Strange Duality Research Articles

Sheaves on abelian surfaces and strange duality

We formulate three versions of a strange duality conjecture for sections of the Theta bundles on the moduli spaces of sheaves on abelian surfaces. As supporting evidence, we check the equality of dimensions on dual moduli spaces, answering a question raised by Göttsche et al. (K-theoretic Donaldson invariants via instanton counting. arXiv:math/0611945).

Duality of linking pairing in Arnold’s singularities

As a new aspect of the Arnold strange duality among 14 unimodal singularities, we point out that there exists a duality in linking pairing on Seifert manifolds associated with singularities.

Duality of Weights, Mirror Symmetry and Arnold's Strange Duality

A duality of weight systems which corresponds to Batyrev's toric mirror symmetry is given. It is shown that Arnold's strange duality for exceptional unimodal singularities reduces to this duality.

The strange duality conjecture for generic curves

Let X X be a smooth connected projective algebraic curve of genus g ≥ 1 g\geq 1 . The strange duality conjecture connects non-abelian theta functions of rank r r and level k k and those of rank k k and level r r on X X (for S U ( r ) SU(r) and U ⁡ ( k ) \operatorname {U}(k) , respectively). In this paper we prove this conjecture for X X generic in the moduli space of curves of genus g g .

The level-rank duality for non-abelian theta functions

We prove that the strange duality conjecture of Beauville–Donagi–Tu holds for all curves. We establish first a more extended level-rank duality, interesting in its own right, from which the standard level-rank duality follows by restriction.

Mirror symmetry, Kobayashi's duality, and Saito's duality

M. Kobayashi introduced a notion of duality of weight systems. We tone this notion slightly down to a notion called coupling. We show that coupling induces a relation between the reduced zeta functions of the monodromy operators of the corresponding singularities generalizing an observation of K. Saito concerning Arnold's strange duality. We show that the weight systems of the mirror symmetric pairs of M. Reid's list of 95 families of Gorenstein K3 surfaces in weighted projective 3-spaces are strongly coupled. This includes Arnold's strange duality where the corresponding weight systems are strongly dual in Kobayashi's original sense. We show that the same is true for the extension of Arnold's strange duality found by the author and C. T. C. Wall.

On $SL(2)-GL(n)$ strange duality

We prove $SL(2)-GL(n)$ strange duality conjecture for a general smooth projective irreducible curve.

Involutions and linear systems on holomorphic symplectic manifolds

A K3 surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that similar statement holds for the generic couple (X, H) with X a deformation of (K3)[n] and H an ample divisor of square 2 for Beauville’s quadratic form. If n = 2 then according to the conjecture X is a double cover of a singular) sextic 4-fold in \(\mathbb{P}^{5} .\) It follows from the conjecture that a deformation of (K3)[n] carrying a divisor (not necessarily ample) of degree 2 has an anti-symplectic birational involution. We test the conjecture. In doing so we bump into some interesting geometry: examples of two antisymplectic involutions generating an interesting dynamical system, a case Strange duality and what is probably an involution on the moduli space degree-2 quasi-polarized (X, H) where X is a deformation of (K3)[2].

The anatomy of folly in Shakespeare's “Henriad”

Abstract Folly, as Johnson said of comedy, has been “particularly unpropitious to definers,” struggling to conceive a notoriously indeterminate term. “Folly” is usually derogatory, pinned on any disbeliever or adversary, telling us as much about the judge as the judged, or ironically laudatory. While “folly” flaunts its maddening elusiveness, fools will rush in where wise men fear to tread. This essay explores the concept of folly, investigates its pertinence for literary criticism, and tests its usefulness in a consideration of literature's greatest fool, Falstaff. The fool's cross-eyed vision always threatens to become a revelation; what starts as impish play may end as prophecy. Fools divide and confuse us, so that we either scant or privilege folly by reducing it to diverting babble or magnifying it into encoded prophecy. A great fool has amazing powers of disorientation: he is an avatar of disequilibrium. Disabled yet enabled, invincible yet particularly vulnerable, the fool is always double, both lightning bolt and lightning rod: his bad luck might bring me good luck, so we make room for fools but keep our distance: there with/but for the grace of God go I. The Fool has a strange duality, like the medieval monarch, two separate “bodies,” one enduring, potent, capable of revival, personifying survival and adaptability; the other marginal, susceptible, provisional, easily hurt. Shakespeare's most majestic fool dramatizes folly's powers, perils, and paradoxes. Foolishly immersed in the “lower bodily element,” Falstaff imagines himself somehow freed from natural law; simultaneously Caliban and Ariel, he is enmired yet aloft, immanent yet transcendent, that quality wonderfully characterized by Bradley as Falstaff 's “inexplicable touch of infinity.” When “Falstaff riseth up” from playing possum, his comic resurrection seems the definitive triumph, “the true and perfect image of life indeed.” This “great fool” not only affirms life but outrageously redeems it with “counterfeit” or bogus scriptural idiom. Falstaff robustly embodies the power of folly and dimly, occasionally perceives its limits. In 2Henry IV, Falstaff, obviously enfeebled, becomes more the object than the source of humor. The banishment of Falstaff is not humorous and it hurts. As a seemingly imperishable fool, exuberantly enacting folly, Falstaff liberates life from fact, in defiance of reason and pursuit of joy. Falstaff's force draws us all into the field of folly, so that the great fool is our double whose loss we deplore.

