Yang–Mills/Gromov–Witten Duality

• Yang–Mills/Gromov–Witten duality is a correspondence linking 2D gauge theory and enumerative geometry, equating partition functions and correlators with Hurwitz numbers and GW invariants.
• It employs large-N asymptotic expansions and explicit combinatorial bijections to rigorously match gauge theory observables with geometric invariants on elliptic curves.
• The duality establishes a mirror-symmetry type link between the Gross–Taylor expansion in 2D Yang–Mills and the stationary sector of topological GW theory, with implications in string theory and algebraic geometry.

Yang–Mills/Gromov–Witten duality is a bridge between two-dimensional gauge theory—specifically the large-$N$ expansion of Yang–Mills theory on Riemann surfaces—and the enumerative geometry of Gromov–Witten (GW) invariants, particularly on elliptic curves. The duality relates the partition functions, correlators, and algebraic structures of 2D Yang–Mills theory, Hurwitz theory of ramified covers, and the stationary sector of topological GW theory, providing explicit and rigorous correspondences between gauge theory observables and geometric invariants in the string-theoretic framework.

1. Central Heat Trace and Its Large-$N$ Expansion

The foundation of the duality is the "central" heat trace on a classical compact group $G_N \subset \mathrm{GL}_N(\mathbb{C})$—including $\mathrm{U}(N)$, $\mathrm{SU}(N)$, $\mathrm{O}(N)$, and symplectic groups—given by

$\Tr\bigl(e^{\frac{t}{2}\Delta_{G_N}}\bigr) = \sum_{\lambda \in \widehat{G}_N} e^{-\frac{t}{2}\,c_2(\lambda)}\,,$

where $c_2(\lambda)$ is the quadratic Casimir associated to the highest weight $\lambda$. The Laplacian $\Delta_{G_N}$ is induced by the invariant inner product $\langle X,Y\rangle = \frac{\beta n}{2} \Tr(X^* Y)$. The heat kernel acts on the space of central, square-integrable functions, effectively encoding the partition function of 2D Yang–Mills theory on a genus one surface.

In the large-$N$ limit, this trace admits an asymptotic expansion either in $1/N$ or as a "genus" expansion: $\Tr\bigl(e^{-t\Delta}\bigr) = \sum_{g=0}^{\infty} \sum_{d=0}^{\infty} N^{2-2g}\, a_{g,d}(t)\,,$ with only finitely many $a_{g,d}(t)$ nonzero at each order. These coefficients encapsulate the entire genus expansion structure familiar from topological string theory (Lemoine et al., 11 Nov 2025).

2. Combinatorial Framework and Bijections

A key insight is a chain of explicit bijections: $$\widehat G_N \longleftrightarrow \{\text{highest weights}\} \longleftrightarrow \{\text{integer partitions}\} \longleftrightarrow \{\text{permutations}\} \longleftrightarrow \{\text{ramified coverings of the torus}\}$$ For $\mathrm{U}(N)$ and $\mathrm{SU}(N)$, highest weights are encoded by pairs of partitions; for other groups, by a single partition. The Casimir element becomes a quadratic function in the size and content of these partitions. Averaging over partitions with geometric weight $q_t = e^{-t/2}$, the trace transforms into an expectation under a $q$-uniform measure, permitting analytic control and Taylor expansion in $1/N$. Large deviations and tail estimates guarantee exponential smallness of error terms outside of controlled regimes (Lemoine et al., 11 Nov 2025).

3. Hurwitz Interpretation and Ramified Coverings

Hurwitz theory interprets terms in the heat trace expansion in terms of the enumeration of degree-$n$ ramified covers of the torus with $2k$ branch points. The (unnormalized) Hurwitz number is

$$H_1(n,2k) = \sum_{\alpha \vdash n} [K(\alpha)]^{2k}\,,$$

where $K(\alpha)$ denotes the total content of the Young diagram of $\alpha$. By integrating over Hurwitz spaces $\mathcal{H}_1(n,2k)$ with measure

$$\rho_t = \sum_{n\geq 1} q_t^n \sum_{k\geq 0} \frac{t^{2k}}{(2k)!} \sum_{[X]\in\mathcal{H}_1(n,2k)} \delta_{[X]}\,,$$

the trace acquires the structure of a sum over ramified covers, with each configuration weighted by $N^{\chi(X)}$, where $\chi(X) = 2 - 2g(X)$ is the Euler characteristic. This matches the `$N$ to the Euler characteristic' scaling known from large-$N$ gauge/string duality (Lemoine et al., 11 Nov 2025).

4. Gauge/String and Yang–Mills/Gromov–Witten Correspondence

For 2D Yang–Mills theory, the partition function on the torus takes the explicit form: $Z_{G_N}(1,t) = \sum_{\lambda \in \widehat{G}_N} (\dim \lambda)^{2-2g}\, e^{-\frac{t}{2}c_2(\lambda)} = \Tr\bigl(e^{\frac{t}{2}\Delta_{G_N}}\bigr)\,,$ which is identified as a sum over branched covers weighted by $N^{\chi(X)}$. For the string side, the Gross–Taylor expansion realizes these as sums over random surfaces (ramified coverings) with $N$-dependent weights.

