Calogero-Moser Polynomial Approximations and Dwork Congruences for Complete Flags

Introduction

Enumerative geometry of Nakajima quiver varieties relates quantum cohomology and quantum $K$-theory to integrable systems [Nak94, MO19, KPSZ21, K23]. For the cotangent bundle to a complete flag variety,

the $K$-theoretic vertex, with its scalar character factor, is an eigenfunction of the trigonometric Ruijsenaars–Schneider (tRS) Hamiltonians in the Kähler parameters [K21, Din21]. Its cohomological limit solves the trigonometric Calogero–Moser (tCM) equations, which also describe the quantum differential module of the Springer resolution [BMO11]. The recent work [K26] establishes compatibility of the complete-flag tRS eigenvalue problem with Frobenius action at roots of unity: partial shifts cancel in Newton–Adams combinations of the Hamiltonians, leaving the operators obtained by Frobenius action. Here we use the cohomological tCM spectral problem to construct polynomial approximations to the vertex and study their arithmetic properties.

A $K$-theoretic precedent is provided by [K21] (Proposition 2.11), where a discrete specialization of framing characters truncates a normalized complete-flag vertex to a Macdonald polynomial. These polynomials occur in the correspondence between $A$-type quiver varieties and ADHM moduli spaces. In the cohomological setting, polynomial solutions arise by specializing the coupling to negative integers. Choosing these integers to approach a fixed rational coupling $p$-adically produces a family of exact tCM eigenfunctions at varying couplings and polynomial solutions of the fixed system modulo $p^s$. After multiplication by a monomial, these terminating cohomological vertices are symmetric Jack polynomials [Mac95, NS13], the differential counterparts of the symmetric Macdonald polynomials in [K21]. We use these polynomial specializations to construct arithmetic approximations of the quasimap vertex. The spectral problem identifies the terminating vertices with Jack polynomials; our main arithmetic result compares them with the master-polynomial approximations.

A different arithmetic construction was introduced by Smirnov and Varchenko [SV23]. They express a Grassmannian vertex as a period integral of a master function, a product of rational powers of differences of auxiliary variables and Kähler parameters. Such period representations are familiar from mirror symmetry [Hori03]. The construction here uses the $3d$ mirror description of the vertex. To obtain arithmetic approximations, they replace the rational exponents by nonnegative integers approaching them $p$-adically. The integrand becomes a master polynomial. Extracting a prescribed coefficient in its auxiliary variables replaces the period integral and gives a polynomial in the Kähler parameters. These polynomials satisfy Dwork congruences and reconstruct the vertex by an infinite product. We extend this construction to complete flags and prove a sharp prime-power comparison with the regular negative-parameter staircase Jack family, uniformly in rank. The two families have the same coefficientwise limit and satisfy the same full commuting equations modulo $p^s$. Their difference has a degree-dependent valuation bound and explicit nonzero leading residues at every $p$-power scale. These formulas describe the arithmetic of the Jack coefficients in the monomial symmetric basis and distinguish the two canonical approximation procedures.

Integrability and Polynomial Truncation

Write $Q=(Q_1,\ldots,Q_{n-1})$ for the Kähler parameters (Novikov variables). The vertex is a generating series of equivariant quasimap invariants of $X_n$; its coefficient in multidegree $\boldsymbol d$ records the corresponding curve class. Quasimaps are maps from a parametrized rational curve with finitely many base points allowed. We take the vertex with trivial insertion at the identity fixed point and normalize the source-rotation weight to one. The cotangent weight is denoted by $\kappa$. After forming the complete localization sum in each degree, we specialize the additive framing weights to zero. We prove that this specialization is regular and write

for the resulting cohomological vertex. Each coefficient is a polynomial in $\kappa$ over $\mathbb Q$. The tCM equation constructs this series recursively by formal perturbation theory around the free Hamiltonian. At any fixed Kähler degree, only finitely many perturbative terms contribute. Specializing the coupling to a negative integer then gives the terminating polynomials used below; the expansion is given in § 3.2.

Let $Q_i=z_i/z_{i+1}$, $D_i=z_i\partial_{z_i}$, and set

All expressions are expanded in the chamber $Q=0$. The differential tRS limit gives $n$ commuting operators $\mathsf I_{n,k}(\kappa)$ of order $k$, with principal symbols $e_k(\xi_1,\ldots,\xi_n)$, where $e_k$ is the elementary symmetric polynomial of degree $k$. The first is total momentum; on functions of the ratios $Q_i$, the second is $-H_{n,\kappa}$. In vertex gauge we put

The vertex-gauge equation $\mathcal L_{n,\kappa}V=0$, where $\mathcal L_{n,\kappa}=J_{n,\kappa}^{-1}H_{n,\kappa}J_{n,\kappa}$, has a unique normalized formal solution in characteristic zero. We shall identify it with both the geometric vertex and the triangular chamber period. We write $P_\lambda^{(\alpha)}$ for the monic Jack polynomial with parameter $\alpha=1/\kappa$, and set $G_n(z)=\prod_{i=1}^n z_i^{i-1}$.

Equivalently, the vertex is the coefficientwise $p$-adic limit

Both the partition and the Jack polynomial parameter vary with $s$; the normalization makes their limit a series in the fixed Kähler variables. The exact spectral equations at coupling $-A_s$ and the constant coefficient one uniquely determine this polynomial family in characteristic zero. The specialization differs from the framing specialization in [K21]; the common principle is to obtain polynomial eigenfunctions from an enumerative series. For $n=2$ this is the termination of ${}_2F_1(-m,-m;1;Q)$. In higher rank polynomiality follows from the triangular constant term and bounded external flows.

Comparison of the Two Families

The master-polynomial family is constructed from a triangular array with $i$ auxiliary variables in row $i$, for $1\leq i<n$, and the external variables $z_1,\ldots,z_n$ in the final row. The master function multiplies a factor $x^{\omega-1}$ for each auxiliary variable $x$, a factor $(x-y)^{2\omega}$ for each pair in the same row, and a factor $(x-y)^{-\omega}$ for each pair in consecutive rows. A normalized integral over interlacing contours represents the vertex.

Following [SV23], we replace these exponents by nonnegative integers depending on $p^s$ and approaching the original exponents $p$-adically. The resulting master polynomial has integral coefficients. Extracting a prescribed coefficient in its auxiliary variables and removing an external monomial gives $T_s^{(n)}$, normalized by $T_s^{(n)}(0)=1$ and $T_0^{(n)}=1$. Definition 2.4 specifies the exponents and coefficient extraction. The two constructions approximate the same vertex, but their polynomials need not coincide. Define their difference by

For a rational number $c\ne0$, let $v_p(c)$ be the exponent of $p$ in $c$ and set $|c|_p=p^{-v_p(c)}$, with $|0|_p=0$ and $v_p(0)=+\infty$. The field $\mathbb Q_p$ is the completion of $\mathbb Q$ for this norm, and $\mathbb Z_p=\{c\in\mathbb Q_p:|c|_p\leq1\}$ is its ring of integers. On $\mathbb Z_p[[Q]]$ the coefficient norm is $\|\sum c_{\boldsymbol e}Q^{\boldsymbol e}\|_p =\sup_{\boldsymbol e}|c_{\boldsymbol e}|_p$. A congruence modulo $p^s$ means divisibility of every coefficient. Coefficientwise convergence is weaker than convergence in this norm.