SRQ seminar – October 1st, 2018

Notes from a talk in the SRQ series.

INI Seminar 20181001 Levy

2d Yang–Mills theory and the Makeenko–Midgal equations (I)

Goal today: characterize the $2 d$ Yang–Mills theory.

Yang–Mills theory is an example of gauge theory, these were introduced in particle physics to describe interactions between elementary particles.

Interactions are modelled via connections in principal bundle (gauge fields). How and why this models play a fundamental role in physics is kind of amazing.

Maybe better to talk about Quantum $2 d$ Yang–Mills theory to make a difference with the “classical” field theory.

Classical Yang–Mills theory.

Ingredients:

A manifold. To describe the space–time (Riemannian, Lorenzian). An example is $\mathbb{R}^4$ with metric $\mathrm{d} t^2 - \mathrm{d} x^2 - \mathrm{d} y^2 - \mathrm{d} z^2$. Another is a compact Riemanian surface $M$.
A group. $G$ structure group which is a compact Lie group. This corresponds to the interaction we are describing: $U (1)$ elecromagnetism, $S U (2) \times U (1)$ weak interactions, $S U (3)$ strong interactions. In general we can think of $U (N)$ for arbitrary $N$ (which eventually can be taken large for interesting limits). $\mathfrak{g}$ is the Lie algebra: $\mathfrak{u} (N)$: $N \times N$ skew–Hermitian matrices. Scalar product on the Lie algebra $\mathfrak{u} (N)$ which is taken to be:
$\displaystyle \langle X, Y \rangle = N\operatorname{Tr} (X^{\ast} Y) .$
Principal bundle. $M \leftarrow P \rightarrow^G P$. Example: $P = M \times G$. Physicists never name it. (Every representation allows to go from the principal bundle to a specific vector bundle.)

Gauge group. $\mathcal{J} (P) =\operatorname{Aut} (P) = C^{\infty} (M, G)$.

Connections. $\mathcal{A} (P) = \{ \text{connections on $P$} \} .$ $P_x, P_y \cong G$ take a path $\gamma$ from $x$ to $y$ and to each of those paths associate a map from $P_x$ to $P_y$. A connection $\omega \in \mathcal{A} (P)$ produces parallel transports (is equivalent), by describing it in an infinitesimal way.

Holonomies. A loop $\ell$ based at a point $x \in M$. Given a connection $\omega$ we can compute the parallel transport along $\ell$ and we obtain a map $g \in G$ from $P_x$ to $P_x$. This particular element we call the holonomy of $\omega$ along $\ell$ denoted $H_{\ell} (\omega) = g$. A connection is described by the set of all holonomies. Technically a connection is a differential form on $P$ which locally can be seen as a differential 1-form on $M$ with values on the Lie algebra $\mathfrak{g}$.

Curvature of $\omega$. $\Omega = \mathrm{d} \omega + \frac{1}{2} [\omega \wedge \omega]$ so that $\Omega (X, Y) = \mathrm{d} \omega (X, Y) + [\omega (X), \omega (Y)]$. This is a two form on $M$ with values in $\mathfrak{g}$. Curvature is an infinitesimal version of the holonomy: take two vectors $X, Y$ at $x$ and form a “little loop” $\ell_{\varepsilon}$ with edge size $\varepsilon$. If we compute the holonomy $H_{\ell_{\varepsilon}} (\omega)$ we obtain something which looks like $H_{\ell_{\varepsilon}} (\omega) = \exp (\varepsilon^2 \Omega (X, Y)) + o (\varepsilon^2)$.

Curvature is a local measure of non–triviality of the holonomy.

Yang–Mills functional. Make $M$ Riemannian and compact and smooth connections $\mathcal{A} (P)$ on $P$.

$\displaystyle S : \mathcal{A} (P) \longrightarrow [0, + \infty]$ $\displaystyle S (\omega) = \frac{1}{2} \int_M \langle \Omega \wedge (\star \Omega) \rangle$

where $\star \Omega$ is the Hodge dual of $\Omega$.

In local coordinates we have $\omega = A_1 \mathrm{d} x_1 + \cdots + A_n \mathrm{d} x_n$ with $A_1, \ldots, A_n$ matrix valued functions. And

$\displaystyle \Omega = \sum_{1 \leqslant i < j \leqslant n} (- \partial_j A_i + \partial_i A_j + [A_i, A_j]) \mathrm{d} x_i \wedge \mathrm{d} x_j$

and

$\displaystyle S (\omega) = \frac{1}{2} \int_M \sum_{1 \leqslant i < j \leqslant n} \| - \partial_j A_i + \partial_i A_j + [A_i, A_j] \|^2_{\mathfrak{g}} \mathrm{d} x = \frac{1}{2} \| \Omega \|_{L^2}^2$

where $\| X \|^2_{\mathfrak{g}} = N\operatorname{Tr} (X^{\ast} X)$. Classical Yang–Mills theory is the study of this functional on this space

$\displaystyle (M, G, P) \qquad S : \mathcal{A} (P) \rightarrow [0, + \infty] .$

The gauge group acts on $\mathcal{A} (P)$ in a way that preserves the Yang–Mills functional, so we have really $S :$

$\displaystyle S : \mathcal{A} (P) /\mathcal{J} (P) \rightarrow [0, + \infty] .$

The quotient $\mathcal{A} (P) /\mathcal{J} (P)$ is always infinite dimensional when the dimension of $M$ is at least $2$. Interesting questions are about the geometry of the space $\mathcal{A} (P) /\mathcal{J} (P)$.

