K-Theoretic Donaldson Invariants

• K-theoretic Donaldson invariants are defined as holomorphic Euler characteristics of determinant line bundles on moduli spaces of sheaves, providing a K-theoretic refinement of classical intersection numbers.
• They are computed via generating functions, wallcrossing formulas, and blowup structures, yielding universal rational expressions that capture changes in moduli space geometry.
• The framework extends to Higgs sheaves, Donaldson–Thomas theories, and local threefold models, linking gauge theory, enumerative geometry, and categorical refinements.

Searching arXiv for the cited papers and closely related work to ground the article in current arXiv records. K-theoretic Donaldson invariants are holomorphic Euler characteristics of determinant line bundles on moduli spaces of sheaves on surfaces, and more generally arise as virtual holomorphic Euler characteristics associated with virtual structure sheaves on moduli spaces carrying perfect, symmetric perfect, or almost perfect obstruction theories. In the surface-theoretic setting, they refine classical cohomological Donaldson invariants by replacing intersection-theoretic data with classes in $K$-theory, while in Donaldson–Thomas-type settings on Calabi–Yau threefolds and local models such as $\mathbb{A}^3$ they appear as $K$-theoretic refinements of DT counts defined through virtual structure sheaves and equivariant localization (Göttsche et al., 2015). The subject now spans moduli of sheaves on rational and general type surfaces, Higgs sheaves and Vafa–Witten theory, generalized DT theory on Calabi–Yau threefolds, and categorical or higher-rank local theories on $\mathbb{C}^3$ and related spaces (Kiem et al., 2019).

1. Definition on surfaces and determinant line bundles

For a simply connected nonsingular projective surface with anticanonical divisor $-K_X$ ample, fixing an ample divisor $H$, Chern classes $c_1 \in H^2(X,\mathbb{Z})$, $c_2 \in \mathbb{Z}$, and $c=(2,c_1,c_2)$, one writes $M:=M_H(c_1,d)$ for the moduli space of $\mathbb{A}^3$0-semistable sheaves with these Chern classes, where $\mathbb{A}^3$1. For a line bundle $\mathbb{A}^3$2 on $\mathbb{A}^3$3, assuming $\mathbb{A}^3$4 is even, the class

$\mathbb{A}^3$5

determines a determinant line bundle $\mathbb{A}^3$6 via the determinant of cohomology construction for families of sheaves parameterized by $\mathbb{A}^3$7. The K-theoretic Donaldson invariant is then

$\mathbb{A}^3$8

namely the holomorphic Euler characteristic of the determinant line bundle over the moduli space (Göttsche et al., 2015).

A parallel formulation for rational surfaces uses moduli spaces

$\mathbb{A}^3$9

of rank $K$0, $K$1-semistable torsion free sheaves $K$2 with $K$3, $K$4, and $K$5. Given a line bundle $K$6, one again forms the determinant line bundle $K$7 on $K$8, and the invariant is the Euler characteristic

$K$9

The associated generating function is

$\mathbb{C}^3$0

For most cases, higher cohomology vanishes by a general vanishing result, so the Euler characteristic equals the dimension of global sections (Göttsche, 2016).

On more general surfaces with $\mathbb{C}^3$1 and $\mathbb{C}^3$2, K-theoretic Donaldson invariants are described as virtual holomorphic Euler characteristics of determinant line bundles over moduli spaces $\mathbb{C}^3$3 of Gieseker $\mathbb{C}^3$4-semistable torsion-free sheaves of rank $\mathbb{C}^3$5. In the formulation highlighted for arbitrary rank, they are written as

$\mathbb{C}^3$6

for suitable determinant line bundle $\mathbb{C}^3$7 (Göttsche, 2021).

The central distinction from classical Donaldson theory is therefore not the moduli problem itself, but the replacement of cohomological intersection numbers by holomorphic Euler characteristics of determinant line bundles. This gives a genuine $\mathbb{C}^3$8-theoretic refinement rather than merely a repackaging of the classical invariants.

