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SRQ seminar – December 5th, 2018

Notes from a talk in the SRQ series.

SRQ Seminar – 200181205 – Chandra

Stochastic quantisation of Yang–Mills.

(joint work with Hairer and Shen)

See also previous work of Shen on \(U (1) +\)Higgs in \(2 d\) finite lattice, arXiv 2018.

Setting and notations.

\(N > 1\). Lie group \(G =\operatorname{SU} (N)\). Lie algebra \(\mathfrak{g}=\mathfrak{s}\mathfrak{u} (N)\), skew–adjoint traceless \(N \times N\) matrices.

Introduce a \(\mathfrak{g}\)-valued connection on \(\mathbb{T}^n\) with \(n = 2, 3\):

\(\displaystyle A (x) = \sum_{j = 1}^n A_j (x) \mathrm{d} x_j \in \Omega^1 [\mathfrak{g}] .\)

From the connection we construct a covariant exterior derivative

\(\displaystyle \mathrm{D}_A : \Omega^r [\mathfrak{g}] \rightarrow \Omega^{r + 1} [\mathfrak{g}],\)

with

\(\displaystyle \mathrm{D}_A B = \mathrm{d} B + [A, B],\)

where \([,]\) is a wedge product in the forms and a Lie bracket in the \(\mathfrak{g}\) components.

Inner product.

For \(a, b \in \mathfrak{g}\) let \(\langle a, b \rangle =\operatorname{Tr} (a^{\ast} b) = -\operatorname{Tr} (a b)\). This product makes sense ofr \(\mathfrak{g}^n\) valued functions in space or space–time by taking the \(L^2\) contraction in the spatial variables. And gives also an inner product on differential forms \(\Omega^r [\mathfrak{g}]\) which allows to defined the adjoint

\(\displaystyle \mathrm{D}_A^{\ast} : \Omega^{r + 1} [\mathfrak{g}] \rightarrow \Omega^r [\mathfrak{g}] .\)

In general we do not have \(\mathrm{D}_A^2 \neq 0\). A simple calculation gives

\(\displaystyle \mathrm{D}_A^2 B = [F_A, B],\)

with \(F_A = \mathrm{D}_A A - \frac{1}{2} [A, A] \in \Omega^2 [\mathfrak{g}]\).

Yang–Mills measure.

\(\displaystyle \frac{1}{Z} e^{- \frac{1}{2} \langle F_A, F_A \rangle} \mathcal{D}A\)

where \(\mathcal{D}A\) is “Lebesgue measure” on connections. The main problem is that this measure, even formally, is invariant under an infinite dimensional group of transformations. Even after “gauge fixing” (choosing a representative for each orbit), this representative is quite rough and we do not expect to make easily sense of the non-linear expressions involved in the exponent \(\langle F_A, F_A \rangle\). So we have a problem with gauge freedom and a problem of renormalization of small scale singularities.

Gauge transformations.

Take \(g : \mathbb{T}^n \rightarrow G\) (e.g. \(C^1\)). \(g\) acts on connection as

\(\displaystyle g \circ A = \underbrace{g A g^{- 1}}_{=: g (A)} + g \mathrm{d} g^{- 1} .\)

We have covariance under this action for forms:

\(\displaystyle \mathrm{D}_{g \circ A} g (B) = g (\mathrm{D}_A B), \qquad g (F_A) = F_{g \circ A}\)

and same for \(\mathrm{D}^{\ast}\).

Stochastic quantisation.

Consider

\(\displaystyle \mu (\mathrm{d} x) = \frac{1}{Z} e^{- V (x)} \mathrm{d} x\)

on \(\mathbb{R}^M\) and a choice of \(\langle \cdot, \cdot \rangle\) on \(\mathbb{R}^M\). An equation which leaves this measure invariant is

\(\displaystyle \partial_t X_i = - \partial_i V (X) + \sqrt{2} \eta_i\)

where \(\eta\) is a \(N\) vector of white noises, defined in terms of \(\langle \cdot, \cdot \rangle\).

