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>>11425260
Indeed they can. Given a link [math]L[/math], we can form a 2-fold [math]M[/math] by gluing [math]f: \coprod \partial D^2 \rightarrow K[/math] some collection of discs to it, whose world-volume is a 3-manifold classified topologically (surgery theory) by [math]L[/math] modulo ambient isotopy and Kirby moves. Just speaking in terms of the underlying TQFT, which maps [math]M[/math] to a state vector in the topological theory, the ground state [math]|L\rangle = |\operatorname{AFB}(L)\rangle[/math] formed this way define a topological vacuum labeled by link invariants [math]\operatorname{AKB}(L)[/math]; in addition, cobordant links define unitarily
equivalent ground states (in the TQFT).
In the string theory itself, the ground states in fact depend on more intricate geometric structures on [math]M[/math], such as the conformal class of the metric on it, but this is determined by much more than just the embedded links inside.

>>11408780
Vasiliev invariants can be used to label the topological vacuua in (2D+1)CFT.
https://www.semanticscholar.org/paper/Vassiliev-invariants-and-de-Rham-complex-on-the-of-Kohno/cc9d910fe4836cb6bbb4015758ceef8e7e2628a9
This labeling allows a state-sum construction for the partition function [math]Z[/math] in a TQFT, since then you may transform summation over coloured ribbon graphs into a summation over these invariants classifying such graphs. By treating [math]Z[/math] as a trace of the Gibbs measure [math]\exp -H[/math] over the ground states, each invariant then labels distinct ground state in the quantum theory.
In this interpretation, more sophisticated invariants such as Chern, Postnikov, Atiyah-Bott-Shapiro, Pontrjagyn and Arf-Kervaire-Brown can all be realized as topological vacuua.

>>11319090
>justify their calculus of variations
They don't. If they do their books would be quadruple their volume.
The mathematical foundation of classical mechanics is symplectic geometry, that of quantum mechanics is von Neumann algebras and that of field theory is jets. There are texts on these topics with a physics slant here and there. Feel free to look up names such as Guillemin, Woodhouse, Von Neumann, Strocchi, Brezis, Sardanashvily, etc.

>>11260486
I find your last statement dubious. It's true that 2D QED is conformal in the IR and 2D conformal field theories can host anyons as well as fractionalized excitations (c.f. I/FQHE), not just fermions and bosons. However, this does not imply that atom-like structures cannot be formed. If we take "atom" to mean an excitation with hidden/internal structure (partons), then Jain's composite fermion description of FQHE is exactly the kind of "atom" that can exist perfectly reasonably in 2D. Sure the chemistry would be very different but that doesn't mean there wouldn't be any. In fact it'd probably be much richer due to the existence of non-Abelian mutual statistics.

>>11105976
>Is this a correct way to think about it?
It's [math]one[/math] way to think about it, though not essentially correct. What's correct from an a priori way is to characterize the irreps with the root system and Weyl groups, then construct singular vectors. This is what ladders are doing, then you find the matrix elements from the action of the ladders on the singular vectors.
>But I still don't know what they look like explicitly
Again, given the action of the ladders on the [math]2j+1[/math] vectors, you can write down the matrix elements. The former characterizes the latter, not the other way around.
>it seems to contradict the idea that the vectors have 2 entries.
For which [math]j[/math]? If [math]j = 1[/math], say, are the eigenvectors still 2-component (as in, are the Pauli matrices still [math]2\times 2[/math])?
>different representations (the different values of m)
Irreps are labeled by [math]j[/math], not [math]m[/math]. The [math]m[/math]'s label the singular vectors in a given irrep.

>>11008798
Of course you'll have to know the physics first, starting with Baez and Polchinsky.
Assuming you know the basics (classical alg. top. and alg. geo.), it'd be good to know about CFTs, SUSY, HYMs, symplectic geo. of hyperkahlers, modern alg. geo (mainly over [math]\mathbb{C}P^n[/math]), and a bit of arith. geo. Also read aout homological mirror symmetry since that's where string theory is most useful.
I'm sure I've missed some stuff since my area of expertise isn't on string theory. Others can fill in the blanks.

