Unoriented Donaldson–Thomas Invariants

• Unoriented Donaldson–Thomas invariants are a generalization of DT invariants that count self-dual complexes equipped with orthogonal or symplectic structures.
• They use the motivic Hall algebra and its orthosymplectic module to derive explicit wall-crossing formulae and compute both numerical and motivic invariants.
• Key examples include orthogonal/symplectic complexes on Calabi–Yau 3-folds and self-dual quiver representations, providing actionable insights into enumerative geometry.

Unoriented Donaldson–Thomas (DT) invariants generalize the theory of DT invariants from counting stable objects with structure group $\mathrm{GL}(n)$ to counting objects equipped with orthogonal or symplectic structures, corresponding to the groups $\mathrm{O}(n)$ and $\mathrm{Sp}(2n)$, respectively. These invariants arise naturally in the enumerative geometry of Calabi–Yau threefolds and linear categories, extending the classical motivic DT framework to include self-dual complexes and representations. The theory is formulated using the motivic Hall algebra and its orthosymplectic Hall module counterpart, enabling the construction and computation of numerical and motivic DT invariants in the unoriented setting, with explicit wall-crossing formulae and several families of concrete examples (Bu, 26 Mar 2025).

1. Framework: Motivic Hall Algebra and Orthosymplectic Module

The construction of unoriented DT invariants begins with the study of algebraic stacks $\mathcal X$ over a field $K$ equipped with:

• A commutative monoid law $$\oplus\colon \mathcal X \times \mathcal X \rightarrow \mathcal X$$,
• A $\mathbb{G}_m$-action,
• An involution $\vee\colon \mathcal X \rightarrow \mathcal X$ (covering inversion on $\mathbb{G}_m$), whose fixed locus $\mathcal X^{sd} = \mathcal X^{\mathbb{Z}_2}$ encodes self-dual (orthogonal or symplectic) objects.

The motivic Hall algebra $M(\mathcal X)$ is defined as a (completed) ring of motives over $\mathcal X$, with the Hall product $a * b$ given by a composition of pullback along the associated-graded filtration and pushforward by the “total” map. The algebraic structure is associative with a unit.

For the self-dual part, $M(\mathcal X^{sd})$ becomes a left $M(\mathcal X)$-module, via the product $a \diamond m$, ensuring the compatibility of mixed-filtration phenomena. This extension to the orthosymplectic context encodes the enumerative geometry of objects with extra symmetry—specifically, stable objects equipped with nondegenerate forms invariant under $\vee$.

2. Stability Conditions and $\epsilon$-Motives

Stability conditions $$\tau \colon \pi_0(\mathcal X) \setminus \{0\} \to T$$ (where $T$ is totally ordered) carve out semistable loci inside $\mathcal X$. Following the Joyce–Song approach, the $\epsilon$-motives are constructed as specific linear combinations of semistable strata: $$\epsilon_\alpha(\tau) = \sum_{n>0,\,\alpha_1+\dots + \alpha_n=\alpha,\,\tau(\alpha_i)\equiv \mathrm{const}} \frac{(-1)^{n-1}}{n}\, [\mathcal X^\mathrm{ss}_{\alpha_1}]*\cdots *[\mathcal X^\mathrm{ss}_{\alpha_n}] \in M(\mathcal X_\alpha).$$ In the self-dual context, similar expressions $\epsilon^{sd}_\theta(\tau)$ are formed using “diamond” products and binomial coefficients $\binom{-1/2}{n}$, capturing sums over self-dual decompositions with controlled types.

A crucial structural result is the no-pole theorem (virtual-rank purity), which implies that $\epsilon$-motives and their self-dual analogs lie in the appropriate summands such that numerical invariants (Euler characteristics) are well-defined.

3. Definition of Unoriented Donaldson–Thomas Invariants

Given a $(-1)$-shifted symplectic stack $\mathcal X$ with orientation and Behrend function $\nu$, the (numerical) DT invariants are defined for nonzero $\alpha$ as: $\DT_\alpha(\tau) = \int_{\mathcal X_\alpha} (1-\mathbb L) \,\epsilon_\alpha(\tau)\, \nu_\mathcal X\, d\chi \in \mathbb Q$ with analogous definitions for the self-dual locus: $\DT^{sd}_\theta(\tau) = \int_{\mathcal X^{sd}_\theta} \epsilon^{sd}_\theta(\tau)\,\nu_{\mathcal X^{sd}}\,d\chi \in \mathbb Q.$ There are motivic lifts where $(1-\mathbb L)\cdots d\chi$ is replaced by $$(\mathbb L^{1/2} - \mathbb L^{-1/2})\cdot (-)\,\nu_\mathrm{mot}$$, yielding values in the ring of monodromic motives.

