/sci/ - Science & Math

>>12566576
To answer your question, homotopies of maps from one sphere to the other can be interpreted as a (pinned; i.e. satisfying certain homotopy conditions) path into the wedge product [math]S^2\wedge S^2[/math]. This path "drags" one sphere into another, so to speak.
>>12566684
Meant [math]f=0[/math] on open sets around [math]\infty[/math].

>>12501231
WAIT ITS THE REAL ONE I THINK!

We've missed you Yukari, welcome back, I unironically look forward to seeing completely excessive Ph.D. level explanations to junior undergrad questions again

>>12142652
Are you aware of what Google is?

Formerly >>11834614
Where did Yukari go? edition

>>11531805
Notice that vectors [math]v=\{v_i\}_{i\leq k}[/math] with [math]k>2[/math] satisfying [math]\sum_i v_i = 0[/math] cannot be colinear, hence they span a (possibly concave) polygon. Denote this polygon by [math]D_v[/math], the quantity defined by [math]f_j(v) = ||\sum_{i\leq j}v_i||[/math] then computes the length of a vertex-to-vertex chord within [math]D_v[/math], hence [math]f(v)=\max_{j\leq k}f_j(v)[/math] computes the length of the longest vertex-to-vertex chord in [math]D_v[/math].
Now permutation of the vertices constitute a homeomorphism of the polygon, and we know that polygons are contractible so we can suppress [math]\operatorname{vol}D_v[/math] to however small value we like with a homeomorphism. In particular, for any [math]C'>0[/math] we can find a simplicial homeomorphism [math]h[/math] such that [math]\operatorname{vol}(hD_v) \leq C'[/math]; this automatically shrinks the length [math]f(v)[/math] of your maximal vertex-to-vertex chord. The problem then becomes showing that, for any such [math]h[/math], we can find [math]\sigma \in S_k[/math] such that [math]h D_v = D_{\sigma(v)}[/math].

Bump with an interesting article https://arxiv.org/abs/1811.08182..
>Moreover, the Rashba SOC can produce a topological phase rather than hinder it, in contrast to the honeycomb lattice.
Quite surprising, since a honeycomb of 1/2-spins is lattice homotopy equivalent to a Kagome of 1-spins; I wouldn't expect the topology hosted by a given lattice homotopy class [math][\Lambda][/math] to change just because the Rashba SOC parameter is tuned, but [math][\Lambda][/math] doesn't contain such data anyway so I can't say for sure.

>>11468579
Yep. Geometrically your fields are sections of the principal adjoint bundle [math]P=\operatorname{ad}\mathfrak{g} \rightarrow M[/math] and the covariant derivative [math]D = d+A[/math] acts as [math]D=d+\operatorname{ad}_A[/math] on the associated vector bundle [math]P\times_G V [/math].
Or alternatively think of [math]\Sigma: M\rightarrow \mathfrak{su}(5)[/math] as a NLSM and pull-back the principal gauge [math]SU(5)[/math]-gauge bundle [math]P\rightarrow \mathfrak{su}(5)[/math] defined with the canonical adjoint representation to [math]\Sigma^* P\rightarrow M[/math].
>>11467112
[math]\Lambda \in SO(1,3)[/math] and [math]L\in\mathfrak{so}(1,3)[/math] right? Then [math][\eta,\Lambda] = [\eta,1+\epsilon L] = \epsilon[\eta,L] = 0[/math]. I don't know how you got [math]\{\eta,L\}=0[/math], unless I misunderstood what your [math]\epsilon[/math] means.

>>11429082
There's like 4 simultaneous structures on Kahler-folds.
>>11429109
Euclidean means the metric is [math]\delta_{ij}[/math] while flat geometry has vanishing curvature. The Euclidean metric condition is much stronger.
>>11429457
It can't do integration on infinite-dimensional spaces for one.

