Previous progress (types A and C). Iritani and Koto proved formal generic semisimplicity for every ordinary partial flag variety $\operatorname{Fl}(k_1,\ldots,k_\ell;n)$ and every symplectic partial flag variety $\operatorname{IFl}(k_1,\ldots,k_\ell;2n)$. Thus the conjecture was already known for all rational homogeneous spaces of Dynkin types $A$ and $C$. More generally, their projective-bundle decomposition shows that, after localization, $\operatorname{BQH}(\mathbb P(V))$ is generically semisimple if and only if $\operatorname{BQH}(B)$ is. Their statement is formal; convergence is not asserted [11, Corollary 1.8, Proposition 5.10, and Remark 1.9].

Preliminary general proof. A temporary manuscript by Changzheng Li, Zhaoyang Liu, Tony Yue Yu, and Chi Zhang proves formal generic semisimplicity for arbitrary flag varieties. Using the formal big Landau–Ginzburg mirror theorem of Hinault–Li–Yu–Zhang–Zhang, the argument shows that the total quantum spectral cover is regular and that its generic fiber is finite étale. The manuscript is still being checked and revised; comments are welcome.

Generic Semisimplicity of the Formal Big Quantum Cohomology of Flag Varieties

Changzheng Li, Zhaoyang Liu, Tony Yue Yu, and Chi Zhang

Temporary version for circulation, 13 August 2026. This manuscript will be further checked and revised before formal submission.

Abstract

Let $X=G/P$, where $G$ is complex, simple and simply connected, and put $r=\dim H^2(X)$ and $N=\dim H^{\ast}(X)$. Using the formal big Landau–Ginzburg mirror theorem of Hinault–Li–Yu–Zhang–Zhang in the completed form recalled below, we prove that the formal big quantum cohomology of $X$ is generically semisimple over the fraction field of

$$\mathbb C(q_1,\ldots,q_r)[[\tau_1,\ldots,\tau_N]].$$

More strongly, the coordinate algebra of the total formal quantum spectral cover is regular. The main geometric ingredient is a general transversality criterion for formal deformations of functions: if the central Jacobian algebra is finite-dimensional and the central Kodaira–Spencer classes span it, then the total critical scheme is regular, its relative Jacobian algebra is finite free, and its generic fiber is finite étale. For a flag variety, the divisor directions are already present through $q_i\mapsto q_i e^{y_i}$; together with the newly adjoined directions they form the full Kodaira–Spencer basis. The result is formal and does not require convergence of the big quantum product. Generic semisimplicity for arbitrary complex semisimple $G$ follows by quantum Künneth.

Keywords. Big quantum cohomology; generic semisimplicity; flag varieties; Landau–Ginzburg mirrors; unfoldings; spectral covers.

2020 Mathematics Subject Classification. Primary 14N35; Secondary 14M15, 32S30, 53D45.

1. Introduction

Quantum cohomology associates to a smooth projective variety $X$ a deformation of its ordinary cohomology algebra governed by genus-zero Gromov–Witten invariants. Even when the small quantum cohomology is not semisimple, its big deformation may be generically semisimple. For Fano varieties, this property is central to the qualitative part of Dubrovin’s conjecture, which relates generic semisimplicity of big quantum cohomology to the existence of a full exceptional collection in the bounded derived category; see [5] and the discussion in [10].

Let $G$ be a complex semisimple algebraic group and let $P\subset G$ be a parabolic subgroup. We recall the following folklore expectation in the form recorded by Perrin and Smirnov [10, Conjecture 1.1].

Conjecture 1.1 (Generic semisimplicity for homogeneous spaces). The big quantum cohomology of $G/P$ is generically semisimple.

Replacing $G$ by its simply connected cover does not change $G/P$. We first work in the simple, simply connected setting of [8]; the passage to a semisimple group is recorded after Theorem 1.2.

