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We introduce the condensed fiber product of two $G$-crossed braided fusion categories, generalizing existing constructions in the literature. We show that this product is closely related to the cohomological construction known as zesting. Furthermore, the condensed fiber product defines a monoidal structure on the 2-category of $G$-crossed and braided extensions of braided categories with a fixed transparent symmetric pointed subcategory.

We study the boundaries of the positroid cells which arise from N = 4 super Yang Mills theory. Our main tool is a new diagrammatic object which generalizes the Wilson loop diagrams used to represent interactions in the theory. We prove conditions under which these new generalized Wilson loop diagrams correspond to positroids and give an explicit algorithm to calculate the Grassmann necklace of said positroids. Then we develop a graphical calculus operating directly on noncrossing generalized Wilson loop diagrams. In this paradigm, applying diagrammatic moves to a generalized Wilson loop diagram results in new diagrams that represent boundaries of its associated positroid, without passing through cryptomorphisms. We provide a Python implementation of the graphical calculus and use it to show that the boundaries of positroids associated to ordinary Wilson loop diagram are generated by our diagrammatic moves in certain cases., Comment: 80 pages

Quantum topology provides various frameworks for defining and computing invariants of manifolds. One such framework of substantial interest in both mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum construction, which uses the data of a spherical fusion category to define topological invariants of triangulated 3-manifolds via tensor network contractions. In this work we consider a restricted class of state sum invariants of 3-manifolds derived from Tambara-Yamagami categories. These categories are particularly simple, being entirely specified by three pieces of data: a finite abelian group, a bicharacter of that group, and a sign $\pm 1$. Despite being one of the simplest sources of state sum invariants, the computational complexities of Tambara-Yamagami invariants are yet to be fully understood. We make substantial progress on this problem. Our main result is the existence of a general fixed parameter tractable algorithm for all such topological invariants, where the parameter is the first Betti number of the 3-manifold with $\mathbb{Z}/2\mathbb{Z}$ coefficients. We also explain that these invariants are sometimes #P-hard to compute (and we expect that this is almost always the case). Contrary to other domains of computational topology, such as graphs on surfaces, very few hard problems in 3-manifold topology are known to admit FPT algorithms with a topological parameter. However, such algorithms are of particular interest as their complexity depends only polynomially on the combinatorial representation of the input, regardless of size or combinatorial width. Additionally, in the case of Betti numbers, the parameter itself is easily computable in polynomial time., Comment: 24 pages, including 3 appendices

For a finite group $G$, a $G$-crossed braided fusion category is $G$-graded fusion category with additional structures, namely a $G$-action and a $G$-braiding. We develop the notion of $G$-crossed braided zesting: an explicit method for constructing new $G$-crossed braided fusion categories from a given one by means of cohomological data associated with the invertible objects in the category and grading group $G$. This is achieved by adapting a similar construction for (braided) fusion categories recently described by the authors. All $G$-crossed braided zestings of a given category $\mathcal{C}$ are $G$-extensions of their trivial component and can be interpreted in terms of the homotopy-based description of Etingof, Nikshych and Ostrik. In particular, we explicitly describe which $G$-extensions correspond to $G$-crossed braided zestings.

We show that the ribbon zesting construction can produce modular isotopes -- different modular fusion categories with the same modular data. The result relies on the observation that the Reshetikhin-Turaev invariants of framed links associated to a ribbon fusion category satisfy a factorization property under zesting. This gives a new perspective on using topological invariants to classify topological order in light of modular data not being a complete invariant.

We give a rigorous development of the construction of new braided fusion categories from a given category known as zesting. This method has been used in the past to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of super-modular categories. Here we provide a complete obstruction theory and parameterization approach to the construction and illustrate its utility with several examples., Comment: 54 pages

We give a construction and algorithmic description of the fusion ring of permutation extensions of an arbitrary modular tensor category using a combinatorial approach inspired by the physics of anyons and symmetry defects in bosonic topological phases of matter. The definition is illustrated with examples, namely bilayer symmetry defects and $S_3$-extensions of small modular tensor categories like the Ising and Fibonacci theories. An implementation of the fusion algorithm is provided in the form of a Mathematica package. We introduce the notions of confinement and deconfinement of anyons and defects, respectively, which develop the tools to generalize our approach to more general fusion rings of $G$-crossed extensions.

