Mirror Symmetric Gamma Conjecture

• Mirror Symmetric Gamma Conjecture posits that the Gamma class governs the asymptotics of quantum differential equations and oscillatory mirror periods.
• It unifies diverse models by linking Dubrovin connections and oscillatory integrals across Fano, Calabi–Yau, and Landau–Ginzburg frameworks.
• Recent proofs combine methods from inverse Mellin transforms, cluster theory, and arithmetic refinements to verify cases in toric, flag, and del Pezzo settings.

The mirror symmetric Gamma conjecture is a family of statements asserting that the Gamma class, or more generally Gamma-modified Chern characters and Gamma-integral structures, governs the asymptotics of quantum differential equations and the leading behavior of exponential periods on the mirror side. In the Fano setting, it identifies the principal asymptotic class of the small quantum connection with the Gamma class and realizes that class by oscillatory integrals of a mirror Landau–Ginzburg potential; in Calabi–Yau settings, it identifies the A-model Gamma-integral structure with natural integral homology or with asymptotics of mirror periods (Galkin et al., 2015, Iritani, 2023, Iritani, 2023).

1. Foundational formulation

For a smooth complex Fano manifold $X$, the small quantum cohomology defines the Dubrovin connection along the $c_1(X)$-direction,

$z\frac{d}{dt}s(t,z)=c_1(X)\star s(t,z),$

and the dominant exponential mode of its flat sections determines the principal asymptotic class $A_X$. Equivalently, $A_X$ is recovered from the small $J$-function by

$$A_X=\lim_{t\to +\infty}\frac{J_X(t)}{\langle[\mathrm{pt}],J_X(t)\rangle}.$$

The Gamma class is

$$\widehat{\Gamma}_X=\prod_{i=1}^{\dim X}\Gamma(1+\delta_i),$$

with $\delta_i$ the Chern roots of $T_X$. Gamma Conjecture I states that $c_1(X)$0 when the relevant semisimplicity and spectral hypotheses hold (Chow, 22 Jan 2025, Galkin et al., 2015).

The mirror-symmetric form replaces the asymptotic analysis of flat sections by oscillatory integrals. In the Fano mirror picture one considers a Landau–Ginzburg potential $c_1(X)$1 and a holomorphic volume form $c_1(X)$2, and the distinguished solution is represented by an integral of the form

$c_1(X)$3

A basic formulation used in the relative and toric literature is

$c_1(X)$4

and its variants with K-theoretic insertions and deformed potentials $c_1(X)$5 (Iritani, 2023, You, 8 Aug 2025).

Across the literature, the conjecture therefore has two mutually reinforcing faces. One is asymptotic and differential-equation theoretic: the Gamma class is the principal asymptotic class of the quantum connection. The other is period-theoretic: a distinguished mirror cycle produces an oscillatory integral whose leading term, after the standard normalization, is the Gamma-calibrated flat section. This dual description is the structural reason the conjecture is called mirror symmetric.

2. Proven Fano cases and geometric models

Several major geometric families now admit proofs or mirror-symmetric realizations of the conjecture.

Setting	Mirror object	Result
Projective space, toric Fano manifolds, toric complete intersections, Grassmannians	Laurent polynomial or Hori–Vafa mirror	Gamma conjecture follows from mirror symmetry (Galkin et al., 2015)
Del Pezzo surfaces	Landau–Ginzburg mirrors; Gross–Hacking–Keel mirror in non-toric cases	Gamma Conjecture I for all del Pezzo surfaces; mirror-symmetric version for degree $c_1(X)$6 (Hu et al., 2019, Fang et al., 2023)
Flag varieties $c_1(X)$7	Rietsch mirror and totally positive part $c_1(X)$8	Gamma conjecture I for any flag variety (Chow, 22 Jan 2025)
Toric Fano orbifolds as toric GIT quotients	Landau–Ginzburg mirror treated by inverse Mellin/Fourier transform	New proof of the mirror symmetric Gamma conjecture (Aleshkin et al., 24 Jan 2025)

For toric Fano manifolds, the decisive mechanism is positivity of the Laurent polynomial mirror and the existence of a unique positive critical point. Galkin–Iritani show that projective spaces, toric Fano manifolds, toric complete intersections, and Grassmannians fit this paradigm, and that Gamma Conjecture I is compatible with taking hyperplane sections via a quantum Lefschetz argument (Galkin et al., 2015). In this framework, the dominant saddle of the oscillatory integral reproduces the dominant eigenvalue of $c_1(X)$9, and the one-loop $z\frac{d}{dt}s(t,z)=c_1(X)\star s(t,z),$0-factors match $z\frac{d}{dt}s(t,z)=c_1(X)\star s(t,z),$1.

