Quadratic Gromov–Witten Invariants

• Quadratic Gromov–Witten invariants are refined counts in enumerative geometry that incorporate quadratic structures from Hirota equations, quadric surfaces, and bilinear forms.
• They connect methods across diverse settings, such as orbifold descendant potentials, type-D monodromy in complete intersections, and Grothendieck–Witt valued arithmetic invariants.
• This framework unifies integrable hierarchies and reconstruction techniques, enabling precise curve counts in complex, real, and arithmetic geometries.

Searching arXiv for recent and foundational papers on quadratic Gromov–Witten invariants and related quadratic structures. Searching "quadratic Gromov-Witten invariants quadrics Welschinger Witt Hirota equations". Quadratic Gromov–Witten invariants are not a single universally fixed object but a family of quadratic structures that recur in Gromov–Witten theory. In current usage, the phrase can refer to quadratic Hirota relations satisfied by descendant potentials of orbifold curves, to enumerative formulas in which curve counts are reduced to geometry on quadric surfaces, to the exceptional genus-$0$ theory of complete intersections of two quadrics, and to Grothendieck–Witt or Witt valued refinements of genus-$0$ curve counts over general fields (Cheng et al., 2019, Brugalle et al., 2015, Hu, 2021, Brugallé et al., 4 Sep 2025).

1. Terminological scope and principal meanings

The literature uses the adjective “quadratic” in several distinct but related senses. In one direction, it refers to quadratic equations for generating functions: Hirota quadratic equations impose bilinear constraints on a tau-function, and therefore quadratic relations on Gromov–Witten correlators. In another, it refers to the geometry of quadrics: curve counts on a target such as $\mathbb{CP}^3$ are expressed through counts on smooth quadrics $\mathbb{CP}^1\times\mathbb{CP}^1$. In a third, it refers to targets defined by two quadratic equations, namely complete intersections of two quadrics. In a fourth, it denotes genuinely quadratic-form-valued invariants in $GW(k)$ or $W(k)$, refining numerical curve counts over nonclosed fields.

Setting	Quadratic aspect	Representative source
Fano orbifold line $\mathbb{P}_{n-2,2,2}^1$	Hirota quadratic equations for the descendant potential	(Cheng et al., 2019)
$\mathbb{CP}^3$ and pencils of quadrics	Reduction of curve counts to smooth quadrics and an elliptic base curve	(Brugalle et al., 2015)
Even-dimensional $X_n(2,2)\subset \mathbb{P}^{n+2}$	Genus-$0$ GW theory of intersections of two quadrics	(Hu, 2021, Gubarevich, 21 Dec 2025)
Rational del Pezzo surfaces over $0$0	$0$1- and $0$2-valued refinements of genus-$0$3 counts	(Brugallé et al., 4 Sep 2025, Brugallé et al., 21 Jun 2025)

This multiplicity of meanings is substantive rather than accidental. Each version isolates a quadratic structure already present in the geometry: bilinear residue identities, quadric fibrations, quadratic defining equations, or quadratic forms attached to local enumerative data.

2. Hirota quadratic equations and descendant potentials

For the orbifold line $0$4, the state space is the Chen–Ruan cohomology

$0$5

with untwisted basis classes $0$6, $0$7, twisted-sector basis elements $0$8, Chen–Ruan degrees

$0$9

and orbifold Poincaré pairing

$\mathbb{CP}^3$0

Its descendant theory is encoded in the total descendant potential

$\mathbb{CP}^3$1

After the dilaton shift $\mathbb{CP}^3$2, the potential is viewed in Givental’s symplectic formalism on

$\mathbb{CP}^3$3

The key statement is that, after an explicit linear change of variables from the $\mathbb{CP}^3$4-coordinates to hierarchy times $\mathbb{CP}^3$5 and a spatial variable $\mathbb{CP}^3$6, the function

$\mathbb{CP}^3$7

is a tau-function of the extended $\mathbb{CP}^3$8-Toda, equivalently extended type-$\mathbb{CP}^3$9 Kac–Wakimoto, hierarchy (Cheng et al., 2019).