Here one must distinguish the small and big quantum products. A semisimple specialization of small quantum cohomology immediately supplies a semisimple point of the big theory. This applies to many important examples. In particular, Chaput, Manivel and Perrin proved semisimplicity of the small quantum cohomology of every minuscule or cominuscule homogeneous space after specializing the quantum parameter to $q=1$ [2]. For many other homogeneous spaces, however, the small quantum ring is genuinely non-semisimple, and one must use the big deformation. Early results in this direction include $\operatorname{IG}(2,6)$ [6] and the family $\operatorname{IG}(2,2n)$ [4]. More recently, Perrin and Smirnov proved generic semisimplicity for all coadjoint homogeneous varieties, including cases in which the small quantum ring has a nonreduced factor of ADE type [10]. They also proposed the stronger regularity conjecture for the total big quantum cohomology ring [10, Conjecture 1.7].

The formal big Landau–Ginzburg mirror theorem of Hinault–Li–Yu–Zhang–Zhang [8] constructs, for every $G/P$ in the simple, simply connected setting, the big quantum $F$-bundle from a maximal unfolding of its small mirror superpotential. Its small input is Chow’s $D_{\hbar}$-module mirror theorem [3]; its output identifies the formal big quantum $F$-bundle with the $F$-bundle of a full Landau–Ginzburg unfolding. The purpose of this note is to extract the resulting semisimplicity statement and to isolate the elementary geometric mechanism behind it. The underlying maximal-unfolding philosophy goes back to Hertling and Manin [7].

Set

$$N=\dim_{\mathbb C}H^{\ast}(G/P,\mathbb C), \qquad r=\dim_{\mathbb C}H^2(G/P,\mathbb C).$$

The relative unfolding in [8] explicitly adjoins only $N-r$ functions. The remaining $r$ parameters have not been omitted: they are the logarithmic divisor directions already contained in the small family through $q_i\mapsto q_i e^{y_i}$. Thus the central Kodaira–Spencer classes are

$$[q_i\partial_{q_i}W]\quad(1\leq i\leq r), \qquad [f_j]\quad(r<j\leq N),$$

and the combined list is a basis of $\operatorname{Jac}(W)$.

Our main observation is independent of flag varieties. For a formal deformation $F$ of a function $W$ on a smooth affine variety, differentiation induces, at every critical point $p$, a natural surjection

$$\operatorname{Jac}(W)\otimes\kappa(p) \longrightarrow \operatorname{coker}\bigl(\operatorname{Hess}_p(W)\bigr).$$

Consequently, a Kodaira–Spencer basis fills every cokernel of the Hessian. The full Jacobian matrix of the relative critical equations has maximal rank, so the total critical scheme is regular. A formal Nakayama argument and Auslander–Buchsbaum then show that the relative Jacobian algebra is finite free; its generic fiber is therefore a zero-dimensional regular algebra, hence finite étale in characteristic zero. Big mirror symmetry transfers this algebra to big quantum cohomology.

Our main result is the following.

Theorem 1.2. Let $G$ be a complex simple, simply connected algebraic group, let $P\subset G$ be a parabolic subgroup, and set $X=G/P$. Put

$$S_X=\mathbb C(q_1,\ldots,q_r) [[\tau_1,\ldots,\tau_N]], \qquad K_X=\operatorname{Frac}(S_X).$$

Write $\operatorname{BQH}^{\ast}(X)=H^{\ast}(X,\mathbb C)\otimes_{\mathbb C}S_X$, equipped with the formal big quantum product. Then

$$\operatorname{BQH}^{\ast}(X)\otimes_{S_X}K_X$$

is a finite étale $K_X$-algebra. After extension to an algebraic closure $\overline K_X/K_X$, it is isomorphic to $\overline K_X^{N}$. Moreover, the coordinate algebra of the quantum spectral cover over the full formal big base is regular.

Corollary 1.3. Let $G$ be a complex semisimple algebraic group and $P\subset G$ a parabolic subgroup. Then the formal big quantum cohomology of $G/P$ is generically semisimple.