Strange Duality and Polar Duality

A relation is described between Arnold's strange duality and a polar duality between the Newton polytopes which is mostly due to M. Kobayashi. It is shown that this relation continues to hold for the extension of Arnold's strange duality found by C. T. C. Wall and the author. By a method of Ehlers–Varchenko, the characteristic polynomial of the monodromy of a hypersurface singularity can be computed from the Newton diagram. This method is generalized to the isolated complete intersection singularities embraced in the extended duality. This is used to explain the duality of characteristic polynomials of the monodromy discovered by K. Saito for Arnold's original strange duality and extended by the author to the other cases.

Duality Construction of Moduli Spaces

The duality of moduli spaces of coherent sheaves on curves and surfaces gives a GIT free construction of these spaces. It is a suitable generalization for topics as: strange duality, jumping lines of vector bundles on projective space, and Faltings' construction of the moduli space of semistable vector bundles on a complex curve given in his article ‘Stable G-bundles and projective connections’. Using this duality we propose a construction for the moduli space of semistable sheaves on a surface.

Duality for regular systems of weights

Regular systems of weights are certain combinatorial and arithmetic objects related to a generalization of Coxeter elements [S6,7,8 and 11], and introduced in motivation to understand the flat structure for primitive forms for isolated hypersurface singularities [S3] (cf. [Man],[S11]). In the present article, the theory is applied to explain the self-duality of ADE (=simply laced Dynkin diagrams) and the strange duality of Arnold. Beyond the original applications, the study gives further class of dual weight systems, which, for instance, has close connection with Conway group and seems interesting to be studied yet further. On the other hand, the duality of weight systems has an interpretation in terms of certain products of Dedekind eta functions. We give a conjecture on the non-negativity of the Fourier coefficients of the eta-products. The conjecture is solved affirmatively for the cases corresponding to elliptic root systems [S45]. But the meaning is not yet clear. Recently, one finds an equivalence between the duality in the present article and certain string duality in mathematical physics [T]. 0. Introduction. The present article gives a general frame work on the duality of regular systems of weights. For a sake of self-containedness, all proofs are given or sketched except for some basic facts. For simplicity, we shall call a regular system of weights a weight system unless otherwise is stated. A weight system W := (a, b, c;h) is a system of 4 positive integers with some arithmetic constraint (see (1.0)). To W , we attach a cyclotomic polynomial φW , called the characteristic polynomial of the weight system (see (2.1)). The duality we study in the present article appears as a duality between the cyclotomic polynomials. Let us explain this by examples. Let h be a positive integer and let φ(λ) and φ∗(λ) be cyclotomic polynomials whose roots are h-th roots of unity. The polynomials can be decomposed in the form:

Landau-Ginzburg theories as orbifolds

It is shown that subclasses of a recently constructed set of Landau-Ginzburg/Calabi-Yau string compactification can be described as orbifolds of simpler superpotentials. This construction allows for a systematic identification of dual pairs that appear naturally in this class of string vacua. The strange duality of Arnold appears as a special case of these constructions.

Landau-Ginzburg theories as orbifolds

It is shown that subclasses of a recently constructed set of Landau-Ginzburg/Calabi-Yau string compactifications can be described as orbifolds of simpler superpotentials. This construction allows for a systematic identification of dual pairs that appear naturally in this class of string vacua. The strange duality of Arnold appears as a special case of these constructions.

A new relation among Cartan matrix and Coxeter matrix

Here pqrfd is a numerical invariant concerning about the Cartan matrix of R defined in Section 1 and u/x/h is a numerical invariant concerning about a Coxeter iransformatioI1 of R defined in Section 2. We give a proof of the equality in Section 4 using the self-intersection number of a Weil divisor on a rational surface. The argument is a version of that for the strange duality on the 14 exceptional unimodular singularities of Arnold due to Pinkham (cf. [ 1, 61). It might be quite interesting to find a purely arithmetic proof of the equality. The author would like to express his glatitude to his colleages I. Naruki and M. Tomari for their interest and helpful1 discussions and also to K. Ueno, who has noticed the author on some works on anti-canonical divisors.

Strange duality revisited

We give a proof of the strange duality or rank-level duality of the WZW models of conformal blocks by extending the genus-0 result, obtained by Nakanishi-Tsuchiya in 1992, to higher genus curves via the sewing procedure. The new ingredient of the proof is an explicit use of the branching rules of the conformal embedding of the affine Lie algebras sl(r) x sl(l) in sl(rl). We recover the strange duality of spaces of generalized theta functions obtained by Belkale, Marian-Oprea, as well as by Oudompheng in the parabolic case.