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Stringy invariants for abelian character varieties

This paper computes all stringy invariants for $G$ -character varieties of free abelian groups and $G$ -Higgs bundle moduli spaces on abelian varieties, providing a direct proof of topological mirror symmetry for Langlands dual groups and establishing that symplectic resolutions of these moduli spaces exist only for groups of Dynkin types $A$ , $B$ , and $C$ .

Original authors: Carlos Florentino, Ángel González-Prieto, Alfonso Zamora

Published 2026-09-15

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CC BY 4.0

Original authors: Carlos Florentino, Ángel González-Prieto, Alfonso Zamora

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern mathematics, there is a field dedicated to understanding the shapes formed by solutions to complex equations. These shapes, known as varieties, can be smooth and simple, or they can be twisted and riddled with sharp points and tears. When these shapes arise from the study of symmetry groups—collections of transformations that leave certain structures unchanged—they become central to understanding everything from the behavior of subatomic particles to the topology of knots. One particularly rich area of study involves "character varieties," which are spaces that organize all the ways a group can act on a specific type of geometric object called an abelian variety, a shape that generalizes the familiar torus or doughnut. These spaces are notoriously difficult to analyze because they often possess singularities, points where the geometry breaks down and standard tools fail to provide a clear picture. To navigate these rough spots, mathematicians have developed a specialized set of tools called "stringy invariants." Think of these invariants as a sophisticated way of counting the holes and twists in a shape, even when that shape is crumpled or broken, by borrowing concepts from string theory to smooth out the rough edges mathematically.

The researchers Carlos Florentino, Ángel González-Prieto, and Alfonso Zamora have tackled the problem of calculating these invariants for a specific and important class of these spaces: those associated with free abelian groups and moduli spaces of Higgs bundles on abelian varieties. Their work provides a complete and explicit formula for these counts, allowing them to map the geometry of these complex spaces with unprecedented precision. Instead of relying on vague approximations, they derived a method that works for any connected reductive group, a broad category of symmetry groups that includes the classical groups used in physics as well as the rare and complex "exceptional" groups. By breaking down the problem into the fundamental building blocks of these groups—their root systems and the way their symmetries permute them—they were able to compute the exact number of stringy invariants for groups of various ranks, including the exceptional groups of rank up to four.

One of the most striking outcomes of their calculations is a direct proof of a phenomenon known as topological mirror symmetry. This concept suggests that two seemingly different mathematical worlds, defined by groups that are "Langlands dual" to one another, actually share the same underlying topological structure when viewed through the lens of these stringy invariants. While this equality had been suspected and partially verified in simpler cases, this paper demonstrates it directly and comprehensively for all stringy invariants, not just a single simplified version. The authors show that the intricate details of the geometry for one group are perfectly mirrored in its dual, confirming a deep and elegant connection between these mathematical structures.

However, the paper also draws a firm line in the sand regarding the existence of "symplectic resolutions" for these spaces. A symplectic resolution is a way of replacing a singular, broken shape with a smooth, perfect one that retains the essential geometric properties of the original. The authors prove that such a smooth replacement can only exist if the underlying symmetry group belongs to specific families, known as Dynkin types A, B, or C. For the exceptional groups and other types, no such smooth resolution exists. This is a crucial finding because it means that for these groups, the stringy invariants are not just counting the features of a hidden smooth shape; they are capturing the true, "virtual" geometry of the singular space itself. The invariants reveal a subtle reality that cannot be smoothed away, highlighting the unique and irreducible complexity of these exceptional cases.

To reach these conclusions, the team developed a highly efficient algorithm that they implemented in a symbolic algebra system. This allowed them to compute the invariants for groups of rank up to four in less than a minute, a task that would have been impossible by hand. They tested their formulas on specific examples, such as the groups SO7 and G2, and found that the resulting numbers were always positive, a property that aligns with a famous conjecture by Batyrev regarding the positivity of these invariants. While they could not prove this positivity for every possible case, their extensive computations provide strong evidence for it, particularly for the groups associated with elliptic curves. They also discovered that the resulting polynomials describing these spaces possess a beautiful symmetry, reading the same forwards and backwards, which further underscores the deep order hidden within these complex geometric structures.

The work stands as a significant step forward in understanding the geometry of character varieties. By providing explicit formulas and proving the existence of mirror symmetry across all stringy invariants, the authors have turned a murky area of research into a clear, calculable landscape. Their results confirm that while some of these spaces can be smoothed out, others possess an intrinsic, unresolvable complexity that is faithfully recorded by their stringy invariants. This distinction not only clarifies the nature of these mathematical objects but also reinforces the power of mirror symmetry as a guiding principle in modern geometry, showing that even in the most singular and broken shapes, there is a profound and symmetric order waiting to be discovered.

Technical Summary: Stringy Invariants for Abelian Character Varieties

Problem Statement
The paper investigates the geometry and topology of character varieties associated with free abelian groups and moduli spaces of Higgs bundles on abelian varieties. Specifically, given a complex connected reductive group $G$ with maximal torus $T$ and Weyl group $W$ , the authors study the quotient variety $T^r/W$ , where $W$ acts diagonally. This variety serves as the normalization of the identity component of the $G$ -character variety $X_r(G) = \text{Hom}(\mathbb{Z}^r, G) // G$ . While the structure of $T^r/W$ appears simple, its singularities are intricate, particularly for $r > 2$ . The primary goal is to compute all stringy invariants (specifically stringy mixed Hodge polynomials) for these spaces for arbitrary $r$ and all complex connected reductive groups $G$ , including exceptional types. These invariants encode orbifold cohomology numbers and are crucial for understanding the geometry of singular symplectic varieties and testing mirror symmetry conjectures.