Proof. After passage to the simply connected cover, write $G/P=\prod_a X_a$ with $X_a=G_a/P_a$ and $G_a$ simple. The big quantum product is independent of the unit coordinate, so omit that coordinate for each factor. Inside $H^{\ast}(\prod_aX_a)$ consider the formal subspace parametrized by $\sum_a\operatorname{pr}_a^{\ast}t_a$, with $t_a\in H^{\ast}(X_a)/\mathbb C1$. The genus-zero product formula [1, 9] identifies the restriction of the product’s big quantum algebra to this subspace with the completed tensor product of the factor big quantum algebras. By Theorem 1.2, each factor is generically finite étale. After a common field extension the factors are split étale, so their tensor product is split étale. The discriminant of the full big quantum algebra therefore has nonzero restriction and hence is not identically zero. $\square$

This proves the formal generic semisimplicity conjecture for arbitrary rational homogeneous spaces and gives the full-base form of the stronger regularity expectation for each simple factor. To our knowledge, this consequence of the big Landau–Ginzburg mirror has not previously been recorded in this generality.

This note uses the formal big-mirror construction of [8] as its principal input. Since that work is currently available as a preprint, Section 2 records precisely the consequences used below. The finite-freeness and regularity of the relative Jacobian algebra are proved directly in Section 3. The completed-lattice input needed for the transfer to quantum cohomology is the reduction-modulo-$u$ identification (3), together with the non-equivariant big-mirror isomorphism. All statements here are formal. No convergence of the Gromov–Witten potential or of the mirror map is assumed. This is a temporary version intended to record the argument and invite comments; it will be further checked and revised before formal submission.

2. The formal big-mirror input

Let $X=G/P$ and retain $N$ and $r$ from the introduction. We work over

$$k=\mathbb C(q_1,\ldots,q_r), \qquad S=k[[y_1,\ldots,y_N]], \qquad K=\operatorname{Frac}(S).$$

Let $Y=\operatorname{Spec}B$ be the smooth affine Landau–Ginzburg mirror over $k$, and let $W\in B$ be the small superpotential. In invariant notation,

$$\operatorname{Jac}(W)=B\big/dW\bigl(\operatorname{Der}_k(B)\bigr).$$

Chow’s small mirror theorem, in the form used in [8], identifies $\operatorname{Jac}(W)$ with the localized small quantum cohomology of $X$. In particular,

$$\dim_k\operatorname{Jac}(W)=N,$$

so the critical scheme of $W$ is finite over $k$.

Choose a Schubert basis $\phi_1,\ldots,\phi_N$ such that $\phi_1,\ldots,\phi_r$ are the divisor Schubert classes. The construction of the full potential $F\in B[[y_1,\ldots,y_N]]$ has two parts:

1. the already existing small parameters are pulled back by $q_i\mapsto q_i e^{y_i}$ for $1\leq i\leq r$;

2. functions $f_j$ representing the remaining Schubert classes are adjoined in the directions $y_j$ for $r<j\leq N$.

Consequently, if

$$g_i:=\left.\frac{\partial F}{\partial y_i}\right|_{y=0},$$

then

$$\tag{1} g_i=q_i\partial_{q_i}W\quad(1\leq i\leq r), \qquad g_i=f_i\quad(r<i\leq N).$$

The small mirror identification and the choice of the $f_j$ give

$$\tag{2} \operatorname{KS}_0:k^N\longrightarrow\operatorname{Jac}(W), \qquad e_i\longmapsto[g_i],$$

and this map is an isomorphism. This is the maximality statement needed in the sequel. Notice that it uses the old logarithmic directions and the new unfolding directions together.

The second input is the formal big-mirror theorem [8, Theorems 4.35 and 4.38]. It provides a formal isomorphism of bases

$$\mathfrak m:\operatorname{Spf}S\xrightarrow{\sim} \operatorname{Spf}k[[\tau_1,\ldots,\tau_N]]$$

and an isomorphism between the associated B-model and A-model $F$-bundles. We use this theorem together with its compatibility with completion, reduction modulo $u$, and non-equivariant specialization. After reduction modulo $u$, the B-model module is the relative Jacobian module

$$\tag{3} \operatorname{Jac}_{Y/S}(F)\Omega,$$

where $\Omega$ is the nowhere-vanishing volume form used in the Landau–Ginzburg model, and the residue in a base direction $v$ is multiplication by $[v(F)]$. On the A-model side the corresponding residue is big quantum multiplication by the tangent vector $d\mathfrak m(v)$. The mirror isomorphism intertwines these residue operators. These statements, rather than a pairing or a primitive form, are all that will be used below.

