/sci/ - Science & Math

>>10798290
No I haven't, those sound very interesting! I am familiar with a bit of integrable systems from the quantum cohomology/Gromov-Witten perspective and have looked around rep theory of quantum/cluster algebras, ultimately to see how these AG/SymG concepts can be applied to refine different types of TQFTs.
>>10797448
Representations of an affine Lie group form Hilbert spaces for the quasiparticles of a CFT. More generally, particles transform (are "charged") under certain representations of the gauge group. An understanding of the representations of e.g. [math]SU(2)[/math] allowed one to generalize the study fermions to spin manifolds which is able to produce novel topological invariants of 3-folds, like the Seiberg-Witten invariant.

>>10788471
https://arxiv.org/abs/0707.1889
https://arxiv.org/abs/1303.1202

>>10768649
The mistaken, my child, is in thinking [math]\delta[/math] arises as a singular entity defined [math]specifically[/math] to have the property of evaluation when integrated against a test function. [math]\delta[/math] is an element of the space of distributions [math]\mathcal{S}'[/math] which arises as the dual space under the non-degenerate [math]L^2[/math]-pairing [math](\cdot,\codot)[/math] on test functions [math]\mathcal{S}[/math], for which the Fourier transform is a unitary isometric automorphism between them. Unless the domain is compact, [math]\mathcal{S}[/math] doesn't even include constant functions and hence the Fourier transform of, say, [math]1[/math] isn't well-defined. The space of distribution fixes this problem by taking advantage of the Faltung theorem.
Why is this useful, you may ask? Well, as you should know the Fourier transform can convert a DE into an AE, which is exponentially easier to solve. However, before you perform a Fourier transform on the PDE you must ensure that you are seeking solutions in a function space on which the Fourier transform is well-defined, else you run into nonsense. By enlarging the admissable space into distributions, we can solve even the non-linear time-dependent KdV equation in [math]\mathcal{S}'[/math].

>>10759966
>because nothing prohibits theta from just randomly happening to be a fine tuned constant very close to 0
And that is dishonest. You are no longer doing science if you take it on faith that "[math]\theta[/math] just happens to be 0", or literally any other statement for that matter. If it is 0 (which it probably is) then something must happen beyond the effective theory of QCD to explain it, and symmetry breaking from [math]SO(10)[/math]-GUT is one such explanation; the upshot is that strong CP violation must be resolved by parity-partners and you must need some big gauge group to embed those partners into.
Of course there are problems with [math]SO(10)[/math] just like any other GUT wannabes, but the only reason I like it is because there are topological defects which can yield actual observable evidence, albeit indirectly.
Personally I'm just a mathematical physicist doing TQFT, so I'm not very heavily invested in this. I'm much more interested in classification problems than actual physics. Regardless, it was nice talking to you anon, though I wish I could have gotten more out of our discussion.

>>10734086
It's the graded algebra over [math]\mathbb{Z}_2[/math] of stable cohomotopy operations.

>>10711687
>do you have a reference on hand for the 4D classification problem?
Literally no one knows how to properly define state sums in the 4D case due to the existence of exotic 4-spheres and the fact that handlebody decompositions don't work. This means that intuitions from the 2D and 3D case fail dramatically, and tackling classification without state sums means you need to find a general way to classify the 2-category of 4D TQFTs without using the cobordism hypothesis which I personally don't recommend.
If we try to construct 4D TQFTs by generalizing the 3D Reshetikhin-Turaev state sum construction, this paper https://arxiv.org/abs/math/0503054 shows how one obtains non-positive partition functions [math]Z(M)[/math] for Mazur [math]M[/math]. In other words, embedded knots are not enough to distinguish smooth 4-folds and the entire program of writing state sums as colourings of certain embedded combinatorial objects in [math]M[/math] has to be completely revamped in the 4D case.

I'm rooting for you if you think you're up to the challenge, however, anon.

>>10597902
>It's entirely mathematical.
Haha no, I wish. There's at the moment no definitive answer to the Riemann-Hilbert correspondence and the question of [math]why[/math] it works in the first place is barely tackled at all.
There have been some work towards grounding the RG flow method on sound mathematical foundations using geometry (Brown-Connes cosmic Galois group) and analysis (Hairer regularity structures) but they both have their own respective problems. The Brown-Connes construction only (explicitly) holds for the RG of scalar field theories and only proves minimal subtraction RG with power counting, while Hairer's regularity structure, despite being more general and only assumes integrability via Wick ordering, contains a whole bunch of non-canonical constructions (the structures considered must be modified realized differently for each theory effective theory). Besides, we have yet to understand how these two completely different pictures give rise to the same answers in RG.
They may be promising directions but there's still much work to be done to say "RG is mathematical". So far all of FRG techniques a la Wetterlich still relies on some dubious handwaving manipulations that are not mathematical in nature.