4. Wall-Crossing Formulae

A core feature of DT theory is the wall-crossing phenomenon, where invariants depend on the choice of stability condition. For two permissible stability conditions $\tau_\pm$ dominated by a common $\tau_0$, the $\epsilon$-motives satisfy combinatorial re-expansions: $$\epsilon_\alpha(\tau_-) = \sum_{\alpha_1+\cdots+\alpha_n=\alpha} U(\alpha_1,\ldots,\alpha_n;\tau_+,\tau_-) \,\epsilon_{\alpha_1}(\tau_+)*\cdots*\epsilon_{\alpha_n}(\tau_+),$$ and similarly for their self-dual analogs.

After integrating against the Behrend function and using a motivic Behrend-integral identity, wall-crossing for DT invariants takes the explicit form: $\DT_\alpha(\tau_-) = \sum_{\alpha=\sum\alpha_i} \tilde U(\alpha_1,\ldots,\alpha_n;\tau_+,\tau_-) \,\ell(\alpha_1,\ldots,\alpha_n)\DT_{\alpha_1}(\tau_+)\cdots\DT_{\alpha_n}(\tau_+),$ with explicit rational and integral coefficients depending on the phases $\tau_\pm$, and similarly for the self-dual case. These wall-crossing structures mirror those in the classical Hall algebra context, adapted to the orthosymplectic module formalism (Bu, 26 Mar 2025).

5. Principal Examples: Calabi–Yau 3-folds and Self-Dual Quivers

Two primary families instantiate the theory:

A) Orthogonal/Symplectic Perfect Complexes on a Calabi–Yau 3-fold $Y$:

• $\mathcal X = \mathrm{Perf}(Y)$, with the self-dual locus determined by duality $D(E) = \mathrm{RHom}(I^*E, L)[s]$.
• Self-dual Bridgeland stability slices $\tau \in \mathrm{Stab}^{sd}(Y)$ enable the computation of $\DT_\alpha(\tau)$ and its motivic analogs.
• Generating series (partition functions) can be assembled; for trivial involution one recovers the ordinary DT series, while in the orthosymplectic case one obtains an “orientifold” modification depending on the geometric data.

B) Self-Dual Representations of a Self-Dual Quiver $(Q,W)$:

• For $W=0$, moduli stacks $\mathcal X_Q$, $\mathcal X_Q^{sd}$ parametrize ordinary and self-dual representations.
• Slope stability for $Q$ induces a corresponding stability condition.
• In explicit computations:
  • For the one-vertex quiver: $\sum_n q^n \DT^{sd}_{B_n} = (1-q)^{-1/4}$ and $\sum_n q^n \DT^{sd}_{D_n} = (1-q)^{+1/4}$.
  • The affine $A_1$ quiver matches the $\mathbb{P}^1$-Higgs bundle example under the identified derived equivalence, reproducing the generating functions for Higgs moduli (Bu, 26 Mar 2025).

6. Motivic Vafa–Witten Invariants for Surfaces

Extending the framework, motivic versions of Vafa–Witten invariants are defined for orthogonal and symplectic Higgs complexes on algebraic surfaces $S$ with $\mathcal{K}_S^{-1} \succeq 0$ (including del Pezzo, K3, and abelian surfaces). This involves:

• The open substack $\mathrm{T}^*[-1]\mathrm{Perf}(S)_J$ determined by a Bridgeland slice $J$ of length $<1$.
• Definitions:

$\vw_\alpha(\tau) = \int_{(\bm T^*[-1]\mathrm{Perf})_\alpha} (1-\mathbb L)\epsilon_\alpha \nu\,d\chi$

$\vw^{sd}_\theta(\tau) = \int_{(\bm T^*[-1]\mathrm{Perf})^{sd}_\theta} \epsilon^{sd}_\theta \nu^{sd} d\chi$

with similar motivic refinements. For generic Bridgeland stability $\tau$ on K3 or abelian $S$, the invariants remain constant under deformations of $\tau$.

7. Connections and Foundational References

Unoriented DT invariants realize a special case of the intrinsic Donaldson–Thomas theory developed by Bu, Halpern-Leistner, Ibáñez Núñez, and Kinjo, building on the foundational motivic DT theory of Joyce–Song and Kontsevich–Soibelman. For detailed construction, definitions, and proofs, see "Orthosymplectic Donaldson-Thomas theory" by Bu (Bu, 26 Mar 2025) and supplementary work (Lee et al., 2021) and updates), as well as the referenced intrinsic DT and motivic Hall algebra literature. These developments establish explicit, computable invariants with wall-crossing and stability dependence that extend DT theory to new symmetry classes.