For orientation, the logarithmic substitutions appear in §4.2.2 of [8], the complementary functions are chosen in Construction 4.28, maximality is established in Proposition 4.34, and the big-mirror isomorphisms are Theorems 4.35 and 4.38. We apply the construction after the non-equivariant specialization; equivariantly, the divisor residues contain the familiar equivariant shifts, which disappear at $\lambda=0$.

3. A transversality criterion for formal unfoldings

We now prove the general commutative-algebra statement underlying the application. Let $k$ be a field of characteristic zero, let $Y=\operatorname{Spec}B$ be a smooth affine $k$-variety of pure dimension $n$, and set

$$S=k[[t_1,\ldots,t_m]], \qquad \mathfrak m=(t_1,\ldots,t_m).$$

For $F\in B[[t_1,\ldots,t_m]]$, write

$$W=F\bmod\mathfrak m, \qquad g_i=\left.\frac{\partial F}{\partial t_i}\right|_{t=0}.$$

The relative critical ideal is

$$I_F=d_YF\bigl(\operatorname{Der}_k(B) \otimes_BB[[t]]\bigr),$$

and the relative Jacobian algebra is

$$A_F=B[[t]]/I_F.$$

Proposition 3.1 (Spanning Kodaira–Spencer criterion). Assume that $\operatorname{Jac}(W)$ is finite-dimensional over $k$ and that the classes $[g_1],\ldots,[g_m]$ span $\operatorname{Jac}(W)$. Then:

1. $A_F$ is a finite free $S$-algebra of rank $\dim_k\operatorname{Jac}(W)$;

2. $A_F$ is a regular ring, of dimension $m$ at every maximal ideal;

3. $A_F\otimes_S\operatorname{Frac}(S)$ is finite étale.

In geometric terms, the total formal relative critical scheme is regular and its generic fiber is reduced and zero-dimensional.

The proof rests on the following local observation.

Lemma 3.2 (Jacobian classes and the Hessian cokernel). Let $p$ be a closed point of $\operatorname{Crit}(W)$, with residue field $\kappa(p)$. The Hessian at $p$ is an intrinsic map

$$H_p=\operatorname{Hess}_p(W):T_pY\longrightarrow T_p^{\ast}Y.$$

There is a natural surjection

$$\tag{4} \rho_p:\operatorname{Jac}(W)\otimes_k\kappa(p) \twoheadrightarrow\operatorname{coker}(H_p), \qquad [h]\otimes c\longmapsto c\cdot dh(p)\bmod\operatorname{im}(H_p).$$

Proof. The Hessian is intrinsic because $dW(p)=0$. Choose étale local coordinates $x_1,\ldots,x_n$ after passing to an étale neighborhood of $p$. If a function belongs to the Jacobian ideal, write locally

$$h=\sum_{a=1}^n a_a\frac{\partial W}{\partial x_a}.$$

At the critical point,

$$dh(p)=\sum_{a=1}^n a_a(p) d\left(\frac{\partial W}{\partial x_a}\right)(p) \in\operatorname{im}(H_p).$$

Thus (4) is well defined. The $k$-differentials of elements of $B$ generate $\Omega^1_{B/k}\otimes_B\kappa(p)$. Since $p$ is a closed point of a finite-type $k$-scheme, $\kappa(p)/k$ is finite; it is separable in characteristic zero. Hence $\Omega^1_{\kappa(p)/k}=0$, and the standard cotangent sequence identifies $\Omega^1_{B/k}\otimes_B\kappa(p)$ with $T_p^{\ast}Y$. Thus the displayed map is surjective. $\square$

Proof of Proposition 3.1. First we prove finiteness. The ring $B[[t]]$ is Noetherian and $\mathfrak m$-adically complete. The ideal $I_F$ is finitely generated and hence closed, so $A_F$ is complete and separated. Moreover,

$$A_F/\mathfrak mA_F\cong\operatorname{Jac}(W).$$

Lift a $k$-basis $\overline a_1,\ldots,\overline a_\mu$ of $\operatorname{Jac}(W)$ to elements $a_1,\ldots,a_\mu\in A_F$, where $\mu=\dim_k\operatorname{Jac}(W)$. Successive approximation modulo $\mathfrak m,\mathfrak m^2,\ldots$, followed by completeness, expresses every element of $A_F$ as

$$\sum_{i=1}^{\mu}s_i a_i, \qquad s_i\in S.$$

Thus $S^\mu\twoheadrightarrow A_F$, and $A_F$ is finite over $S$.

