arXiv subjects

Jim Bryan

Publications and source records attributed to Jim Bryan.

At least 19 recordsLinked to original sources

arXiv2026

Gopakumar-Vafa Invariants for Local Calabi-Yau Orbifolds of $A_{N}$-type

Let $\mathcal{X}$ be a local orbifold Calabi-Yau threefold whose coarse space $X$ has transverse $A_{N}$ singularities along a smooth non-compact curve. We define orbifold Gopakumar-Vafa invariants, and we prove they are integers and satisfy a finiteness property. We compute our invariants for local orbifold $K3$ surfaces, where we prove an orbifold version of the classical Yau-Zaslow formula: we show that for an orbifold $K3$ surface $\mathcal S$, the genus zero Gopakumar-Vafa invariant of $\mathcal{S}$ in a class of square $2n$ is the Euler characteristic of the Hilbert scheme of $n+1$ points on the singular surface $S$.

math.AG↗

arXiv2026

Based maps to Lagrangian Grassmannians, Quivers, and Bott Periodicity

We give a quiver description of the space of based algebraic maps from $\mathbb{P}^{1}$ to the Lagrangian Grassmannian (and its orthogonal counterpart). We show our descriptions lead to an algebro-geometric refinement of some of the homotopy equivalences in real Bott periodicity. In particular, we get an isomorphism in Larson and Vakil's ``naive algebro-geometric homotopy category'' whose topological realization (after specializing to $\mathbb{C}$) recovers the classical homotopy equivalences $\Omega^{2}(Sp/U) \simeq BO\times \mathbb{Z}$ and $\Omega^{2}(O/U) \simeq BSp\times \mathbb{Z}$.

math.AG↗

arXiv2026

The motivic class of the space of genus $0$ maps to the flag variety

Let $\operatorname{Fl}_{n+1}$ be the variety of complete flags in $\mathbb{A}^{n+1}$ and let $\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})$ be the space of based maps $f:\mathbb{P}^{1}\to \operatorname{Fl}_{n+1}$ in the class $f_{*}[\mathbb{P}^{1}]=\beta$. We show that under a mild positivity condition on $\beta$, the class of $\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})$ in $K_{0}(\operatorname{Var})$, the Grothendieck group of varieties, is given by \[ [\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})] = [\operatorname{GL}_{n}\times \mathbb{A}^{a}]. \] The proof of this result was obtained in conjunction with Google Gemini and related tools. We briefly discuss this research interaction, which may be of independent interest. However, the treatment in this paper is entirely human-authored (aside from excerpts in an appendix which are clearly marked as such).

math.AG↗

arXiv2025

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

cs.CL↗

arXiv2024

The Enumerative Geometry and Arithmetic of Banana Nano-Manifolds

A banana manifold is a Calabi-Yau threefold fibered by Abelian surfaces whose singular fibers contain banana configurations: three rational curves meeting each other in two points. A nano-manifold is a Calabi-Yau threefold $X$ with very small Hodge numbers: $h^{1,1}(X)+h^{2,1}(X)\leq 6$. We construct four rigid banana nano-manifolds $\tilde{X}_N$, $N\in \{5,6,8,9 \}$, each with Hodge numbers given by $(h^{1,1},h^{2,1})=(4,0)$. We compute the Donaldson-Thomas partition function for banana curve classes and show that the associated genus $g$ Gromov-Witten potential is a genus 2 meromorphic Siegel modular form of weight $2g-2$ for a certain discrete subgroup $P^{*}_{N} \subset Sp_{4}(\mathbb{R})$. We also compute the weight 4 modular form whose $p$th Fourier coefficient is given by the trace of the action of Frobenius on $H^{3}_{et }(\tilde{X}_N ,{\mathbb{Q}}_{l})$ for almost all prime $p$. We observe that it is the unique weight 4 cusp form on $\Gamma_{0}(N)$.

math.AG↗

arXiv2022

Counting Invariant Curves: a theory of Gopakumar-Vafa invariants for Calabi-Yau threefolds with an involution

