Grassmannian Flop Example

• Grassmannian flops are birational transformations between vector bundle spaces over Grassmannians, characterized by window subcategories and derived equivalences.
• They leverage algebraic GIT constructions with Kapranov’s exceptional collections to establish explicit functorial equivalences across flops.
• Applications include non-abelian GLSMs, string theory wall-crossing, and the birational geometry of Calabi–Yau varieties.

A Grassmannian flop is a birational transformation between total spaces of vector bundles over Grassmannians, equipped with derived equivalences governed by the structure of exceptional collections and spherical functors. This class of flops generalizes the standard threefold flop by replacing the flopping locus with a Grassmannian—frequently appearing in non-abelian gauged linear sigma models and string theory wall-crossing phenomena. The analysis of window subcategories, construction of derived autoequivalences, and their explicit computation all illustrate deep interactions between algebraic geometry, representation theory, and mathematical physics.

1. Algebraic Construction: Windows, GIT Quotients, and Derived Equivalences

The core local model begins with an Artin stack

$$\mathcal{E} = [\mathrm{Hom}(S,V) \oplus \mathrm{Hom}(V,S) / \mathrm{GL}(S)]$$

where $V$ is a $d$-dimensional complex vector space and $S$ is an $r$-dimensional vector space ($0 < r < d$). Variation of GIT yields two smooth loci:

• $X^+ =$ total space of $\mathrm{Hom}(V,S)$ over $\mathrm{Gr}(r,V)$; this corresponds to the full-rank locus for $S \to V$
• $X^- =$ total space of $\mathrm{Hom}(S,V)$ over $\mathrm{Gr}(d-r,V^\vee)$; similarly, the full-rank locus for $V \to S$

Kapranov's full strong exceptional collection on $\mathrm{Gr}(r,V)$ gives a basis to construct subcategories $W_k$:

$$W_k = \langle \mathbb{S}^\delta S \otimes \mathcal{O}(k) \mid \delta \in I_{d,r} \rangle$$

where $I_{d,r}$ is the set of Young diagrams with height $\leq r$ and width $\leq d-r$, $\mathbb{S}^\delta$ is the Schur functor, and $\mathcal{O}(1) = \det S$ is the tautological line bundle.

Restriction functors

$i_+^* : W_k \longrightarrow D^{\mathrm{b}}(X^+)$

are equivalences for each $k$, yielding derived equivalences across the flop via window shifts indexed by $k$. The window shift autoequivalences

$w_{k,\ell} := u_k^{-1} \circ u_\ell$

encode the categorical effect of the flop and are realized as explicit functorial compositions—see the formulas in (Donovan et al., 2012).

2. Geometric Construction: Spherical Functors and Twists

A geometric perspective uses spherical functors to relate $X^+$ and auxiliary stack/space $Y^+$ via

$$F = j_* \circ \pi^* : D^{\mathrm{b}}(Y^+) \rightarrow D^{\mathrm{b}}(X^+)$$

where the correspondence is modeled by an auxiliary vector space $H$ with $\dim H = r-1$, and

$$Y^+ = [\mathrm{Hom}(H,V) \oplus \mathrm{Hom}(V,H) / \mathrm{GL}(H)]$$

is involved in the diagram $$Y^+ \stackrel{\pi}{\leftarrow} Z \xrightarrow{j} X^+$$.

The spherical functor $F$ admits both adjoints, and thus the spherical twist autoequivalence

$T_F = \mathrm{Cone}(F R \rightarrow \mathrm{id})$

reconstructs a window shift $w_{0,1}$, while the inverse cotwist reconstructs $w_{-1,0}$ (possibly up to shift). The equivalence between algebraic and geometric constructions is established via "transfer" functors and commutative diagrams, with properties ensuring the categorical behavior matches the geometry of the flopping locus.

3. Spherical Twists Beyond the Standard Case

In the standard $3$-fold flop ($r=1$, $d=2$), the locus is a projective space, and the spherical twist is taken around the skyscraper sheaf $\mathcal{O}_{\mathbb{P}^1}$,

$$T_{\mathcal{O}_{\mathbb{P}^1}}(E) := \mathrm{Cone}(\mathrm{RHom}(\mathcal{O}_{\mathbb{P}^1}, E) \otimes \mathcal{O}_{\mathbb{P}^1} \to E)$$

Grassmannian flops generalize this: Instead of a single spherical object, one has a family of "spherical functors" associated to variable Schur-functor-generated bundles. Spherical twists now encode "family-level" monodromy and allow for non-abelian analogs in GLSM derived categories. For $r>1$, the twist is around $F: D^{\mathrm{b}}(Y^+) \to D^{\mathrm{b}}(X^+)$, as above.

4. Applications: Birational Geometry, GLSMs, and SKMS Monodromy

Grassmannian flop autoequivalences model derived equivalence of birational Calabi-Yau varieties, string-theoretic monodromy of B-branes, and wall-crossing phenomena in stringy Kähler moduli spaces. The window constructions match the grade-restriction rules in GLSMs [Herbst-Hori-Page], integrating into a physical framework for brane transport. Additionally, the mathematical techniques developed (e.g. locally-free resolutions, Eagon-Northcott and Buchsbaum-Rim complexes) are instrumental in analyzing determinantal varieties and moduli of sheaves.

5. Explicit Examples and Calculations

Explicit calculations for $r=1$ recover Atiyah flops; for higher $r$, carrying through on Kapranov generators for $\mathrm{Gr}(r, d)$ (e.g., $d=4$, $r=2$ with bundles $$\mathcal{O}, S, \mathrm{Sym}^2 S, \mathcal{O}(1), S(1)$$), the window shift $w_{-1,0}$ is determined by exact sequences:

• Eagon-Northcott complex resolves $\mathrm{Sym}^2 S$
• Buchsbaum-Rim complex resolves $S(1)$
• Higher generators use generalized Koszul complexes

Typical results:

$$w_{-1,0}(\mathcal{O}(1)) \simeq \{ \mathcal{O}^{\oplus 2} \to \mathcal{O}(-1) \} \ w_{-1,0}(\mathrm{Sym}^2 S(1)) \simeq \cdots \to \mathbb{S}^\delta S(-1)$$

On generators common to both $W_0$ and $W_1$ the autoequivalence acts as the identity.

6. Relation to Semi-Orthogonal Decompositions and Vanishing

Semi-orthogonal decompositions arise naturally: window restriction maps are equivalences, and the associated kernels induce decomposition of the derived category. The canonical idempotent kernel (Ballard et al., 2019) restricts to open a window matching Kapranov’s collection. Any summand with a Young diagram having a full row of length $d$ has vanishing invariants—crucial for ensuring suitable semi-orthogonal decompositions.

7. Philosophy and Future Directions

Grassmannian flops serve as paradigmatic examples where categorical derived equivalence, geometric wall-crossing, and representation-theoretic constructions intersect. The framework accommodates non-abelian geometric invariant theory and provides a blueprint for further exploration of moduli of sheaves and birational geometry in the presence of complicated base loci and parameter spaces.

The methodology developed for window shifts and spherical functors continues to underpin results on derived categories for more general flips and flops, including Mukai, Abuaf and relative settings, with ongoing advances in semiorthogonal decompositions, mutation techniques, and applications in homological mirror symmetry and enumerative geometry (Donovan, 2013, Ballard et al., 2019).