Symplectic duality and hypertoric varieties Tom Braden

(joint work with Anthony Licata, Nicholas Proudfoot and Ben Webster)

Symplectic duality is a (still largely conjectural) set of relations between certain pairs of symplectic algebraic varieties. More precisely, it involves data (𝔐, T, η, ξ), where 𝔐 is a smooth variety over ℂ with a holomorphic symplectic form ω_{ℂ}, T is an algebraic torus which acts with finite fixed-point set, η is a class in H^{2}(𝔐; ℤ) and ξ: ℂ^{∗} → T is a cocharacter of T. These are required to satisfy:

• The affinization map α: 𝔐 → 𝔐_{0} = Spec Γ(𝔐, O_{𝔐}) is a resolution of singularities,

• η is the first Chern class of a line bundle on 𝔐 which is relatively ample for α, and

• the image ξ(ℂ^{∗}) has the same fixed point set as T.

Symplectic duality is a relation between pairs (𝔐, T, η, ξ), (𝔐^{∨}, T^{∨}, η^{∨}, ξ^{∨}) of data of this type. These pairs have appeared in mathematical physics, as Higgs branches of dual N = 4, d = 3 sypersymmetric gauge theories [6] (classical mirror symmetry involves d = 2 theories). However, our conjectures go beyond what is currently predicted in the physics literature. Examples of symplectic dual pairs include the following.

(1) 𝔐 = T^{∗}(G/B), the cotangent bundle to the flag variety of a semisimple algebraic group G. The torus T is a Cartan subgroup contained in B, the parameter η is given by a dominant weight, and ξ comes from a dominant coweight. The symmetric dual variety is T^{∗}(^{L}G/^{L}B), where ^{L}G is the Langlands dual group.

(2) For G = SL_{n}(ℂ), a resolution of \overline{O_{λ}} ∩ N_{μ} is dual to a resolution of \overline{O_{μ^{t}}} ∩ N_{λ^{t}}, where O_{λ} denotes the nilpotent orbit with Jordan form λ and N_{λ} is a transverse slice to O_{λ}.

(3) Certain pairs of Nakajima quiver varieties for the quiver of affine type A should be symplectic dual. As a special case, the Hilbert scheme of n points in ℂ^{2} should be self-dual.

(4) A hypertoric variety defined by a hyperplane arrangement (see the discussion below) is symplectic dual to the hypertoric variety defined by the Gale dual arrangement.

There are a number of relations that should be satisfied by symplectic dual pairs. They are all known in case (1), and by our recent work [2, 3, 4], in case (4). Specifically, symplectic dual pairs should have:

(a) natural isomorphisms H^{2}(𝔐; ℂ) ≅ Lie T^{∨} and H^{2}(𝔐^{∨}; ℂ) ≅ Lie T, under which the parameter η is sent to ξ^{∨} and ξ is sent to η^{∨},

(b) a natural bijection between the fixed point sets 𝔐^{T} and (𝔐^{∨})^{T^{∨}}, inducing a bijection between the components of the Lagrangian varieties 𝒳 = {x ∈ 𝔐 | lim_{t→∞} ξ(t) · x exists} and 𝒳^{∨} = {y ∈ 𝔐^{∨} | lim_{t→∞} ξ^{∨}(t) · y exists}, (c) a natural order-reversing bijection between strata of the affine varieties 𝔐_{0} and 𝔐_{0}^{∨} (they are Poisson varieties, and so have natural stratifications by Poisson leaves),

(d) a perfect pairing between compactly supported cohomology groups H_{c}^{∗}(𝒳) and H_{c}^{∗}(𝒳^{∨}) which induces perfect pairings on the associated graded pieces for the filtrations induced by the decomposition theorem applied to the resolutions 𝔐 → 𝔐_{0} and 𝔐^{∨} → 𝔐_{0}^{∨} (note that these groups are nonzero in only one degree),

(e) a perfect pairing between the equivariant homology spaces H_{2}^{T} (𝔐) = Spec(Sym(H_{T}^{2} (𝔐)) and H_{2}^{T^{∨}}(𝔐^{∨}) = Spec(Sym(H_{T^{∨}}^{2}(𝔐^{∨})), under which the images of the affine varieties Spec(H_{T}^{∗} (𝔐)) and Spec(H_{T^{∨}}^{∗}(𝔐^{∨})) are orthogonal arrangements of subspaces,

(f) and finally, certain abelian categories associated to the data (𝔐, T, η, ξ) and (𝔐^{∨}, T^{∨}, η^{∨}, ξ^{∨}) should be Koszul dual. These are categories of modules over quantizations of the structure sheaves of 𝔐 and 𝔐^{∨}, supported on the Lagrangians 𝒳 and 𝒳^{∨}.

All of these properties are known in the case (1) of cotangent bundles to flag varieties. In this case, the category that appears in (f) is a regular block of Bernstein-Gelfand-Gelfand category O, and Koszul duality was established by Beilinson, Ginzburg, and Soergel [1]. The linear duality (e) was first noticed by Goresky and MacPherson [7], where they established it in case (2) for μ = (n).

