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Section 2.12 Dessins in Physics (Jim)

Physics.

Let \(M\) be a manifold with a metric \(g\text{.}\) We call the pair \((M,g)\) a “spacetime manifold”. Let \(\mathcal E\) be a “space of fields”, either \(\cinf (M)\text{,}\) sections of some \(E\to M\text{,}\) connections, or similar.

\begin{equation*} S(\phi) = \int_M \mathcal L (\phi) \end{equation*}

for \(\phi\in \mathcal E\) and \(\mathcal L\) the Lagrangian. “Physically realisable states” are then fields \(\phi\) that minimise \(S(\phi)\text{.}\) \(W\) is a superpotential, this is a term in \(\mathcal L\) that satisfies some special symmetries. E.g. we could also have

\begin{equation*} S(\phi_1, \phi_2) = \int_M \mathcal L(\phi_1,\phi_2) \end{equation*}

the \(W\) might satisfy \(W(\phi_1, \phi_2) = W(\phi_2, \phi_1)\text{.}\)

Definition 2.12.1. Gauge transformations.

Let \(G \acts E \xrightarrow p M\) be an action s.t. each fibre \(E_x = p \inv(x)\) is a representation of \(G\text{.}\) A gauge is a section \(s(x)\) of \(E \to M\text{.}\) A gauge transformation is a map \(g\colon M \to G\) s.t.

\begin{equation*} g(x) s(x) \end{equation*}

is another section, call \(G\) the gauge group. The important gauge transformations are the ones that fix the set of physically realisable states (i.e. fixes the subset of \(\mathcal E\) that minimise \(S\)).

Quivers and dessins.

Let's now study the relationship between quivers and dessins.

Example 2.12.2.

\(\mathcal N = 4\) SYM (supersymmetric Yang-Mills) (Gauge symmetries given by some product of \(\specialunitary (N)\)) .

A quiver is a directed graph, possibly with self-loops. Here we think of the nodes as corresponding to factors of the gauge group. And the arrows as fields, so in a bouquet with 3 petals we have three fields, and only \(G = \specialunitary (N)\text{.}\)

There is also the notion of a periodic quiver (a tiling of the plane). We can take the triangular lattice and consider its dual, this is a hexagonal tiling with a bicolouring corresponding to the fact we had upwards pointing and downwards pointing triangles. This is a Dimer model.

Relating the Dimer model back to physics: We have hexagonal faces in correspondence with factors of the Gauge group, and edges fields, with vertices terms in \(W\text{.}\)

So one distinct face gives one factor in the gauge group so \(G = \specialunitary (N)\text{.}\) 3 distinct edges give 3 fields \(X_1, X_2, X_3\text{.}\) To recover \(W\) consider the permutation arising from reading the edges around the vertices counterclockwise. A black vertex \((1,2,3)\) gives \(\sigma_B\) corresponding to a positive term in \(W\text{.}\) A white vertex \((1,2,3)\) gives \(\sigma_W\) corresponding to a negative term in \(W\text{.}\) Then \(\sigma_\infty = (\sigma_B\sigma_W)\inv = (123)\) \(\sigma_i\) gives a term for each cycle. Each cycle in \(\sigma_B\) gives a product of fields indexed by the cycle, e.g. in this example \(\sigma_B\) gives \(X_1X_2X_3\text{.}\) Each cycle in \(\sigma_W\inv\) gives a product of fields indexed by the cycle, e.g. in this example \(\sigma_W\) gives \(X_1X_3X_2\text{.}\) Then

\begin{equation*} W = \trace((\text{sim of }\sigma_B\text{ terms}) - (\text{sim of }\sigma_W\text{ terms})) \end{equation*}
\begin{equation*} = \trace(X_1X_2X_3 - X_1X_3X_2)\text{.} \end{equation*}
\begin{equation*} \Aut( \{\sigma_B,\sigma_W,\sigma_\infty\}) = \{ \gamma\in S_3: \gamma\sigma_i \gamma\inv = \sigma_i) \end{equation*}
\begin{equation*} = \{1,(123), (132)\} \end{equation*}
\begin{equation*} = \ZZ/3\ZZ\text{.} \end{equation*}