Yang–Milles SQ.

Using the Yang–Mills measure and out choice of inner product we obtain the equation

\(\displaystyle \partial_t A = - \mathrm{D}_A^{\ast} F_A + \sqrt{2} \eta\)

where \(\eta\) is a \(\mathfrak{g}^n\)–valued space–time white noise, i.e. \(\mathbb{E} [\langle \eta, f \rangle \langle \eta, g \rangle] = \langle f, g \rangle\). This will be called YM stochastic gradient flow YMSGE.

If we write \(B = g \circ A\) where \(g\) is a time–dependent gauge transformation. Then

\(\displaystyle \partial_t B = g (\partial_t A) + \underbrace{(\partial_t g) A g^{- 1} + g A \partial_t g^{- 1} + \partial_t (g \mathrm{d} g^{- 1})}_{=: \mathrm{I}} .\)

Note that \(\mathrm{I} = 0\) if \(\partial_t g = 0\). If \(A\) satisfy the gradient flow we have

\(\displaystyle \partial_t B = g \left( - \mathrm{D}_A^{\ast} F_A + \sqrt{2} \eta \right) + \mathrm{I} = - \mathrm{D}_B^{\ast} F_B + \sqrt{2} g (\eta) + \mathrm{I}\)

and we observe that \(g (\eta)\) is equal in distribution to \(\eta\). So \(B\) satisfy in law the same equation, provided the gauge transformation is constant in time.

Issue: we cannot make sense of the YMSGE using the theory of singular SPDEs, because the equation is not parabolic.

We have to modify the equation in order to gain parabolicity and consider

\(\displaystyle \partial_t A = - \mathrm{D}_A^{\ast} F_A - \mathrm{D}_A \mathrm{D}^{\ast}_A A + \eta \qquad \text{(YMSHE)}\)

with the additional term \(- \mathrm{D}_A \mathrm{D}^{\ast}_A A\). YMSHE=Yang–Mills stochastic heat equation.

\(\displaystyle \partial_t A_j = \Delta A_j + 2 [A_i, \partial_i A_j] - [A_i, \partial_j A_i] + [A_i, [A_i, A_j]] + \eta_j\)

with \(i, j = 1, \ldots, N\).

If \(A\) solves YMSHE, what does \(B = g \circ A\) solves?

\(\displaystyle \partial_t B = g (\partial_t A) + (\mathrm{I})\)
\(\displaystyle = - \mathrm{D}_B^{\ast} F_B + g (\eta) - g (\mathrm{D}_A \mathrm{D}^{\ast}_A A)\)

with

\(\displaystyle - g (\mathrm{D}_A \mathrm{D}^{\ast}_A A) = - \mathrm{D}_B \mathrm{D}^{\ast}_B B + \underbrace{\mathrm{D}_B \mathrm{D}_B^{\ast} (g \mathrm{d} g^{- 1})}_{=: (\mathrm{I} \mathrm{I})}\)

If we enforce a gauge transformation such that \((\mathrm{I}) = (\mathrm{I} \mathrm{I})\) then we recover the same dynamics (De–Turck trick). So we consider the system of equations for \((g, A)\)

\(\displaystyle \left\{ \begin{array}{l} \partial_t A = - \mathrm{D}_A^{\ast} F_A - \mathrm{D}_A \mathrm{D}^{\ast}_A A + \eta\\ g^{- 1} \partial_t g = - \mathrm{D}_A^{\ast} (g \mathrm{d} g^{- 1}) \end{array} \right.\)

then \(B = g \circ A\) satisfies

\(\displaystyle \partial_t B = - \mathrm{D}_B^{\ast} F_B - \mathrm{D}_B \mathrm{D}^{\ast}_B B + g (\eta)\)

If \(g\) is adapted one can hope that \(g (\eta)\) has still the same law as \(\eta\) by using Ito integral (this requires a suitable regularization).

Goal post

We want a Markov process on connections such that the following holds

\(\displaystyle \raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=8.92450478814115cm]{image-1.pdf}}\)

At least locally in time.

