Prakash Belkale
For X a compact Riemann surface of positive genus, the strange duality conjecture predicts that the space of sections of certain theta bundle on moduli of bundles of rank r and level k is naturally dual to a similar space of sections of rank k and level r. We prove this conjecture for X generic in the moduli space of curves of a given genus.
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We consider a mirror symmetry between invertible weighted homogeneous polynomials in three variables. We define Dolgachev and Gabrielov numbers for them and show that we get a duality between these polynomials generalizing Arnold's strange duality between the 14 exceptional unimodal singularities.
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.
For a compact Riemann surface X of positive genus, the space of sections of certain theta bundle on moduli of bundles of rank r and level k admits a natural map to (the dual of) a similar space of sections of rank k and level r (the strange duality isomorphism). Both sides of the isomorphism carry projective connections as X varies in a family. We prove that this map is (projectively) flat.
A basis for the space of generalized theta functions of level one for the spin groups, parameterized by the theta characteristics (the even theta characteristcs for the odd spin groups) on a curve, is shown to be projectively flat over the moduli space of curves (for Hitchin's connection). The symplectic strange duality conjecture, conjectured by Beauville is shown to hold for all curves of genus at least two, by using Abe's proof of the conjecture for generic curves, and the above monodromy result.
We describe a relation between Arnold's strange duality and a polar duality between the Newton polytopes which is mostly due to M.~Kobayashi. We show 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. We generalize this method to the isolated complete intersection singularities embraced in the extended duality. We use this to explain the duality of characteristic polynomials …
There is a strange duality between the quadrangle isolated complete intersection singularities discovered by the first author and C.T.C.Wall. We derive this duality from the mirror symmetry, the Berglund-Hübsch transposition of invertible polynomials, and our previous results about the strange duality between hypersurface and complete intersection singularities using matrix factorizations of size two.
The more recent paper "Generic strange duality for K3 surfaces" by the authors contains stronger results.
We prove that the pull back of the canonical theta divisor for E_8-bundles at level one induces a strange duality between Verlinde spaces for G_2 and F_4 at level one on smooth curves of genus g. We also prove a parabolic generalization in terms of conformal blocks and write down identities between conformal blocks divisors in the Picard group of \bar-M_{g,n}.
C.T.C. Wall and the first author 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-Hübsch transposition of invertible polynomials.
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 Gottsche-Nakajima-Yoshioka.
We show that the failure of the usual Verdier duality on Bun(G) leads to a new duality functor on the category of D-modules, and we study its relation to the operation of Eisenstein series.
A notion of duality of weight systems which corresponds to Batyrev's toric mirror symmetry is given. Explicit duality on the (1,1)-cohomology of K3 surfaces which are minimal models of toric hypersurfaces is constructed using monomial divisor mirror map of Aspinwall-Greene-Morrison. It is shown that Arnold's strange duality for exceptional unimodal singularities reduces to this duality.
K-theoretic Donaldson invariants are holomorphic Euler characteristics of determinant line bundles on moduli spaces of sheaves on surfaces. We compute generating functions of K-theoretic Donaldson invariants on the projective plane and rational ruled surfaces. We apply this result to prove some cases of Le Potier's strange duality.
We extend results on generic strange duality for K3 surfaces by showing that the proposed isomorphism holds over an entire Noether-Lefschetz divisor in the moduli space of quasipolarized K3s. We interpret the statement globally as an isomorphism of sheaves over this divisor, and also describe the global construction over the space of polarized K3s.
We study Le Potier's strange duality conjecture for moduli spaces of sheaves over generic abelian surfaces. We prove the isomorphism for abelian surfaces which are products of elliptic curves, when the moduli spaces consist of sheaves of equal ranks and fiber degree 1. The birational type of the moduli space of sheaves is also investigated. Generalizations to arbitrary product elliptic surfaces are given.