We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions of unitary modular tensor categories. The algebraic data of these categories afford projective unitary representations of the braid group. Elements in the image of such representations correspond to quantum gates arising from exchanging anyons and symmetry defects in topological phases of matter with symmetry. We provide some small examples that highlight features of practical interest for quantum computing. In particular, symmetry defects show the potential to generate non-abelian statistics from abelian topological phases and to be used as as a tool to enlarge the set of quantum gates accessible to an anyonic device., Comment: Accepted for publication in Proceedings of the 2016 AMS Special Session on Topological Phases of Matter and Quantum Computation

The smallest known example of a family of modular categories that is not determined by its modular data are the rank 49 categories $\mathcal{Z}(\text{Vec}_G^{\omega})$ for $G=\mathbb{Z}_{11} \rtimes \mathbb{Z}_{5}$. However, these categories can be distinguished with the addition of a matrix of invariants called the $W$-matrix that contains intrinsic information about punctured $S$-matrices. Here we show that it is a common occurrence for knot and link invariants to carry more information than the modular data. We present the results of a systematic investigation of the invariants for small knots and links. We find many small knots and links that are complete invariants of the $\mathcal{Z}(\text{Vec}_G^{\omega})$ when $G=\mathbb{Z}_{11} \rtimes \mathbb{Z}_{5}$, including the $5_2$ knot.

We study novel invariants of modular categories that are beyond the modular data, with an eye towards a simple set of complete invariants for modular categories. Our focus is on the $W$-matrix--the quantum invariant of a colored framed Whitehead link from the associated TQFT of a modular category. We prove that the $W$-matrix and the set of punctured $S$-matrices are strictly beyond the modular data $(S,T)$. Whether or not the triple $(S,T,W)$ constitutes a complete invariant of modular categories remains an open question., Comment: 24 pages; references to arXiv:1606.04378 and arXiv:1806.02843 are added

We provide an elementary introduction to topological quantum computation based on the Jones representation of the braid group. We first cover the Burau representation and Alexander polynomial. Then we discuss the Jones representation and Jones polynomial and their application to anyonic quantum computation. Finally we outline the approximation of the Jones polynomial and explicit localizations of braid group representations.

Following Manin's approach to renormalization in the theory of computation, we investigate Dyson-Schwinger equations on Hopf algebras, operads and properads of flow charts, as a way of encoding self-similarity structures in the theory of algorithms computing primitive and partial recursive functions and in the halting problem., Comment: 26 pages, LaTeX, final version, in "Feynman Amplitudes, Periods and Motives", Contemporary Mathematics, AMS 2015

This paper defines a generalization of the Connes-Moscovici Hopf algebra, $\mathcal{H}(1)$ that contains the entire Hopf algebra of rooted trees. A relationship between the former, a much studied object in non-commutative geometry, and the later, a much studied object in perturbative Quantum Field Theory, has been established by Connes and Kreimer. The results of this paper open the door to study the cohomology of the Hopf algebra of rooted trees.

We present initial results from a new high-contrast imaging program dedicated to stars that exhibit long-term Doppler radial velocity accelerations (or "trends"). The goal of the TRENDS (TaRgetting bENchmark-objects with Doppler Spectroscopy and) imaging survey is to directly detect and study the companions responsible for accelerating their host star. In this first paper of the series, we report the discovery of low-mass stellar companions orbiting HD 53665, HD 68017, and HD 71881 using NIRC2 adaptive optics (AO) observations at Keck. Follow-up imaging demonstrates association through common proper-motion. These co-moving companions have red colors with estimated spectral-types of K7--M0, M5, and M3--M4 respectively. We determine a firm lower-limit to their mass from Doppler and astrometric measurements. In the near future, it will be possible to construct three-dimensional orbits and calculate the dynamical mass of HD 68017 B and possibly HD 71881 B. We already detect astrometric orbital motion of HD 68017 B, which has a projected separation of 13.0 AU. Each companion is amenable to AO-assisted direct spectroscopy. Further, each companion orbits a solar-type star, making it possible to infer metallicity and age from the primary. Such benchmark objects are essential for testing theoretical models of cool dwarf atmospheres., Comment: Accepted to ApJ, comments welcome