Conference

This is a joint work with T. Pantev. In this talk, we will discuss moduli of flat bundles on smooth algebraic varieties, with possibly irregular singularities at infinity. For this, we use the notion of “formal boundary”, previously studied by Ben Bassat-Temkin, Efimov and Hennion– Porta–Vezzosi, as well as the moduli of flat bundles at infinity. We prove that the fibers of the restriction map to infinity are representable. We also prove that this restriction map has a canonical Lagrangian structure in the sense of shifted symplectic geometry.

It is conventional wisdom among physicists that anomalies of fermionic theories measure an obstruction to the existence of a well-defined (gauge-invariant) partition function. The aim of this talk is to use the formalism of Costello and Gwilliam to show how this wisdom is instantiated for perturbative anomalies of the massless free fermion. We will show how an action of a dg Lie algebra L on the massless free fermion theory gives rise to a line bundle over the formal moduli problem corresponding to L; the anomaly is precisely the failure of this line bundle to be trivial. Our running example will be the axial symmetry of the massless free fermion.

The generalized double semion model, introduced by Freedman and Hastings, is a lattice field theory similar to the toric code, with a gapped Hamiltonian whose space of ground states depends on the topology of the ambient manifold. In this talk, I’ll explain how to calculate its low-energy limit, which forms part of a topological field theory, in terms of characteristic classes of the ambient manifold.

The goal of my talk will be to discuss the relation between two approaches to the geometric Langlands program. The first has been proposed by Beilinson and Drinfeld, using ideas and methods from conformal field theory (CFT). The second was initiated by Kapustin and Witten based on a topological version of four-dimensional maximally supersymmetric Yang–Mills theory and its reduction to a two-dimensional topological sigma model. After discussing some issues complicating a direct comparison we will formulate a proposal for a precise relation between two main ingredients in the two approaches.

I will describe examples of holomorphic N=1 super-symmetric vertex algebras with small (non-zero) values of the elliptic genus. I will speculate on a relation to certain patterns in the theory of topological modular forms.

Let us suppose that we are trying to build a physical theory of the universe, in order to do so, we have to introduce some primitive notions, on which the theory will be based upon. We explore possible candidates that can be considered to be such "primitives": for example, the structure of the spacetime, or quantum states. However, the examples can be given such that show that these notions are not as objective as we would want them to be. The concept of objectivity, on the other hand, is closedly linked to that one of "an observer", thus, we can at least assign it as a primitive of the theory. Now agents are themselves physical systems, and we should take this into account when we specify the ground rules of what they can do. On the one hand, we take agents and their communication as a primitive of the theory and then see which concepts can be derived from there. On the other hand, we treat agents as quantum systems themselves and investigate what kind of logic applies to their interpersonal reasoning; for that, as a guiding example we use the Frauchiger-Renner thought experiment {1,2}.

Possibility to communicate between spatially separated regions, without even a single photon passing between the two parties, is an amazing quantum phenomenon. The possibility of transmitting one value of a bit in such a way, the interaction-free measurement, was known for quarter of a century. The protocols of full communication, including transmitting unknown quantum states were proposed only few years ago, but it was shown that in all these protocols the particle was leaving a weak trace in the transmission channel, the trace larger than the trace left by a single particle passing through the channel. However, a simple modification of these recent protocols eliminates the trace in the transmission channel and makes all these protocols truly counterfactual.

It was recently found that it is theoretically possible for there to exist higher-order quantum processes in which the operations performed by separate parties cannot be ascribed a definite causal order. Some of these processes are believed to have a physical realization in standard quantum mechanics via coherent control of the times of the operations. A prominent example is the quantum SWITCH, which was recently demonstrated experimentally. However, up until now, there has been no rigorous justification for the interpretation of such an experiment as a genuine realization of a process with indefinite causal structure as opposed to a simulation of such a process. Where exactly are the local operations of the parties in such an experiment? On what spaces do they act given that their times are indefinite? Can we probe them directly rather than assume what they ought to be based on heuristic considerations? How can we reconcile the claim that these operations really take place, each once as required, with the fact that the structure of the presumed process implies that they cannot be part of any acyclic circuit? Here, I offer a precise answer to these questions: the input and output systems of the operations in such a process are generally nontrivial subsystems of Hilbert spaces that are tensor products of Hilbert spaces associated with different times—a fact that is directly experimentally verifiable. With respect to these time-delocalized subsystems, the structure of the process is one of a circuit with a cycle, which cannot be reduced to a (possibly dynamical) probabilistic mixture of acyclic circuits. This provides, for the first time, a rigorous proof of the existence of processes with indefinite causal structure in quantum mechanics. I further show that all bipartite processes that obey a recently proposed unitary extension postulate, together with their unitary extensions, have a physical realization on such time-delocalized subsystems, and provide evidence that even more general processes may be physically admissible. These results unveil a novel structure within quantum mechanics, which may have important implications for physics and information processing.

Models that have some but not all features of standard quantum theory can be valuable in several ways, as Bell, Ghirardi-Rimini-Weber-Pearle, Hardy, Spekkens and many others have shown. One is to illuminate quantum theory and shed light on possible reaxiomatisations or reformulations. Another is to suggest experiments that might confirm some untested aspect of quantum theory or point the way to a new theory. I discuss here some models that combine quantum theory and gravity and experimental tests.

For well over a decade, we developed an entirely pictorial (and formally rigorous!) presentation of quantum theory [*]. At the present, experiments are being setup aimed at establishing the age at which children could effectively learn quantum theory in this manner. Meanwhile, the pictorial language has also been successful in the study of natural language, and very recently we have started to apply it to model cognition, where we employ GPT-alike models. We present the key ingredients of the pictorial language language as well as their interpretation across disciplines. [*] B. Coecke & A. Kissinger (2017) Picturing Quantum Processes. A first course on quantum theory and diagrammatic reasoning. Cambridge University Press.