Khyathi Komalan

Khyathi Komalan

category theory & physics enthusiast interested in the arts and humanities

Posts by Collection

portfolio

publications

Constructing TQFTs Using Non-Hermitian Ribbon Fusion Categories

Published in arXiv, 2024

In this paper, we provide a construction of a Topological Quantum Field Theory from a Non-Hermitian Ribbon Fusion Category. This is a simple method that does not involve enriching over Fusion Categories, or using other complicated structures. To substantiate this construction, we also prove theorems on the Müger center, braiding, and spherical structure of such a fusion category. [Submitted for Publication]

Recommended citation: Komalan, K. (2024). Constructing TQFTs Using Non-Hermitian Ribbon Fusion Categories. arXiv preprint arXiv:2410.16993.
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Quantum Barcodes: Persistent Homology for Quantum Phase Transitions

Published in arXiv, 2025

We introduce “quantum barcodes,” a theoretical framework that applies persistent homology to classify topological phases in quantum many-body systems. By mapping quantum states to classical data points through strategic observable measurements, we create a “quantum state cloud” analyzable via persistent homology techniques. Our framework establishes that quantum systems in the same topological phase exhibit consistent barcode representations with shared persistent homology groups over characteristic intervals. We prove that quantum phase transitions manifest as significant changes in these persistent homology features, detectable through discontinuities in the persistent Dirac operator spectrum. Using the SSH model as a demonstrative example, we show how our approach successfully identifies the topological phase transition and distinguishes between trivial and topological phases. While primarily developed for symmetry-protected topological phases, our framework provides a mathematical connection between persistent homology and quantum topology, offering new methods for phase classification that complement traditional invariant-based approaches. [Submitted for Publication]

Recommended citation: Komalan, K. (2025). Quantum Barcodes: Persistent Homology for Quantum Phase Transitions. arXiv preprint arXiv:2504.10468.
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Brushstrokes and Tensor Products: Painting with a Monoidal Category

Published in arXiv, 2025

This article offers an intuitive introduction to monoidal categories through the lens of painting, presenting abstract mathematical concepts with visual and tactile analogies. Aimed at curious undergraduates and non-specialists, it seeks to demystify category theory by showing how ideas like the tensor product, associators, and braidings can be understood as compositional tools on a canvas. [Submitted for Publication]

Recommended citation: Komalan, K. (2025). Brushstrokes and Tensor Products: Painting with a Monoidal Category. arXiv preprint arXiv:2508.05482.
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talks

Non-Hermitian Ribbon Fusion Categories

Published:

Talk Abstract: Several open problems have been asked and are being solved on the nature of dagger structures that can be defined on Fusion Categories. Using previous work that has been done investigating the Hermitian Dagger case, we now shift our focus to dealing with Fusion Categories that have a Non-Hermitian Dagger structure on them. What can we do with these categories? Do they yield any interesting properties? We discuss the outline of proofs of theorems on the Müger center, braiding, and spherical structure of such a fusion category, and discuss a potential construction of a TQFT, and what it could mean for us.

Some Results on Non-Hermitian Ribbon Fusion Categories

Published:

Several open problems have been asked and are being solved on the nature of dagger structures that can be defined on Fusion Categories, and many results have been shown for Hermitian (positive dagger) Fusion Categories. However, Non-Hermitian Fusion Categories do exist, the most prominent example of them being the Yang-Lee Category. Using properties of Fusion Categories, previous results that has been found about Fusion Categories with a Hermitian Dagger Structure, and properties that arise from the existence of a Ribbon Structure and the ability to define a braiding, we now shift our focus on investigating Ribbon Fusion Categories that have a Non-Hermitian Dagger structure on them. Utilizing the rich and complicated structure of these categories, we discuss the outline of proofs of theorems on the Muger center, braiding, and spherical structure of such a fusion category, and discuss a potential construction of a TQFT using these results. slides

There is No Band (AQFT and Double Categories)

Published:

David Lynch’s neo-noir classic Mulholland Drive challenges our sense of continuity, identity, and causality. In this talk, I use its fragmented narrative structure as a metaphorical guide to construct an algebraic quantum field theory using a double functor between free globularly generated double categories.

teaching

Teaching experience 1

Undergraduate course, University 1, Department, 2014

This is a description of a teaching experience. You can use markdown like any other post.

Teaching experience 2

Workshop, University 1, Department, 2015

This is a description of a teaching experience. You can use markdown like any other post.