>>10981230
Here's how an analytic geometer might interpret divisors. They are "bad points" in the sense that they form the zero loci of some section [math]s\in \mathcal{O}[/math] in the sheaf of holomorphic germs on a Riemann surface [math]M[/math]. In neighborhoods away from these points, the Deligne exact sequence [math]1\rightarrow \mathbb{Z}(1)\rightarrow \underline{\mathbb{C}} \xrightarrow[]{\exp}\underline{\mathbb{C}}^*\rightarrow 0[/math] splits and [math]\ln s[/math] is locally well-defined. This interpretation also relates to Riemann-Roch I believe.
In the context of CFT whose partition function [math]Z[/math] is a holomorphic section of a V-bundle on the moduli space of Riemann surfaces [math]\mathcal{M}[/math], Riemann surfaces with nodes (i.e. those obtained from [math]M\in\mathcal{M}[/math] via some "singular" shrinking/pinching operation) form the compactification divisor. You can open the nodes by unpinching the surfaces with a nome [math]q[/math], and sections [math]Z\in\Gamma(\overline{\mathcal{M}},V)[/math] around these compactification divisors achieve singularities when you close the node at [math]q = 0[/math]. So my general understanding is that divisors form no-no points we wish to stay away from but also encodes much topological/geometric data. I can see why (bi)tangent points would fit this interpretation, since non-transversal intersections usually leads to some bifurcation (Morse/Cerf theory).
>>10981351
Complete opposite to my experience.

>>10851599
>methods in TQFT and Geometric Quantization in order to study quantum gravity I think
The invariants developed with TQFT is not strong enough to detect the geometric details required for quantum gravity. Geometric quantization works well for systems with finite-dimensional gauge groups [math]G[/math] as long as
1. the symplectic manifold is Kahler or hyper-Kahler so you have a holomorphic structure compatible with the symplectic structure and you can do holomorphic quantization, and
2. you can find a moment map [math]M\rightarrow \operatorname{Lie}G[/math] that generates local operators under Kostant central extension.
Both of these fail for gravity outside of dimension 2.
>Mostly because physicists are pretty satisfied with their QFT already
Not when the failure of Borel summability of renormalization series is instantiated in an actual real system like fractional spin-Hall with defect impurities.
>>10853316
Treat the integral as an evaluation of a distribution [math]p \in\mathcal{S}'[/math] on test functions [math]f \in\mathcal{S}[/math]. Use a sequence of [math]C^\infty[/math] functions to approximate [math]p[/math] then prove that [math]p \rightharpoonup \delta[/math] weakly to the Dirac delta.
Alternatively you can use the heat kernel.

>>10707973
https://arxiv.org/abs/hep-th/9912092
https://arxiv.org/abs/1006.4064
https://www.math.toronto.edu/drorbn/papers/weights/weights.pdf
https://arxiv.org/abs/1108.3103

>>10547534
A modular tensor category is a monoidal symmetric tensor semisimple ribbon fusion category. A ribbon category is a monoidal tensor braided category equipped with twists upon which the braiding satisfies the pentagon/hexagon equations and the twist is compatible with the braiding through the ribbon equations. Fusion means that the semisimple family of objects in the category satisfy groupoid-like fusion relations compatible with the braiding and twisting morphisms, usually expressed through modularity conditions on the fusion matrices, hence the name.
>>10548054
This is the only correct answer in the thread, though the section defines a tensor field and its evaluation at each point is the thing that gives a tensor.
>the Jacobean matrix assigns a linear transformation to each point of space, i.e a rank (1,1) tensor
Careful, this is only when you have a chart specified and when the map between manifolds is an immersion.

>>10385604
Verma modules are representation spaces of the Viasoro algebra. It's the Hilbert space of the conformal blocks in conformal field theory; all correlation functions can be expressed as a bilinear quadratic combination of conformal blocks.
It's really impossible to know what Verma modules are without knowing how it applies to physics since that's where it came from.