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Topological quantum field theory.

12 votes
0 answers
426 views

What is the motivation for a Frobenius manifold?

A Frobenius manifold is a type of manifolds with extra structure. The main examples are quantum cohomology (viewed as a space itself), GBV algebras, the ``Saito'' examples arising from singularities ( …
Pulcinella's user avatar
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8 votes
0 answers
230 views

$U_q(\mathfrak{g})$ is to knot theory as $U_q(\hat{\mathfrak{g}})$ is to $?$

Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over the complex numbers, e.g. $\mathfrak{sl}_n$. Then every representation $\DeclareMathOperator\Rep{Rep}V\in \Rep U_q(\mathfrak{g})$ …
Pulcinella's user avatar
  • 5,701
6 votes
1 answer
513 views

Defining extended TQFTs *with point, line, surface, … operators*

$\newcommand\Cob{\mathrm{Cob}}\newcommand\Vect{\mathrm{Vect}}\DeclareMathOperator\Rep{Rep}$The ordinary definition of a TQFT is: Defnition: A $d$-dimensional TQFT is a symmetric monoidal functor $\Cob …
Pulcinella's user avatar
  • 5,701
1 vote
0 answers
71 views

Reshetikhin-Turaev invariants from extended 3d TQFTs

Attached to any object $V\in \mathcal{C}$ of a ribbon category $\mathcal{C}$, Reshetikhin and Turaev have defined knot invariants $$\tau_V(K)\ \in\ \text{End}_{\mathcal{C}}(1_{\mathcal{C}})$$ for inst …
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