Strange Duality Research Articles

Strange duality at level one for alternating vector bundles

In this paper, we show a strange duality isomorphism at level one for the space of generalized theta functions on the moduli spaces of alternating anti-invariant vector bundles in the ramified case. These anti-invariant vector bundles constitute one of the non-trivial examples of parahoric G-torsors, where G is a twisted (not generically split) parahoric group scheme.

Rigid local systems and the multiplicative eigenvalue problem

We give a construction that produces irreducible complex rigid local systems on $\mathbb{P}_C^1-\{p_1,\ldots ,p_s\}$ via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups $\mathrm{SU}(n)$ (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals of the simplest vertices of these polytopes give all possible unitary irreducible rigid local systems. As a consequence we obtain that the ranks of unitary irreducible rigid local systems, including those with finite global monodromy, on $\mathbb{P}^1-S$ are bounded above if we fix the cardinality of the set $S=\{p_1,\ldots ,p_s\}$ and require that the local monodromies have orders that divide $n$ for a fixed $n$. Answering a question of N.~Katz, we show that there are no irreducible rigid local systems of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing $n$, when $n$ is a prime number. We also show that all unitary irreducible rigid local systems on $\mathbb{P}_C^1-S$ with finite local monodromies arise as solutions to the Knizhnik-Zamalodchikov equations on conformal blocks for the special linear group. Along the way, generalizing previous works of the author and J.~Kiers, we give an inductive mechanism for determining all vertices in the multiplicative eigenvalue problem for $\mathrm{SU}(n)$.

Rank-one sheaves and stable pairs on surfaces

Finite Quot schemes were used by Bertram, Johnson, and the first author to study Le Potier's strange duality conjecture on del Pezzo surfaces when one of the moduli spaces is the Hilbert scheme of points. In order to rigorously enumerate the finite Quot scheme, we study the moduli space of limit stable pairs in which the target has rank one on a smooth complex projective surface. We obtain an embedding of this moduli space into a smooth space that induces a perfect obstruction theory. This obstruction theory yields a virtual fundamental class that can be computed explicitly. Because the moduli space coincides with the Quot scheme when they have dimension 0, this makes the desired count of the finite Quot scheme in Bertram et al. rigorous. As another application, we obtain a universality result for tautological integrals on the moduli space of stable pairs.

Moduli spaces of stable sheaves over quasi-polarized surfaces, and the relative Strange Duality morphism

The main result of the present paper is a construction of relative moduli spaces of stable sheaves over the stack of quasipolarized projective surfaces. For this, we use the theory of good moduli spaces, whose study was initiated by Alper. As a corollary, we extend the relative Strange Duality morphism to the locus of quasipolarized K3 surfaces.

An Affine Approach to Peterson Comparison

The Peterson comparison formula proved by Woodward relates the three-pointed Gromov-Witten invariants for the quantum cohomology of partial flag varieties to those for the complete flag. Another such comparison can be obtained by composing a combinatorial version of the Peterson isomorphism with a result of Lapointe and Morse relating quantum Littlewood-Richardson coefficients for the Grassmannian to k-Schur analogs in the homology of the affine Grassmannian obtained by adding rim hooks. We show that these comparisons on quantum cohomology are equivalent, up to Postnikov’s strange duality isomorphism.

Strange Duality Between the Quadrangle Complete Intersection Singularities

There is a strange duality between the quadrangle isolated complete intersection singularities discovered by the first author and Wall. We derive this duality from a variation of the Berglund–Hübsch transposition of invertible polynomials introduced in our previous work about the strange duality between hypersurface and complete intersection singularities using matrix factorizations of size two.

Today Notes and Re-Reading ‘Upon a Common Language Related to Interdisciplinarity and Knowledge Transfer’

The today heroic RESPONSIBILITY within the global and dramatic fight on the pandemic Coronavirus is constituted by risky and uncertain awareness, insights, knowledge, and action nearly done by ourselves, communities of human beings, within an also heroic but sometime strange duality between EVOLUTION (active cycles of ‘resources-objectives’ – dynamics – non-linear resilience) and SECURITY (‘homeostasis’ – stability – conservative resilience). Parallel to the crucial taken up actions, the probabilistic, fuzzy and subtle basis of the operative events (with highly unknown elements) will evolve within a step by step larger information/knowledge (as a newly comprehended ratio and respectively resulting insight and knowledge). But, as the XXI century would contain and must sensitively prevail, there may be innovatively contemporary cognitive items within expected inter/trans/co-disciplinary approaches; also, it may be elicited a new kind of intercultural multi-open society. There are mostly non-systemic tasks on.The above “must sensitively prevail” is possible received today as a pattern of our humankind civilization; it is and exists inward our contemporary heritage, challenge, and perspectives within our multivalent legacies and multi-orientated futures. The proposed today re-reading on the 1966 study.