2. Generating functions, wallcrossing, and blowup structure

The surface theory is organized by generating functions. In one standard normalization,

$\mathbb{C}^3$9

with additional modifications for $-K_X$0 to accommodate wallcrossing and blowup compatibility (Göttsche et al., 2015). For rational surfaces, a principal structural result is that the generating functions are rational functions of a special form: for $-K_X$1, $-K_X$2, or a blowup, there is a polynomial $-K_X$3 and integer $-K_X$4 such that

$-K_X$5

for all sufficiently ample $-K_X$6. These expressions are described as Verlinde-type formulas for surfaces (Göttsche, 2016).

Wallcrossing across chambers in the ample cone is described by explicit formulas in Jacobi theta functions and modular forms. In one formulation,

$-K_X$7

where $-K_X$8 is a combination of theta functions depending on a class $-K_X$9, the Chern classes, and $H$0. A crucial result is that the wallcrossing is always a polynomial in $H$1, and the space of allowed jumps is finite (Göttsche et al., 2015). An analogous wallcrossing formula for rational surfaces takes the form

$H$2

with $H$3 again constructed from theta functions (Göttsche, 2016).

Blowup formulas provide another organizing principle. For the blowup $H$4 at a point with exceptional divisor $H$5, conjectural blowup formulas relate K-theoretic Donaldson invariants on $H$6 and $H$7. For $H$8, the case corresponding to K-theoretic Donaldson invariants in the notation of Verlinde numbers, Conjecture 1.3 gives relations of the form

$H$9

for specified ranges of $c_1 \in H^2(X,\mathbb{Z})$0, together with more refined relations for special values of $c_1 \in H^2(X,\mathbb{Z})$1. These formulas are encoded by universal power series $c_1 \in H^2(X,\mathbb{Z})$2, $c_1 \in H^2(X,\mathbb{Z})$3, indexed by subsets $c_1 \in H^2(X,\mathbb{Z})$4, satisfying functional equations such as

$c_1 \in H^2(X,\mathbb{Z})$5

$c_1 \in H^2(X,\mathbb{Z})$6

The dependence on the surface enters entirely through Seiberg–Witten basic classes and their invariants, the quantities $c_1 \in H^2(X,\mathbb{Z})$7, $c_1 \in H^2(X,\mathbb{Z})$8, intersection theory involving $c_1 \in H^2(X,\mathbb{Z})$9 and $c_2 \in \mathbb{Z}$0, and the virtual dimension

$c_2 \in \mathbb{Z}$1

(Göttsche, 2021).

This wallcrossing-and-blowup package is a recurrent feature of the subject. It governs computational access, rationality properties, and the relation between explicit formulas and conjectural universal structures.

3. Explicit formulas on rational surfaces and strange duality

For $c_2 \in \mathbb{Z}$2, rational ruled surfaces, and blowups, explicit formulas are known in rank $c_2 \in \mathbb{Z}$3. On rational ruled surfaces $c_2 \in \mathbb{Z}$4 and $c_2 \in \mathbb{Z}$5, the generating functions for natural line bundles are given by closed expressions. Representative examples include

$c_2 \in \mathbb{Z}$6

$c_2 \in \mathbb{Z}$7

and

$c_2 \in \mathbb{Z}$8

These formulas are stated to be universal for $c_2 \in \mathbb{Z}$9 and $c=(2,c_1,c_2)$0 and for all ample divisors $c=(2,c_1,c_2)$1 with $c=(2,c_1,c_2)$2 (Göttsche et al., 2015).

For $c=(2,c_1,c_2)$3, the blowup formulas allow one to deduce explicit generating functions from those on the blowup at a point. The cited formulas include

$c=(2,c_1,c_2)$4

$c=(2,c_1,c_2)$5

$c=(2,c_1,c_2)$6

$c=(2,c_1,c_2)$7

In the rationality framework of Verlinde-type formulas, one also obtains examples such as

$c=(2,c_1,c_2)$8

and explicit polynomials $c=(2,c_1,c_2)$9 up to $M:=M_H(c_1,d)$0 (Göttsche et al., 2015, Göttsche, 2016).

