/sci/ - Science & Math

>>11490705
The pion vertex in QED [math]\propto\int_M F^2[/math], the chiral anomaly [math]\propto \int_D F^-[/math]. Anomalies in general come about when your partition function [math]Z[\psi^g] \neq Z[\psi][/math] in the thermodynamic limit, where [math]g\in\operatorname{Map}(M,G)[/math] is a gauge transformation. In terms of the normalized invariant functional measure [math]d\Psi[/math], the partition function [math]Z[\psi^g] = \int d\Psi e^{-S[\psi^g]}[/math] picks up an extra functional Jacobian [math]|J_{g^{-1}}(\psi)| = e^{-S_g[\psi]}[/math] corresponding to the gauge transformation [math]g[/math] when you send [math]\psi \rightarrow \psi^{g^{-1}}[/math] in the functional integral. This functional Jacobian can be cancelled only if you can renormalize your action such that [math]S \rightarrow S
+ S_\text{ren}[/math] where [math]S_\text{ren}[\psi^g] - S_\text{ren}[\psi] = S_g[\psi][/math]. When you can't, there's typically a topological reason for it: for instance, if [math]S[/math] is an NLSM wth [math]\psi:M \rightarrow \Sigma[/math] then an anomaly occurs if holomorphic bundles [math]V\rightarrow \Sigma[/math] encounters an obstruction to a spin structure in the pullback; i.e. [math]\psi^*V \ominus \mathcal{S}[/math] is non-trivial, where [math]\mathcal{S}[/math] is the spinor bundle on [math]M[/math].

>>11484719
>H is precisely the kernel
Oh? Normal subgroups [math]N[/math] of [math]G[/math] satisfies [math][g,N] = 1[/math] for every [math]g\in G[/math] hence [math]p_a[/math] is trivial for every [math]a\in N[/math], so [math]h(N) = 1[/math]. Normal subgroups in fact fit into the kernel of [math]h[/math].
>>11484752
>Any element is written uniquely as a sum of finitely many elements of the base
Really? For [math]every[/math] vector space? Think harder sweetie.

>>11388946
A graviton is an excitation in the gravitational field. Similar to how the EM field is quantized by promoting 4-gauge [math]A[/math] satisfying the wave equation [math]\Delta A = 0[/math] to operator-valued distributions/generalized [math]L^2[/math] sections on a principal [math]U(1)[/math]-bundle over Minkowski space, the gravitational field is quantized by promoting the metric [math]g[/math] satisfying the Einstein field equations [math]R - \kappa g = c T[/math] to an operator-valued generalized symmetric bilinear form on a Riemannian manifold. Let [math]S[g][/math] denote the classical Riemann-Hilbert action, this promotion [math]g\mapsto \hat{g} = g + \hat{h}[/math] leads to quantum corrections [math]\sim \langle \hat{h}\rangle + \dots[/math] in the free energy [math]F =-\ln \int dg \exp(-S[g])[/math], which constitute vertices of graviton exchange that mediate gravitational interactions. Objects feel gravity [math]precisely[/math] because they exchange gravitons.
>>11379671
Because compact universes must necessarily have non-trivial mean curvature by the geometric characterization of Riemannian spaces. By classical GR the curvature contributes to observable gravitational phenomena which, so far, do [math]not[/math] at all suggest any curvature even throughout the entire observable universe.
>>11386103
The Casimir force is caused by expectations of the Casimirs of the field algebra. You can study its effect in a cavity or region [math]\Omega[/math] as long as you can consistently quantize (find a *-irrep of) and diagonalize [math]at ~least[/math] the Poisson subalgebra [math]\mathcal{B}\subset\mathcal{P}(\Omega)[/math] at most linear in the momenta [math]p[/math]. A strip geometry turned out to be the simplest, since we can use Fourier coefficients.

