connections on gauge groupoids, based on a work in progress with S. Azzali, Y. Boutaib, A. Garmendia and S. Paycha."