These computations are closely tied to Le Potier’s strange duality. The duality predicts, for suitable orthogonal Chern data $M:=M_H(c_1,d)$1, a canonical isomorphism

$M:=M_H(c_1,d)$2

Specific cases are proved for $M:=M_H(c_1,d)$3, $M:=M_H(c_1,d)$4, and $M:=M_H(c_1,d)$5, including

$M:=M_H(c_1,d)$6

and further cases with $M:=M_H(c_1,d)$7 or $M:=M_H(c_1,d)$8, $M:=M_H(c_1,d)$9 (Göttsche et al., 2015). The underlying mechanism is that vanishing of higher cohomology converts Euler characteristics into dimensions of spaces of sections, allowing dimension counts from K-theoretic Donaldson series to feed directly into strange duality arguments.

A common misconception is that the explicit formulas on rational surfaces exhaust the subject. In fact, they represent the best-understood computational sector, but the theory extends well beyond rational surfaces through Seiberg–Witten expansions, Higgs-sheaf refinements, and generalized DT constructions.

4. Verlinde-type formulas, Higgs sheaves, and interpolation with Vafa–Witten theory

For surfaces with a holomorphic $\mathbb{A}^3$00-form, the subject expands from moduli of sheaves to moduli of Higgs sheaves. Let

$\mathbb{A}^3$01

be the moduli space of rank $\mathbb{A}^3$02 Gieseker stable torsion-free sheaves, and

$\mathbb{A}^3$03

the moduli space of rank $\mathbb{A}^3$04 Gieseker stable Higgs sheaves $\mathbb{A}^3$05 with $\mathbb{A}^3$06 and $\mathbb{A}^3$07. These spaces carry perfect or symmetric perfect obstruction theories, hence virtual fundamental classes and virtual structure sheaves (Göttsche et al., 2019).

In this framework, K-theoretic Donaldson invariants are written as virtual holomorphic Euler characteristics

$\mathbb{A}^3$08

where $\mathbb{A}^3$09 is a determinant line bundle over $\mathbb{A}^3$10 constructed from a universal sheaf. The Higgs-sheaf theory carries a $\mathbb{A}^3$11-action scaling the Higgs field, whose fixed locus decomposes into an instanton branch $\mathbb{A}^3$12, isomorphic to $\mathbb{A}^3$13, and a monopole branch described in terms of nested Hilbert schemes. For K-theoretic Vafa–Witten invariants, a Nekrasov–Okounkov twisting is introduced: $\mathbb{A}^3$14 in order to ensure symmetry in the refinement parameter $\mathbb{A}^3$15 (Göttsche et al., 2019).

The central conjectural structure is a Verlinde-type formula interpolating between K-theoretic Donaldson invariants and K-theoretic Vafa–Witten invariants. Conjecture 1.2 gives a refined invariant $\mathbb{A}^3$16 as the coefficient of $\mathbb{A}^3$17 in an explicit modular expression involving infinite products, theta functions, the Dedekind eta function, and a sum over Seiberg–Witten basic classes. The stated interpretation is:

• $\mathbb{A}^3$18: recovers the K-theoretic Donaldson invariant formula.
• $\mathbb{A}^3$19, general $\mathbb{A}^3$20: gives the K-theoretic Vafa–Witten formula.
• General $\mathbb{A}^3$21: yields a refined interpolating theory (Göttsche et al., 2019).

For surfaces of simple Seiberg–Witten type, including K3 and minimal general type surfaces, the formulas simplify. In the arbitrary-rank setting, K-theoretic Donaldson invariants are computed as the coefficient of $\mathbb{A}^3$22 in a sum over subsets $\mathbb{A}^3$23 weighted by Seiberg–Witten data and universal series $\mathbb{A}^3$24, $\mathbb{A}^3$25. When only $\mathbb{A}^3$26 and $\mathbb{A}^3$27 are basic classes with Seiberg–Witten invariants $\mathbb{A}^3$28 and $\mathbb{A}^3$29, the formula simplifies to an expression of the form

$\mathbb{A}^3$30

in the notation of the paper (Göttsche, 2021).

This interpolation suggests that K-theoretic Donaldson invariants should be regarded not as an isolated surface theory, but as one limiting face of a broader $\mathbb{A}^3$31-theoretic enumerative package on surfaces that includes sheaf, Higgs-sheaf, and Vafa–Witten-type moduli.