Strange duality at level one for alternating vector bundles
该研究在代数几何领域证明了分歧情形下交错反不变向量丛模空间上广义θ函数空间的一级奇异对偶同构，指出这类反不变向量丛是扭仿射群概形对应的仿射G-旋子的非平凡实例。
The main result of the present paper is the proof of the Strange Duality for elliptic surfaces -- a duality between global sections of determinantal line bundles on moduli spaces of stable sheaves on a fixed elliptic surface. For this, we employ the "Marian-Oprea trick": using Bridgeland's birational isomorphisms, we reduce the problem from a pair of general moduli spaces to a pair of Hilbert schemes. The latter case is a theorem by Marian-Oprea.
We study Le Potier's strange duality conjecture on $\mathbb{P}^2$. We focus on the strange duality map $SD_{c_n^r,d}$ 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 $\mathcal{O}_{\mathbb{P}^2}(d)$ and Euler characteristic 0. By using tools in quiver representation theory, we show that $SD_{c^r_n,d}$ is an isomorphisms for $r=n$ or $r=n-1$ or $d\leq 3$, and in general $SD_{c^r_n,d}$ is injective for any $n\geq r>0$ and $d>0$.
Strange duality is shown to hold over generic $K3$ surfaces in a large number of cases. The isomorphism for elliptic $K3$ surfaces is established first via Fourier-Mukai techniques. Applications to Brill-Noether theory for sheaves on $K3$s are also obtained. The appendix written by Kota Yoshioka discusses the behavior of the moduli spaces under change of polarization, as needed in the argument.
We give a survey on old and new results concerning Arnold's strange duality. We show that most of the features of this duality continue to hold for the extension of it discovered by C. T. C. Wall and the author. The results include relations to mirror symmetry and the Leech lattice.
We study Le Potier's strange duality conjecture on $\mathbb{P}^2$. 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.
We study Le Potier's strange duality on del Pezzo surfaces using quot schemes to construct independent sections of theta line bundles on moduli spaces of sheaves, one of which is the Hilbert scheme of n points. For n at most 7, we use multiple point formulas to count the length of the quot scheme, which agrees with the dimension of the space of sections on the Hilbert scheme. When the surface is the projective plane and n is arbitrary, we use nice resolutions of general stable sheaves to show that the quot schemes that arise are finite and reduced. Combining our results, we obtain a lower boun…
Moduli spaces of stable sheaves over quasi-polarized surfaces, and the relative Strange Duality morphism
本文利用 Alper 的好模空间理论，在拟极化射影曲面栈上构造稳定层的相对模空间，并将相对奇异对偶态射推广至拟极化 K3 曲面轨迹。
The moduli stack M_X(E_8) of principal E_8-bundles over a smooth projective curve X carries a natural divisor Delta. We study the pull-back of the divisor Delta to the moduli stack M_X(P), where P is a semi-simple and simply connected group such that its Lie algebra Lie(P) is a maximal conformal subalgebra of Lie(E_8). We show that the divisor Delta induces "Strange Duality"-type isomorphisms between the Verlinde spaces at level one of the following pairs of groups (SL(5), SL(5)), (Spin(8), Spin(8)), (SL(3), E_6) and (SL(2), E_7).
