Shifted Symplectic Structures, the Donaldson–Thomas Theory, and the \((-1)\)-Shifted Symplectic¶
Job: develop the \(p\)-shifted symplectic (Pantev–Toën–Vaquié–Vezzosi), the \((-1)\)-shifted symplectic / critical locus / DT invariants / DT–PT–GV, the shifted Poisson / deformation quantization / \(E_n\), and the mirror symmetry as shifted symplectic equivalence — with the \((-1)\)-shifted symplectic / DT of a CY 3-fold as the worked example.
Core¶
Shifted symplectic structures (Pantev–Toën–Vaquié–Vezzosi)¶
A \(p\)-shifted symplectic structure on a derived stack \(X\): a symplectic form \(\omega\in\mathrm{Ext}^{2p}(\mathbb{L}_X,\mathcal{O})\), the \(p\)-shifted, the non-degeneracy the \(\omega:\mathbb{L}_X\xrightarrow{\sim}\mathbb{L}_X^\vee[2p]\) (the isomorphism of the cotangent with its dual shifted by \(2p\)).
| \(p\)-shifted symplectic | Description |
|---|---|
| \(\omega\in\mathrm{Ext}^{2p}(\mathbb{L}_X,\mathcal{O})\) | the \(p\)-shifted symplectic form (the degree \(2p\)) |
| Non-degeneracy | \(\omega:\mathbb{L}_X\xrightarrow{\sim}\mathbb{L}_X^\vee[2p]\) (the isomorphism, the shifted) |
| \(0\)-shifted | the smooth symplectic variety (the classical symplectic) |
| \((-1)\)-shifted | the critical locus / the derived moduli (the DT) |
| \((-2)\)-shifted | the shift / the higher (the \((-2)\) the higher symplectic) |
The \((-1)\)-shifted symplectic and the Donaldson–Thomas¶
Theorem (Pantev–Toën–Vaquié–Vezzosi). The critical locus \(\mathrm{Crit}(f)\) (the derived zero locus of \(df:X\to T^*X\), the derived critical locus) has a \((-1)\)-shifted symplectic.
| \((-1)\)-shifted / DT | Description |
|---|---|
| Critical locus \(\mathrm{Crit}(f)\) | the derived zero locus of \(df\) (the \((-1)\)-shifted symplectic, the PTVV) |
| DT invariants | the Behrend–Fantechi / the virtual count: \(\mathrm{DT}(X,\beta,n)=\int_{[\mathrm{Perf}_X]^{\mathrm{vir}}}1\) (the count of the stable sheaves / the perfect complexes) |
| DT / PT / GV | the Donaldson–Thomas / the Pandharipande–Thomas / the Gopakumar–Vafa (the mirror symmetry / GW the GW / the DT the BPS / the GV the physical) |
| MNOP | the Maulik–Nekrasov–Okounkov–Pandharipande: the GW / DT the equivalence for the CY 3-fold (the conjecture / the partial) |
The shifted Poisson and the deformation quantization¶
| Shifted Poisson / quantization | Description |
|---|---|
| Shifted Poisson \(\pi\) | \(\pi\in\mathrm{Ext}^{-2p}(\mathrm{Sym}^2\mathbb{L}_X,\mathcal{O})\) (the Poisson the shifted / the degenerate symplectic) |
| Deformation quantization | the \(p\)-shifted Poisson \(\to\) the \(E_n\)-algebra / the deformation quantization (the Pantev–Safronov–Toën) |
| \(E_n\)-algebra | the deformation quantization the \(E_n\) (the \(n\)-fold loop, the higher / the shifted Poisson the \(E_n\)) |
| NCG / free probability | the shifted Poisson / the free Poisson / the noncommutative (the connection to the free probability / the NCG) |
The shifted symplectic and the mirror symmetry¶
| Shifted symplectic / mirror | Description |
|---|---|
| Shifted symplectic of \(\mathrm{Perf}_X\) | the \((-1)\)-shifted symplectic of the derived moduli (the DT) |
| Shifted symplectic of \(\mathrm{Fuk}\) | the \((-1)\)-shifted symplectic of the wrapped Fukaya / the HMS (the mirror symmetry / Fukaya — the B-side and the A-side the same shifted symplectic) |
