Topological Defects 5 credits

Time

Period 2-3, Academic Year 2024/25

Short Description

A defect in a quantum field is an inhomogeneity localised on a submanifold of positive codimension. For codimension 1 this includes interfaces between different theories. A defect is topological if the precise location of the defect submanifold does not matter in the sense that it can be deformed without affecting correlation functions, as long as it does not cross other defects or field insertions.

Conceptually, topological defects generalise the notion of a group symmetry. An element g of the symmetry group corresponds to a topological defect of codimension 1 across which the value of field operators jumps by the action of g. In this generalisation, the mathematical language of group theory is upgraded to that of (higher) category theory. Namely, in an n-dimensional field theory or statistical model, topological defects are expected to form an n-category, with defects localised on k-manifolds providing the (n − k)-morphisms.

Thinking of a group as a category with one object where all morphisms are invertible illustrates that the n-category of topological defects provides much finer information, since it includes p-morphism with p >1 (corresponding to lower-dimensional defects joining those of higher dimensions into defect networks), and it allows for non-invertible morphisms (corresponding to defects of a given dimension which cannot be fused with another defect to arrive at the trivial defect).

Topological defects appear in a variety of settings and languages. It is remarkable that the same mathematical structure – higher categories – appears in all of these descriptions. This course presents some of these settings in their language of choice – from lattice models, to topological field theory, as well as quantum field theories described by action functionals – and it sketches how topological defects and higher categories arise in each case.

Learning Outcomes

Aspects of higher categories and their counterparts in topological field theories • Symmetry categories of lattice models (state sum models) • Symmetries in quantum field theories in various dimensions and their applications

Preliminary Bibliography

N. Carqueville, M. Del Zotto, I. Runkel, “Topological Defects”, https://arxiv.org/abs/ 2311.02449 • D. S. Freed, "Lectures on Field Theory and Topology", CBMS Regional Conference Series in Mathematics, Number 133, AMS 2019, https://web.ma.utexas.edu/users/dafr/ CBMS133.pdf • S.-H. Shao, “What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetry”, https://arxiv.org/abs/2308.00747 During the course the students will be provided further materials and texts.

Lecturer

For further information about the course, please feel free to contact Michele Del Zotto, michele.delzotto@math.uu.se