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 one using the representation theory of Heisenberg groups. For product abelian surfaces, the isomorphism is then shown to hold for sheaves with fiber degree 1 vi…
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 th…
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 $Σ_e:=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(e))$ except for $e=1$. When $e=1$, our result applies to $L=aG+bF$ with $b\geq a+[a/2]$, where $F$…
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 st…
Generalized theta functions, strange duality, and odd orthogonal bundles on curves
本文研究了曲线上具非平凡Stiefel-Whitney类的奇正交丛的广义theta函数及扭曲旋量丛，证明了Verlinde型公式与Oxbury-Wilson维数猜想，确立了模空间上的平坦射影联络，揭示了奇异对偶朴素猜想的失效性，并解决了带spin权重的共形块秩-水平对偶问题。
We study Le Potier's strange duality conjecture on a rational surface. We focus on the strange duality map $SD_{c_n^r,L}$ 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 $L$ 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 $r\geq1$ under some conditions on $X$ and $L$ which applies to a large number of cases on $\p^2$ or Hirzebruch surfaces . Also on $\mathbb{P}^2…
Over the projective plane and at most two-step blowups of Hirzebruch surfaces, where there are strong full exceptional sequences of line bundles, we obtain foundational results about Gaeta resolutions of coherent sheaves by these line bundles. Under appropriate conditions, we show the locus of semistable sheaves not admitting Gaeta resolutions has codimension at least 2. We then study Le Potier's strange duality conjecture. Over these surfaces, for two orthogonal numerical classes where one has rank one and the other has sufficiently positive first Chern class, we show that the strange morphis…
We examine a sequence of examples of pairs of moduli spaces of sheaves on $\mathbb P^2$ where Le Potier's strange duality is expected to hold. One of the moduli spaces in these pairs is the Hilbert scheme of two points. We compute the sections of the relevant theta bundle as a representation of $SL(X)$, where $\mathbb P^2=\mathbb P(X)$. For the higher rank space, we construct a moduli space using the resolution of exceptional bundles from Coskun, Huizenga, and Woolf. We compute a subspace of the sections of the theta bundle which is dual to the sections on the Hilbert scheme. In the second par…
Let $\mhu$ be the moduli space of semi-stable pure sheaves of class $u$ on a smooth complex projective surface $X$. We specify $u=(0,L,χ(u)=0),$ i.e. sheaves in $u$ are of dimension $1$. There is a natural morphism $π$ from the moduli space $\mhu$ to the linear system $\ls$. We study a series of determinant line bundles $\lcn$ on $\mhu$ via $π.$ Denote $g_L$ the arithmetic genus of curves in $\ls.$ For any $X$ and $g_L\leq0$, we compute the generating function $Z^r(t)=\sum_{n}h^0(\mhu,\lcn)t^n$. For $X$ being $\mathbb{P}^2$ or $\mathbb{P}(\mo_{\pone}\oplus\mo_{\pone}(-e))$ with $e=0,1$, we com…
A refinement of the strange duality conjecture on K3 surfaces
针对K3曲面上奇异对偶猜想，本文构造三类反例说明原猜想不成立，并提出改进猜想，要求相关theta线丛为大且数值有效。
We prove that the strange duality conjecture of Beauville-Donagi-Tu holds for all curves. We establish first a more extended rank-level duality, interesting in its own right, from which the standard rank-level duality follows by restriction.
Seiberg Duality conjecture for star-shaped quivers and finiteness of Gromov-Witten thoery for D-type quivers
本文证明星形箭图的Seiberg对偶猜想成立，并进一步证明D_3型箭图簇代数簇的Gromov-Witten理论及Kähler变量在有限次突变后恢复原状。
There are strange dual pairs of bimodal singularities that are not assigned an invertible projectivisation in EbelingPloog. We study families of $K3$ surfaces associated to such pairs.
We determine the splitting type of the Verlinde vector bundles in higher genus in terms of simple semihomogeneous factors. In agreement with strange duality, the simple factors are interchanged by the Fourier-Mukai transform, and their spaces of sections are naturally dual.
In [GM1], we defined a G_R-K_C invariant subset C(S) of G_C for each K_C-orbit S on every flag manifold G_C/P and conjectured that the connected component C(S)_0 of the identity would be equal to the Akhiezer-Gindikin domain D if S is of nonholomorphic type. This conjecture was proved for closed S in [WZ2,WZ3,FH,M4] and for open S in [M4]. It was proved for the other orbits in [M5] when G_R is of non-Hermitian type. In this paper, we prove the conjecture for an arbitrary non-closed K_C-orbit when G_R is of Hermitian type. Thus the conjecture is completely solved affirmatively.