Strange duality on [formula omitted] via quiver representations

We study Le Potier's strange duality conjecture on P2. We focus on the strange duality map SDcnr,d which involves the moduli space of rank r sheaves with trivial first Chern class and second Chern class n, and the moduli space of 1-dimensional sheaves with determinant OP2(d) and Euler characteristic 0. By using tools in quiver representation theory, we show that SDcnr,d is an isomorphism for r=n or r=n−1 or d≤3, and in general SDcnr,d is injective for any n≥r>0 and d>0.

Strange duality on rational surfaces

We study Le Potier’s strange duality conjecture on a rational surface. We focus on the case involving the moduli space of rank $2$ sheaves with trivial first Chern class and second Chern class $2$, and the moduli space of $1$-dimensional sheaves with determinant $L$ and Euler characteristic $0$. We show the conjecture for this case is true under some suitable conditions on $L$, which applies to $L$ ample on any Hirzebruch surface $\Sigma_e := \mathbb{P} (\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1} (e))$ except for $e = 1$. When $e = 1$, our result applies to $L = aG + bF$ with $b \geq a + [a/2]$, where $F$ is the fiber class, $G$ is the section class with $G^2 = -1$ and $[a/2]$ is the integral part of $a/2$.

Verlinde-type formulas for rational surfaces

K -theoretic Donaldson invariants are holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 sheaves on surfaces. We develop an algorithm which determines the generating functions of the K -theoretic Donaldson invariants on the projective plane and more general rational surfaces, and apply it in many cases to get explicit formulas. We relate the results to Le Potier’s strange duality conjecture.

O’Grady’s birational maps and strange duality via wall-hitting

We prove that O’Grady’s birational maps [K. G O’Grady, The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface, J. Algebr. Geom. 6(4) (1997) 599–644] between moduli of sheaves on an elliptic K3 surface can be interpreted as intermediate wall-crossing (wall-hitting) transformations at so-called totally semistable walls, studied by Bayer and Macrì [A. Bayer and E. Macrì, MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, Inventiones Mathematicae 198(3) (2014) 505–590]. As a key ingredient, we describe the first totally semistable wall for ideal sheaves of [Formula: see text] points on the elliptic [Formula: see text]. As an application, we give new examples of strange duality isomorphisms, based on a result of Marian and Oprea [A. Marian and D. Oprea, Generic strange duality for K3 surfaces, with an appendix by Kota Yoshioka, Duke Math. J. 162(8) (2013) 1463–1501].

Generalized Theta Functions, Strange Duality, and Odd Orthogonal Bundles on Curves

This paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial Stiefel–Whitney class and the associated space of twisted spin bundles. In particular, we prove a Verlinde type formula and a dimension equality that was conjectured by Oxbury–Wilson. Modifying Hitchin’s argument, we also show that the bundle of generalized theta functions for twisted spin bundles over the moduli space of curves admits a flat projective connection. We furthermore address the issue of strange duality for odd orthogonal bundles, and we demonstrate that the naive conjecture fails in general. A consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the Hitchin connection for these moduli spaces. Finally, we answer a question of Nakanishi–Tsuchiya about rank-level duality for conformal blocks on the pointed projective line with spin weights.

Bundles of generalized theta functions over abelian surfaces

We study the Verlinde bundles of generalized theta functions constructed from moduli spaces of sheaves over abelian surfaces. In degree 0, the splitting type of these bundles is expressed in terms of indecomposable semihomogeneous factors. Furthermore, Fourier–Mukai symmetries of the Verlinde bundles are found consistently with strange duality. Along the way, a transformation formula for the theta bundles is derived, extending a theorem of Drézet–Narasimhan from curves to abelian surfaces.

A note on strange duality for holomorphic triples on a projective line

We propose a strange duality conjecture for holomorphic triples $$(E_{2}\rightarrow E_{1})$$ on $$\mathbb {P}^{1}$$ . As an affirmative evidence, we prove the conjecture in the case $$\mathrm {rank}\, E_{1}=1$$ .

