SRQ seminar – November 23th, 2018 — Massimiliano Gubinelli

\(\displaystyle \chi := \{ \ell \subset \mathbb{T}^2 : \ell = \{ x + c e_{\mu} : c \in [0, \lambda] \}, x \in \mathbb{T}^2, \lambda \in [0, 1], \mu \in \{ 1, 2 \} \}\)

the space of line segments. For \(\ell \in \chi\) we let \(| \ell | = \lambda\) and the direction of \(\ell\) is \(\mu\). Two line segments \(\ell, \bar{\ell} \in \chi\) are parallel if \(\pi_{\mu} (\ell) = \pi_{\mu} (\bar{\ell})\) for \(\mu = 1\) or \(\mu = 2\). Here \(\pi_{\mu} : \mathbb{T}^2 \rightarrow [0, 1]\) is the projection on the \(\mu\)-th coordinate. The distance between two parallel segments is denoted by \(d (\ell, \bar{\ell})\).

Definition 1. For \(\alpha \in [0, 1]\), let \(\Omega_{\alpha}^1\) denote the closure of bounded measurable \(\mathfrak{g}\)–valued one–forms under the norm

\(\displaystyle | A |_{\alpha} := \sup_{| \ell | > 0} \frac{| A (\ell) |}{| \ell |^{\alpha}} + \sup_{\ell \| \bar{\ell}} \frac{| A (\ell) - A (\bar{\ell}) |}{d (\ell, \bar{\ell})^{\alpha / 2} | \ell |^{\alpha / 2}}\)

\(\displaystyle A = \sum_{\mu = 1}^2 A_{\mu} \mathrm{d} x_{\mu}, \qquad A_{\mu} : \mathbb{T}^2 \rightarrow \mathfrak{g}\)

\(\displaystyle A (\ell) := \sum_{\mu = 1}^2 \int_0^1 A_{\mu} (x + \lambda t e_{\mu}) \lambda \mathrm{d} t.\)

Note that \(| A (\ell) | \leqslant | A |_{\infty} | \ell |\). Moreover \(A (\ell) - A (\bar{\ell})\) is associated to the holonomy of a loop made by \(\ell, \bar{\ell}\) (and the orthogonal segments) and we expect by Brownian behaviour of the holonomy that the fluctuations of \(| A (\ell) - A (\bar{\ell}) |^2\) are of the size of the area of the loop, which is \(d (\ell, \bar{\ell}) | \ell |\). As usual we can take any \(\alpha < 1\) to account for the loss given by controlling the pathwise behaviour.

Remark 2. If \(\alpha \in (1 / 2, 1]\) then \(\ell_A : [0, 1] \rightarrow \mathfrak{g}\)

\(\displaystyle \ell_A (t) := \int_0^t A_{\mu} (x + s \lambda e_{\mu}) \lambda \mathrm{d} s\)

\(\displaystyle | \ell_A |_{\alpha \text{-Höl}} \leqslant | \ell |^{\alpha} | A |_{\alpha} .\)

In particular \(\operatorname{hol} (A, \gamma)\) for \(\gamma\) an axis path is well–defined for \(A \in \Omega^1_{\alpha}\) by Young integration. Here \(\operatorname{hol} (A, \gamma)\) is the development of \(\ell_A\) from the Lie algebra \(\mathfrak{g}\) to the Lie group \(G\) by solving an ODE which makes sense here by Young integration.

Theorem 3. Suppose \(G\) is either abelian or simply connected. For all \(\alpha \in (1 / 2, 1)\) there exists (and its definitely not unique) a \(\Omega^1_{\alpha}\)–valued r.v. \(A\) such that for all axis loops \(\gamma_1, \ldots, \gamma_n\), representation \(\varphi : G \rightarrow \operatorname{Mat}_{m \times m}\),

\(\displaystyle (\operatorname{Tr} [\varphi \operatorname{hol} (A, \gamma_1)], \ldots \operatorname{Tr} [\varphi \operatorname{hol} (A, \gamma_n)])\)

Remark 4. (motivation) \(\Omega^1_{\alpha}\) embeds into \(\mathcal{C}^{\alpha - 1} = B^{\alpha - 1}_{\infty, \infty}\) (but the reverse in not true). One can use \(A \in \Omega^1_{\alpha}\) as initial condition of an SPDE in order to perform the stochastic quantisation of the YM measure.

The method of proof of the theorem is to take lattice approximations and fix a gauge by choosing the Landau gauge. Some preliminary notations:

\(\mathfrak{A}^{(N)}\) is the space where the discrete Yang–Mills measure \(U^{(N)}\) lives. Our goal is to show that

\(\displaystyle (| U^{(N), g} |_{\alpha}^{(N)})_{N \geqslant 0}\)

is a tight sequence by appropriately choosing a sequence of gauge transformation \(g = g_N (U^{(N)})\).

I'm going to look for a gauge invariant quantity on \(U^{(N)}\) which can help to control the \(| \cdot |_{\alpha}\) norm after a gauge transformation.