We develop a theory of Gopakumar-Vafa (GV) invariants for a Calabi-Yau threefold (CY3) $X$ which is equipped with an involution $\imath$ preserving the holomorphic volume form. We define integers $n_{g,h}(\beta) $ which give a virtual count of the number of genus $g$ curves $C$ on $X$, in the class $\beta \in H_{2}(X)$, which are invariant under $\imath$, and whose quotient $C/\imath$ has genus $h$. We give two definitions of $n_{g,h}(\beta) $ which we conjecture to be equivalent: one in terms of a version of Pandharipande-Thomas theory and one in terms of a version of Maulik-Toda theory. We compute our invariants and give evidence for our conjecture in several cases. In particular, we compute our invariants when $X=S\times \mathbb{C}$ where $S$ is an Abelian surface with $\imath (a)=-a$ or a $K3$ surface with a symplectic involution (a Nikulin $K3$ surface). For these cases, we give formulas for our invariants in terms of Jacobi modular forms. For the Abelian surface case, the specialization of our invariants $n_{g,h}(\beta) $ to $h=0$ recovers the count of hyperelliptic curves on an Abelian surface first computed by Bryan-Oberdieck-Pandharipande-Yin.

math.AG↗

arXiv2019

$G$-fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta products

Let $X$ be a complex $K3$ surface with an effective action of a group $G$ which preserves the holomorphic symplectic form. Let $$ Z_{X,G}(q) = \sum_{n=0}^{\infty} e\left(\operatorname{Hilb}^{n}(X)^{G} \right)\, q^{n-1} $$ be the generating function for the Euler characteristics of the Hilbert schemes of $G$-invariant length $n$ subschemes. We show that its reciprocal, $Z_{X,G}(q)^{-1}$ is the Fourier expansion of a modular cusp form of weight $\frac{1}{2} e(X/G)$ for the congruence subgroup $\Gamma_{0}(|G|)$. We give an explicit formula for $Z_{X,G}$ in terms of the Dedekind eta function for all 82 possible $(X,G)$. The key intermediate result we prove is of independent interest: it establishes an eta product identity for a certain shifted theta function of the root lattice of a simply laced root system. We extend our results to various refinements of the Euler characteristic, namely the Elliptic genus, the Chi-$y$ genus, and the motivic class.

math.AG↗

arXiv2019

The Donaldson-Thomas partition function of the banana manifold

A banana manifold is a compact Calabi-Yau threefold, fibered by Abelian surfaces, whose singular fibers have a singular locus given by a "banana configuration of curves". A basic example is given by $X_{ban}$, the blowup along the diagonal of the fibered product of a generic rational elliptic surface $S\to \mathbb{P}^{1}$ with itself. In this paper we give a closed formula for the Donaldson-Thomas partition function of the banana manifold $X_{ban }$ restricted to the 3-dimensional lattice $\Gamma$ of curve classes supported in the fibers of $X_{ban}\to \mathbb{P}^{1}$. It is given by \[ Z_{\Gamma}(X_{ban}) = \prod_{d_{1},d_{2},d_{3}\geq 0} \prod_{k} \left(1-p^{k}Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}}\right)^{-12c(||\mathbf{d} ||,k)} \] where $||\mathbf{d} || = 2d_{1}d_{2}+ 2d_{2}d_{3}+ 2d_{3}d_{1}-d_{1}^{2}-d_{2}^{2}-d_{3}^{2}$, and the coefficients $c(a,k)$ have a generating function given by an explicit ratio of theta functions. This formula has interesting properties and is closely realated to the equivariant elliptic genera of $\operatorname{Hilb} (\mathbb{C}^{2})$. In an appendix with S. Pietromonaco, it is shown that the corresponding genus $g$ Gromov-Witten potential $F_{g}$ is a genus 2 Siegel modular form of weight $2g-2$ for $g\geq 2$; namely it is the Skoruppa-Maass lift of a multiple of an Eisenstein series: $\frac{6|B_{2g}|}{g(2g-2)!} E_{2g}(\tau )$.

math.AG↗

arXiv2018

CHL Calabi-Yau threefolds: Curve counting, Mathieu moonshine and Siegel modular forms

