Mathematical Physics

In 2003 Witten introduced twistor string theory as a novel description of the scattering matrix of the maximally supersymmetric Yang-Mills theory in four dimensions. In these lectures I will give an introduction to the developments that have led to new formulations, also based on Riemann surfaces, of a large variety of theories, with and without supersymmetry, in arbitrary space-time dimensions.

I will give a brief overview of Topological Recursion and present the general setting and our contribution to this field via geometry and topology techniques. In particular, I will discuss the toplogical recursion applied to the family of spectral curves of Hitchen modulo spaces of Higgs bundles over a smooth base curve C. We study meromorphic Higgs fields of rank two and we realized their spectral curves as divisors in the compactifed cotangent bundle. Topological recursion gives a way to quantize the spectral curve of a Higgs bundle. I will present as typical examples of our theory, some well-known constructions as the recursion of Witten-Kontsevich intersection numbers and the recursion of Catalan numbers, that count the number of cellular graphs on a Riemann Surfaces. In particular, we present a model for the twisted version of the Topological Recursion via Cohomological Field Theory for Mg;n(BG). We prove tat edge contraction axioms of cellular graphs leads to a TQFT.

Buildings are higher dimensional analogues of trees. The goal of these lectures is to explain how the theory of harmonic maps to buildings affords a new perspective on certain aspects of the WKB analysis of differential equations that depend on a small parameter. We will also touch upon some motivation for developing this perspective, which derives from questions about compactifications of higher Teichmüller spaces, stability in Fukaya categories, and the work of Gaiotto, Moore and Neitzke on spectral networks and wall-crossing phenomena. These talks are based on joint work with Ludmil Katzarkov, Alexander Noll and Carlos Simpson. A central role in our discussion will be played by the notion of a versal pre-building associated with a given spectral cover of a Riemann surface. This notion generalizes to higher rank the leaf space of the foliation defined by a quadratic differential. We will see that spectral networks are closely related to the singular loci of versal buildings, and that distances in these buildings encode information about the asymptotic behavior at infinity of the Riemann-Hilbert correspondence.

Exact WKB analysis, developed by Voros et.al., is an effective method for the global study of differential equations (containing a large parameter) defined on a complex domain. In the first and second lecture I'll give an introduction to exact WKB analysis, and recall some basic facts about WKB solutions, Borel resummation, Stokes graphs etc. On the other hand, cluster algebras are a particular class of commutative subalgebras of the field of rational functions with distinguished generators. I'll explain about a hidden cluster algebraic structure in exact WKB analysis in the third lecture. This is a joint work with Tomoki Nakanishi (Nagoya).

In 2003 Witten introduced twistor string theory as a novel description of the scattering matrix of the maximally supersymmetric Yang-Mills theory in four dimensions. In these lectures I will give an introduction to the developments that have led to new formulations, also based on Riemann surfaces, of a large variety of theories, with and without supersymmetry, in arbitrary space-time dimensions.

Perhaps the first use of the mathematical theory of heat to develop another theory was Thomson’s use of Fourier’s equations to formulate equations for electrostatics in the 1840s. After extracting a lesson from this historical case, I will fast forward more than a century to examine the relationship between classical statistical mechanics and QFT that is induced by analytic continuation. While there is no doubt that this mathematical relationship has been heuristically useful in guiding developments in both statistical mechanics and QFT, this is a case in which the physical interpretation of the mathematics does not carry over from one theory to the other.

It is sometimes envisaged that the behaviour of elementary particles can be characterised by the information content it carries, and that exchange of energy and momentum, or more generally the change of state through interactions, can likewise be characterised in terms of its information content. But exchange of information occurs only in the context of a (typically noisy) communication channel, which traditionally requires a transmitter and a receiver; whereas particles evidently are not equipped with such devices. In view of this a new concept in communication theory is put forward whereby signal processing is carried out in the absence of a transmitter; hence mathematical machineries in communication theory serves as new powerful tools for describing a wide range of observed phenomena. In the quantum context, this leads to a tentative—and perhaps speculative—idea that the dynamical evolution of the state of a quantum particle is such that the particle itself acts as if it were a "signal processor", trying to identify the stable configuration that it should settle, and adjusts its own state accordingly. It will be shown that the mathematical scheme of such a hypothesis works well for a broad class of noise structures having stationary and independent increments. (The talk will be based on work carried out in collaboration with L. P. Hughston.)

The third law of thermodynamics has a controversial past and a number of formulations due to Planck, Einstein, and Nernst. It's most accepted version, the unattainability principle, states that "any thermodynamic process cannot reach the temperature of absolute zero by a finite number of steps and within a finite time". Although formulated in 1912, there has been no general proof of the principle, and the only evidence we have for it is that particular cooling methods become less efficient as a the temperature lowers. Here we provide the first derivation of a general unattainability principle, which applies to arbitrary cooling processes, even those exploiting the laws of quantum mechanics or involving an infinite-dimensional reservoir. We quantify the resources needed to cool a system to any particular temperature, and translate these resources into a minimal time or number of steps by considering the notion of a Cooling Machine which obeys similar restrictions to universal computers. We generally find that the obtainable temperature scales as an inverse power of the cooling time, and place ultimate bounds on the speed at which information can be erased.

As is well known, time plays a special role in the standard formulation of quantum theory, bringing the latter into severe conflict with the principles of general relativity. This suggests the existence of a more fundamental and (as it turns out) covariant and timeless formulation of quantum theory. A conservative way to look for such a formulation would be to start from quantum theory as we know it, taken in its experimentally most successful form of quantum field theory, and try to uncover structure in the formalism made for actual physical predictions. A radical way to look for such a formulation would be to forget the standard formulation, take only a few first principles (locality and operationalism turn out to be good ones) and try to construct things from there. Remarkably, approaches following these apparently opposite paths have recently been shown to converge in a single framework. In this talk I want to provide an overview of the current understanding of the resulting "positive formalism", its implications, and the paths that led to it. This includes relations to works of Witten and Segal in mathematical physics and of Aharonov, Hardy and others in quantum foundations.

The theory of causal fermion systems is an approach to describe fundamental physics. It gives quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. Instead of introducing physical objects on a preexisting space-time manifold, the general concept is to derive space-time as well as all the objects therein as secondary objects from the structures of an underlying causal fermion system. The dynamics of the system is described by the causal action principle. I will give a non-technical introduction, with an emphasis on conceptual issues related to information theory.

The CA interpretation presents a view on the origin of quantum mechanical behavior of physical degrees of freedom, suggesting that, at the Planck scale, bits and bytes are processed, rather than qubits or qubites, so that we are dealing with an ordinary classical cellular automaton. We demonstrate how this approach naturally leads to Born's expression for probabilities, shows how wave functions collapse at a measurement, and provides a natural resolution to Schroedinger's cat paradox without the need to involve vague decoherence arguments. We then continue to discuss the implications of Bell's inequalities, and other issues.