Hypertoric varieties

The papers [2, 3, 4] show that all of these aspects of symplectic duality also hold for hypertoric varieties, whose very explicit geometry makes direct calculations possible. A hypertoric variety 𝔐 = 𝔐(V, η) is determined by giving a linear subspace V ⊂ ℝ^{n}, defined over ℚ, and a vector η ∈ ℝ^{n}/V which lies in the image of the lattice ℤ^{n}. The torus (ℂ^{∗})^{n} acts on T^{∗}ℂ^{n} preserving the natural holomorphic symplectic form, with moment map

Φ(z, w) = (z_{1}w_{1}, …, z_{n}w_{n}).

The hypertoric variety 𝔐 is defined to be the GIT quotient

Φ^{−1}(V)//_{η} G,

where G ⊂ (ℂ^{∗})^{n} is the subtorus with Lie algebra V^{⊥} ⊗_{ℝ} ℂ. It carries an induced symplectic form which is preserved by the induced action of the quotient torus T := (ℂ^{∗})^{n}/G.

Just as a semiprojective toric variety can be described by its moment polyhedron, the resulting hypertoric variety is governed by the hyperplane arrangement in the affine space V_{η} (the linear space V translated away from the origin by η) whose hyperplanes are H_{i} = {x ∈ V_{η} | x_{i} = 0}, i = 1, …, n. For instance, 𝔐 has at worst orbifold singularities if and only if the arrangement is simple, which happens for generic values of the parameter η. For each sign vector α ∈ {±1}^{n}, the toric variety X_{α} with moment polyhedron

Δ_{α} := {x ∈ V_{η} | α(i)x_{i} ≥ 0 for all 1 ≤ i ≤ n}

appears as a (possibly empty) T -invariant Lagrangian subvariety of 𝔐.

A covector ξ ∈ V^{∗} which takes integral values on V ∩ ℤ defines a cocharacter ℂ^{∗} → T. For generic ξ it will have the same fixed point set as T, and the components of the Lagrangian 𝒳 are just the toric varieties X_{α} for which the moment polyhedron Δ_{α} is nonempty and bounded above under the linear function ξ.

For such parameters V, η, ξ, the symplectic dual variety 𝔐^{∨} is defined by the Gale dual triple (V^{⊥}, −ξ, −η). Properties (a) and (e) of symplectic dual varieties follow essentially by definition, using natural identifications H^{2}(𝔐; ℝ) ≅ ℝ^{n}/V and H_{T}^{2} (𝔐; ℝ) ≅ ℝ^{n}. Properties (b) and (c) are easy consequences of combinatorial facts about Gale duality.

Property (f) (Koszul duality) is the subject of [3], which gives two presentations of an algebra in terms of the data (V, η, ξ), so that modules over this algebra give the required category O. The first presentation is by generators and relations, while the second is geometric, putting a convolution product on the vector space

⨁_{α,β} H^{∗}(X_{α} ∩ X_{β})[−d_{αβ}],

where the sum is over α and β with X_{α}, X_{β} ⊂ 𝒳, and d_{αβ} is the codimension of Δ_{α} ∩ Δ_{β} in V_{η}. Combinatorial arguments show that this ring is Koszul, quasihereditary, and Koszul dual to the same ring for the Gale dual arrangement. A forthcoming paper [5] will show that modules over this ring are equivalent to modules over a quantization of O_{𝔐} supported on 𝒳.

We speculate that the Koszul duality property (f) is the most fundamental aspect of symplectic duality. To that end, we show [2] with Chris Phan that suitable pairs of Koszul dual algebras define orthogonal subspace arrangements as in (e), without reference to any variety. The paper [4] will treat the duality (d). Under appropriate hypotheses, it can be deduced from Koszul duality as well.

References

[1] Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527.

[2] Tom Braden, Anthony Licata, Christopher Phan, Nicholas Proudfoot, and Ben Webster, Goresky-MacPherson duality and deformations of Koszul algebras, In preparation.

[3] Tom Braden, Anthony Licata, Nicholas Proudfoot, and Ben Webster, Gale duality and Koszul duality, preprint arXiv:0806.3256.

[4] Tom Braden, Anthony Licata, Nicholas Proudfoot, and Ben Webster, Cohomological symplectic duality, In preparation.

[5] Tom Braden, Anthony Licata, Nicholas Proudfoot, and Ben Webster, Categorical symplectic duality, In preparation.

[6] Jan de Boer, Kentaro Hori, Hirosi Ooguri, and Yaron Oz, Mirror symmetry in threedimensional gauge theories, quivers and D-branes, Nuclear Phys. B 493 (1997), no. 1-2, 101–147.

[7] Mark Goresky and Robert MacPherson, On the spectrum of the equivariant cohomology ring, preprint available at http://www.math.ias.edu/∼goresky/pdf/Spec6.pdf.