If $\mu=0$, topological Nakayama gives $A_F=0$ and the assertions are vacuous; hence assume $\mu>0$.

Let $\mathfrak n$ be a maximal ideal of $A_F$. Since $A_F$ is finite over the local ring $S$, its contraction is $\mathfrak m$. Hence $\mathfrak n$ corresponds to a closed critical point $p$ of the central function $W$. Choose étale local coordinates $x_1,\ldots,x_n$ at $p$. The total relative critical locus is locally cut out by

$$G_a(x,t)=\frac{\partial F}{\partial x_a}(x,t)=0, \qquad 1\leq a\leq n.$$

At $(p,0)$, the differential of these equations is, up to transpose, the block matrix

$$\tag{5} \left[\operatorname{Hess}_p(W)\mid dg_1(p)\quad\cdots\quad dg_m(p)\right].$$

Equivalently, it is the map

$$T_pY\oplus\kappa(p)^m\longrightarrow T_p^{\ast}Y, \qquad (v,c_1,\ldots,c_m)\longmapsto H_p(v)+\sum_{i=1}^m c_i\cdot dg_i(p).$$

The classes $[g_i]$ span $\operatorname{Jac}(W)$, and Lemma 3.2 shows that their differentials span the cokernel of $H_p$. Hence (5) is surjective.

The completed ambient local ring at $(p,0)$ is regular of dimension $n+m$. Surjectivity says that the $n$ critical equations have independent linear parts; they may therefore be completed to a regular system of parameters. Their quotient is a regular local ring of dimension $m$. Regularity is detected after completion, so $(A\_F)\_{\mathfrak n}$ is regular for every maximal ideal $\mathfrak n$. Consequently $A_F$ is regular.

It remains to prove flatness. At each maximal ideal $\mathfrak n$ of $A_F$, the images of $t_1,\ldots,t_m$ form a system of parameters of the regular local ring $(A\_F)\_{\mathfrak n}$: their quotient is zero-dimensional and this local ring has dimension $m$. Since a regular local ring is Cohen–Macaulay, they form an $(A\_F)\_{\mathfrak n}$-regular sequence. Hence the positive Koszul homology modules $H_j(t_1,\ldots,t_m;A_F)$ vanish after localization at every maximal ideal of $A_F$, and therefore vanish. Thus $\operatorname{depth}_S A_F=m$. The regular local ring $S$ has finite global dimension $m$, so the Auslander–Buchsbaum formula gives $\operatorname{pd}_S A_F=0$. Consequently $A_F$ is finite projective, and hence free because $S$ is local. Its rank is read from the central fiber and equals $\mu$.

Finally, regularity is preserved under localization. Thus

$$A_F\otimes_S\operatorname{Frac}(S)$$

is a zero-dimensional regular algebra. It is a finite product of finite field extensions of $\operatorname{Frac}(S)$; these extensions are separable in characteristic zero. The generic algebra is therefore finite étale. $\square$

Remark 3.3 (Total regularity versus relative smoothness). Surjectivity of the full matrix (5) proves that the total critical scheme is regular. It does not say that its projection to $\operatorname{Spf}S$ is smooth at the central fiber. Relative smoothness there would require the Hessian block itself to be invertible. For example, the total curve $x^2=t$ is regular even though its projection to the $t$-line is ramified at the origin. This is exactly how a nonreduced central critical scheme can deform to a reduced generic one.