>>11289770
>everything is pointwise
Well, in that case we can endow even more structure to our [math]\chi_A[/math]. For instance, if we send [math]S\mapsto \left(y \mapsto \int_S dP_A(x) (x-yi)\right)[/math] for [math]S\subset \Omega[/math], then [math]\chi_A(S)(y) = \chi_A(S;y)[/math] is a nice parameterization of [math]A[/math] away from its Cayley transform [math]A \pm i = \int_\Omega dP_A(x)(x\pm i) = \chi_A(\Omega;\pm 1)[/math]. Since the denseness of [math]\operatorname{im}(A\pm i)[/math] gives us the self-adjointedness of [math]A[/math], regularity/analyticity (or lack thereof) of [math]\chi_A[/math] can detect obstructions to [math]A[/math] being self-adjoint.
If we in addition endow [math]\Omega[/math] with the structures of a continuum, we can perform calculus and relax our partial ordering to the [math]L^p[/math] sense: [math]S\subset T \implies |\chi_A(S)|_{L^p(\Omega)} \leq |\chi_A(T)|_{L^p(\Omega)}[/math]. If we also have [math]p=2[/math], then we can even relax axiom 2 to the convolution product: [math]\chi_A(S\cup T) = \chi_A(S) \ast \chi_A(T)[/math]. This allows us to treat distributions properly, and also gives you more elbow room for the characterization of such [math]\chi[/math]'s.

>>11234121
Techmuller space is the space of differential structures on a Riemann surface modulo conformal transformations. "Inter-universal" is just a buzzword for describing how the analytic geometry is related to arithmetic geometry.
>>11236487
On a topological vector space [math]V[/math], the determinant on [math]TV[/math] determines the determinant line bundle of [math]V[/math]. The determinant line bundle then determines the orientability of [math]V[/math] in the sense that all principal [math]O(d)[/math]-bundles must factor through the determinant line bundle.
>>11236881
That's due to Kirchoff's loop rules, which is part of graph theory.
>>11238456
First we notice that [math]\ln[/math] maps [math]\mathbb{C}\setminus\{0\} \cong\mathbb{C}^* \rightarrow S^1 \times (0,\infty)= C_\infty[/math] to the semiinfinite cylinder on the principal branch, hence we see that [math]f: \mathbb{C}^*\rightarrow \mathbb{C}^*[/math] defined by [math]f(z) = z^z[/math] pulls back to [math]\ln f : C_\infty\rightarrow C_\infty [/math] as [math]\ln f(z) = z \ln z = z(\ln r + i\theta)[/math]. Hence holomorphicity of [math]f[/math] on [math]\mathbb{C}^*[/math] is equivalent to holomorphicity of [math]\ln f[/math] on [math]C_\infty[/math]. We can then develop the Laurent series for [math]z\ln z[/math] on [math]C_\infty[/math] then push it forward back down to [math]\mathbb{C}^*[/math] to obtain a Laurent series for [math]z^z[/math].
>>11241084
>Does anyone know anything better?
Look at linear representations of the semisimple multi-fusion category [math]\mathcal{M}_{n\times n}(\mathbb{R})[/math] of [math]n\times n[/math] matrices (hint: look at how the identity decomposes on the units).

>>11184687
Oh it's no joke sweetie. It's reality.

>>11154778
Show their orthogonality by integration. [math](T_n,T_m) \propto \delta_{nm}[/math].
>>11154813
Anon meant linearly independent over [math]L^2([-1,1],d\mu)[/math] where [math]d\mu(x) = \frac{1}{\sqrt{1+x^2}}dx[/math] is the half-circle measure, not over [math]\mathbb{R}[x][/math].

>>11109299
>but isn't it described in terms of Chern-Simons theories
Yep, effectively at least, with a statistical gauge field.
>Is this all that holography in condensed matter is?
Topologically speaking yes, but there are more cond mat-centred considerations that are less well understood like FQH states, SET phases and effects of impurities. I'm not very familiar with what the holography people are doing but I suspect they're looking for more than the AdS/CFT duality.
>>11109308
Among the ideas there I'm mainly interested in the HMS bit, but studying TQFT doesn't give me enough geometric background to substantially look further than topological Fourier-Mukai.
From what I know, most of my colleagues in HEP are migrating towards PPEFT and applications (one of which is fitting CERN data), and it does seem quite barren.