Sixty-Sixth Meeting of the North British Mathematical Physics Seminar

I will make some comments on \(SU(N)\) QCD theory with a single adjoint majorana fermion, and fundamental scalars or fermions. The flavor group contains a baryon symmetry \(U(1)\). I will show that the presence of the majorana fermion causes the system to have a mixed ’t Hooft anomaly between the \(U(1)\) baryon number and the time reversal symmetry – the so-called parity anomaly – enforcing the vacuum to be non-trivial. Some scenarios will be proposed about how one can match the anomalies.

Higher forms arise as gauge potentials in a number of contexts within physics. The prime example is the Kalb–Ramond B-field of string theory, which is found also in the low-energy supergravity limits. Mathematically, these gauge potentials are connections on higher or categorified principal bundles also known as gerbes. In the Abelian case, the theory of gerbes is well-established and used in many contexts.

Here, I will talk about how one can extend this theory to the non-Abelian setting, where the situation is much more subtle. Although the topological construction of non-Abelian gerbes is relatively straightforward (one just lifts the cocycle relations for transition functions of an ordinary principal bundle to hold up to homotopies, which are then encoded in higher components of the cocycle), defining a connection on such a gerbe requires much more work. After formulating the general construction I will illustrate it by lifting the principal bundle corresponding to an instanton--anti-instanton pair to a string 2-group bundle. Such ``string structures'' are believed to play a role in the dynamics of M5-branes.

This talk is based on our recent work arXiv:2203.00092.

We produce novel non-involutive solutions of the Yang-Baxter equation coming from (skew) braces. These solutions are generalisations of the known ones coming from the standard braces and skew braces, and surprisingly in the case of braces they are not necessarily involutive. Novel bijective maps associated to the inverse solutions are also introduced. Moreover, we show that the recently derived Drinfeld twists of the involutive case are still admissible in the non-involutive frame, and we identify the twisted r-matrices and twisted copoducts. We observe that as in the involutive case the underlying quantum algebra is not a quasi-triangular bialgebra, as one would expect, but a quasi-triangular quasi-bialgebra.

Recently, there has been much interest in the study of symmetries in physical systems as they characterise topological aspects of quantum field theories. An important realisation is that symmetries, their anomalies and BF theory in a given QFT may be characterised in one higher dimension by a symmetry topological field theory (SymTFT) via inflow mechanics. Such descriptions can naturally emerge whenever we can construct quantum field theories from string theory. In this talk, I will introduce SymTFTs and explain how to construct them from geometry.

It has been argued that \(PSU(N)\) bundles could play a role in \(SU(N)\) theories since they may provide the means to understand the microscopic dynamics. I present a serious examination of this proposal by computing the gaugino condensate in super Yang-Mills theory on a small four-torus with 't Hooft twists in a weakly-coupled regime. By taking the torus very large, approaching \(R^4\), we obtain N-times the value of the condensate calculated from other methods on \(R^4\). I discuss possible loopholes in the continuity approach that may lead to this unexpected result.