For del Pezzo surfaces, Hu–Ke–Li–Yang prove Conjecture $z\frac{d}{dt}s(t,z)=c_1(X)\star s(t,z),$2 and Gamma Conjecture I for all two-dimensional Fano manifolds by combining a generalized Perron–Frobenius theorem with vanishing statements for certain Gromov–Witten invariants (Hu et al., 2019). A complementary mirror-symmetric result treats non-toric del Pezzo surfaces of degree at least $z\frac{d}{dt}s(t,z)=c_1(X)\star s(t,z),$3: the Gross–Hacking–Keel mirror is equipped with explicit cycles mirror to line bundles, and the leading-order oscillatory integrals on those cycles recover the Gamma-modified central charge for arbitrary $z\frac{d}{dt}s(t,z)=c_1(X)\star s(t,z),$4-group insertions generated by line bundles (Fang et al., 2023).

For flag varieties $z\frac{d}{dt}s(t,z)=c_1(X)\star s(t,z),$5, the conjecture acquires a distinctly cluster-theoretic form. The theorem of “Gamma conjecture I for flag varieties” proves that the small quantum cohomology is semisimple, $z\frac{d}{dt}s(t,z)=c_1(X)\star s(t,z),$6 has a simple spectral radius with Perron–Frobenius positivity in the Schubert basis, and the totally positive part $z\frac{d}{dt}s(t,z)=c_1(X)\star s(t,z),$7 of the Rietsch mirror yields the principal asymptotic flat section through the positive thimble integral

$z\frac{d}{dt}s(t,z)=c_1(X)\star s(t,z),$8

The unique non-degenerate positive critical point $z\frac{d}{dt}s(t,z)=c_1(X)\star s(t,z),$9 governs stationary phase, and under the mirror identification one has $A_X$0 for every flag variety (Chow, 22 Jan 2025).

A different proof strategy appears for toric Fano orbifolds presented as toric GIT quotients. There the $A_X$1-equivariant quantum cohomology central charge on $A_X$2 is subjected to an inverse Mellin, or Fourier-type, transform. On the A-side this produces the quantum cohomology central charge of the associated line bundle on the quotient orbifold; on the B-side it becomes the oscillatory integral on the Landau–Ginzburg mirror, and deforming the parameters to real values simultaneously deforms the cycle to the SYZ mirror cycle of the line bundle (Aleshkin et al., 24 Jan 2025).

3. Calabi–Yau, relative, and tropical extensions

The mirror symmetric Gamma conjecture is not confined to compact Fano quantum cohomology. A central extension, due to Iritani, starts from a Fano manifold $A_X$3 satisfying the Fano-side mirror-symmetric identity and applies a Laplace transform to the oscillatory integrals. This yields corresponding statements for the total space $A_X$4 of the canonical bundle and for anticanonical hypersurfaces $A_X$5, as well as more general sums of anti-nef line bundles and nef complete intersections. The resulting formulas express relative or fiber periods on the mirror side in terms of the $A_X$6-functions of $A_X$7 or $A_X$8, together with the appropriate Gamma classes (Iritani, 2023).

A Calabi–Yau variation emphasizes integral structures rather than only dominant asymptotics. In that setting, mirror symmetry identifies the A-model variation of Hodge structure of quantum cohomology with the B-model Gauss–Manin variation, and the Mirror Symmetric Gamma Conjecture states that Iritani’s Gamma-integral structure matches the natural integral local system $A_X$9 of the mirror family. The tropical approach of Abouzaid–Ganatra–Iritani–Sheridan, as explained in “Gamma conjecture and tropical geometry,” shows that $A_X$0-values arise as “error terms of tropicalization” in mirror period asymptotics. In dimension $A_X$1, the universal correction is $A_X$2; in dimension $A_X$3, local vertex contributions produce $A_X$4, and these match the coefficients appearing in the Gamma class expansion (Iritani, 2023).

Relative mirror symmetry furnishes another extension. For a Fano variety $A_X$5 with an snc anticanonical divisor $A_X$6, the mirror is constructed from the degree-zero part of relative quantum cohomology, with theta functions $A_X$7 and superpotential

$A_X$8

The paper “Relative mirror symmetry, theta functions and the Gamma conjecture” proves mirror-symmetric Gamma identities for $A_X$9 and $J$0, equating oscillatory integrals over a real Lefschetz thimble or a compact cycle with pairings of $J$1 against $J$2. It also states that this picture is consistent with counterexamples to the non-mirror-symmetric Gamma conjecture: the mirror-symmetric version remains valid in that relative setting (You, 8 Aug 2025).