A CHL model is the quotient of $\mathrm{K3} \times E$ by an order $N$ automorphism which acts symplectically on the K3 surface and acts by shifting by an $N$-torsion point on the elliptic curve $E$. We conjecture that the primitive Donaldson-Thomas partition function of elliptic CHL models is a Siegel modular form, namely the Borcherds lift of the corresponding twisted-twined elliptic genera which appear in Mathieu moonshine. The conjecture matches predictions of string theory by David, Jatkar and Sen. We use the topological vertex to prove several base cases of the conjecture. Via a degeneration to $\mathrm{K3} \times \mathbb{P}^1$ we also express the DT partition functions as a twisted trace of an operator on Fock space. This yields further computational evidence. An extension of the conjecture to non-geometric CHL models is discussed. We consider CHL models of order $N=2$ in detail. We conjecture a formula for the Donaldson-Thomas invariants of all order two CHL models in all curve classes. The conjecture is formulated in terms of two Siegel modular forms. One of them, a Siegel form for the Iwahori subgroup, has to our knowledge not yet appeared in physics. This discrepancy is discussed in an appendix with Sheldon Katz.

math.AG↗

arXiv2018

Locally Maximally Entangled States of Multipart Quantum Systems

For a multipart quantum system, a locally maximally entangled (LME) state is one where each elementary subsystem is maximally entangled with its complement. This paper is a sequel to arXiv:1708.01645, which gives necessary and sufficient conditions for a system to admit LME states in terms of its subsystem dimensions $(d_1, d_2, \dots, d_n)$, and computes the dimension of the space ${\cal H}_{LME}/K$ of LME states up to local unitary transformations for all non-empty cases. In this paper, we provide a pedagogical overview and physical interpretation of the the underlying mathematics that leads to these results and give a large class of explicit constructions for LME states. In particular, we construct all LME states for tripartite systems with subsystem dimensions $(2,A,B)$ and give a general representation-theoretic construction for a special class of stabilizer LME states. The latter construction provides a common framework for many known LME states. Our results also give the dimension of the space of SLOCC equivalence classes for states with "generic" entanglement for all multipart systems since this space is equivalent to ${\cal H}_{LME}/K$. Finally, we give the dimension of the stabilizer subgroup $S \subset SL(d_1, \mathbb{C}) \times \cdots \times SL(d_n, \mathbb{C})$ for a generic state in an arbitrary multipart system and identify all cases where this stabilizer is trivial.

quant-ph↗

arXiv2017

Existence of locally maximally entangled quantum states via geometric invariant theory

We study a question which has natural interpretations in both quantum mechanics and in geometry. Let $V_1,..., V_n$ be complex vector spaces of dimension $d_1,...,d_n$ and let $G= SL_{d_1} \times \dots \times SL_{d_n}$. Geometrically, we ask given $(d_1,...,d_n)$, when is the geometric invariant theory quotient $\mathbb{P}(V_1 \otimes \dots \otimes V_n)// G$ non-empty? This is equivalent to the quantum mechanical question of whether the multipart quantum system with Hilbert space $V_1\otimes \dots \otimes V_n$ has a locally maximally entangled state, i.e. a state such that the density matrix for each elementary subsystem is a multiple of the identity. We show that the answer to this question is yes if and only if $R(d_1,...,d_n)\geqslant 0$ where \[ R(d_1,...,d_n) = \prod_i d_i +\sum_{k=1}^n (-1)^k \sum_{1\leq i_1<\dotsb <i_k\leq n} (\gcd(d_{i_1},\dotsc ,d_{i_k}) )^{2}. \] We also provide a simple recursive algorithm which determines the answer to the question, and we compute the dimension of the resulting quotient in the non-empty cases.

math.AG↗

arXiv2016

Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex

We compute the Donaldson-Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in arXiv:1603.05271, we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the Katz-Klemm-Vafa formula for primitive curve classes which is independent of the computation of Kawai-Yoshioka.

math.AG↗

arXiv2016

Trace Identities for the Topological Vertex

The topological vertex is a universal series which can be regarded as an object in combinatorics, representation theory, geometry, or physics. It encodes the combinatorics of 3D partitions, the action of vertex operators on Fock space, the Donaldson-Thomas theory of toric Calabi-Yau threefolds, or the open string partition function of $\mathbb{C}^3$. We prove several identities in which a sum over terms involving the topological vertex is expressed as a closed formula, often a product of simple terms, closely related to Fourier expansions of Jacobi forms. We use purely combinatorial and representation theoretic methods to prove our formulas, but we discuss applications to the Donaldson-Thomas invariants of elliptically fibered Calabi-Yau threefolds at the end of the paper.