Example 3.4 (The $A_2$ model). Let

$$W(x)=\frac{x^3}{3}, \qquad F(x;t,s)=\frac{x^3}{3}-tx+s.$$

Then $\operatorname{Jac}(W)=k[x]/(x^2)$. Think of $t$ as an already existing small direction and $s$ as a newly adjoined direction. The new class $[\partial_sF]=[1]$ alone does not smooth the fat point, whereas

$$[\partial_tF]=[-x],\qquad[\partial_sF]=[1]$$

form a basis. The relative Jacobian algebra is

$$k[[t,s]][x]/(x^2-t).$$

Its central fiber is nonreduced, its total space is regular, and its generic fiber is a product of two fields after adjoining $\sqrt t$. This is the local mechanism of the proof.

4. Application to flag varieties

Return to the notation of Section 2. Define

$$A:=\operatorname{Jac}_{Y/S}(F).$$

The central critical scheme is finite and the $N$ classes in (1) form a basis of $\operatorname{Jac}(W)$. Proposition 3.1, with $m=N$, therefore gives the following.

Corollary 4.1. The algebra $A$ is finite free of rank $N$ over $S$ and is a regular ring. Moreover,

$$A_K:=A\otimes_SK$$

is finite étale over $K$, and

$$A_K\otimes_K\overline K\cong\overline K^{N}$$

for an algebraic closure $\overline K/K$.

For clarity, the full linearization at a central critical point $p$ is

$$\tag{6} \left[\operatorname{Hess}\_p(W)\mid d(q\_1\partial\_{q\_1}W)(p)\quad\cdots\quad d(q\_r\partial\_{q\_r}W)(p)\mid df\_{r+1}(p)\quad\cdots\quad df\_N(p)\right].$$

The old logarithmic columns in (6) are essential. The new columns alone need not fill the Hessian cokernel.

We next transfer the algebra through the $F$-bundle mirror isomorphism. Let $\mathcal Q$ denote the pullback, along the formal mirror map $\mathfrak m$, of the big quantum algebra to $S$. Modulo $u$, the B-model $F$-bundle is $A\Omega$. For a vector field $v$ on the base, set

$$C_v^B=(u\nabla_v^B)\bmod u, \qquad C_{d\mathfrak m(v)}^A=(u\nabla_{d\mathfrak m(v)}^A)\bmod u.$$

The residue formulas are

$$\tag{7} C_v^B=m_{[v(F)]}, \qquad C_{d\mathfrak m(v)}^A=m^{\star}_{d\mathfrak m(v)}.$$

By Corollary 4.1, both $T\operatorname{Spf}S$ and $A$ are free of rank $N$. The Kodaira–Spencer map

$$\operatorname{KS}_F:T\operatorname{Spf}S\longrightarrow A, \qquad v\longmapsto[v(F)]$$

reduces modulo $\mathfrak m$ to the isomorphism (2); hence it is an isomorphism by Nakayama’s lemma. Thus the B-model residues in (7) span the full regular-representation algebra

$$m_A(A)\subset\operatorname{End}_S(A\Omega).$$

On the A-model side, tangent vectors in flat coordinates identify with cohomology classes, and the residues span the regular representation $m_{\mathcal Q}(\mathcal Q)$.

Let $\Phi_0$ be the reduction modulo $u$ of the big-mirror $F$-bundle isomorphism. Compatibility with the connections gives

$$\Phi_0 C_v^B\Phi_0^{-1}=C_{d\mathfrak m(v)}^A.$$

Since $d\mathfrak m$ is an isomorphism, conjugation identifies the two regular-representation subalgebras:

$$\Phi_0m_A(A)\Phi_0^{-1}=m_{\mathcal Q}(\mathcal Q).$$

Both regular-representation homomorphisms

$$m\_A:A\longrightarrow\operatorname{End}\_S(A\Omega),\qquad m\_{\mathcal Q}:\mathcal Q\longrightarrow\operatorname{End}\_S(\mathcal Q)$$

are faithful, and their images are the two subalgebras above. Therefore

$$\tag{8} \alpha:=m_{\mathcal Q}^{-1}\circ\operatorname{Ad}_{\Phi_0}\circ m_A :A\xrightarrow{\sim}\mathcal Q$$

is a well-defined unital $S$-algebra isomorphism. This argument uses only the Higgs residues; no pairing or primitive form is required.