Local mirror symmetry admits a related but noncompact formulation. For the canonical bundle $J$3 of a smooth toric Fano variety, one can lift tropical curves associated to holomorphic curves $J$4 to Lagrangian cycles $J$5 in the Gross–Siebert mirror family. The resulting equality

$J$6

matches the B-side central charge defined using the $J$7-class with the A-side period of the holomorphic volume form over $J$8 (Wang, 2020).

A higher-dimensional tropical incarnation appears in “Lifts of cycles in tropical hypersurfaces and the Gamma conjecture.” There, torus-fibered lifts of tropical $J$9-cycles in toric hypersurfaces yield period asymptotics

$$A_X=\lim_{t\to +\infty}\frac{J_X(t)}{\langle[\mathrm{pt}],J_X(t)\rangle}.$$0

which is explicitly presented as a mirror symmetric Gamma conjecture formula (Yamamoto, 9 Feb 2026).

4. Landau–Ginzburg and FJRW analogues

A parallel Gamma-conjectural structure exists in Fan–Jarvis–Ruan–Witten theory for quasi-homogeneous polynomials of general type. In this setting, Shen–Zhang define the index

$$A_X=\lim_{t\to +\infty}\frac{J_X(t)}{\langle[\mathrm{pt}],J_X(t)\rangle}.$$1

so that $$A_X=\lim_{t\to +\infty}\frac{J_X(t)}{\langle[\mathrm{pt}],J_X(t)\rangle}.$$2 is the general-type case, and study the restricted Dubrovin-type connection determined by the small $$A_X=\lim_{t\to +\infty}\frac{J_X(t)}{\langle[\mathrm{pt}],J_X(t)\rangle}.$$3-function and the specialization $$A_X=\lim_{t\to +\infty}\frac{J_X(t)}{\langle[\mathrm{pt}],J_X(t)\rangle}.$$4. The operator

$$A_X=\lim_{t\to +\infty}\frac{J_X(t)}{\langle[\mathrm{pt}],J_X(t)\rangle}.$$5

plays the role of quantum multiplication by $$A_X=\lim_{t\to +\infty}\frac{J_X(t)}{\langle[\mathrm{pt}],J_X(t)\rangle}.$$6, and the paper formulates an LG Quantum Spectrum Conjecture together with LG Gamma Conjectures I and II (Shen et al., 2023).

The LG Gamma class is defined by a Gamma map on the FJRW state space,

$$A_X=\lim_{t\to +\infty}\frac{J_X(t)}{\langle[\mathrm{pt}],J_X(t)\rangle}.$$7

and for the stabilization of the residue field and its twists this yields explicit classes

$$A_X=\lim_{t\to +\infty}\frac{J_X(t)}{\langle[\mathrm{pt}],J_X(t)\rangle}.$$8

These classes are the LG analogues of $$A_X=\lim_{t\to +\infty}\frac{J_X(t)}{\langle[\mathrm{pt}],J_X(t)\rangle}.$$9, and the mirror dictionary is stated explicitly: $$\widehat{\Gamma}_X=\prod_{i=1}^{\dim X}\Gamma(1+\delta_i),$$0 The weak version identifies $$\widehat{\Gamma}_X=\prod_{i=1}^{\dim X}\Gamma(1+\delta_i),$$1 as weak asymptotic classes; the strong version matches Stokes data with Euler pairings in $$\widehat{\Gamma}_X=\prod_{i=1}^{\dim X}\Gamma(1+\delta_i),$$2 (Shen et al., 2023).

The paper proves the spectrum conjecture for mirror simple singularities of ADE type and for Fermat homogeneous polynomials $$\widehat{\Gamma}_X=\prod_{i=1}^{\dim X}\Gamma(1+\delta_i),$$3 with $$\widehat{\Gamma}_X=\prod_{i=1}^{\dim X}\Gamma(1+\delta_i),$$4. It also proves the weak Gamma Conjecture for Fermat polynomials $$\widehat{\Gamma}_X=\prod_{i=1}^{\dim X}\Gamma(1+\delta_i),$$5 and the strong Gamma Conjecture for Fermat homogeneous polynomials. In this form, the mirror symmetric Gamma conjecture becomes an interface between asymptotic analysis of the FJRW quantum connection, the categorical theory of matrix factorizations, and Orlov’s semiorthogonal decompositions (Shen et al., 2023).