The Gromov–Witten interpretation proceeds via the Okounkov–Pandharipande GW/Hurwitz correspondence, which links GW invariants of the elliptic curve $E$ to completed cycles (linear combinations of partial permutations) on partitions. The generating function for stationary GW invariants is

$$\mathcal{Z}_q^E(z_1,\dots,z_n) = \sum_{d\geq 1} q^d \sum_{k_1,\dots,k_n \geq 0} \langle \tau_{k_1}(\omega)\cdots\tau_{k_n}(\omega)\rangle^E_d\, z_1^{k_1+1}\cdots z_n^{k_n+1}$$

and the coefficients $a_{g,d}(t)$ in the heat trace genus expansion become explicit functionals of $\mathcal{Z}_{q_t}^E$, establishing a dictionary: $$a_k(t) = \sum_{r+2s=k} \kappa_{r,s}(t) [\mathcal{D}^r \mathcal{F}_{1,2s}](q_t)$$ where $$\mathcal{F}_{1,2s}(q) = \sum_{d\geq 0} H_1(d,2s)\,q^d$$, and $\mathcal{F}_{1,2s} \leftrightarrow \mathcal{Z}_q^E$ (Lemoine et al., 11 Nov 2025).

5. Frobenius Algebra Structure and Deformed Theories

To organize the algebraic data efficiently, the duality employs a partial permutation Frobenius algebra $\mathcal{A}_\infty$, whose orbits $A_r$ encode coverings of all degrees via partial permutations—pairs $(d,\rho)$ with $d \subset \mathbb{N}$ and $\rho \in \mathrm{Sym}(d)$. This algebra is endowed with a commutative Frobenius structure via one‐point functions on the moduli space $\mathcal{M}_G$.

The chiral 2D Yang–Mills/Gromov–Witten duality involves deformations:

• Transposition (area-dependent) deformation: Adding $-g_s t C_2$ to the effective Lagrangian, with $C_2$ the sum of transpositions.
• $\Omega$-operator (compactification) deformation: Inserting $\Omega_{g_s u}^{2-2G}$, where

$$\Omega_{g_s u} = \sum_{\eta \vdash d}(g_s u)^{|\eta|} C_\eta$$

modifies the contribution of branched points and provides regularization/compactification of Hurwitz spaces (Benizri et al., 4 Feb 2025).

These deformations are mirrored in the GW framework by inserting $\Omega$-type operators and $\tau_1$ insertions (corresponding to area dependence) in the generating functionals of the topological sigma model.

6. Partition Functions, Correlator Correspondence, and the Chiral Limit

The full correspondence manifests as the equivalence, order by order in $g_s$ and $q$, of partition functions and correlators on both sides. For Yang–Mills with $k$ boundary holonomies $U_i$: $$Z^+_G[U_i] = \sum_{d\geq 0} (q\,g_s^{2G-2+k})^d \langle \prod_{i=1}^k \bigl( \sum_{\alpha \vdash d} p_\alpha(U_i) \hat{C}_\alpha \bigr) \rangle_d^{t,u}$$ maps precisely to deformed GW correlators of the form

$$\langle \hat V_{k_1} \cdots \hat V_{k_n} \rangle^{\mathrm{def}} = \sum_{g,d}g_s^{2g-2}q^d \langle (\prod_i \tau_{k_i})\,e^{-t\tau_1}\,\widetilde{\Omega}_u^{2-2G-n} \rangle_{g,d}$$

with inverse correspondences allowing for GW invariants to be written in terms of Yang–Mills observables (Benizri et al., 4 Feb 2025).

The genus-zero free energy,

$$F_0(q, t, u) = \sum_d q^d \sum_{\alpha,\beta \vdash d} (-1)^{\ell(\alpha)+\ell(\beta)-2} \frac{t^{\ell(\alpha)+\ell(\beta)-2}u^{|\alpha|+|\beta|}}{(\ell(\alpha)+\ell(\beta)-2)!} H^\circ_{0,d}(C_\alpha, C_\beta, C_2^{\ell(\alpha)+\ell(\beta)-2})$$

displays phase structure reproducing known phenomena such as the Douglas–Kazakov transition for $u=0$ and large $t$.

7. Physical and Mathematical Implications

The duality provides a rigorous realization of the Gross–Taylor string expansion (including the full, non-chiral expansion) for 2D Yang–Mills on the torus and extends to all compact classical groups (Lemoine et al., 11 Nov 2025). The GW/Hurwitz correspondence yields an explicit mirror-symmetry-type duality in genus one, mapping 2D gauge theory partition functions and Wilson loop observables to topological sigma model correlators on the elliptic curve. The factor $N^{\chi}$ in large-$N$ expansions is identified with the traditional ’t Hooft string coupling, with the combinatorics of partitions unifying the combinatorial and enumerative geometric frameworks.

All correlators, partition functions, and free energies coincide termwise between Yang–Mills and the deformed stationary GW theory, with completed cycles in the algebra of partial permutations corresponding to closed-string states and area dependence translated into $\tau_1$ insertions. The approach permits detailed control of subleading corrections in the $1/N$ expansion, provides a systematic treatment of Wilson loop observables, and allows for the analysis of higher-genus and compactification effects via the $\Omega$-operator formalism (Lemoine et al., 11 Nov 2025, Benizri et al., 4 Feb 2025).

The duality thus constructs an explicit and quantitative dictionary between 2D gauge theory, Hurwitz theory, and Gromov–Witten theory, cementing a detailed bridge in the genus one case and opening possibilities for higher-genus extensions, deepening the interface of gauge/string duality and algebraic geometry.