5. Virtual structure sheaves and generalized Donaldson–Thomas theory

A major extension of the subject replaces the classical perfect obstruction theory framework with almost perfect obstruction theory on Deligne–Mumford stacks. An almost perfect obstruction theory consists of an étale covering $\mathbb{A}^3$32, local perfect obstruction theories $\mathbb{A}^3$33, gluing isomorphisms

$\mathbb{A}^3$34

satisfying cocycle conditions and local compatibility on overlaps. The hierarchy

$\mathbb{A}^3$35

is explicitly stated (Kiem et al., 2019).

When a global perfect obstruction theory exists with two-term locally free presentation, the virtual structure sheaf is classically

$\mathbb{A}^3$36

and defines the virtual Euler characteristic

$\mathbb{A}^3$37

For an almost perfect obstruction theory, the obstruction sheaves $\mathbb{A}^3$38 glue to a sheaf stack $\mathbb{A}^3$39, into which the coarse intrinsic normal cone $\mathbb{A}^3$40 embeds. The virtual structure sheaf is then defined by

$\mathbb{A}^3$41

where $\mathbb{A}^3$42 is a $\mathbb{A}^3$43-theoretic Gysin map for sheaf stacks constructed via local charts, Koszul complexes, and descent for Koszul homology sheaves (Kiem et al., 2019).

This yields K-theoretic generalized Donaldson–Thomas invariants

$\mathbb{A}^3$44

The theory applies to Gieseker or slope semistable sheaves, simple perfect complexes, PT-semistable and Bridgeland-semistable complexes, and moduli spaces arising from d-critical structures, derived stacks, or Kirwan partial desingularization (Kiem et al., 2019).

A central theorem is deformation invariance: $\mathbb{A}^3$45 is deformation invariant, and the formation of the virtual structure sheaf commutes with base change in families $\mathbb{A}^3$46 with $\mathbb{A}^3$47 smooth. This is technically realized through deformation to the normal cone, double deformation spaces, and compatibility of the Gysin maps with pullback and specialization (Kiem et al., 2019).

The significance for K-theoretic Donaldson invariants is structural. Surface invariants defined as holomorphic Euler characteristics of determinant line bundles belong to a wider framework in which $\mathbb{A}^3$48-theoretic virtual structure sheaves are the natural receptacle for enumerative data on singular moduli spaces. This does not identify surface Donaldson invariants with threefold DT invariants, but it places them in a common formalism of virtual $\mathbb{A}^3$49-theory.

6. Local threefold models, higher rank, and categorical refinements

Local models on $\mathbb{A}^3$50 and $\mathbb{A}^3$51 provide explicit $\mathbb{A}^3$52-theoretic DT partition functions. For the Quot scheme

$\mathbb{A}^3$53

which parametrizes torsion quotients of length $\mathbb{A}^3$54, the critical locus description furnishes a symmetric perfect obstruction theory, and the torus-fixed points are classified by $\mathbb{A}^3$55-colored plane partitions. The rank $\mathbb{A}^3$56 K-theoretic DT partition function is

$\mathbb{A}^3$57

where $\mathbb{A}^3$58 is the twisted virtual structure sheaf (Fasola et al., 2020).

The main result is the plethystic formula

$\mathbb{A}^3$59

with $\mathbb{A}^3$60 and $\mathbb{A}^3$61, together with the product factorization

$\mathbb{A}^3$62

A nontrivial theorem states that these invariants do not depend on the framing torus equivariant parameters $\mathbb{A}^3$63 (Fasola et al., 2020).

A cohomological reduction yields

$\mathbb{A}^3$64

where $\mathbb{A}^3$65 is the MacMahon function. The same paper also defines elliptic DT invariants via a virtual chiral elliptic genus (Fasola et al., 2020).

A related physics-driven formulation studies higher-rank equivariant K-theoretic Donaldson–Thomas invariants on $\mathbb{A}^3$66 through elliptic genera and plethystic exponentials. The conjectural higher-rank grand canonical partition function is

$\mathbb{A}^3$67

and the rational case factorizes as

$\mathbb{A}^3$68

(Benini et al., 2018).

