On the 2D Yang-Mills/Hurwitz Correspondence

Let $\mathrm{G}$ be a connected finite simple graph with vertices $\pi,\rho,\sigma,\dots$. Let $\mathbb{C}\mathrm{G}$ be the free Hilbert space over the vertices of $\mathrm{G}$ modeled as the space of kets $|\pi\rangle,|\rho\rangle,|\sigma\rangle,\dots$. The adjacency operator $K\in\operatorname{End}\mathbb{C}\mathrm{G}$ encodes the edges of $\mathrm{G}$,

Instead of asking how many walks there are between two given vertices in $\mathrm{G}$, we may ask how far apart they are. This information is encoded by the distance operator $D\in\operatorname{End}\mathbb{C}\mathrm{G}$, whose matrix

Another way to encode distance in graphs is to use linear operators $L_{1},L_{2},L_{3},\dots$ corresponding to metric level sets,

The general study of exponential distance operators in graph theory is just beginning; see [10]. To the best of this author’s knowledge, the first appearance of $\Omega_{q}$ was in Zagier’s study of deformed commutation relations in quantum physics [58], where the exponential distance operator on the Cayley graph of the symmetric group as generated by the set of Coxeter transpositions $(i\ i+1)$ plays a pivotal role. The analysis in [58] culminates in an explicit formula for the determinant of $\Omega_{q}$ showing that its zeros lie on the unit circle, implying the existence of a Hilbert space representation of the $q$-commutation relations for all $-1<q<-1$, interpolating between bosons and fermions.

The setting of this paper is the Cayley graph of the symmetric group as generated by the full conjugacy class of transpositions, which has the feature that its exponential adjacency and distance operators $\Psi_{t}$ and $\Omega_{q}$ are simultaneously and explicitly diagonalizable. This is not the case for the Coxeter-Cayley graph of the symmetric group. On the other hand, in the all-transpositions case the zeros of $\det\Omega_{q}$ are unstable: they converge on the origin as the degree of the symmetric group increases.

The all-transpositions Cayley graph is intimately linked to Hurwitz theory, a classical branch of enumerative geometry concerned with counting maps between compact Riemann surfaces [11], and we therefore refer to it as the Hurwitz-Cayley graph. Cover counting is associated with the exponential adjacency operator $\Psi_{t}$ on this graph. For example, loop enumeration on the Hurwitz-Cayley graph is algebraically equivalent to computing a diagonal matrix element of $\Psi_{t}$ and geometrically equivalent to counting simple branched covers of the Riemann sphere, a classical and much-studied problem [21].

It is a special feature of the Hurwitz-Cayley graph that matrix elements of the exponential distance operator $\Omega_{q}$ and its inverse also count walks [48]. The walks enumerated by $\langle\sigma|\Omega_{q}^{\pm 1}|\rho\rangle$ are “monotone” with respect to an edge labeling of the Hurwitz-Cayley graph related to the representation theory [46] and order theory [45] of the symmetric group. Thus $\Omega_{q}$ corresponds to a monotone version of Hurwitz theory [24, 25, 26, 27, 28] which turns out to have many applications in random matrix theory and related areas.

The idea that two-dimensional Yang-Mills theory with $\mathrm{U}_{N}$ gauge group becomes Hurwitz theory in the large $N$ limit [33] has been intensively studied by physicists [14, 15, 35, 36, 54], who noticed that eigenvalues of the exponential adjacency operator $\Psi_{t}$ of the Hurwitz-Cayley graph appear in the partition function of $YM_{2}$. The Yang-Mills/Hurwitz correspondence works perfectly in the Calabi-Yau case, where spacetime is a torus, even manifesting mirror symmetry [19]. For non-torus spacetimes, Hurwitz theory is not an exact match for the large $N$ limit of $YM_{2}$ due to the presence of certain “$\Omega$-factors” which appear in the partition function with multiplicity equal to the Euler characteristic of spacetime but have no obvious Hurwitz-theoretic meaning. The large $N$ limit of $YM_{2}$ is necessarily a completion of classical Hurwitz theory which accounts for these $\Omega$-factors.