The fundamental domain of the Dimer gives a dessin on the torus with two vertices of degree 3. This corresponds to the Belyi pair \((\Sigma, \beta)\) where

\begin{equation*} \Sigma \colon y^2 = x^3 + 1 \end{equation*}
\begin{equation*} \beta \colon \Sigma \to \PP^1 \end{equation*}
\begin{equation*} (x,y) \mapsto \frac{y+1}{2}\text{.} \end{equation*}
\begin{equation*} \Aut(\Sigma, \beta) \simeq \Aut(\{\sigma_B,\sigma_W, \sigma_\infty\}) \end{equation*}

\(\Aut(\Sigma, \beta)\) is generated by

\begin{equation*} (x,y) \mapsto (w^3 x,y) \end{equation*}

where \(w^3 = 1\text{.}\)

Example 2.12.3.

Take the quiver with two vertices and two edges in each direction connecting them. This has 4 fields and two factors of \(G\) (i.e. \(G = \specialunitary (N) \times \specialunitary (N)\)). The dimer is a square lattice alternately coloured, with \(\sigma_B = \sigma_W = (1234), \sigma_\infty = (13)(24)\text{.}\)

\begin{equation*} W = \trace (X_1X_2X_3X_4 - X_1 X_4 X_3 X_2)\text{.} \end{equation*}

In this case the Belyi pair is

\begin{equation*} \Sigma \colon y^2= x(x-1)(x-\frac12) \end{equation*}
\begin{equation*} \beta = \frac{x^2}{2x-1}\text{.} \end{equation*}
\begin{equation*} \Aut(\{\sigma_B,\sigma_W, \sigma_\infty\}) = \langle (1234) \rangle \simeq \ZZ/4\ZZ \end{equation*}
\begin{equation*} \phi_\pm \colon (x,y) \mapsto \left(\frac{x}{2x -1},\frac{\pm i}{(2x-1)^2} \right) \end{equation*}
\begin{equation*} \phi_+^2 = \phi_-^2 \colon (x,y) \mapsto (x,-y) \end{equation*}
\begin{equation*} \phi_+^3 = \phi_+\inv = \phi_- \end{equation*}
\begin{equation*} \phi_+^4 = 1 \end{equation*}

so

\begin{equation*} \Aut(\Sigma, \beta) \simeq \ZZ/4\ZZ \end{equation*}
\begin{equation*} \beta\inv(0)= \{(0,0)\} \end{equation*}
\begin{equation*} \beta\inv(1)= \{(1,0)\} \end{equation*}
\begin{equation*} \beta\inv(\infty)= \{(\frac 12,0), (\infty,\infty)\} \end{equation*}

on the Dimer we have the square lattice so taking a fundamental domain containing of the vertices we see the torus as a topology.

Example 2.12.4. Final example.

Let's jump straight to the Dimer the hexagonal lattice with fundamental domain containing 6 vertices. We have 9 fields and three factors in the gauge group \(G = \specialunitary (N)^2\text{.}\)

\begin{equation*} \sigma_B = (147)(258)(369) \end{equation*}
\begin{equation*} \sigma_W = (123)(456)(789) \end{equation*}
\begin{equation*} \sigma_\infty = (195)(276)(384) \end{equation*}

so

\begin{equation*} W = \trace \sum_{i,j,k} X_{12}^iX_{23}^j X_{31}^k \epsilon_{ijk} \end{equation*}

where

\begin{equation*} \epsilon_{ijk} = \begin{cases} \sgn(ijk) \amp\text{if }i,j,k \text{ distinct}\\ 0 \amp\text{otw}\end{cases} \end{equation*}

\(X_{12}^i\) acts on the \(i\)th field by \(N, \bar N, 1\) where \(N\) is the canonical representation, \(\bar N\) the anticanonical and \(1\) is trivial.

\begin{equation*} \Aut(\{\sigma_B,\sigma_W, \sigma_\infty\}) \simeq \ZZ/3\ZZ\times \ZZ/3\ZZ \end{equation*}

now the Belyi pair

\begin{equation*} \Sigma = \text{projective closure of } F = \{(x,y) : x^3+ y^3 =1\} \end{equation*}
\begin{equation*} \beta(x,y) = x^3 \end{equation*}
\begin{equation*} \gamma_1(x,y) = (w_1 x, y) \end{equation*}
\begin{equation*} \gamma_2(x,y) = (x,w_2 y) \end{equation*}
\begin{equation*} w_i^3 = 1\text{.} \end{equation*}