There is also a categorical refinement. The DT category

$\mathbb{A}^3$69

is defined using matrix factorizations on the non-commutative Hilbert scheme with super-potential

$\mathbb{A}^3$70

whose critical locus is $\mathbb{A}^3$71. Semiorthogonal decompositions of $\mathbb{A}^3$72 are interpreted as categorical wall-crossing formulas of the framed triple loop quiver, built from quasi-BPS categories $\mathbb{A}^3$73. The torus localized $\mathbb{A}^3$74-theory of DT categories has a basis whose cardinality is the number of plane partitions, giving a K-theoretic analogue of MacMahon’s formula (Pădurariu et al., 2022).

These local models do not redefine surface K-theoretic Donaldson invariants, but they supply a local laboratory in which the virtual $\mathbb{A}^3$75-theoretic mechanisms, higher-rank behavior, factorization phenomena, and categorical refinements become completely explicit.

7. Physical derivations, anomalies, and broader enumerative context

A five-dimensional gauge-theoretic interpretation realizes K-theoretic Donaldson invariants as partition functions of $\mathbb{A}^3$76 $\mathbb{A}^3$77 super Yang–Mills theory on $\mathbb{A}^3$78, where $\mathbb{A}^3$79 is a closed smooth four-manifold. A partial topological twisting along $\mathbb{A}^3$80 renders the theory formally independent of the metric on $\mathbb{A}^3$81. The coefficients of the $\mathbb{A}^3$82-expansion of the partition function are Witten indices, identified with $\mathbb{A}^3$83-indices of Dirac operators on moduli spaces of instantons, and these indices are described as special cases of K-theoretic Donaldson invariants (Kim et al., 27 Sep 2025).

For $\mathbb{A}^3$84, the partition function can be derived from integration over the Coulomb branch of the effective $\mathbb{A}^3$85 low-energy theory, while for toric $\mathbb{A}^3$86 one can use equivariant localization with respect to the $\mathbb{A}^3$87 symmetry. The two methods lead to the same results for the wall-crossing formula. When the 't Hooft flux is nonzero and $\mathbb{A}^3$88 is not spin, the $\mathbb{A}^3$89 theory can be anomalous; the anomaly is canceled by coupling to a line bundle with connection for the global $\mathbb{A}^3$90 instanton number symmetry. The anomaly-free condition is

$\mathbb{A}^3$91

If the fluxes are anomalous, the partition function vanishes (Kim et al., 27 Sep 2025).

This physical derivation matches formulas for algebraic surfaces due to Göttsche, Kool, Nakajima, Yoshioka, and Williams, while extending to a larger class of manifolds. It also underscores a point that is sometimes obscured in purely algebro-geometric treatments: K-theoretic Donaldson invariants are not only Euler characteristics on projective moduli spaces but also Witten indices and Dirac indices in a $\mathbb{A}^3$92 topologically twisted gauge theory (Kim et al., 27 Sep 2025).

A related extension appears in the theory of tetrahedron instantons. The moduli space

$\mathbb{A}^3$93

is described as both a Quot scheme and a moduli space of representations of a framed four-loop quiver. It is realized as the zero locus of an isotropic section of a special orthogonal bundle over a smooth ambient non-commutative Quot scheme, which yields a symmetric three-term obstruction theory and a virtual structure sheaf

$\mathbb{A}^3$94

The equivariant K-theoretic invariant is

$\mathbb{A}^3$95

with partition function

$\mathbb{A}^3$96

The explicit formula factorizes into rank-one DT partition functions, and when $\mathbb{A}^3$97 the tetrahedron instanton invariants reduce to rank-one DT invariants for $\mathbb{A}^3$98, recovering Okounkov’s partition function (Fasola et al., 2023).

A plausible implication is that the modern theory of K-theoretic Donaldson invariants is best understood as a network of closely related virtual $\mathbb{A}^3$99-theoretic constructions. Surface Donaldson invariants, Verlinde-type series, Higgs-sheaf refinements, generalized DT invariants on Calabi–Yau threefolds, higher-rank local models, and gauge-theoretic partition functions are not interchangeable objects, but the cited work shows that they are organized by a common set of mechanisms: determinant line bundles, virtual structure sheaves, wallcrossing, blowup formulas, equivariant localization, and modular or plethystic generating functions.