The quadratic content appears in the Hirota quadratic equations. Using calibrated periods $\mathbb{CP}^1\times\mathbb{CP}^1$0, Givental quantization, and vertex operators $\mathbb{CP}^1\times\mathbb{CP}^1$1, one obtains the condition that for every $\mathbb{CP}^1\times\mathbb{CP}^1$2,

$\mathbb{CP}^1\times\mathbb{CP}^1$3

evaluated at

$\mathbb{CP}^1\times\mathbb{CP}^1$4

is regular in $\mathbb{CP}^1\times\mathbb{CP}^1$5. Because the equations are written on $\mathbb{CP}^1\times\mathbb{CP}^1$6, their coefficient expansion yields quadratic relations among descendant correlators in all genera and degrees. The paper’s Theorem 5 states that the total descendant potential of $\mathbb{CP}^1\times\mathbb{CP}^1$7 satisfies these HQEs, so the corresponding Gromov–Witten theory is governed by a coherent quadratic integrable structure (Cheng et al., 2019).

This framework is tied to type-$\mathbb{CP}^1\times\mathbb{CP}^1$8 representation theory. The orbifold $\mathbb{CP}^1\times\mathbb{CP}^1$9 is the $GW(k)$0 member of the ADE classification of Fano orbifold lines; its reflection data form a type-$GW(k)$1 root system, and the monodromy acts by a product of simple reflections. In this usage, “quadratic Gromov–Witten invariants” means quadratic relations on generating functions rather than quadratic-form-valued counts.

3. Quadrics, pencils of quadrics, and curve counts on $GW(k)$2

A different use of “quadratic” occurs in enumerative geometry of $GW(k)$3. Brugallé and Georgieva study genus-$GW(k)$4 Gromov–Witten and Gromov–Witten–Welschinger invariants of $GW(k)$5 through a pencil of quadric surfaces with base locus a nondegenerate elliptic curve $GW(k)$6 of degree $GW(k)$7. Every smooth quadric in the pencil is isomorphic to $GW(k)$8, and a degree-$GW(k)$9 rational curve in $W(k)$0 through $W(k)$1 suitably chosen points on $W(k)$2 is forced to lie on one of these quadrics (Brugalle et al., 2015).

The complex formula is

$W(k)$3

Here $W(k)$4 counts degree-$W(k)$5 rational curves in $W(k)$6 through $W(k)$7 generic points, while $W(k)$8 counts rational curves of bidegree $W(k)$9 through $\mathbb{P}_{n-2,2,2}^1$0 generic points. The square factor $\mathbb{P}_{n-2,2,2}^1$1 comes from the number of quadrics in the pencil associated with a given bidegree, via the elliptic group law on $\mathbb{P}_{n-2,2,2}^1$2 (Brugalle et al., 2015).

In the real case, with $\mathbb{P}_{n-2,2,2}^1$3 odd and $\mathbb{P}_{n-2,2,2}^1$4 conjugate pairs among the point constraints, the corresponding formula is

$\mathbb{P}_{n-2,2,2}^1$5

The coefficient is now $\mathbb{P}_{n-2,2,2}^1$6, reflecting both the number of real quadrics in the pencil and the comparison of Welschinger signs in $\mathbb{P}_{n-2,2,2}^1$7 and $\mathbb{P}_{n-2,2,2}^1$8 (Brugalle et al., 2015).

In this setting the word “quadratic” has two precise meanings. Geometrically, the computation proceeds through quadric surfaces. Algebraically, the contribution of a bidegree $\mathbb{P}_{n-2,2,2}^1$9 is weighted by a square $\mathbb{CP}^3$0 in the complex count and by a linear factor $\mathbb{CP}^3$1 with sign in the real count. The resulting formulas realize Kollár’s suggestion that enumerative invariants of $\mathbb{CP}^3$2 can be expressed in terms of invariants of $\mathbb{CP}^3$3 and the elliptic base curve of a pencil of quadrics (Brugalle et al., 2015).

