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Symmetry underpins all physics research. We look for fundamental and beautiful patterns to describe and explain the laws of nature. One way of explaining symmetry is to ask: "what is the full set of operations I can do to my real-world experiment or abstract theory written on paper that doesn't change any physical measurements or predictions?'' There are simple symmetries we are perhaps already familiar with. For example, lab-based physics experiments usually don't care if you wait an hour to do the experiment or if you rotate your apparatus by 90 degrees. Beyond these everyday symmetries, theoretical physicists think about all kinds of symmetries including "global" and "gauge'' symmetries, supersymmetry and most recently "generalized'' symmetries.

My research lies at the intersection of three areas: generalized symmetries, quantum field theory (QFT) and string theory. The term "generalized symmetries'' originates from a paper in which it was demonstrated that the concept of a "symmetry'' of a QFT is much richer than previously thought! The authors tell us that we should think not only of symmetries as transformations which act on local point-like operators, but also those which act on higher-dimensional operators. For a longer-form introduction to this topic, see this very nice article recently published in Quanta magazine.

My PhD focuses primarily on understanding generalized symmetries in QFTs which are constructed using string theory. In a certain sense which will be explained below, string theory provides a dictionary between physical QFT features of interest and mathematics. One of the aims of my work is to understand how we can utilize geometry and topology to probe features of generalized symmetries.

Quantum Field Theory. QFT is a powerful theoretical construct which can be used to describe the interactions of real-world particles with one another. It combines quantum mechanics with special relativity in describing these particles as excitations of underlying fundamental fields.

QFT comes hand-in-hand with the study of symmetries: specifying the symmetry which you'd like to keep can highly constrain the types of fields you can have and their allowed interactions. Theorists aim to learn more about QFTs by studying broad classes of them in a wide range of spacetime dimensions with varying amounts of symmetry.

Generalized Global Symmetries. An "ordinary symmetry" is some group-like action on point-like operators which leaves the theory invariant. It would be difficult to overstate how influential these symmetries have been in the development of physics as we know it today.

A "higher-form'' symmetry is the first generalization of this idea: it is a symmetry which acts on higher-dimensional operators (1-dimensional lines, 2-dimensional surfaces etc.). Furthermore, these higher-form symmetries need not combine in a trivial way: one should in fact generically consider "higher-group'' symmetries which are non-trivial combinations of more than one type of higher-form symmetry. Most recently, all of this structure was generalized further to "non-invertible/ categorical'' symmetries which are those which cannot be mathematically encoded by groups. The key technical insight of all of these developments was the observation that symmetries are described by topological operators in the QFT.

Given how powerful symmetry has already been, the discovery that the theories we know and love may possess extra symmetry is incredibly tantalizing!

Generalized Symmetries in String Theory. String theory is a remarkable proposal which combines quantum physics with gravitational physics in a framework defined in 11 spacetime dimensions. It provides a vast playground in which theorists can lose themselves studying QFT from a unique and intrinsically mathematical perspective.

For my own work, string theory is most useful for its ability to probe QFTs of dimension closer to our own world (4!). One can construct such lower-dimensional QFTs in several different ways. Two cases of interest in my research are holography and geometric engineering. Common to both approaches is the idea that in order to reconcile the 11 spacetime dimensions of string theory with the $d<11$ dimensions of the QFT of interest, we must "wrap up'' some of the dimensions. The physics of the $d$-dimensional QFT depends intricately on the geometry of these wrapped up directions. To give a concrete example: the higher-form symmetry group of a QFT constructed in string theory can often be read off directly from the homology groups of the "wrapped up'' manifold. It is in this sense that mathematical tools can be used to probe physical quantities of QFTs.

Now suppose one of these QFTs has a generalized symmetry. My research asks: what is the imprint of this symmetry in the string theory construction? Can string theory help us understand these symmetries?

There has been a lot of exciting work in this direction, with our understanding developing as I write this. In holographic solutions, I have explored how higher-form symmetries and their anomalies are encoded in string theory (e.g. in 3 dimensions and 4 dimensions). In other work, we found that particularly special classes of (super-conformal) QFTs in 6 dimensions can have a so-called 2-group symmetry which combines an ordinary symmetry with a 1-form symmetry which acts on line operators. Related to this we studied how finite ordinary symmetries can act on continuous ordinary symmetries and 2-groups in a non-trivial way. In upcoming work we also explore how one can study the action of non-invertible symmetries on higher-dimensional operators using branes in string theory (short for "membrane'' - these are higher-dimensional analogues of the string, which extend in multiple dimensions).

Dewi Gould is a postgraduate student in the Mathematical Physics Group in Oxford Mathematics.

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