Strategy: mollify to get

\(\displaystyle \left\{ \begin{array}{l} \partial_t A_{\varepsilon} = - \mathrm{D}_{A_{\varepsilon}}^{\ast} F_{A_{\varepsilon}} - \mathrm{D}_{A_{\varepsilon}} \mathrm{D}^{\ast}_{A_{\varepsilon}} A + J_{\varepsilon} \eta\\ g_{\varepsilon}^{- 1} \partial_t g_{\varepsilon} = - \mathrm{D}_{A_{\varepsilon}}^{\ast} (g_{\varepsilon} \mathrm{d} g_{\varepsilon}^{- 1}) \end{array} \right. \) (1)

where \(J_{\varepsilon}\) is a non-anticipative mollifier. Thanks to this \(g_{\varepsilon}\) is adapted. Now after tranformation we have

\(\displaystyle \partial_t B_{\varepsilon} = - \mathrm{D}_{B_{\varepsilon}}^{\ast} F_{B_{\varepsilon}} - \mathrm{D}_{B_{\varepsilon}} \mathrm{D}^{\ast}_{B_{\varepsilon}} B_{\varepsilon} + g_{\varepsilon} (J_{\varepsilon} \eta) . \) (2)

Introduce a new equation

\(\displaystyle \partial_t \tilde{B}_{\varepsilon} = - \mathrm{D}_{\tilde{B}_{\varepsilon}}^{\ast} F_{\tilde{B}_{\varepsilon}} - \mathrm{D}_{\tilde{B}_{\varepsilon}} \mathrm{D}^{\ast}_{\tilde{B}_{\varepsilon}} \tilde{B}_{\varepsilon} + g_{\varepsilon} (J_{\varepsilon} g_{\varepsilon}^{- 1} (\eta)) . \) (3)

By Ito isometry eq. (3) is equal in distribution to eq. (2) for \(\varepsilon > 0\). But using the theory of regularity structures we can show that eq. (2) is equal in distribution to eq. (1) because we can see explicitly the cancellations in \(g_{\varepsilon} (J_{\varepsilon} g_{\varepsilon}^{- 1} (\eta))\) as the mollifier is removed. This comes from the BPHZ solution theory.

BPHZ theorem for regularity structures.

In this context we have an abstract equation and an approximation of the noise give a corresponding BPHZ solution where there is continuity in the law of the noise wrt. these approximations.

Renormalization.

We have to be sure that renormalization do not spoil the gauge covariance. We have to add the following counterterms

\(\displaystyle \left\{ \begin{array}{l} \partial_t A_{\varepsilon} = - \mathrm{D}_{A_{\varepsilon}}^{\ast} F_{A_{\varepsilon}} - \mathrm{D}_{A_{\varepsilon}} \mathrm{D}^{\ast}_{A_{\varepsilon}} A + J_{\varepsilon} \eta - \alpha_{\varepsilon} A_{\varepsilon}\\ g_{\varepsilon}^{- 1} \partial_t g_{\varepsilon} = - \mathrm{D}_{A_{\varepsilon}}^{\ast} (g_{\varepsilon} \mathrm{d} g_{\varepsilon}^{- 1}) \end{array} \right.\)

and

\(\displaystyle \partial_t \tilde{B}_{\varepsilon} = - \mathrm{D}_{\tilde{B}_{\varepsilon}}^{\ast} F_{\tilde{B}_{\varepsilon}} - \mathrm{D}_{\tilde{B}_{\varepsilon}} \mathrm{D}^{\ast}_{\tilde{B}_{\varepsilon}} \tilde{B}_{\varepsilon} + g_{\varepsilon} (J_{\varepsilon} g_{\varepsilon}^{- 1} (\eta)) - \alpha_{\varepsilon} \tilde{B}_{\varepsilon} + \beta_{\varepsilon} g_{\varepsilon} \mathrm{d} g_{\varepsilon}^{- 1}\)

in order to be able to show the limits \(\varepsilon \rightarrow 0\) exists. But at this point we do not know that \(g \circ A \xequal{d} B\) in the limit. We would need \(\beta_{\varepsilon} = \alpha_{\varepsilon}\) since we have the correspondence