5. Categorical, Stokes-theoretic, and integral-structure aspects

The mirror symmetric Gamma conjecture is also a statement about integral structures and Stokes data. In the toric Fano case, Fang–Zhou prove Gamma II by showing that oscillatory integrals on Lefschetz thimbles represent the same relative homology classes as characteristic cycles of constructible sheaves under the coherent–constructible correspondence and the Ganatra–Pardon–Shende equivalence. Through Iritani’s theorem, these oscillatory integrals equal genus-zero descendant potentials twisted by the Gamma class, and the asymptotic basis is identified with the classes $$\widehat{\Gamma}_X=\prod_{i=1}^{\dim X}\Gamma(1+\delta_i),$$6 of a full exceptional collection, with Stokes matrix

$$\widehat{\Gamma}_X=\prod_{i=1}^{\dim X}\Gamma(1+\delta_i),$$7

(Fang et al., 2019).

The paper “Gamma conjecture II via global Gamma-I” refines this picture on the A-side. It introduces the Gamma-I property at points satisfying the SR condition, proves that this property propagates across connected components of the SR-region, and derives a strategy theorem reducing Gamma conjecture II to Gamma-I at a possibly non-semisimple point plus a small-quantum-cohomology analysis. The resulting application proves Gamma conjecture II for all del Pezzo surfaces, with the Stokes matrix identified with the Euler form and the central connection matrix identified with $$\widehat{\Gamma}_X=\prod_{i=1}^{\dim X}\Gamma(1+\delta_i),$$8-modified Chern characters (Hu et al., 5 Jun 2026). This removes a common simplification in earlier discussions: semisimplicity at the starting point is not required.

On the Calabi–Yau side, the integral-structure formulation is explicit. The A-model bundle carries Iritani’s Gamma-integral structure

$$\widehat{\Gamma}_X=\prod_{i=1}^{\dim X}\Gamma(1+\delta_i),$$9

while the B-model side carries the natural integral local system $\delta_i$0. The mirror symmetric Gamma conjecture states that these two $\delta_i$1-structures correspond under the mirror isomorphism of variations of Hodge structure (Iritani, 2023). This suggests that the conjecture is not merely an asymptotic statement about a preferred flat section, but a structural assertion about how K-theory, cohomology, monodromy, and relative cycles are intertwined by mirror symmetry.

6. Arithmetic, modularity, and current directions

A striking arithmetic refinement appears for Picard rank-$\delta_i$2 Fano threefolds. The Borel transform of the irregular A-side quantum differential equation produces a Picard–Fuchs operator for a modular pencil of K3 surfaces, and the corresponding “quantum differential operator” $\delta_i$3 has virtual Yukawa coupling

$\delta_i$4

For the modular levels $\delta_i$5, the virtual instanton numbers are periodic with period $\delta_i$6, and the $\delta_i$7 case is tied to the Apéry–Beukers–Peters pencil for $\delta_i$8. In this framework, the $\delta_i$9 term in the degree-six part of $T_X$0 is mirrored by the Eichler-integral constants on the B-side, and this arithmetic mechanism underlies Golyshev–Zagier’s proof of Gamma I and II for the $T_X$1 Iskovskikh classes (Malmendier et al., 2024).

A different arithmetic manifestation is provided by Frobenius constants. Bloch–Vlasenko show that for a Picard–Fuchs differential operator with a special reflection point, the generating series of Frobenius constants is the Taylor expansion of a generalized gamma function built from Mellin transforms of solutions of the adjoint equation. As a consequence, the Frobenius constants of Picard–Fuchs operators are periods (Bloch et al., 2019). In the geometric cases emphasized by mirror symmetry, this gives a motivic explanation for the appearance of $T_X$2-phenomena and $T_X$3-values in canonical solution bases near maximally unipotent monodromy points.

Current directions are explicit across the recent literature. For flag varieties, open problems include extending Gamma II to general $T_X$4, describing all Stokes matrices via exceptional collections, and refining the mirror picture to integral structures, equivariant settings, quantum $T_X$5-theory, and cluster-algebraic constructions (Chow, 22 Jan 2025). In relative mirror symmetry, the natural next step is a version of Gamma II for full exceptional collections and a direct comparison with punctured log invariants (You, 8 Aug 2025). In the Fano-to-Calabi–Yau Laplace-transform framework, it is natural to seek broader mirror constructions beyond Laurent polynomial mirrors and to relate the vanishing cycles more precisely to large-complex-structure asymptotics (Iritani, 2023).

Taken together, these developments show that “Mirror Symmetric Gamma Conjecture” no longer denotes a single isolated assertion. It names a program in which the Gamma class controls dominant asymptotics, integral lattices, oscillatory integrals, tropical period corrections, categorical Stokes data, and arithmetic constants, across Fano, Calabi–Yau, relative, and Landau–Ginzburg theories.