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The Memory of Solitons

Periodic Reporting for period 2 - MEMO (The Memory of Solitons)

Reporting period: 2021-08-01 to 2023-01-31

The quantum theory of fields is a formalism providing a universal language for a large part of contemporary physics. Originally successfully developed to model fundamental interactions of elementary particles, the quantum theory of fields has then found applications in condensed matter, statistical physics, and cosmology, in each case leading to profound and novel results that considerably improved our understanding of Nature. Not a single experiment has ever falsified a prediction of QFT. As an aside, the formalism also has deep connections with pure mathematics, leading to surprising and ground-breaking discoveries.

In a quantum field theory (QFT), the dynamics of a given microscopical system is described in terms of correlations among fluctuations of a medium that fills space and time, the quantum field. It is convenient to organize quantum fields according to the strength of their interactions, from weakly-interacting to strongly-interacting ones. Nature provides numerous realizations of both.

Weakly-interacting QFTs can be analyzed using perturbation theory, and the predictions of weakly interacting QFTs are among the most striking successes of modern theoretical physics. Strongly interacting QFTs are intrinsically non-perturbative, meaning that their dynamics differs significantly from that of a free field, and hence perturbation theory does not apply. Non-perturbative regimes are among the most challenging ones: phase transitions, the physics of confined quarks within protons, and the dynamics near the horizons of black holes are all examples of non-perturbative phenomena. Despite their importance, we lack tools to characterize non-perturbative quantum fields: any progress in this direction is a ground-breaking achievement with far-reaching interdisciplinary consequences in many areas of physics and mathematics.

A hallmark of non-perturbative QFTs is the presence of solitons in the spectrum of the quantum field excitations. Solitons are very special ripples of a given field, lumps of energy held together by the strength of field interactions that cannot be described perturbatively and are often protected by topological quantum numbers, meaning that their features are robust against continuous deformations of the parameters of the theory at hand. An important feature of quantum fields is given by the fact that their behavior changes as a function of energy. Such changes are captured by the so-called renormalization flow. Along a renormalization flow from a given phase to another, relativistic QFTs often develop solitonic sectors which are typically massive and decouple at low energies. In the context of various toy models obtained by constraining the quantum field imposing it has extra symmetries that allows to have a better control over its dynamics, one can see that it is possible to reconstruct some properties of the original highly energetic, ultraviolet phase exactly from the knowledge of the spectrum of decoupled solitons along the renormalization flow, including the values of certain observables. This effect is the memory of solitons. The main objective of the memory of solitons project is to develop this effect into a series of tools to constrain the non-perturbative regimes of various quantum fields and compute their properites. More concretely we will determine the full spectrum of excitations of given quantum fields in non-perturbative regimes (objective I: spectral problem), and we will establish dictionaries between the properties of said excitations and the properties of the non-perturbative quantum field of interest, thus providing a different pathway towards extracting the values of several observables of interest (objective II: reconstruction of observables).
In order to obtain concrete results towards understanding the memory of solitons, we have identified a class of models which have various features of interest but are also so constrained that non-perturbative regimes are accessible. The most powerful way to constrain the dynamics of quantum fields is provided by imposing that it is organised by symmetries, a fact we have used to determine our theoretical laboratory. The classes of models we have focused on to develop and refine our techniques is given by systems which have the same amount of fermionic and bosonic excitations as well as conformal invariance. Such systems can be realised in all spacetime dimensions strictly lower than seven, and we have been working on characterising them in terms of their solitonic properties in all such dimensions. As expected, models in higher dimensions are more constrained that lower dimensional ones, and for this reason during the first phase of the project we have focused our attention on theories in spacetime dimensions four, five and six.

One of the first major achievements during the first phase of the project was a better understanding of novel quantum numbers for higher dimensional operators. It is well-known that quantum fields can have interesting extended operators, which describe coherent field configurations occupying entire regions of spacetime, like lines, surfaces or various kinds of higher dimensional membranes. The quantum numbers of such field configurations can be captured by exploiting suitable generalizations of the notion of symmetries. The solitonic excitations for such quantum fields are also organized in terms of higher dimensional structures. As a first application of the memory of solitons we have developed a technique to describe how these extended solitons can screen the generalized symmetry charges, thus giving rise to a generalization of the famous ’t Hooft screening mechanism to higher dimensional field configurations. We have used this effect to capture the generalized symmetry properties of various models with and without a perturbative formulation in four, five and six dimensions (part of objective I). Models with a perturbative formulation serve as a consistency check: in all cases, our results agrees and generalises the known ones.

A second milestone that was obtained consisted of the ability of computing exactly the value of some observables for four and five dimensional theories exploiting the memory of solitons (part of objective II). This is a groundbreaking result that relies on quantum correspondences. A convenient way of thinking about correspondences is as follows: two quantum fields are in correspondence whenever there is a class of observables that, despite being defined very differently on the two sides, happen to have coinciding values. Whenever a correspondence is established, this gives a new tool to compute observables for quantum fields, which we have exploited to obtain new explicit results for the partition functions on gravitational instantons, a rich class of observables that can be studied in infinitely many classes of examples in four and five dimensions.
The memory of solitons project has already advanced considerably the state of the art as far as its main objectives are concerned. During this first reporting period, we have focused on the problem of reconstructing observables from the solitonic structures of non-perturbative quantum fields (which required progress both in the direction of objectives I and II). We have successfully developed a novel technique which allows to understand the quantum numbers of extended operators in quantum fields without a perturbative formulation. This goes beyond the state of the art, as these properties were computable only for theories with a perturbative formulation, while now these are explicitly computable for a variety of non-perturbative quantum fields with a number of applications in four, five, and six dimensions thanks to our results.

In the next reporting period, we plan to continue developing our results in the context of applications of the memory of solitons to the main objectives of our project. In particular, we expect to be able to use it to understand new quantum correspondences as well as the algebraic structure governing the category of generalized symmetries for a given quantum field, both problems of great interest for the community in line with objective II. We plan to extend these techniques to generalize of our results to systems without superconformal invariance. We are after a novel class of constraints on the renormalization flow, and ultimately on the structure of phases: we expect this novel class of constraints will have broader applications also to systems without superconformal symmetry.
Diagram encoding the algebra of solitons of a five-dimensional field theory