>>11403691
Yep that's right. This is the basis of Floer-Morse theory in which we assign to each smooth manifold [math]M[/math] with a Morse [math]f:M\rightarrow \mathbb{R}[/math] the index [math]\operatorname{ind}df = n_+ - n_-[/math] at each critical point, where [math]n_\pm[/math] are the number of positive/negative eigenvalues. Morse theorem states that this is a topological invariant.
>>11403709
At 4.

>>11383569
First note that projections [math]p_A:V\rightarrow V/A[/math] preserves the norm on a normed linear space [math]V[/math] for each [math]A\subset V[/math]. If [math]V[/math] is in addition Hilbert, then each projection is orthogonal whence [math]V = \bigoplus_{A\in\mathcal{A}} \operatorname{im}p_A[/math] and [math]|x|^2 = |p_A x|^2 + |p_{A^\perp} x|^2[/math] by the polarization identity for any mutually orthogonal collection [math]\mathcal{A}[/math] of subsets [math]A[/math]. If [math]V[/math] is in addition separable then we may fix a complete ONB for which we have a one-to-one correspondence between [math]\mathcal{A}[/math] and partitions [math]\mathfrak{A}[/math] of [math][\infty] = \{1,\dots,\infty\}[/math]. Denote this correspondence by [math]\pi:(\mathcal{A},\bigoplus)\rightarrow (\mathfrak{A},\coprod)[/math], which sends [math]0\mapsto \emptyset[/math].
Now for any collection of partitions [math]\mathfrak{a} = \{a_\alpha\}_\alpha[/math] put [math]P_\mathfrak{a}(x,|x|)^2 = \sum_\alpha |p_{\pi^{-1}a_\alpha}x|^2= |p_{\bigoplus_\alpha \pi^{-1}a_\alpha}x|^2[/math] by the polarization identity, hence [math]P_\mathfrak{a}(x,|x|)[/math] is merely the norm-squared [math]|p_A x|^2[/math] of [math]x \in V[/math] on the subset [math]A = \bigoplus_\alpha \pi^{-1}a_\alpha[/math]. In other words, [math]P[/math] is your quantity [math]S[/math] with the sum passed into the square root.
Now since [math]\sqrt{\cdot}[/math] is convex on [math][0,1)[/math] and concave on [math](1,\infty)[/math], we have the inequalities [eqn]S_\mathfrak{a}(x,|x|) \begin{cases} \geq |p_A x| &; |x|<1 \\ \leq |p_A x| &; |x|>1\end{cases}.[/eqn]
Hence for [math]|x|<1[/math] the minimization occurs for [math]\mathfrak{a} = [\infty][/math] while for [math]|x|>1[/math] the minimization occurs for [math]\mathfrak{a} = a[/math]. Constraints the partitions can be simply encoded by a map [math]b:\alpha\mapsto \beta_\alpha[/math] and taking [math]b^*\mathfrak{A} = \{a_{\beta_\alpha}\}_\alpha[/math].

>>11287990
>and your complaint would be gone
That'd be fine with me if that "some dynamical process" is well-justified. Sweeping this argument under a rug by calling it, or subsequent arguments against the validity of said dynamical processes, a nitpick isn't going to satisfy anyone.
>CM will revolutionize QG
Who said anything about QG? I've always focused my points on string theory. There's a reason for that, which I hope haven't escaped you. If you see all the interesting new directions that cond. mat. considerations took T-duality and HMS, you'd doubt it's a pipe dream too.
>i would probably recommend you
Despite cond. mat. being my background, my current field is math.TQFT with a cond. mat. slant. But I'll take your advice into cooperation.

>>11153675
CW's in [math]h{\bf Spec}[/math] are dualizable so using Brown representability all cohomology theories [math]E^n(X) = [X,\Sigma^n E][/math] with [math]X[/math] weakly-CW are dualizable. Given a weakly CW-fold, you can use Poincare duality to move between homology and cohomologies.

>>11101230
Careful. The weak topology is the weakest topology on which the linear functionals are continuous. For linear [math]operators[/math] on a Banach, being continuous is equivalent to being bounded, but this may not be true once you take away the "Banach" qualifier.
In addition, weak-* topologies also have to respect the involutive [math]*[/math]-structure.