Let $Bun_G(X)$ be the moduli stack of $G$-torsors on a smooth projective curve $X$ for a reductive group $G$. We prove a conjecture made by Drinfeld-Wang and Gaitsgory on the Deligne-Lusztig duality for D-modules on $Bun_G(X)$. This conjecture relates Drinfeld-Gaitsgory's pseudo-identity functors to the enhanced Eisenstein series and geometric constant term functors on $DMod(Bun_G(X))$. We also prove a "second adjointness" result for these enhanced functors.
In [GM1], we defined a G_R-K_C invariant subset C(S) of G_C for each K_C-orbit S on every flag manifold G_C/P and conjectured that the connected component C(S)_0 of the identity will be equal to the Akhiezer-Gindikin domain D if S is of nonholomorphic type. This conjecture was proved for closed S in [WZ1,WZ2,FH,M6] and for open S in [M6]. In this paper, we prove the conjecture for all the other orbits when G_R is of non-Hermitian type.
This is the first part of two papers. In this part, we prove the blowup formulae for virtual Hodge polynomials of Gieseker moduli spaces of rank-2 stable sheaves and Uhlenbeck compactification spaces over algebraic surfaces. In particular, we verify the blowup formulae for the S-duality conjecture of Vafa-Witten, i.e. the blowup formulae for the Euler numbers of instanton moduli spaces over algebraic surfaces.
We prove the transformation formula of Donaldson-Thomas (DT) invariants counting two dimensional torsion sheaves on Calabi-Yau 3-folds under flops. The error term is described by the Dedekind eta function and the Jacobi theta function, and our result gives evidence of a 3-fold version of Vafa-Witten's S-duality conjecture. As an application, we prove a blow-up formula of DT type invariants on the total spaces of canonical line bundles on smooth projective surfaces. It gives an analogue of the similar blow-up formula in the original S-duality conjecture by Yoshioka, Li-Qin and Göttsche.
On Hypergeometric Duality Conjecture
本文证明 Horja–Borisov 的 GKZ 超几何对偶猜想，给出显式公式，并借助 Γ-级数上同调解将其与环面 Deligne–Mumford 栈的 Euler 特征配对逆联系起来。
It is conjectured that the dual variety of every smooth nonlinear subvariety of dimension $> \frac{2N}{3}$ in projective $N$-space is a hypersurface, an expectation known as the duality defect conjecture. This would follow from the truth of Hartshorne's complete intersection conjecture but nevertheless remains open for the case of subvarieties of codimension $> 2$. A combinatorial approach to proving the conjecture in the codimension $2$ case was developed by Holme, and following this approach Oaland devised an algorithm for proving the conjecture in the codimension $3$ case for particular $N$…
This is the second part of two papers. In this part, we establish closed formulae for the universal functions in the blowup formulae for virtual Hodge polynomials of Gieseker moduli spaces of rank-2 stable sheaves and Uhlenbeck compactification spaces over algebraic surfaces.
Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of PDEs. We explain these conjectures and verify them in some cases.
In this paper we deal with algebro-geometrical problems connected with testing S-duality conjecture for super-symmetric Yang-Mills quantum field theories in four dimensions. We describe all field configurations such that beta function coefficient for gauge groups of rank 1 and 2 is zero or negative. We also give special series of such field configurations for gauge groups of an arbitrary rank. Realization of one of the series discovers a connection with exceptional bundles. That points again at relation between S-duality and string duality.
Seiberg duality conjecture asserts that the Gromov-Witten theories (Gauged Linear Sigma Models) of two quiver varieties related by quiver mutations are equal via variable change. In this work, we prove this conjecture for $A_n$ type quiver varieties.
A formula for the first Chern class of the Verlinde bundle over the moduli space of smooth genus g curves is given. A finite-dimensional argument is presented in rank 2 using geometric symmetries obtained from strange duality, relative Serre duality, and Wirtinger duality together with the projective flatness of the Hitchin connection. A derivation using conformal-block methods is presented in higher rank. An expression for the first Chern class over the compact moduli space of curves is obtained.