| Mirror as shifted symplectic | the HMS the equivalence of the shifted symplectic derived stacks (the B-side the \(\mathrm{Perf}\) / the A-side the \(\mathrm{Fuk}\), the shifted symplectic the same) |
See it¶
\((-1)\)-shifted symplectic of the derived critical locus and the DT invariants of a CY 3-fold. For a CY 3-fold \(X\) (the Calabi–Yau, \(\omega_X\simeq\mathcal{O}_X\), \(\dim X=3\)), the derived moduli \(\mathrm{Perf}_X\) at \([E]\) (a stable sheaf / a perfect complex) has the \((-1)\)-shifted symplectic (the Pantev–Toën–Vaquié–Vezzosi): the \(\mathrm{Ext}^1(E,E)\) the symplectic, the \(\mathrm{Ext}^0(E,E)\) the Poisson, the \(\mathrm{Ext}^2(E,E)\) the obstruction, the CY the Serre duality \(\mathrm{Ext}^i(E,E)\cong\mathrm{Ext}^{3-i}(E,E)^*\) giving the symplectic pairing on \(\mathrm{Ext}^1\) (the \(\mathrm{Ext}^1\times\mathrm{Ext}^1\to\mathrm{Ext}^3\cong\mathbb{C}\) the Serre duality, the symplectic / the \((-1)\)-shifted from the \(\mathrm{Ext}^0\) the degree \(-1\)). The DT invariant: the virtual count \(\mathrm{DT}(X,\beta,n)=\int_{[\mathrm{Perf}_X]^{\mathrm{vir}}}1\) (the VFC / the \((-1)\)-shifted symplectic, the Behrend–Fantechi the virtual count, the DT the count of the stable sheaves). The mirror: the GW / the DT the BPS / the GV (the mirror symmetry / GW the GW / the DT the same BPS invariants, the GV the physical, the MNOP the GW / DT the equivalence for the CY 3-fold). The key point: the derived moduli \(\mathrm{Perf}_X\) the \((-1)\)-shifted symplectic, the DT invariants the virtual count, the mirror symmetry the shifted symplectic equivalence — the derived AG / shifted symplectic the framework for the DT / the mirror.
Check¶
Show that the \((-1)\)-shifted symplectic of \(\mathrm{Perf}_X\) for a CY 3-fold \(X\) comes from the Serre duality, and identify the symplectic pairing.
The CY 3-fold \(X\): \(\omega_X\simeq\mathcal{O}_X\), \(\dim X=3\). The derived moduli \(\mathrm{Perf}_X\) at \([E]\): the tangent \(T=R\mathrm{Hom}(E,E)[1]\), the \(\mathrm{Ext}^i(E,E)\) the cohomology (\(\mathrm{Ext}^0\) the automorphisms / degree \(-1\), \(\mathrm{Ext}^1\) the deformations / degree \(0\), \(\mathrm{Ext}^2\) the obstructions / degree \(1\), \(\mathrm{Ext}^3\) the Serre dual of \(\mathrm{Ext}^0\) / degree \(2\)). The Serre duality: \(\mathrm{Ext}^i(E,E)\cong\mathrm{Ext}^{3-i}(E,E)^*\) (the CY, the \(\omega=\mathcal{O}\), the Serre duality the \(\mathrm{Ext}^i\cong\mathrm{Ext}^{3-i*}\)). The symplectic pairing: the \(\mathrm{Ext}^1\times\mathrm{Ext}^1\to\mathrm{Ext}^3\cong\mathbb{C}\) (the Yoneda / the Serre duality, the \(\mathrm{Ext}^1\times\mathrm{Ext}^1\to\mathrm{Ext}^2\)... the correct: the pairing \(\mathrm{Ext}^1\times\mathrm{Ext}^1\to\mathrm{Ext}^3\cong\mathbb{C}\) via the Yoneda / the composition, the \((-1)\)-shifted: the \(\mathrm{Ext}^1\) the degree \(0\) of the tangent, the symplectic the degree \(0\) the \((-1)\)-shifted from the \(\mathrm{Ext}^0\) the degree \(-1\)). So the \((-1)\)-shifted symplectic from the Serre duality (the CY the \(\mathrm{Ext}^i\cong\mathrm{Ext}^{3-i*}\)), the pairing on \(\mathrm{Ext}^1\) the symplectic, the \(\mathrm{Ext}^0\) the Poisson (the \((-1)\)-shifted: the symplectic the degree \(0\), the Poisson the degree \(-1\)).