Let \(r \subset \Lambda_N\) be a rectange of size \(2^{- N} \times k 2^{- N}\) or \(k 2^{- N} \times 2^{- N}\), \(k \in \{ 1, \ldots, 2^N - 1 \}\). For \(U \in \mathfrak{A}^{(N)}\) we define \(U (\partial r) \in G\) as ordered product along the bonds of \(\partial r\) (say we start from the bottom left corner of \(r\) and we proceed clockwise).

Define a sequence of subrectangles \(r_1 \subset r_2 \subset \cdots \subset r_k = r\) by increasing slowly the size and we define \(X = (X_i \in \mathfrak{g})_{i = 0, \ldots, k}\) (antidevelopment of \(U\) along \(r\)) by

\(\displaystyle X_0 = 0, \qquad X_{i + 1} = X_i + \log (U (\partial r_i)^{- 1} U (\partial r_{i + 1})),\)

where \(\log : G \rightarrow \mathfrak{g}\) is the left–inverse of the exponential map \(\exp : \mathfrak{g} \rightarrow G\) satisfying \(\exp (\log x) = x\). (we choose a, non unique, version of this map). We assume that it is the canonical log in the neighborhood of the identity. It is not globally continuous, but is a diffeo near the identity.

where \(| r | = k 2^{- 2 N}\) is the area of \(r\) and \(| X |_{q \text{-var}}\) is the \(q\)–variation of \(X\), namely

\(\displaystyle | X |_{q \text{-var}} := \sup_{\{ t_i \}} \left[ \sum_i | X_{t_{i + 1}} - X_{t_i} |^q \right]^{1 / q}\)

Fix \(N_1 \geqslant 1\) and \(U \in \mathfrak{A}^{(N_1)}\). Note that \(U\) defines an element of \(\mathfrak{A}^{(N)}\) for all \(N \leqslant N_1\).

Theorem 5. (Landau) Suppose there exists \(\alpha \in (2 / 3, 1)\), \(C \geqslant 0\), \(N_0 \leqslant N_1\) \(q \in [1, (1 - \alpha)^{- 1})\) such that (1) holds for all \(r \in \Lambda_N\), \(N \in \{ N_0, \ldots, N_1 \}\) and

\(\displaystyle C 2^{- N_0 \alpha} + \max_{(x, y) \in B_{N_0}} | \log U (x, y) | \leqslant a\)

where \(a\) depends only on \(G\) (and \(a = \infty\) if \(G\) is Abelian). Then for \(\bar{\alpha} < \alpha\), there exists \(K > 0\), not depending on \(N_1\) (!) and \(g \in G^{(N_1)}\) so that, for \(A := \log U^g\) we have

\(\displaystyle \sup_{\ell \in \chi^{(N_1)}} \frac{| A (\ell) |}{| \ell |^{\bar{\alpha}}} + \sup_{\ell \| \bar{\ell}} \frac{| A (\ell) - A (\bar{\ell}) |}{| \ell |^{\bar{\alpha} / 2} d (\ell, \bar{\ell})^{\bar{\alpha} / 2}} \leqslant K.\)

In order to control the large gauge fields we need to use a different gauges (an axial type one):

Lemma 6. (Axial) If eq. \(\textrm{(\hyperref[eq:star-2]{2})}\) holds for all rectanges in \(\mathcal{A}^{(M)}\) and \(G\) is simply connected, then there exists \(K\) not depending on \(M\) and \(g \in G^{(M)}\) such that

\(\displaystyle \max_{(x, y) \in B_M} | \log U^g (x, y) | \leqslant K 2^{- M \alpha / 2} .\)

Consider a box \(B\) in \(\mathbb{R}^3\) and \(A \in \Omega^1_{\infty}\) on \(B\), \(A = \sum_{\mu} A_{\mu} \mathrm{d} x_{\mu}\) \(\sum_{\mu} \partial_{\mu} A_{\mu} = 0\) (Landau condition) and on \(\partial_{\mu} B\) it satisfies \(\sum_{\nu \neq \mu} \partial_{\nu} A_{\nu} = 0\) then we can recover \(A\) from the curvature

\(\displaystyle F_{\mu, \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} + [A_{\mu}, A_{\nu}] .\)

Here \(\partial_{\mu} B\) is the component of the boundary orthogonal to \(e_{\mu}\). Then inside \(B\) we have

\(\displaystyle \sum_{\nu} \partial_{\nu} F_{\mu, \nu} = \Delta A_{\mu} + \sum_{\nu} [\partial_{\nu} A_{\mu}, A_{\nu}] .\)

The conditions on the faces implies \(\partial_{\mu} A_{\mu} = 0\) on \(\partial_{\mu} B\). If I know \(A\) on the complement of \(\partial_1 B\) in \(\partial B\) then I can recover approximatively \(A_1\) inside by letting

\(\displaystyle A_{\mu} (x) =\mathbb{E} \left[ A_{\mu} (W_{\tau}) + \int_0^{\tau} \sum_{\nu} F_{\nu \mu} (W_s) \mathrm{d} s \right]\)

where \(W\) is a Brownian motion starting in \(x\) and conditioned to exit \(\partial B\) at \(\partial B\backslash \partial_{\mu} B\) at the random time \(\tau\). So if I know \(A\) on the boundary then I can control \(A\) inside. We can implement this in discrete and produce refined version of the field in finer and fined grids.