4. Complete intersections of two quadrics and type-$\mathbb{CP}^3$4 monodromy

For an even-dimensional smooth complete intersection

$\mathbb{CP}^3$5

the genus-$\mathbb{CP}^3$6 Gromov–Witten theory is exceptional among complete intersections. The cohomology splits into ambient and primitive parts,

$\mathbb{CP}^3$7

with $\mathbb{CP}^3$8. Deligne monodromy identifies the primitive lattice with type $\mathbb{CP}^3$9, and the primitive sector carries a Weyl-group $X_n(2,2)\subset \mathbb{P}^{n+2}$0 symmetry rather than full orthogonal symmetry (Hu, 2021).

Hu computes all length-$X_n(2,2)\subset \mathbb{P}^{n+2}$1 genus-$X_n(2,2)\subset \mathbb{P}^{n+2}$2 invariants and proves that, except for one special primitive correlator, all genus-$X_n(2,2)\subset \mathbb{P}^{n+2}$3 primary invariants can be reconstructed from length-$X_n(2,2)\subset \mathbb{P}^{n+2}$4 data by WDVV and monodromy considerations. In an orthonormal primitive basis $X_n(2,2)\subset \mathbb{P}^{n+2}$5, the basic four-point primitive invariants satisfy

$X_n(2,2)\subset \mathbb{P}^{n+2}$6

The remaining undetermined datum is the special correlator

$X_n(2,2)\subset \mathbb{P}^{n+2}$7

The same work proves that the genus-$X_n(2,2)\subset \mathbb{P}^{n+2}$8 potential has positive radius of convergence and that, although the small quantum cohomology is not semisimple, the associated big Frobenius manifold is generically tame semisimple (Hu, 2021).

A later paper computes the remaining primitive invariant by Jun Li’s degeneration formula. For an even-dimensional complete intersection of two quadrics

$X_n(2,2)\subset \mathbb{P}^{n+2}$9

it constructs a semistable degeneration to a union of two quadrics meeting along a smooth divisor and shows that there exists an orthonormal basis $0$0 of $0$1 such that

$0$2

The argument reduces the invariant to relative invariants on the two components of the degeneration and then excludes all contributions by virtual-dimension estimates (Gubarevich, 21 Dec 2025).

These two descriptions make clear that the quadratic character of the theory lies in both the target geometry and the symmetry group. The target is cut out by two quadratic equations, while the primitive sector is governed by a type-$0$3 Weyl group. The sources also indicate a subtle basis dependence of the exceptional primitive multilinear form. Since the theory is not forced to be fully $0$4-invariant, a plausible implication is that comparing the distinguished primitive correlators in different papers requires careful attention to basis choice and normalization rather than only to degree or length (Hu, 2021, Gubarevich, 21 Dec 2025).

5. Quadratic-form-valued Gromov–Witten invariants over general fields

A more literal meaning of “quadratic Gromov–Witten invariant” appears in arithmetic enumerative geometry. For an $0$5-connected del Pezzo surface $0$6 over a perfect field $0$7 of characteristic $0$8, an effective divisor class $0$9, and

$0$00

one considers the evaluation morphism on an open locus of genus-$0$01 stable maps with smooth source and ordinary double point image singularities. The evaluation map carries a canonical double-point orientation, and its $0$02-degree defines a Grothendieck–Witt valued invariant

$0$03

for each field extension $0$04 and each degree-$0$05 étale $0$06-algebra $0$07; its image in the Witt ring is

$0$08

The rank of $0$09 is the ordinary genus-$0$10 Gromov–Witten invariant, so these are genuine quadratic refinements rather than unrelated replacements (Brugallé et al., 4 Sep 2025).