The purpose of this paper is to point out that the $\Omega$-factors in $YM_{2}$ are exactly the eigenvalues of the exponential adjacency operator $\Omega_{q}$ of the Hurwitz-Cayley graph, and that consequently the large $N$ limit of $YM_{2}$ is the union of classical and monotone Hurwitz theory: mixed Hurwitz theory. The mixed Hurwitz theory of a genus zero target was introduced in [27] as a way to interpolate between results in Hurwitz theory [50] and random matrix theory [60]. It was generalized to higher genus targets in [40].

In Section 2 we introduce the Hurwitz-Cayley graph and explain what is perhaps its most important feature: a combinatorial duality called the Jucys-Murphy correspondence. In Section 3 we derive the Gross-Taylor formula for the chiral partition function of $YM_{2}$ on compact orientable surfaces [14, 35], and explain its relationship to the graph operators $\Psi_{t}$ and $\Omega_{q}$, which constitutes the Yang-Mills/Hurwitz correspondence.

The remaining sections of the paper illustrate the Yang-Mills/Hurwitz correspondence for specific choices of spacetime. This is done first for the finitely many cases which can be described using only classical Hurwitz theory: the cylinder and surfaces obtained from it by identifying boundaries or plugging holes (torus, disc, sphere). We then consider the three-holed sphere and one-holed torus as representative examples of the infinitely many remaining compact orientable spacetimes, where monotone Hurwitz theory is inescapable. In fact, the area zero limit of $YM_{2}$, a much-studied degeneration [14, 15, 54], is pure monotone Hurwitz theory. This implies a relationship between monotone Hurwitz numbers and Euler characteristics of Hurwitz moduli spaces [22] which seems to be very interesting, but which we do not explore in detail here.

Let $\mathrm{S}^{d}=\mathrm{Aut}\{1,\dots,d\}$ be the symmetric group of degree $d\in\mathbb{N}$, where we define the group law as $\pi_{1}\pi_{2}=\pi_{2}\circ\pi_{1}$ so that permutations are multiplied left to right. Let $K=\{(i\ j)\colon 1\leq i<j\leq d\}$ be the conjugacy class of transpositions in $\mathrm{S}^{d}$. The Hurwitz-Cayley graph has vertex set $\mathrm{S}^{d}$, with $\rho,\sigma$ adjacent if and only if $\rho\tau=\sigma$ for $\tau\in K$. We henceforth identify $\mathrm{S}^{d}$ with the Hurwitz-Cayley graph.

Geodesic distance in Cayley graphs is implemented by word norm. For the Hurwitz-Cayley grpah, $\mathrm{d}(\rho,\sigma)=|\rho^{-1}\sigma|$ where $|\pi|$ is the minimal length of a factorization of $\pi$ into transpositions. The Join-Cut Lemma [29] gives $|\pi|=d-\ell(\pi)$, where $\ell(\pi)$ is the number of factors in the unique decomposition of $\pi$ as a product of disjoint cycles. Accordingly, the Hurwitz-Cayley graph decomposes as

The Cayley graph of any group with respect to any symmetric generating set decomposes into independent sets made up of identity-centered spheres, giving a graded graph structure. The Hurwitz-Cayley graph has the non-generic feature that its levels are unions of conjugacy classes: we have

There are in general many geodesics between permutations $\rho,\sigma\in\mathrm{S}^{d}$. The number of geodesics $\rho\to\sigma$ equals the number of minimal transposition factorizations of $\rho^{-1}\sigma$, and this number was computed independently by Hurwitz and Cayley; see [44]. It is

A fundamental feature of the Hurwitz-Cayley graph is that it can be Euclideanized by choosing a distinguished geodesic between every pair of points in a systematic way. This construction involves an edge labeling of the symmetric group introduced, implicitly and independently, by Jucys [41] and Murphy [47]. Let us mark each edge corresponding to the transposition $(i\ j)$ with $j$, the larger of the two numbers interchanged. Thus, emanating from every vertex of $\mathrm{S}^{d}$ we have one $2$-edge, two $3$-edges, three $4$-edges, etc. See Figure 1 for the case $d=4$.

A walk on $\mathrm{S}^{d}$ is said to be strictly monotone if the labels of the edges it traverses form a strictly increasing sequence. Clearly the length of any such walk is at most $d-1$, the number of edge labels and the diameter of $\mathrm{S}^{d}$.

