All Questions
Tagged with qa.quantum-algebra quantum-field-theory
22 questions
10
votes
1
answer
477
views
Where does the definition of ($\infty$-)groupoid cardinality come from?
The cardinality of a finite set $X$ is the number of its elements. Once you know that, you would define the groupoid cardinality of a $\infty$-groupoid $X$ as the quantity
$$\lvert X\rvert := \sum_{[x]...
- 609
0
votes
0
answers
105
views
A variant of quantum harmonic oscillators
We have the following variant of harmonic oscillators.
$$
\left\{
\begin{array}{**lr**}
T = a + a^\dagger\\
a | n \rangle = \sqrt{[n]} |n-1 \rangle \\
a^\dagger |n\rangle = \sqrt{[n+1]} |n+1\...
- 105
0
votes
0
answers
120
views
Some version of non-commutative Wick formula
Let $V$ be a vertex algebra. The traditional non-commutative Wick formula is a tool to calculate term like $[a_\lambda:bc:]$. However, I need to calculate terms of the form $[:ab:_\lambda c]$. I found ...
- 1,391
1
vote
0
answers
79
views
Ordering in Cobordism Category
Let $Cob^{3}$ denote the cobordism category of $1$ dimensional manifolds i.e the objects are finite disjoint union of circles and morphisms are represented by surfaces.
Is it possible to treat the ...
- 341
7
votes
1
answer
296
views
Affine Kac-Moody algebra from quantum group exchange algebra
In `Hidden Quantum Groups Inside Kac-Moody Algebra', by Alekseev, Faddeev, and Semenov-Tian-Shansky, a relationship between quantum groups and affine Kac-Moody algebras is shown for the WZW model.
...
- 1,155
7
votes
0
answers
221
views
Representations of 2-groups and quantum double constructions
Let $G$ be a finite group. The category of its representations (complex linear, finite dimensional, throughout this whole question) is equivalent to $\mathbb{C}[G]$-modules. V. Drinfeld constructed a ...
- 5,230
4
votes
0
answers
278
views
Are vertex operator algebras ever conspiratorial?
I have a vertex operator algebra (VOA) $V$ with all niceness properties (unitary, rational, CFT type, etc). Its Lie algebra $\mathfrak{g} = V_1$ of spin-$1$ fields is large, and I understand how the ...
- 54.6k
17
votes
2
answers
2k
views
Quantum corrections to geometry
In this video Alain Connes made a comment about the ,,quantum corrections'' of the geometry. I would like to understand this notion in some details since I haven't found anything about this in the ...
- 9,330
9
votes
0
answers
445
views
Lagrangian subgroups/submanifolds, 2d topological boundary and 3d "non-abelian" Chern–Simons theory
This post is meant to ask for proper references to fill a gap in the literature.
My short question is that are there known and precise ways to formulate 2d topological boundary conditions" for ...
- 10.5k
9
votes
2
answers
3k
views
The Fock space vs the Hilbert space in the context of quantum field theory
Mathematically the definitions are as follows : if $H_n$ is a $n$-dimensional complex Hilbert space then its two different corresponding ``Fock space"(s) are often denoted as $F_{1}$ and $F_{-1}$ ...
- 2,246
5
votes
0
answers
114
views
Differential form TQFT for Walker-Wang model?
In terms of the TQFTs in continuous differential form gauge fields, what would the Walker-Wang lattice model describe? Obviously, there is a $BF$ theory part:
$$\frac{N}{2 \pi}\int B dA$$
if it ...
- 755
12
votes
0
answers
333
views
Does the $(\mathbb Z/2)$-graded isomorphism $E_n \cong E_{n+2}$ have any nice properties?
This question assumes everything is dg. Let's decide to work over the "field" $\mathbb Q[\mu,\mu^{-1}]$ where $\mu$ has homological degree $+2$. Then chain complexes are just $\mathbb Z/2$-graded. ...
- 54.6k
21
votes
1
answer
2k
views
Gauss linking integral and quadratic reciprocity
In the setting of Mazur's "primes and knots" analogy, prime ideals $\mathfrak p\subset\mathcal O_K$ correspond to "knots" $\operatorname{Spec}\mathcal O_K/\mathfrak p$ inside a "3-manifold" $\...
- 18.7k
13
votes
1
answer
848
views
Are Turaev--Viro invariants secretly a discretized path integral?
Turaev--Viro http://www.ams.org/mathscinet-getitem?mr=1191386 defined an invariant of three-manifolds $M$ denoted $TV(M)$, which was subsequently shown by Kevin Walker to coincide with $\left|WRT(M)\...
- 18.7k
67
votes
4
answers
9k
views
Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?
The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first ...
- 18.7k
13
votes
0
answers
591
views
Generalization of Witten's computation of the volume of moduli space
Let $\Sigma$ be a Riemann surface, and let $X:=\operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(2))/\operatorname{SU}(2)$ be the $\operatorname{SU}(2)$ character variety of $\pi_1(\Sigma)$.
There ...
- 18.7k
4
votes
0
answers
358
views
Do real polarization and Kahler polarization of character varieties of closed surfaces give equivalent representations of the Mapping Class Group?
This is a question about the Witten--Reshetikhin--Turaev representations of the mapping class group of a closed surface $\Sigma_g$. For simplicity, we'll stick to the case $G=SU(2)$.
These ...
- 18.7k
7
votes
2
answers
654
views
What is the state in the WRT TQFT associated to a handlebody?
Let $Y^3$ be a handlebody with boundary $\Sigma$. By definition, there is some associated vector $v_{WRT}(Y^3)\in Z(\Sigma)$, the (finite dimensional) Hilbert space associated to $\Sigma$ by the ...
- 18.7k
3
votes
1
answer
569
views
What if I change field in a Topological Quantum Field Theory?
Of course I'm talking about the algebraic notion of field.
In a few words, if a TQFT consists of a functor $Z\colon Cob(n)\to \mathbf{Vec}_k$, I'm wondering if there are sensible relations among ...
- 13.6k
10
votes
3
answers
2k
views
In the dictionary between Poisson and Quantum, what corresponds to Coisotropic?
I work entirely over a field of characteristic $0$, in case it matters.
Recall that a Poisson algebra is a commutative algebra $A$ with a bracket $\lbrace,\rbrace : A^{\wedge 2} \to A$ which is (1) a ...
- 54.6k
11
votes
1
answer
2k
views
Spectral theory for self-adjoint field operators on a symmetric Fock space
Background
Suppose we have a finite-dimensional Hilbert space $H = \mathbb{C}^s$ (for a natural number s) and we construct the symmetric (or bosonic) Fock space built from it: $$F(H):= \mathbb{C} \...
- 195
12
votes
2
answers
1k
views
Turaev-Viro extended TQFT
Hi I am looking for any papers which extends the Turaev-Viro TQFT to a 3-2-1 theory (i.e. allows manifolds with corners) . I know this construction is known, but I cannot find a source. Please help.
...
- 121