\(\displaystyle A \xrightarrow{g} B - g \mathrm{d} g^{- 1} .\)

But we do not need so much, indeed we need only that along some subsequence of \(\varepsilon\)'s we have

\(\displaystyle \lim_{\varepsilon} (\alpha_{\varepsilon} - \beta_{\varepsilon}) = \theta < \infty\)

where \(\theta\) can be a finite number. Then this means we just pick the wrong abstract equation and consider instead the system

\(\displaystyle \left\{ \begin{array}{l} \partial_t A_{\varepsilon} = - \mathrm{D}_{A_{\varepsilon}}^{\ast} F_{A_{\varepsilon}} - \mathrm{D}_{A_{\varepsilon}} \mathrm{D}^{\ast}_{A_{\varepsilon}} A + J_{\varepsilon} \eta - \alpha_{\varepsilon} A_{\varepsilon} + \theta A_{\varepsilon}\\ g_{\varepsilon}^{- 1} \partial_t g_{\varepsilon} = - \mathrm{D}_{A_{\varepsilon}}^{\ast} (g_{\varepsilon} \mathrm{d} g_{\varepsilon}^{- 1}) \end{array} \right.\)
\(\displaystyle \partial_t \tilde{B}_{\varepsilon} = - \mathrm{D}_{\tilde{B}_{\varepsilon}}^{\ast} F_{\tilde{B}_{\varepsilon}} - \mathrm{D}_{\tilde{B}_{\varepsilon}} \mathrm{D}^{\ast}_{\tilde{B}_{\varepsilon}} \tilde{B}_{\varepsilon} + g_{\varepsilon} (J_{\varepsilon} g_{\varepsilon}^{- 1} (\eta)) + \theta \tilde{B}_{\varepsilon} - \alpha_{\varepsilon} \tilde{B}_{\varepsilon} + \beta_{\varepsilon} g_{\varepsilon} \mathrm{d} g_{\varepsilon}^{- 1} .\)

If we do not have even this subsequence, then we need to play with the intensity of the noise by using the equation

\(\displaystyle \partial_t A_{\varepsilon} = - \mathrm{D}_{A_{\varepsilon}}^{\ast} F_{A_{\varepsilon}} - \mathrm{D}_{A_{\varepsilon}} \mathrm{D}^{\ast}_{A_{\varepsilon}} A + \sigma_{\varepsilon} J_{\varepsilon} \eta - \alpha_{\varepsilon} A_{\varepsilon} \sigma_{\varepsilon}^2 + \theta_{\varepsilon} \sigma_{\varepsilon}^2 A_{\varepsilon}\)
\(\displaystyle \partial_t \tilde{B}_{\varepsilon} = - \mathrm{D}_{\tilde{B}_{\varepsilon}}^{\ast} F_{\tilde{B}_{\varepsilon}} - \mathrm{D}_{\tilde{B}_{\varepsilon}} \mathrm{D}^{\ast}_{\tilde{B}_{\varepsilon}} \tilde{B}_{\varepsilon} + \sigma_{\varepsilon} g_{\varepsilon} (J_{\varepsilon} g_{\varepsilon}^{- 1} (\eta)) - \alpha_{\varepsilon} \sigma_{\varepsilon}^2 \tilde{B}_{\varepsilon} + \beta_{\varepsilon} \sigma_{\varepsilon}^2 g_{\varepsilon} \mathrm{d} g_{\varepsilon}^{- 1} + \theta_{\varepsilon} \sigma_{\varepsilon}^2 \tilde{B}_{\varepsilon}\)

and we set \(\theta_{\varepsilon} = \alpha_{\varepsilon} - \beta_{\varepsilon}\) and we can choose \(\sigma_{\varepsilon} = (| \theta_{\varepsilon} |^{1 / 2})^{- 1}\) and looking at the limit we generate a contradiction since the limit is not gauge invariant.

\(\displaystyle \)