Theorem 2.1 says that for any permutation $\rho$, the points of the sphere of radius $r$ centered at $\rho$ correspond bijectively to endpoints of strictly monotone $r$-step walks emanating from $\rho$. We refer to this bijection the Jucys-Murphy correspondence; see Figure 2.

Instead of defining a strictly monotone walk by the condition that the labels of its edges are strictly increasing, we could have stipulated that they strictly decrease along the walk. Since path reversal defines a bijection between strictly decreasing walks $\rho\to\sigma$ and strictly increasing walks $\sigma\to\rho$, Theorem 2.1 implies that there is one and only one strictly decreasing walk between any two vertices of the Hurwitz-Cayley graph, and that it is a geodesic. Theorem 2.1 is thus valid in both senses of the word “monotone.”

Theorem 2.1 allows us to view an arbitrary product $\pi_{1}\pi_{2}\dots\pi_{k}$ of permutations graphically, as a concatenation of strictly monotone walks

Since $\mathrm{S}^{d}$ is a group, $\mathbb{C}\mathrm{S}^{d}$ is an algebra, the product of kets being $|\pi_{1}\rangle|\pi_{2}\rangle=|\pi_{1}\pi_{2}\rangle$. The group algebra $\mathbb{C}\mathrm{S}^{d}$ acts on itself by right multiplication: every $|A\rangle\in\mathbb{C}\mathrm{S}^{d}$ gives a linear operator $A\in\operatorname{End}\mathbb{C}\mathrm{S}^{d}$ defined by

The map $|A\rangle\mapsto A$ is a faithful linear representation of $\mathbb{C}\mathrm{S}^{d}$, the right regular representation. By abuse of notation, if $|A\rangle\in\mathbb{C}\mathrm{S}^{d}$ has $\{0,1\}$-coefficients in the permutation basis we write $A$ for both the subset of $\mathrm{S}^{d}$ determined by the nonzero coefficients of $|A\rangle$, and for the operator on $\mathbb{C}\mathrm{S}^{d}$ which is the image of $|A\rangle$ in the regular representation. Thus $K$ denotes both the conjugacy class of transpositions in $\mathrm{S}^{d}$ and the adjacency operator on the Hurwitz-Cayley graph, and $L_{r}$ denotes both the sphere of radius $r$ centered at $\iota$ in $\mathrm{S}^{d}$ and the operator which maps $|\rho\rangle$ to the sum of all permutations $|\sigma\rangle$ on the sphere of radius $r$ with center $\rho$. In matrix form, the Jucys-Murphy correspondence is as follows.

Let $|\Omega_{q}\rangle\in\mathbb{C}\mathrm{S}^{d}$ be the exponential distance element,

Theorem 2.2 says that

For $r=1$, Theorem 2.4 is the obvious fact that the adjacency operator of the Hurwitz-Cayley graph is the sum of the Jucys-Murphy operators,

An analogous but subtler factorization of the exponential distance operator of the Hurwitz-Cayley graph is obtained by combining Theorem 2.4 with the generating series

We conclude that computing the eigenvalues of the exponential adjacency and distance operators $\Psi_{t}$ and $\Omega_{q}$ on the Hurwitz-Cayley graph reduces to computing the eigenvalues of the Jucys-Murphy operators $J_{1},\dots,J_{d}\in\operatorname{End}\mathbb{C}\mathrm{S}^{d}$. Before addressing this spectral problem, we consider a further implication of the Jucys-Murphy correspondence.

Combining the polynomial form of the Jucys-Murphy correspondence with Newton’s theorem on symmetric polynomials, every symmetric polynomial function of the Jucys-Murphy elements $|J_{1}\rangle,\dots,|J_{d}\rangle\in\mathbb{C}\mathrm{S}^{d}$ is a polynomial function of the levels $|L_{r}\rangle=e_{r}(|J_{1}\rangle,\dots,|J_{d}\rangle)$ of the Hurwitz-Cayley graph. As explained above, each level is a sum of conjugacy classes,

A particularly interesting case is that of complete symmetric polynomials in Jucys-Murphy elements,