>>11004706
We very much know how it [math]should[/math] be defined. In fact, free relativistic QFTs a la Haag-Kastler are mathematically well-defined in terms of local nets of observables and representation theory thereof, or via Baez-Zhang in terms of symplectic nets of probability measures.
The entire problem is with scattering, in which particle interactions may lead to no amplitudes satisfying both the required physical (e.g. crossing symmetry) and mathematical (e.g. analyticity/meromorphicity) conditions.
There are several classes of mathematically well-defined interacting relativistic QFTs, however:
1. QFTs with fields satisfying the Strocchi-Swieca conditions, which have well-defined [math]S[/math]-matrices,
2. QFTs reconstructed from an interacting Euclidean FT, whose amplitudes have relaxed analytic conditions and the relativistic versions can be Wick rotated,
3. QFTs whose (quantum) dynamics can be described by SDEs, and whose renormalization can be captured via recently developed "regularity structures" of Hairer, or
4. QFTs obtained on hyperkahler manifolds that admit a Kostant-Souriau quantization of the jet bundle.
Not to mention relativistic QFTs with mostly topological/geometric data (i.e. very little dynamics), such as TQFTs and CFTs, can be mathematically defined, written down and classified completely categorically. This line of thinking has recently been pushed to study superconformals, SuGras and (SUSY) strings and sees application in homological mirror symmetry and even geometric Langlands.
Even standard texts like Weinberg covers some of the required foundations such as Dirac's quantization of classical holonomic constraints. So no, the problem is with you, not with relativistic QFT.

>>10929749
The complexification of 6-sphere? Yeah

>>10388152
>they appear mostly
They [math]only[/math] explicitly appear in physics, specifically CFT and TQFT. There are some connections to arithmetic geometry and the theory of operads but Verma modules are not strictly studied there since those fields care about the modular algebras themselves, not their representations.
>is there any particular model i could read up on
Any 2D field theory near a critical point will have full conformal symmetry and make use of affine Lie algebras to describe its primary field operators. The representation spaces thereof are generalizations of Verma modules. This is independent of the number of internal degrees of freedom you have in the system, so any [math]O(N)[/math] (or more generally any non-linear [math]\sigma[/math]-model) will make use of Verma modules as Hilbert spaces.
Di Francesco and Heine have a few good books on the topic.

>>10376057
Analo-operator theory proof:
For a bounded Teoplitz operator [math]T(W)[/math] with "kernel" [math]W \in L^2(S^1,\mathbb{R})[/math] we have [math]|T| = \operatorname{sup}\operatorname{Spec}T[/math]. Since [math]W = \text{const.}\in L^2(S^1,\mathbb{R})[/math] trivially we see [math]|T| = 4[/math], the largest eigenvalue.

Geometric proof:
Notice that [math]\underset{{\bf x}\in\mathbb{R}^2}{\operatorname{sup}}\frac{|T{\bf x}|}{|{\bf x}|} = \operatorname{sup}\frac{R}{r}[/math] , where [math]R[/math] is the distance from the origin to the ellipse [math]E_R = \{{\bf x}\mid 4x^2+ y^2 = R^2\}[/math] and [math]r[/math] is that to the circle [math]C_r = \{{\bf x}\mid x^2 + y^2 = r^2\}[/math]. Now since we get [math]\frac{|T{\bf x}|}{|{\bf x}|} = 1[/math] for [math]{\bf x} = (0,y)[/math], we know that [math]E_R[/math] and [math]C_r[/math] intersect tangentially (at two points), and can hence be parameterized by the same variable [math]t[/math], via e.g. rational parameterization. Since [math]R(t)[/math] and [math]r(t)[/math] have the same quadratic scaling through this parameterization, for each [math]t[/math] we see that [math]\underset{{\bf x}\in\mathbb{R}^2}{\operatorname{sup}}\frac{|T{\bf x}|}{|{\bf x}|} = 4[/math], the ratio of the major axis [math]R_x[/math] of the ellipse [math]E_R[/math] to the radius [math]r[/math] of the circle [math]C_r[/math].