Strange Duality on Rational Surfaces II: Higher-Rank Cases

Abstract We study Le Potier’s strange duality conjecture on a rational surface. We focus on the strange duality map $SD_{c_n^r,L}$ that involves the moduli space of rank $r$ sheaves with trivial 1st Chern class and 2nd Chern class $n$, and the moduli space of one-dimensional sheaves with determinant $L$ and Euler characteristic 0. We show there is an exact sequence relating the map $SD_{c_r^r,L}$ to $SD_{c^{r-1}_{r},L}$ and $SD_{c_r^r,L\otimes K_X}$ for all $r\geq 1$ under some conditions on $X$ and $L$ that applies to a large number of cases on $\mathbb{P}^2$ or Hirzebruch surfaces. Also on $\mathbb{P}^2$ we show that for any $r>0$, $SD_{c^r_r,dH}$ is an isomorphism for $d=1,2$, injective for $d=3,$ and moreover $SD_{c_3^3,rH}$ and $SD_{c_3^2,rH}$ are injective. At the end we prove that the map $SD_{c_n^2,L}$ ($n\geq 2$) is an isomorphism for $X=\mathbb{P}^2$ or Fano rational-ruled surfaces and $g_L=3$, and hence so is $SD_{c_3^3,L}$ as a corollary of our main result.

Universal Series for Hilbert Schemes and Strange Duality

Abstract We show how the “finite Quot scheme method” applied to Le Potier’s strange duality on del Pezzo surfaces leads to conjectures (valid for all smooth complex projective surfaces) relating two sets of universal power series on Hilbert schemes of points on surfaces: those for top Chern classes of tautological sheaves and those for Euler characteristics of line bundles. We have verified these predictions computationally for low order. We then give an analysis of these conjectures in small ranks. We also give a combinatorial proof of a formula predicted by our conjectures: the top Chern class of the tautological sheaf on $S^{[n]}$ associated to the structure sheaf of a point is equal to $(-1)^n$ times the nth Catalan number.

Moduli spaces of 1-dimensional semi-stable sheaves and strange duality on [formula omitted

We study Le Potier's strange duality conjecture on P2. We show the conjecture is true for the pair (W(2,0,2), M(d,0)) with d>0, where W(2,0,2) is the moduli space of semistable sheaves of rank 2, zero first Chern class and second Chern class 2, and M(d,0) is the moduli space of 1-dimensional semistable sheaves of first Chern class dH and Euler characteristic 0.

On the strange duality conjecture for abelian surfaces II

In the prequel to this paper, two versions of Le Potier’s strange duality conjecture for sheaves over abelian surfaces were studied. A third version is considered here. In the current setup, the isomorphism involves moduli spaces of sheaves with fixed determinant and fixed determinant of the Fourier-Mukai transform on one side, and moduli spaces where both determinants vary, on the other side. We first establish the isomorphism in rank 1 using the representation theory of Heisenberg groups. For product abelian surfaces, the isomorphism is then shown to hold for sheaves with fiber degree 1 1 via Fourier-Mukai techniques. By degeneration to product geometries, the duality is obtained generically for a large number of numerical types. Finally, it is shown in great generality that the Verlinde sheaves encoding the variation of the spaces of theta functions are locally free over moduli.

SEMISTABLE SHEAVES WITH SYMMETRIC ON A QUADRIC SURFACE

For moduli spaces of sheaves with symmetric $c_{1}$ on a quadric surface, we pursue analogy to some results known for moduli spaces of sheaves on a projective plane. We define an invariant height, introduced by Drezet in the projective plane case, for moduli spaces of sheaves with symmetric $c_{1}$ on a quadric surface and describe the structure of moduli spaces of height zero. Then we study rational maps of moduli spaces of positive height to moduli spaces of representation of quivers, effective cones of moduli spaces, and strange duality for height-zero moduli spaces.

Strange Duality Between Hypersurface and Complete Intersection Singularities

W. Ebeling and C. T. C. Wall discovered an extension of Arnold’s strange duality embracing on one hand series of bimodal hypersurface singularities and on the other, isolated complete intersection singularities. In this paper, we derive this duality from the mirror symmetry and the Berglund–Hubsch transposition of invertible polynomials.