The fact that $|M_{r}\rangle$ is a central element in $\mathbb{C}\mathrm{S}^{d}$ implies that $W^{r}_{\leq}(\iota,\pi)$ depends only on the conjugacy class of $\pi\in\mathrm{S}^{d}$, which is equivalent to the statement that $W^{r}_{\leq}(\rho,\sigma)$ depends only on the cycle type of $\rho^{-1}\sigma$. In particular, we have

We have seen above that the total number of geodesics $\rho\to\sigma$ in the Hurwitz-Cayley graph is given by a product of Cayley numbers along the cycles of $\rho^{-1}\sigma$ times a shuffle factor. We have also seen that the number of strictly monotone geodesics $\rho\to\sigma$ is exactly one. The number of weakly monotone geodesics $\rho\to\sigma$, i.e. the number of minimal weakly monotone factorizations of $\rho^{-1}\sigma$ into transpositions, must lie between these two extremes.

The proof of Theorem 2.6 is again elementary, reducing to counting minimal monotone factorizations of a cycle; that this is a Catalan number was first discovered by Gewurz and Merola [23], and the lack of shuffle factor is again due to monotonicity. A more general result counting weakly monotone factorizations with a given “signature,” corresponding to evaluation of monomial symmetric polynomials in Jucys-Murphy elements, was proved in [46]. This result can in turn be used to develop an enumerative theory of weighted walks in the Hurwitz-Cayley graph [39], which turns out to be very rich [1].

As with the group algebra of any finite group [52], we have an orthogonal decomposition

A remarkable formula for computing the Fourier transform of a symmetric polynomial function in Jucys-Murphy elements was discovered by Jucys and Murphy; see [18]. Recall that the content $c(\Box)$ of a cell $\Box$ in a Young diagram $\lambda$ is defined to be its column index minus its row index. Thus, filling the cells of $\lambda$ with their contents produces a jagged $\lambda$-shaped corner of the infinite Toeplitz matrix $[j-i]_{i,j=1}^{\infty}$.

According to Theorem 2.7, the transposition class $|K\rangle=|J_{1}\rangle+\dots+|J_{d}\rangle$ has Fourier transform

To find the eigenvalues of the exponential distance operator $\Omega_{q}$ of the Hurwitz-Cayley graph, we need to compute the Fourier of every level of the graph. Theorem 2.7 gives the Fourier transform of $|L_{r}\rangle=e_{r}(|J_{1}\rangle,\dots,|J_{d}\rangle)$ as

For $q$ away from the above unit fractions, the exponential distance operator $\Omega_{q}$ of the Hurwitz-Cayley graph is invertible, and its inverse acts in $\mathbf{V}^{\lambda}$ as multiplication by

The exponential adjacency and distance matrices $\Psi_{t}$ and $\Omega_{q}$ of the Hurwitz-Cayley graph are generating functions for walks in $\mathrm{S}^{d}$. The matrix elements of the exponential adjacency matrix are

The partition function of Yang-Mills theory on a compact two-dimensional orientable spacetime of area $t$ with $m$ holes and $n$ handles admits a dual representation in terms of the gauge group, $\mathrm{U}_{N}$. The elegant formula reads [14]

It was perceived by Gross that (44) provides a promising path to a string description of $YM_{2}$ at large $N$, which ought to be Hurwitz theory [33]. This insight was fully developed by Gross and Taylor [35, 36], who argued that as $N\to\infty$ the partition function $\mathcal{Z}_{N}$ factorizes into two copies of a chiral partition function, $Z_{N}$, obtained by restricting the sum (44) to nonnegative $\lambda$. The chiral partition function may be viewed as a sum over the set $\mathsf{Y}_{N}$ of Young diagrams with at most $N$ rows, and thus decomposes as

Gross and Taylor’s fundamental insight [35, 36] is that $Z_{N}^{d}$ can be presented entirely in terms of the representation theory of the symmetric group $\mathrm{S}^{d}$. This is accomplished in three steps.

The Gross-Taylor formula reveals that the existence of a $1/N$ expansion of $Z_{N}^{d}$, and the Hurwitz-theoretic interpretation of its coefficients, is a somewhat subtle issue. The internal sum in Theorem 3.1 can be written