/sci/ - Science & Math

>>11499936
You can get info about the irreps of Lie algebras from them. Specifically since characters are class functions, we can get the irrep characters from the Cartans, which can tell you the rank and module properties of the irrep spaces themselves. By studying the roots and their Weyl groups, you can then get the full ladder algebra structure on the irreps.

>>11255616
1. Did you mean an almost complex structure? Rotations are unimodular so you may also think about possibly the leaves being stretched, but I think your intuition is fine locally.
2.
>point-set topology
Yeah that's because the field is filled with pathologies, same with analysis and OA. E.g. commuting unbounded ESA unitaries [math]U,V[/math] do not necessarily have commuting generators, even if their domains are dense and coincide. A classical example is Nelson's example where [math]U,V[/math] are translations along the real and imaginary axes on the branched Riemann surface [math]M\xrightarrow{\sqrt{\cdot}} \mathbb{C}[/math], respectively.
3. Turn her into your shiki. Works pretty well.
4. It seems to be applicable to Lagrangian field theory but jets are more often used and suffices perfectly fine. Check out https://arxiv.org/abs/hep-th/0612182..

>>10096037
Yes, it's called a CW complex. The Euler polyhedron formula is a special case of Euler characteristic [eqn]\chi(X) = \sum_i (-1)^i\operatorname{rank}H_i(X)[/eqn], which is related to the genus [math]g[/math] via [math]\chi = 2- g[/math] for closed compact orientable [math]2[/math]-manifolds.

>>10056808
The AB effect is nothing but the effect of the first cohomology group on the prequantum bundle. Let [math]L \rightarrow M[/math] be a prequantum bundle on the punctured plane [math]M = \mathbb{R}^2\setminus\{0\}[/math] such that [math]\omega = -\frac{1}{r}(dy \wedge dx)[/math] is the symplectic form. The prequantum bundle then acquires the Hermitian conenction 1-form [math]\theta = -\frac{i\hbar}{2\pi r} (xdy- ydx)[/math] which generates the first de Rham cohomology group [math]\frac{2\pi}{\hbar}H^1(M) \cong \mathbb{Z}[/math]. This means that the prequantum bundles [math](L,A)[/math] on [math]M[/math], with [math]A = \theta - i\frac{\hbar}{2\pi}\lambda[/math], are mutually inequivalent for [math]\lambda \in [0,1)[/math]. Each prequantum bundle are then labeled by an integer for which the local sections [math]\psi\in\Gamma(M,L)[/math] satisfy [math]\exp\left(i\frac{2\pi}{\hbar}\otimes_\gamma \theta_m\right) \psi = \exp\left(i2\pi m\right)\psi[/math] by Weil integrality condition.

>>9842041
>This is the purpose of grad school
Wrong. They're not paying you to find what you want to research, they're paying you to DO research. It's another story if you find a faculty gullible or believes in you enough to pay you when you don't even have a direction, but this is NOT the norm.

>>9533018
>FTC
Fundamental Theorem of Calculus. The reason I say that this should convince you as to why Berezin integration is really an integral is because FTC-like or Stokes-like theorems are basically continuous versions of telescoping sums. They really are implicitly summing things.
>Is there any good reference explaining the relationships between Atiyah-Bott formula and how it reduces to the stationary phase approximation in classical mechanics?
Start here https://arxiv.org/abs/1305.4293..

>>9313000
I like them a lot, anon! Please don't stop posting!

>>9266633
Sorry only got holonomy here.

>>9219642
How do mathematicians cope with the fact that invariants of 4-manifolds are constructed by using quantum field theories?
https://arxiv.org/abs/1705.01645

>>9063028
>say what kind of geometric objects you are dealing with or what kind of properties you want for them
The problem is that that isn't always possible anon. As I've said rigorous mathematical treatment of physical theories can only go so far and does not have nearly as much predictive power as heuristic theories. For example if I want to do Wilson-Kardanoff renormalization group analysis of a lattice ladder model I would need to do calculations that completely obfuscate the overarching group/groupoid structure, all I'd get at the end is a set of Callan-Symanzik equations that tell me how the renomralized order parameters of the theory flows (which is the bit that physicists write into their papers). Even if I understood how RG works at the level at which the mathematics is established I would not be able to tell you how that would give any insight into this particular situation.
At most what mathematics can get you is the big-picture physical intuition that us physicists gloss over, not the details that actually make it into papers.
>ghosts
Ghosts are auxiliary fields that come up when you fix a specific gauge in gauge space. If we have the space of gauge connections [math]\Omega^1(M)\otimes \mathfrak{g}[/math] we can construct a dirac delta function [math]\delta(F)[/math] for e.g. the Lorent gauge [math]F = 0[/math] and put this into the partition function, expressing it in terms of the Fourier transform [math]\int \mathcal{D}c\mathcal{D}\bar{c}\exp(-i\bar{c}Fc)[/math]. These [math]c[/math] fields are the ghosts. This can be circumvented by treating the space of gauge connections as a symplectic vector space and the Lorenz condition as a holonomic constraint, and only defining your path integral over its symplectic reduction by the holonomic constraint. Ghosts are then completely taken into account non-perturbatively in this case. You can find how to do this in detail in Bertlmann.