Combining (8) with Corollary 4.1 proves Theorem 1.2. In particular, after extension to $K$, the big quantum algebra is finite étale, and after algebraic closure its $N$ characters give the decomposition into $N$ one-dimensional factors. Since $\mathfrak m$ is an isomorphism of formal bases, regularity and generic finite étaleness of $\mathcal Q=\mathfrak m^{\ast}\operatorname{BQH}^{\ast}(X)$ are equivalent to the corresponding properties of the original big quantum algebra.

Remark 4.2 (Why the mirror makes transversality visible). On the A-model side, the spectral cover is the joint spectrum of all quantum multiplication operators. The first derivatives of its structure constants at the small point involve four-point Gromov–Witten invariants. Merely indexing the big parameters by a basis of $H^{\ast}(X)$ does not itself prove that these derivatives smooth the small algebra. On the Landau–Ginzburg side, the tangent space to deformations of a function modulo coordinate changes is the Jacobian algebra. A Kodaira–Spencer basis therefore supplies every missing Hessian direction automatically. Big mirror symmetry is the bridge identifying this explicit deformation with the Gromov–Witten deformation.

5. Divisor coordinates and formal scope

Big quantum cohomology is often written without separate variables for $H^2(X)$, because the divisor equation absorbs them into the quantum parameters. This convention does not remove the divisor directions from the transversality calculation. The rank argument must first be performed on the full base using the complete matrix (6).

For a precise comparison, return temporarily to the A-model flat coordinates $\tau_1,\ldots,\tau_N$. Put

$$S_{\mathrm{abs}}=k[[\tau_{r+1},\ldots,\tau_N]], \qquad L=\operatorname{Frac}(S_{\mathrm{abs}}), \qquad K_A=\operatorname{Frac} k[[\tau_1,\ldots,\tau_N]].$$

Define

$$\theta:S_{\mathrm{abs}}\longrightarrow k[[\tau_1,\ldots,\tau_N]], \qquad \theta(q_i)=q_i e^{\tau_i},\quad \theta(\tau_j)=\tau_j\quad(j>r).$$

For a rational function of the $q_i$, the indicated substitution is well defined because its denominator has nonzero constant term. Specialization $\tau_1=\cdots=\tau_r=0$ is a left inverse, so $\theta$ is injective and extends to a field embedding $L\hookrightarrow K_A$. If $\mathcal Q_{\mathrm{abs}}$ denotes the quantum algebra with the divisor variables absorbed, the divisor equation gives

$$\operatorname{BQH}^{\ast}(X)\otimes_{k[[\tau_1,\ldots,\tau_N]]} K_A \cong (\mathcal Q_{\mathrm{abs}}\otimes_{S_{\mathrm{abs}}}L) \otimes_{L,\theta}K_A.$$

The field extension $K_A/L$ is faithfully flat. Since finite étaleness is fpqc-local, finite étaleness of the left-hand side implies that $\mathcal Q_{\mathrm{abs}}\otimes_{S_{\mathrm{abs}}}L$ is finite étale. Thus the usual $H^2$-absorbed convention gives the same generic semisimplicity conclusion. One should not instead repeat the Hessian argument using only $df_{r+1},\ldots,df_N$; those restricted columns need not have full rank.

The theorem is formal in the variables $\tau_i$ and is stated over the rational function field $k=\mathbb C(q_1,\ldots,q_r)$ in the quantum parameters. It does not assert that the small quantum ring is semisimple, that every big fiber is semisimple, or that every numerical specialization of $q$ is covered. It also does not by itself prove convergence of the big Gromov–Witten potential or of the formal mirror map. An analytic version requires an additional convergence argument. The theorem applies uniformly to all simple flag varieties; Corollary 1.3 gives generic semisimplicity for arbitrary semisimple $G$:

$$\boxed{\text{maximal Landau--Ginzburg unfolding} \Longrightarrow\text{regular spectral cover} \Longrightarrow\text{generic semisimplicity}}$$

Acknowledgments and AI-use statement

This temporary draft was prepared with extensive assistance from OpenAI’s ChatGPT and Codex. These systems contributed to proof exploration, commutative-algebra checks, literature review, exposition, and LaTeX editing. Before formal submission, the human authors will independently verify all mathematical claims and will retain responsibility for the final manuscript.

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