Geometric Langlands duality for periods
本文研究Ben-Zvi--Sakellaridis--Venkatesh关于几何朗兰兹对偶中周期与L-函数关系的范畴化猜想，并通过导出傅里叶分析和手征代数在低秩例子中提供证据。
This is a survey of results and conjectures on mirror symmetry phenomena in the non-Abelian Hodge theory of a curve. We start with the conjecture of Hausel-Thaddeus which claims that certain Hodge numbers of moduli spaces of flat SL(n,C) and PGL(n,C)-connections on a smooth projective algebraic curve agree. We then change our point of view in the non-Abelian Hodge theory of the curve, and concentrate on the SL(n,C) and PGL(n,C) character varieties of the curve. Here we discuss a recent conjecture of Hausel-Rodriguez-Villegas which claims, analogously to the above conjecture, that certain Hodge…
A converse to geometric Manin's conjecture for general low degree hypersurfaces and Poincaré duality
该研究通过高亏分数的圆法推广，证明了对于一般低次超曲面，不存在增加Fujita不变量的累积映射，从而在特定范围内建立了这些模空间的庞加莱对偶性。
We study a pair of conjectures on better behaved GKZ hypergeometric systems of PDEs inspired by Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities. We prove the conjectures in the case of dimension two.
V. Drinfeld proposed conjectures on geometric Langlands correspondence and its quantum deformation. We refine these conjectures and propose their relationship with algebraic conformal field theory.
Generalised ordinary vs fully simple duality for $n$-point functions and a proof of the Borot--Garcia-Failde conjecture
本文研究 VEV 形式下 $n$-点函数的普通与完全简单对偶性，推广了地图与完全简单地图间的对偶性，提供转移代数性及分析奇点的工具，并据此证明了 Borot--Garcia-Failde 关于完全简单地图拓扑递归的猜想。
A proof of all ranks S-duality conjecture for K3 surfaces
利用Toda多重覆盖公式计算K3曲面上任意秩的SU(r)/Z_r-Vafa-Witten不变量，并基于Tanaka-Thomas结果证明所有秩的S-对偶猜想。
Motivated by the S-duality conjecture of Vafa-Witten, Tanaka-Thomas have developed a theory of Vafa-Witten invariants for projective surfaces using the moduli space of Higgs sheaves. Their definition and calculation prove the S-duality prediction of Vafa-Witten in many cases in the side of gauge group $SU(r)$. In this survey paper for ICCM-2019 we review the S-duality conjecture in physics by Vafa-Witten and the definition of Vafa-Witten invariants for smooth projective surfaces and surface Deligne-Mumford stacks. We make a prediction that the Vafa-Witten invariants for Deligne-Mumford surface…
The Beilinson-Bloch type conjectures predict that the low degree rational Chow groups of intersections of quadrics are one dimensional. This conjecture was proved by Otwinowska. Making use of homological projective duality and the recent theory of (Jacobians of) noncommutative Chow motives, we give an alternative proof of this conjecture in the case of a complete intersection of either two quadrics or three odd-dimensional quadrics. Moreover, without the use of the powerful Lefschetz theorem, we prove that in these cases the unique non-trivial algebraic Jacobian is the middle one. As an applic…
We study $A$-hypergeometric systems $H_A(β)$ in the sense of Gelfand, Kapranov and Zelevinsky under two aspects: the structure of their holonomically dual system, and reducibility of their rank module. We prove first that rank-jumping parameters always correspond to reducible systems, and we show that the property of being reducible is ``invariant modulo the lattice''. In the second part we study a conjecture of Nobuki Takayama which states that the holonomic dual of $H_A(β)$ is of the form $H_A(β')$ for suitable $β'$. We prove the conjecture for all matrices $A$ and generic parameter $β$, exh…
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