Explain the shifted Poisson / the deformation quantization, and identify the \(E_n\) and the NCG connection.
The \(p\)-shifted Poisson \(\pi\in\mathrm{Ext}^{-2p}(\mathrm{Sym}^2\mathbb{L}_X,\mathcal{O})\) (the Poisson bivector the shifted, the degenerate symplectic — the \(p\)-shifted Poisson the analog of the classical Poisson, the shifted by \(-2p\)). The deformation quantization (Pantev–Safronov–Toën): the \(p\)-shifted Poisson \(\to\) the \(E_n\)-algebra / the deformation quantization (the \(E_n\) the \(n\)-fold loop, the \(n=1\) the associative, the \(n=2\) the braided, the deformation the \(E_n\)-deformation of the Poisson). The \(E_n\): the \(E_n\)-algebra the deformation quantization of the \(p\)-shifted Poisson (the \(E_n\) the operad the little \(n\)-disks, the deformation the \(E_n\)-deformation, the \(p\)-shifted the \(n=1-p\)... the relation \(p\) and \(n\)). The NCG / free probability connection: the shifted Poisson / the free Poisson / the noncommutative (the \(p\)-shifted Poisson the noncommutative, the \(E_n\) the deformation the noncommutative, the free probability the free Poisson / the free cumulants the noncommutative, the NCG the noncommutative the connection). The shifted Poisson the noncommutative / the free, the deformation quantization the \(E_n\), the NCG / the free probability the structural connection.
Pitfalls¶
- The shifted symplectic (the PTVV) the \(p\)-shifted, the \((-1)\)-shifted the DT / the critical locus, the \(0\) the classical, the \((-2)\) the higher — the shift is the degree, not the symplectic (the \(p\)-shifted the degree \(2p\), the \((-1)\)-shifted the degree \(-2\), the symplectic the shifted form).
- The DT invariants the VFC / the \((-1)\)-shifted symplectic (the Behrend–Fantechi / the PTVV), the MNOP the GW / DT the equivalence for the CY 3-fold (the conjecture / the partial — the Maulik–Nekrasov–Okounkov–Pandharipande, the equivalence the conjecture, the partial the proven cases).
- The shifted Poisson / the deformation quantization (the Pantev–Safronov–Toën) the \(E_n\) the deformation, the connection to the NCG / the free probability the structural, not the direct (the shifted Poisson the noncommutative, the \(E_n\) the deformation, the free probability the free Poisson — the structural / the categorical connection, not the direct numerical).
- The mirror symmetry as the shifted symplectic equivalence (the HMS the equivalence of the shifted symplectic derived stacks) — the B-side the \(\mathrm{Perf}\) / the A-side the \(\mathrm{Fuk}\), the shifted symplectic the same — the structural / the categorical, not the numerical (the numerical the GW / the DT the MNOP, the structural the shifted symplectic).
- The critical locus \(\mathrm{Crit}(f)\) the \((-1)\)-shifted symplectic (the PTVV) — the derived zero locus of \(df\), the \((-1)\)-shifted the natural / the intrinsic, not the added structure.
Refs¶
- Pantev–Toën–Vaquié–Vezzosi, "Shifted symplectic structures" (Publ. Math. IHÉS 2013)
- Behrend–Fantechi, "The intrinsic normal cone" (Invent. 1997); the VFC
- Maulik–Nekrasov–Okounkov–Pandharipande, "Gromov–Witten theory and Donaldson–Thomas theory I–II" (2006)
- Pantev–Safronov–Toën, "Shifted symplectic structures and deformation quantization" (2019); the \(E_n\)-deformation
Next¶
DAG index · Foundations · Ind-Coherent Sheaves · Derived Moduli & Perfect · D-Modules & Crystals · Mirror symmetry / GW · Mirror symmetry / Fukaya · Free probability