math.CO↗

arXiv2015

Curve counting on abelian surfaces and threefolds

We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete results in all genera for primitive classes. The generating series are quasimodular forms of pure weight. Conjectures for imprimitive classes are presented. In genus 2, the counts in all classes are proven. Special counts match the Euler characteristic calculations of the moduli spaces of stable pairs on abelian surfaces by G\"ottsche-Shende. A formula for hyperelliptic curve counting in terms of Jacobi forms is proven (modulo a transversality statement). For abelian threefolds, complete conjectures in terms of Jacobi forms for the generating series of curve counts in primitive classes are presented. The base cases make connections to classical lattice counts of Debarre, Goettsche, and Lange-Sernesi. Further evidence is provided by Donaldson-Thomas partition function computations for abelian threefolds. A multiple cover structure is presented. The abelian threefold conjectures open a new direction in the subject.

math.AG↗

arXiv2015

The Donaldson-Thomas theory of $K3\times E$ via the topological vertex

Oberdieck and Pandharipande conjectured that the curve counting invariants of $S\times E$, the product of a $K3$ surface and an elliptic curve, is given by minus the reciprocal of the Igusa cusp form of weight 10. For a fixed primitive curve class in $S$ of square $2h-2$, their conjecture predicts that the corresponding partition functions are given by meromorphic Jacobi forms of weight $-10$ and index $h-1$. We calculate the partition functions for primitive classes of square -2 and of square 0. Our computation uses reduced Donaldson-Thomas invariants which are defined as the Behrend function weighted Euler characteristics of the quotient of the Hilbert scheme of curves in $S\times E$ by the action of $E$. Our technique is a mixture of motivic and toric methods (developed with Martijn Kool) which allows us to express the partition functions in terms of the topological vertex and subsequently in terms of Jacobi forms. We compute the partition functions for both Behrend function weighted Euler characteristics and for unweighted Euler characteristics. The results for the Behrend function weighted case depends on Conjecture 18 from https://arxiv.org/abs/1608.07369

math.AG↗

arXiv2012

Curve-counting invariants for crepant resolutions

We construct curve counting invariants for a Calabi-Yau threefold $Y$ equipped with a dominant birational morphism $\pi:Y \to X$. Our invariants generalize the stable pair invariants of Pandharipande and Thomas which occur for the case when $\pi:Y\to Y$ is the identity. Our main result is a PT/DT-type formula relating the partition function of our invariants to the Donaldson-Thomas partition function in the case when $Y$ is a crepant resolution of $X$, the coarse space of a Calabi-Yau orbifold $\mathcal{X}$ satisfying the hard Lefschetz condition. In this case, our partition function is equal to the Pandharipande-Thomas partition function of the orbifold $\mathcal{X}$. Our methods include defining a new notion of stability for sheaves which depends on the morphism $\pi $. Our notion generalizes slope stability which is recovered in the case where $\pi $ is the identity on $Y$. Our proof is a generalization of Bridgeland's proof of the PT/DT correspondence via the Hall algebra and Joyce's integration map.

math.AG↗

arXiv2012

Motivic Classes of Commuting Varieties via Power Structures

We prove a formula, originally due to Feit and Fine, for the class of the commuting variety in the Grothendieck group of varieties. Our method, which uses a power structure on the Grothendieck group of stacks, allows us to prove several refinements and generalizations of the Feit-Fine formula. Our main application is to motivic Donaldson-Thomas theory.

math.AG↗

arXiv2010

The Orbifold Topological Vertex

We define Donaldson-Thomas invariants of Calabi-Yau orbifolds and we develop a topological vertex formalism for computing them. The basic combinatorial object is the orbifold vertex, a generating function for the number of 3D partitions asymptotic to three given 2D partitions and colored by representations of a finite Abelian group G acting on C^3. In the case where G=Z_n acting on C^3 with transverse A_{n-1} quotient singularities, we give an explicit formula for the vertex in terms of Schur functions. We discuss applications of our formalism to the Donaldson-Thomas Crepant Resolution Conjecture and to the orbifold Donaldson-Thomas/Gromov-Witten correspondence. We also explicitly compute the Donaldson-Thomas partition function for some simple orbifold geometries: the local football and the local BZ_2 gerbe.

math.AG↗