/sci/ - Science & Math

>>11492814
In QM on a lattice, you assign a local finite-dimensional Hilbert space [math]\mathcal{H}_x[/math] to each lattice point [math]x\in\Lambda\subset \mathbb{R}^n[/math], then you construct observables [math]\mathcal{A} = \operatorname{cl}\bigcup_{X\subset \Lambda}\mathcal{B}(\mathcal{H}_X)[/math] as the closure of the direct limit of local nets of bounded linear operators where [math]X\subset \Lambda[/math] is a finite subset and [math]\mathcal{H}_X = \bigotimes_{x\in X}\mathcal{H}_x[/math].
A state [math]\Psi[/math] determines measurements [math]|\langle \Psi,A\Psi\rangle|[/math] for observables [math]A\in\mathcal{A}[/math] only up to its projection [math]\tilde{\Psi}[/math] onto the unit sphere [math]S\mathcal{H}_\Lambda[/math]. For a short-range entangled state [math]\Psi \sim_{\text{unitary equiv.}} \bigotimes_x \psi_x[/math], the measurement clusters [math]|\langle \Psi,A\Psi\rangle| = \prod_{x\in \operatorname{supp}A}|\langle \psi_x , A_x\psi_x\rangle|[/math] and hence is determined by [math]\sum_{x\in\operatorname{supp}A}\operatorname{dim}_\mathbb{C}S\mathcal{H}_x \leq \sum_{x\in\Lambda}\operatorname{dim}S\mathcal{H}_x[/math] number of variables. If [math]\operatorname{dim}_\mathbb{C} \mathcal{H}_x =\frac{1}{2}\operatorname{dim}_\mathbb{R} \mathcal{H}_x= K+1[/math] is uniform, then measurements are determined by at most [math]\sum_{x\in\Lambda}2K = 2K \cdot |\Lambda|[/math] number of real parameters. This is in general untrue if [math]\Psi[/math] is long-range entangled, however.

>>9047188
>I think he mentioned gauge theory in the lecture. How is it related to physics?
Given a manifold [math]M[/math] and a Lie group [math]G[/math], the principal [math]G[/math] bundle [math]P \rightarrow G \rightarrow M[/math] defines a connection [math]A \in \Omega^1(M)\otimes \mathfrak{g}[/math] (and thus a curvature [math]F[/math]) on the manifold [math]M[/math]. The gauge group [math]\mathcal{G} = \operatorname{Map}(M,G)[/math] is the group of smooth maps from [math]M[/math] to [math]G[/math], and the connection [math]A[/math] and the curvature [math]F[/math] transform covariantly under its action as [math]A \rightarrow g^{-1}Ag + g^{-1}dg[/math] and [math]F \rightarrow g^{-1}Fg[/math]. This allows us to define gauge theories with an action [math]S[/math], which is a function (of the quantized sections [math]\psi[/math] of [math]M[/math] and its covariant derivatives, the connection [math]A[/math] and the curvature [math]F[/math]) that satisfies additional symplectic conditions.
A TQFT is a gauge theory with "purely topological data", meaning that the theory has no dynamics (i.e. the Hamiltonian is identically 0 up to gauge choice and all phases are geometric). Chern-Simons and Wess-Zumino-Witten actions are examples of actions that give rise to TQFTs, and they are used mainly to describe defects (see https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.51.591)), among them Dirac monopoles, the Aharonov-Bohm effect, cosmological strings, quantum Hall effect, etc. The surprising thing is that the sequence [math]SU(2)_k[/math] of Chern-Simons theory can be mapped onto a conformal field theory with modular (conformal) group [math]SL(m,\mathbb{Z})_2[/math], i.e. the (universal enveloping algebra of the) Virasoro algebra (see https://arxiv.org/abs/0707.1889).). This is one example of the correspondence I was talking about, and which was made mathematically explicit in the article I linked in the previous post.

Let [math]\Sigma[/math] be a genus [math]g[/math] surface and let [math]\mathcal{M}_g[/math] be its mapping class group. We assign [math]2g[/math] level-[math]k[/math] integrable highest weights [math]\mu_1,\mu_1^*,\dots,\mu_g,\mu_g^* \in P^+(k)[/math], onto [math]2g[/math] points on the Riemann sphere and let [math] V_{\mu,\mu^*} = V_{\mu_1,\mu_1^*,\dots, \mu_g,\mu_g^*}[/math] be the space of conformal blocks of the CFT that satisfy the KZ monodromy. A TQFT functor can be defined that maps the surface [math]\Sigma[/math] to the vector space [eqn]V_\Sigma = \bigoplus_{(\mu),(\mu^*)} V_{\mu,\mu^*}.[/eqn] This shows that a TQFT can be constructed from a CFT.
Now let [math]L_1,L_2[/math] be links whose regular neighborhoods are copies [math]H_1,H_2[/math] of [math]H[/math], where [math]\partial H = \Sigma[/math], and denote by [math]M[/math] the 3-manifold obtained by gluing [math]h:\partial H_1 \rightarrow \partial H_2[/math]. Let [math]L(h)[/math] be the link that gives [math]M[/math] upon Dehn surgering [math]S^3[/math] and does not intersect [math]L_1, L_2[/math]. Take [math]T(h) = L \cup L_1 \cup L_2[/math] and let [eqn]\rho(h)_{\mu\nu} = \sqrt{S_{0\mu}}\sqrt{S_{0\nu}}C^{\sigma(L(h))}\sum_{\lambda:\{1,\dots,m\}\rightarrow P^+(k)} S_{0\lambda}J(T(h);\lambda)_{\mu\nu}[/eqn] via Witten's tangle operator [math]J(T(h);\lambda)_{\mu\nu}: V_{\mu,\mu^*} \rightarrow V_{\nu.\nu^*}[/math], where [math]S_{0\mu} = \prod_{i}S_{0\mu_i}[/math]. The map [math]\rho = \bigoplus_{\mu\nu}\rho_{\mu\nu}: \mathcal{M}_g \rightarrow GL(V_\Sigma)[/math] is a projectively linear representation that satisfies [math]\rho(fg) = \xi(f,g)\rho(f)\rho(g)[/math], where [math]\xi(f,g) = C^{\sigma(L(f)\cup L(g))-\sigma(L(f))-\sigma(L(g))} \in \mathbb{C}^*[/math]. This shows that the TQFT is unitary and not anomaly-free.
This construction gives me an idea of what sort of geometric data can be recovered from a TQFT, and how I can incorporate it into the space structure of a TQFT to describe AdS/CFT.