Over $0$11, the signature of $0$12 recovers the Welschinger invariant with $0$13 conjugate pairs among the point constraints. The same paper constructs Witt-valued “Welschinger–Witt invariants” $0$14 for $0$15, and more generally multivariable invariants $0$16 for blow-ups of $0$17, by packaging entire families of real Welschinger invariants into Witt invariants in the sense of Serre. It proves that quadratic Gromov–Witten invariants are themselves Witt invariants, that they are unramified away from residue characteristics $0$18 and $0$19, and conjectures that for $0$20-rational del Pezzo surfaces the quadratic GW invariants coincide with the corresponding Welschinger–Witt invariants. This conjectural identification is proved for $0$21-rational del Pezzo surfaces of degree at least $0$22 (Brugallé et al., 4 Sep 2025).

The quadratic Abramovich–Bertram formula gives a surgery law for these $0$23-valued counts. For a $0$24-surgery of a 1-nodal Lefschetz fibration of del Pezzo surfaces, with vanishing cycle $0$25, it expresses the invariant of the twisted generic fiber $0$26 in terms of the split fiber $0$27: $0$28 This formula is built from enriched and twisted binomial coefficients in $0$29, and it yields applications to rational del Pezzo surfaces of degree at least $0$30, some cubic surfaces, point constraints over quadratic extensions, and a Dehn-twist invariance statement

$0$31

In this arithmetic usage, “quadratic” refers literally to values in Grothendieck–Witt groups and to local quadratic masses attached to nodes of rational curves (Brugallé et al., 21 Jun 2025).

6. Structural themes, comparisons, and directions

Across these different settings, several common structures recur. Type-$0$32 symmetry is one of them: the orbifold line $0$33 leads to an extended type-$0$34 Kac–Wakimoto hierarchy, while even-dimensional intersections of two quadrics have primitive monodromy of type $0$35. Quadrics themselves recur both as ambient geometric objects, as in pencils of quadrics in $0$36, and as defining equations of exceptional Fano targets. Over general fields, the same adjective “quadratic” shifts from geometry to arithmetic, where curve counts take values in $0$37 and $0$38 (Cheng et al., 2019, Brugalle et al., 2015, Hu, 2021, Brugallé et al., 4 Sep 2025).

The four meanings are therefore best regarded as complementary rather than competing. Hirota quadratic equations organize descendants into bilinear residue identities. Pencils of quadrics reduce difficult threefold counts to surface counts and elliptic curve arithmetic. Complete intersections of two quadrics provide an exceptional primitive sector governed by Weyl-group rather than full orthogonal symmetry. Witt-valued quadratic invariants refine the numerical theory by retaining arithmetic information lost after taking ranks or signatures.

Several research directions are explicitly indicated. For ADE orbifold lines, the explicit construction of extended hierarchies beyond type $0$39, especially in type $0$40, remains open, as does the extension of such integrable descriptions to higher-dimensional semisimple targets (Cheng et al., 2019). For rational surfaces, the conjectural equality between Welschinger–Witt invariants and quadratic Gromov–Witten invariants is proved only in degree $0$41 del Pezzo cases, and its extension to broader classes of targets remains a central problem (Brugallé et al., 4 Sep 2025). The quadratic Abramovich–Bertram formalism suggests further developments for higher genus, more general surgeries, and additional non-toric Fano surfaces and threefolds (Brugallé et al., 21 Jun 2025). For complete intersections of two quadrics, the relation between monodromy symmetry, basis dependence, and the exceptional primitive correlator remains a natural locus for comparison between reconstruction, degeneration, and enumerative descriptions (Hu, 2021, Gubarevich, 21 Dec 2025).

In this broad sense, quadratic Gromov–Witten invariants form a constellation of ideas rather than a single definition. What unifies them is the systematic appearance of quadratic structure—bilinear Hirota identities, quadratic hypersurfaces, type-$0$42 Weyl symmetries, or quadratic forms on arithmetic counts—inside the organization of Gromov–Witten theory.