Quantum field theories usually depend on a set of parameters that are related to physical observables, such as particle masses or coupling constants measured at some given energy. The parameters take value in a certain parameter space, and one can "deform" the theory by varying the parameters continuously within this space. A more complicated kind of "deformation" is the one induced by the renormalization group (RG) flow, which connects a quantum field theory describing a physical system at very high energies (UV) with the one describing it at very low energies (IR). In general, it can be very difficult to determine whether two QFTs are related via a deformation of parameters or an RG flow -- for example, the relevant degrees of freedom in the IR might be completely different from the ones in the UV. A general strategy to attack this problem would be to provide a complete set of invariants, i.e. quantities that can be computed in any QFT (possibly satisfying some conditions), and that do not change under (suitably defined) "continuous deformations". There has been recent progress in implementing this program in certain simple classes of QFTs. In particular, it has been proposed that one- and two-dimensional minimally supersymmetric quantum field theories can be classified, up to deformations, by generalized cohomology theories known as K-theory and topological modular forms, respectively. The goal of this thesis is to describe these proposals and apply them to some simple examples of QFTs.
Topological Modular Forms and Quantum Field Theory
Oreglia, Pietro
2021/2022
Abstract
Quantum field theories usually depend on a set of parameters that are related to physical observables, such as particle masses or coupling constants measured at some given energy. The parameters take value in a certain parameter space, and one can "deform" the theory by varying the parameters continuously within this space. A more complicated kind of "deformation" is the one induced by the renormalization group (RG) flow, which connects a quantum field theory describing a physical system at very high energies (UV) with the one describing it at very low energies (IR). In general, it can be very difficult to determine whether two QFTs are related via a deformation of parameters or an RG flow -- for example, the relevant degrees of freedom in the IR might be completely different from the ones in the UV. A general strategy to attack this problem would be to provide a complete set of invariants, i.e. quantities that can be computed in any QFT (possibly satisfying some conditions), and that do not change under (suitably defined) "continuous deformations". There has been recent progress in implementing this program in certain simple classes of QFTs. In particular, it has been proposed that one- and two-dimensional minimally supersymmetric quantum field theories can be classified, up to deformations, by generalized cohomology theories known as K-theory and topological modular forms, respectively. The goal of this thesis is to describe these proposals and apply them to some simple examples of QFTs.File | Dimensione | Formato | |
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https://hdl.handle.net/20.500.12608/21192