The critical points of $S$ are called Yang–Mills connections. Moreover $\mathcal{M} (M, G, P) = S^{- 1} (0) /\mathcal{J} (P)$: moduli space of flat connection: a finite dimensional object which very rich geometrical structure.

Note: Hodge operator from $2$-forms to $d - 2$ forms. When $d = 4$ this operator is conformally invariant. When $d = 2$ we are using the Hodge dual from top forms to zero degree forms which only uses the Riemannian volume, namely the area. Therefore in $d = 2$ we only need the ability to measure area on surfaces. Any two Riemannian structure on surfaces which gives the same area they define the same YM functional.

Yang–Mills measure.

We want to make sense of the measure

$\displaystyle \mathrm{d} \mu (\omega) = \frac{1}{Z} e^{- S (\omega)} \mathcal{D} \omega$

as a probability measure on $\mathcal{A} (P)$. We want to integrate special functions: Wilson's loops. Take loops on the manifold $M$: $\ell_1, \ldots, \ell_n$ and we want to compute

$\displaystyle \int_{\mathcal{A} (P)} \frac{1}{N} \operatorname{Tr} [H_{\ell_1} (\omega)] \cdots \frac{1}{N} \operatorname{Tr} [H_{\ell_n} (\omega)] \mathrm{d} \mu (\omega) =\mathbb{E} \left[ \frac{1}{N} \operatorname{Tr} (H_{\ell_1}) \cdots \frac{1}{N} \operatorname{Tr} (H_{\ell_n}) \right] .$

$H_{\ell} : \mathcal{A} (P) \ni \omega \mapsto H_{\ell} (\omega) \in G$ is a matrix valued random variable for which we want to describe the joint distributions for different loops. The set of all these Wilson's loops caracterise (as much as possible) the law of the connection “modulo gauge transformations”.

Multiplicativity: $H_{\ell_1 \ell_2} = H_{\ell_2} H_{\ell_1}$ almost surely where $\ell_1 \ell_2$ is the concatenation of the two loops by going around $\ell_1$ first and then around $\ell_2$.

Therefore $\operatorname{Tr} [H_{\ell}^2] =\operatorname{Tr} [H_{\ell \ell}]$. So the Wilson-loops traces give access to the whole distribution of the random matrices $(H_{\ell})_{\ell}$.

When $d = 2$ there exists an “honest” process $(H_{\ell})_{\ell}$ which has the right to be called the Yang–Mills measure.

The YM functional is a generalization of the Lagrangian of electromagnetism. A short story: Between 1700 and 1864 a lot has been done to understand electromagnetism, culminating in Maxwell's equations:

$\displaystyle \begin{array}{lllll} \operatorname{div} \vec{E} & = & \rho & & \text{Gauss--Poisson (1810)}\\ \operatorname{curl} \vec{E} + \partial_t \vec{B} & = & 0 & & \text{Faraday (1830)}\\ \operatorname{div} \vec{B} & = & 0 & & \text{Thomson (lord Kelvin)}\\ \operatorname{curl} \vec{B} - \partial_t \vec{E} & = & \vec{j} & & \text{�rsted--Amp�re/Maxwell (1821)} \end{array}$

This is actually not an equation for vector fields.

Minkowski realized that if one work on $\mathbb{R}^4$ with Minkowski metric things simplified. If we introduce a $2$-form

$\displaystyle F = B_x \mathrm{d} y \wedge \mathrm{d} z + \cdots - E_x \mathrm{d} t \wedge \mathrm{d} x - \cdots \text{}$ $\displaystyle J = \rho \mathrm{d} x \wedge \mathrm{d} y \wedge \mathrm{d} z + \cdots - j_x \mathrm{d} t \wedge \mathrm{d} y \wedge \mathrm{d} z - \cdots$

and then Maxwell's equations have the form

$\displaystyle \left\{ \begin{array}{l} \begin{array}{lll} \mathrm{d} F & = & 0\\ \mathrm{d} \star F & = & J \end{array} \end{array} \right.$

At least locally then $F = \mathrm{d} A$. $F$ is the strenght field and $A$ is the gauge field. $A$ is defined modulo transformations of the form $A \rightarrow A + \mathrm{d} \varphi$ for $\varphi : \mathbb{R}^4 \rightarrow \mathbb{R}$. Lagrangian for the above equations is

$\displaystyle \mathcal{L} (A) = \frac{1}{2} F \wedge \star F + A \wedge J.$

From the electromagnetic Lagrangian to the YM measure one follows Feynman prescription for the path integral in its Euclidean version.