>>10581050
https://arxiv.org/pdf/0912.4706.pdf

>>10375147
That's because you're not doing rep theory correctly, i.e. with applications to QM.

>>10334417
You do know that's literally what the entire field of deformation quantization is based on, right? And you do know absolute units such as Kontsevich, Moyal and Fesodov helped shaped the field as a mathematical study, right?
I highly doubt you have any credentials/achievements that allow you to call whatever Kontsevich works on a "blasphemy". I doubt you even know what constitutes a "mathematical blasphemy" at all to be quite honest.

>>10305420
Yes. Atiyah-Singer index theorem relates the analytic index of the Dirac operator to the topological Chern index of the manifold. The analytic index tells you the number of zero modes and hence those of the topological vacuum states.

>>10280802
Theory of gauge bundles
Symplectic geometry
Geometric/deformation quantization
Equivariant cohomology
Knot/braid theory
Chern-Weil theory
Von Neumann algebras (and rep theory thereof)
Affine Lie algebras (and rep theory thereof)
Kodaira-Spencer theory
K-theory
Stable homotopy theory
Hodge theory
Seiberg-Witten theory
Heegaard-Floer cohomology

>>10221980
>Thoughts?
I have none.
>>10222011
Jacak has a good book on this topic.

>>10212741
>is the lagrangian density also defined on the tangent bundle?
It depends on what "tangent bundle" means here now. In field theory, the configuration space of the Lagrangian is replaced by functional spaces. For instance, suppose we have a scalar field [math]\phi \in \Gamma(M,E)[/math] defined as a section of a (real) line bundle [math]E\rightarrow M[/math], then the Lagrangian is a functional on the tangent bundle [math]T\Gamma(M,E)[/math]of the space of sections [math]\Gamma(M,E)[/math]. The integration [math]\int d^nx L(\phi,\nabla\phi)[/math] is now interpreted as the pullback of the fibre integration on the sections.
Notice here that the "time" variable in regular (non-field) mechanics is a parameter that describes a trajectory [math]t \mapsto (q(t),\dot{q}(t))[/math] in the tangent space, but in relativistic field theory the actual parameter is a "proper time" parameter while actual "time" is part of the Minkowskian configuration space [math]M[/math].

>>10200736

>>10174809
Yeah whatever dude

>>10148531
>>10148701
Correct, in the sense that compactly supported initial conditions will be delocalized, or "smeared out", by the heat equation for any [math]t > 0[/math]. Starting with a localized region of heat, you will get a non-zero heat everywhere in the material. This is mainly due to the fact that the heat kernel [math]K \sim \frac{1}{\sqrt{t}}e^{-\frac{r^2}{t}}[/math] is the solution to the heat equation with an initial condition [math]\delta[/math], hence the solution [math]f[/math] of the heat equation with initial condition [math]g[/math] can be obtained by convolution [math]f \sim (K \ast g)[/math] with [math]K[/math]. Since [math]K \in \mathcal{S}[/math] is Schwarz but not compactly supported, [math]f \in\mathcal{S}[/math] but it cannot be compactly supported.
This is related to the fact that the heat equation is not time-reversal invariant: namely given a solution at time [math]t>0[/math], you cannot time evolve it in the reverse direction to get a unique initial condition: there are several possible configurations of the initial condition that can lead to the same heat distribution at time [math]t[/math]. This means that there is no well-defined notion of a "region of influence" for the heat equation (unlike the wave equation), which means heat in [math]any[/math] region can affect that at any other region in space.
These pathological things happen because you get the heat kernel from distributional techniques instead of by classical PDE. There is no notion of "locality" in the theory of distributions, since [math]\mathcal{S}'[/math] is defined in terms of non-local integrations and convolutions.

>>10117724
>They need to add fermions somehow
No. They added "fermionic charges", i.e. generators of supersymmetry that behave like conformal generators BUT with Grassmannian properties like fermions.
Actual fermions are given by spin (spin-c) structures on the gauge bundle and they can be incorporated into gravity pretty straightforwardly.
>>10117764
>I still don't fully understand the physical picture of why that happens in a superconductor though
The Higgs mechanism is the consequence of Goldstone's theorem via spontaneous symmetry breaking. A superconducting/BEC phase transition is exactly this, a spontaneous breaking of symmetry in which the ground state transitions from a disordered state into a macroscopically ordered state. Everything is analogous; Volovik explains this in a very transparent manner.
>>10117773
The concepts involved in that book are hints of AdS/CFT. In fact Volovik claims that the cosmological constant problem can be resolved with a phenomenological theory based on condensed matter notions.

>>10114179
The topology dictates the closure-ness of your space and hence what elements are in it.
Please actually study functional analysis before posting.