Methodology
The authors employ a combination of Geometric Invariant Theory (GIT), representation theory of Weyl groups, and Chen–Ruan orbifold cohomology.

Orbifold Cohomology Framework: The study utilizes Chen–Ruan cohomology $H^\bullet_{CR}(T^r/W)$ , which decomposes into twisted sectors indexed by conjugacy classes of $W$ . The cohomology of each sector is shifted by a "fermionic shift" (or age) determined by the codimension of the fixed locus $(T^r)^w$ .
Explicit Combinatorial Formulas: The core methodological contribution is the derivation of explicit closed-formulas for the stringy mixed Hodge polynomial $\mu_{str}(T^r/W)$ $μ_{s t r} (T^{r} / W)$ . This involves:
- Decomposing the cokernel of the map $I_\Lambda - w$ (where $\Lambda$ is the character lattice) into free ( $\Phi_w$ ) and torsion ( $D_w$ ) parts.
- Analyzing the action of the centralizer $C(w)$ on these parts via induced morphisms $\phi_w(g)$ and $\tau_w(g)$ .
- Calculating the cardinality of cokernels $n_w(g)$ and determinants of linear maps involving the free part.
Symplectic Resolution Analysis: The paper investigates the existence of symplectic resolutions for these moduli spaces using the Ginzburg–Kaledin local resolution framework. It relates the existence of such resolutions to the Dynkin type of the group $G$ .
Computational Implementation: The theoretical formulas are implemented in SageMath to compute explicit polynomials for classical and exceptional groups (e.g., $SO_7$ , $G_2$ ) up to rank 4.

Key Contributions and Results

Explicit Formula for Stringy Invariants (Theorem A): The authors provide a general formula for the stringy mixed Hodge polynomial of $T^r/W$ purely in terms of the action of $W$ on the character lattice $\Lambda$ . The formula sums over conjugacy classes $[w] \in \text{Conj}(W)$ and involves terms related to the torsion and free parts of the cokernel of $I_\Lambda - w$ .
$\mu_{str}(T^r/W)(t, u, v) = \sum_{[w]} \frac{(t^2uv)^{\frac{r}{2}\text{rk Im}(I_\Lambda-w)}}{|C(w)|} \left( \sum_{g \in C(w)} n_w(g)^r \det(I_{\Phi_w} + tuv \phi_w(g))^r \right)$
An analogous formula is derived for the moduli space of $G$ -Higgs bundles on an abelian variety of dimension $d$ (Theorem B).
Direct Proof of Topological Mirror Symmetry (Theorem 6.1): Using the explicit formulas, the paper provides a direct proof that the stringy mixed Hodge polynomials for a group $G$ and its Langlands dual $L G$ are identical. This extends previous results by Thaddeus, which were limited to the stringy $E$ -polynomial, to the full mixed Hodge structure. The proof relies on the duality of the character and cocharacter lattices, showing that the relevant determinants and cokernel cardinalities are preserved under the transpose map in the dual representation.
Non-Existence of Symplectic Resolutions (Theorem 4.5): The paper establishes that symplectic resolutions for the quotient $T^{2d}/W$ exist only in specific cases.
- For $d \ge 2$ , no symplectic resolution exists for any group with positive semisimple rank.
- For $d=1$ (elliptic curves), a symplectic resolution exists only if the Dynkin diagram of $G$ consists of components of type $A$ , $B$ , or $C$ . No such resolution exists for groups of type $D$ or exceptional groups.
- Consequently, for exceptional groups, the computed stringy invariants represent "virtual" Hodge numbers rather than the cohomology of a smooth resolution.
Positivity and Palindromic Properties:
- The authors propose a Rank 2 Character Variety Positivity Conjecture (Conjecture 8.1), stating that the stringy $E$ -polynomial of $T^2/W$ has non-negative coefficients for any connected semisimple group $G$ . This is verified for all computed examples (including exceptional groups) but fails for non-semisimple groups (e.g., $GL_2$ ) or higher rank abelian varieties (e.g., $SL_3$ on an abelian surface).
- Proposition 8.4 proves that the stringy $E$ -polynomial is always palindromic in the variable $q=uv$ for any reductive group and any dimension $d \ge 1$ .

Significance
The paper claims significance in several areas:

Computational Accessibility: It provides a completely explicit, algorithmic method to compute stringy invariants for arbitrary reductive groups, including exceptional ones, which were previously difficult to access.
Mirror Symmetry: It offers a direct, Lie-theoretic proof of topological mirror symmetry for abelian character varieties, bypassing the need for identifying specific module structures used in earlier proofs.
Geometry of Singularities: By clarifying the conditions under which symplectic resolutions exist (only for types $A, B, C$ in the elliptic case), the paper highlights the "exceptionality" of the positivity and palindromic properties of stringy invariants for other groups. It demonstrates that these properties are intrinsic to the orbifold structure and not merely artifacts of an underlying smooth symplectic resolution.
Batyrev's Conjecture: The work provides strong empirical evidence for a modified version of Batyrev's positivity conjecture in the context of non-projective quotient singularities, while also delineating the sharp boundaries of its validity (failing for non-semisimple groups and higher-dimensional abelian varieties).

The authors emphasize that their results apply verbatim to moduli spaces of Higgs bundles on abelian varieties, extending the scope of these invariants to the Dolbeault side of the non-abelian Hodge correspondence.

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