>>9026691
[math]A_n[/math] is a formal series Lie algebra generated by indeterminants [math]X^{ij}[/math] modulo some relations, and [math]P_n[/math] is the pure braid group, i.e. the group of actions that braids n particles together such that the initial and final configurations the same. [math]\theta[/math] uses the KZ monodromy [math]\omega[/math] (which gives rise to a conformal field theory) to define a representation of [math]P_n[/math] with values in [math]A_n[/math], and this is useful for finding holonomies of the field theory with connection [math]\nabla = d + \omega[/math] (just take the exp).
Holonomies are important because it tells us the particle statistics of the theory, namely how our wavefunction transforms [math]\psi \rightarrow e^{-i\int_\gamma A} \psi[/math] under [math]P_n[/math], and this is directly linked to the Aharonov-Bohm effect. It is also important in constructing a gauge theory equivalent to the CFT due to the equivalence shown here https://arxiv.org/abs/1504.02401..
The symplectic form obtained from the holonomy [math]\exp\left(-i\int_\gamma \theta\right) = \exp\left(-i\int_B \Omega\right)[/math] where [math]\partial B = \gamma[/math] can also tell us quantization conditions that govern the CFT via the first integrality condition.
The rest are just calculations based on these.

>>9021871
>do you visualize these things
In general yes. I sort of have to.
Most of the steps for constructing quantum invariants on 3-manifolds for TQFT's is by doing it through knots first, and the properties of which almost all come from visualizing how knots in [math]S^3[/math] behave. Similarly the axioms of a 2-DMF all come from visualizing cutting up pieces of paper with several holes in them.
On the other hand CFT's are a bit more computational heavy, but if you have the proper physics background you'll be able to assign physical objects (e.g. particle trajectories, Feynman diagrams, gauge bundles, etc.) to quantities that allows you to better understand the CFT.

>>9002320
Very bad OP.
>what are you studying?
Same shit as always.
>any cool problems?
As I've elaborated previously, I may be able to construct a correspondence between TQFTs a la Atiyah with CFTs by cutting up and assigning link components in decorated 3-manifolds to marked points the space conformal blocks are on. However the process of "cutting up" these decorated 3-manifolds wasn't precisely defined in the context of space structures. Turaev was able to precisely define a version of this cutting up called "excision" on 2-manifolds instead, and I intend on working to extending this to 2-surfaces.
>any cool theorems or remarks?
Wentzl's limit for the quantum invariant of unitary TQFTs over a semisimple category [math]\mathscr{V}[/math] with fundamental object [math]V_l[/math]: [eqn]\tau(M) = \Delta^{\sigma(L)}\mathscr{D}^{-\sigma(L) + m -1} \lim_{N\rightarrow \infty}\frac{1}{N^m}\left[\sum_{\lambda \in \{1,\dots,N\}^m}(\operatorname{dim}(l))^{-|\lambda|} F(L^\lambda_l)\right]. [/eqn]
It's very interesting that a mostly algebraic quantity can be calculated with analysis. It has a very nice proof too.
>reference suggestions?
Nayak's paper is very nice for quantum braiding algebras.

>>8990874
>>8990879
Chern-Simons TQFT can be used to describe topological defects and the quantum Hall effect in condensed matter systems. You won't need much more than Chern numbers though.
>>8990899
Here's another one I made.

Fix a strict monoidal tensor category [math]\mathscr{V}[/math] with ground ring [math]K[/math].
Define its Grothendieck algebra [math]K_0(\mathscr{V})[/math] as the [math]K[/math]-module of isomorphism classes of the objects in [math]\mathscr{V}[/math] with addition defined by [math][V] + [W] = [V\oplus W][/math] and multiplication [math][V][W] = [V \otimes W][/math]. On the other hand, define the commutative associative unital Verlinde algebra [math]\mathbb{V}[/math] of [math]\mathscr{V}[/math] as a [math]K[/math]-module generated by the basis [math]\{b_i\}_I[/math] ([math]0 \in I[/math]) such that [math]\forall v \in \mathbb{V}[/math], we have [math]v = \sum_{i\in I} k_i b_i[/math] with [math]k_i \in K[/math]. Multiplication in this module is defined by [math]b_ib_j = \sum_{r\in I} h_r^{ij}b_r[/math], where [math]h_r^{ij} = \operatorname{dim}(\operatorname{Hom}(V_r,V_i \otimes V_j))[/math].
If [math]\mathscr{V}[/math] is semisimple with a dominating family [math]\{V_i\}_I[/math] of objects such that [math]V = \bigoplus_{i\in I} V_i[/math] for every [math]V \in \mathscr{V}[/math], then the maps [math][V] \mapsto \sum_i \operatorname{dim}(\operatorname{Hom}(V_i,V))b_i[/math] and [math]b_i \mapsto [V_i][/math] are mutually inverse, and they define a canonical isomorphism between [math]K_0(\mathscr{V})[/math] and [math]\mathbb{V}[/math].

>>8944606
>more interesting
Symplectic geometry because it's how quantization in physics is done.
>rich
Vector bundles since you're going to mix algebra with topology when you study (co)homology.
>useful
??? None of them lmao try engineering instead.