Braided Frobenius algebras from certain Hopf algebras

A braided Frobenius algebra is a Frobenius algebra with a Yang–Baxter operator that commutes with the operations, that are related to diagrams of compact surfaces with boundary expressed as ribbon graphs. A heap is a ternary operation exemplified by a group with the operation [Formula: see text], that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of the heap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopf algebras that admit integrals and cointegrals. For these Hopf algebras we show that the heap operation induces a Yang–Baxter operator on the tensor product, which satisfies the required compatibility conditions. Diagrammatic methods are employed for proving commutativity between Yang–Baxter operators and Frobenius operations.

CFT Correlators for Cardy Bulk Fields via String-Net Models

We show that string-net models provide a novel geometric method to construct invariants of mapping class group actions. Concretely, we consider string-net models for a modular tensor category C. We show that the datum of a specific commutative symmetric Frobenius algebra in the Drinfeld center Z(C) gives rise to invariant string-nets. The Frobenius algebra has the interpretation of the algebra of bulk fields of the conformal field theory in the Cardy case.

A (dummy’s) guide to working with gapped boundaries via (fermion) condensation

We study gapped boundaries characterized by “fermionic condensates” in 2+1 d topological order. Mathematically, each of these condensates can be described by a super commutative Frobenius algebra. We systematically obtain the species of excitations at the gapped boundary/junctions, and study their endomorphisms (ability to trap a Majorana fermion) and fusion rules, and generalized the defect Verlinde formula to a twisted version. We illustrate these results with explicit examples. We also connect these results with topological defects in super modular invariant CFTs. To render our discussion self-contained, we provide a pedagogical review of relevant mathematical results, so that physicists without prior experience in tensor category should be able to pick them up and apply them readily.

3d $$ \mathcal{N} $$ = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K -theory of Grassmannians

We study a correspondence between 3d $$ \mathcal{N} $$ N = 2 topologically twisted Chern-Simons-matter theories on S1× Σg and quantum K -theory of Grassmannians. Our starting point is a Frobenius algebra depending on a parameter β associated with an algebraic Bethe ansatz introduced by Gorbounov-Korff. They showed that the Frobenius algebra with β = −1 is isomorphic to the (equivariant) small quantum K -ring of the Grassmannian, and the Frobenius algebra with β = 0 is isomorphic to the equivariant small quantum cohomology of the Grassmannian. We apply supersymmetric localization formulas to the correlation functions of supersymmetric Wilson loops in the Chern-Simons-matter theory and show that the algebra of Wilson loops is isomorphic to the Frobenius algebra with β = −1. This allows us to identify the algebra of Wilson loops with the quantum K - ring of the Grassmannian. We also show that correlation functions of Wilson loops on S1× Σg satisfy the axiom of 2d TQFT. For β = 0, we show the correspondence between an A-twisted GLSM, the Frobenius algebra for β = 0, and the quantum cohomology of the Grassmannian. We also discuss deformations of Verlinde algebras, omega-deformations, and the K -theoretic I -functions of Grassmannians with level structures.

Functoriality of Khovanov homology

In this paper, we prove that every Khovanov homology associated to a Frobenius algebra of rank 2 can be modified in such a way as to produce a TQFT on oriented links, that is a monoidal functor from the category of cobordisms of oriented links to the homotopy category of complexes.

Twisted orthogonality relations for certain ℤ/mℤ-graded algebras

In this paper, we will study the notion of a Frobenius ⋆-algebra and prove some orthogonality relations for the irreducible characters of a Frobenius ⋆-algebra. Then we will study [Formula: see text]-graded Frobenius ⋆-algebras and prove some twisted orthogonality relations for them.

The ZX calculus is a language for surface code lattice surgery

A leading choice of error correction for scalable quantum computing is the surface code with lattice surgery. The basic lattice surgery operations, the merging and splitting of logical qubits, act non-unitarily on the logical states and are not easily captured by standard circuit notation. This raises the question of how best to design, verify, and optimise protocols that use lattice surgery, in particular in architectures with complex resource management issues. In this paper we demonstrate that the operations of the ZX calculus --- a form of quantum diagrammatic reasoning based on bialgebras --- match exactly the operations of lattice surgery. Red and green ``spider'' nodes match rough and smooth merges and splits, and follow the axioms of a dagger special associative Frobenius algebra. Some lattice surgery operations require non-trivial correction operations, which are captured natively in the use of the ZX calculus in the form of ensembles of diagrams. We give a first taste of the power of the calculus as a language for lattice surgery by considering two operations (T gates and producing a CNOT) and show how ZX diagram re-write rules give lattice surgery procedures for these operations that are novel, efficient, and highly configurable.

Khovanov homology and diagonalizable Frobenius algebras

We give a short elementary proof that a Khovanov-type link homology constructed from a diagonalizable Frobenius algebra is degenerate.

Decomposing Frobenius Heisenberg categories

We give two alternate presentations of the Frobenius Heisenberg category, [Formula: see text], defined by Savage, when the Frobenius algebra [Formula: see text] decomposes as a direct sum of Frobenius subalgebras. In these alternate presentations, the morphism spaces of [Formula: see text] are given in terms of planar diagrams consisting of strands “colored” by integers [Formula: see text], where a strand of color [Formula: see text] carries tokens labelled by elements of [Formula: see text] In addition, we prove that when [Formula: see text] decomposes this way, the tensor product of Frobenius Heisenberg categories, [Formula: see text] is equivalent to a certain subcategory of the Karoubi envelope of [Formula: see text] that we call the partial Karoubi envelope of [Formula: see text].