Quantum-cohomology, period and mirror-map interface

Let $Y$ be a smooth projective variety over $\mathbb{C}$, let $H=H^{\ast }(Y;\mathbb{Q})$ be its singular cohomology ring with cup product, and equip $H$ with the nondegenerate graded Poincaré pairing $(a,b{)}_Y={\int }_{Y}a\cup b$, using the complex orientation and integrating the top-degree component. Let $\mathrm{NE}(Y)$ be the additive monoid of homology classes represented by effective algebraic $1$-cycles. Fix an ample divisor $A$ and complete $\mathbb{Q}[\mathrm{NE}(Y)]$ with respect to the filtration by $A\cdot \beta$ to obtain the coefficient ring ${\mathrm{\Lambda }}_Y$. Genus-zero Gromov-Witten invariants use the stable-map stack, virtual class, evaluation maps, and cotangent-line classes at marked points; their exact proposed construction interface is recorded by the stable-map supplier. For a universal curve $\pi :\mathcal{C}\to {\bar{\mathcal{M}}}_{0,n}(Y,\beta )$ with marking section $s_i$ and universal map $f:\mathcal{C}\to Y$, set ${\mathrm{ev}}_i=f\circ s_i$ and ${\psi }_i=c_1(s_i^{\ast }{\omega }_{\pi })$.

When the stated virtual classes and cohomological splitting data exist, fix a homogeneous basis $T_1,\dots ,T_N$ of $H$. Write $\mathrm{\mid }{T}_i\mathrm{\mid }\in \{0,1\}$ for cohomological degree modulo two. Introduce formal coordinates $t_i$ of parity $\mathrm{\mid }{T}_i\mathrm{\mid }$, and let ${\mathcal{P}}_Y$ be the graded-commutative polynomial algebra over ${\mathrm{\Lambda }}_Y$ on these coordinates: $t_i{t}_j=(-1{)}^{\mathrm{\mid }{T}_i\mathrm{\mid }\mathrm{\mid }{T}_j\mathrm{\mid }}{t}_j{t}_i$, with $t_i^2=0$ for odd $t_i$. Thus the even coordinates generate a polynomial algebra and the odd coordinates an exterior algebra. Put $\mathfrak{m}=(t_1,\dots ,t_N)$ and . This is the insertion-adic completion with Novikov-completed coefficients; on its coefficients retain the ample-degree topology of ${\mathrm{\Lambda }}_Y$. Use the even universal insertion $\tau ={\sum }_i{t}_i{T}_i\in H\otimes R_Y$, with scalars written on the left. Extend the pairing and correlators by graded multilinearity, using the sign $(-1{)}^{pq}$ whenever factors of parities $p,q$ are interchanged. Define the big genus-zero quantum product on $H\otimes R_Y$ by $(a{\star }_{\tau }b,c{)}_Y={\sum }_{\beta \in \mathrm{NE}(Y)}{\sum }_{m\ge 0}\frac{Q^{\beta }}{m!}{⟨a,b,c,\underset{m}{\underbrace{\tau ,\dots ,\tau }}⟩}_{0,m+3,\beta },$ for $a,b,c\in H$, extending over $R_Y$. At a fixed insertion monomial of total degree $m$, only the summand with $m$ additional markings contributes; curve-class sums are interpreted in ${\mathrm{\Lambda }}_Y$. Thus no infinite scalar sum at one curve class is required. The small product is the specialization $t_i=0$ for every $i$. Other specializations require convergence in the target coefficient topology; strictly positive Novikov valuation of every nonzero substituted coordinate is sufficient. With $z$ an invertible even formal parameter, the quantum connection convention is ${\mathrm{\nabla }}_i=z{\mathrm{\partial }}_{t_i}-T_i{\star }_{\tau }$, using left graded derivatives; a flat section satisfies $z{\mathrm{\partial }}_{t_i}s=T_i{\star }_{\tau }s$. Associativity requires the cohomological genus-zero splitting axiom for all numbers of marked points, with node contractions using the graded inverse of the Poincaré pairing. The earlier associativity supplier treats $\tau =0$; for the universal even insertion the same four-point boundary relation with $m$ additional labelled markings has $\left(\genfrac{}{}{0px}{}{m}{k}\right)$ distributions of $k$ markings to one branch, and $\left(\genfrac{}{}{0px}{}{m}{k}\right)\mathrm{/}m!=1\mathrm{/}(k!(m-k)!)$ gives the two iterated big products. The graded WDVV argument therefore applies coefficientwise in $R_Y$.

For independent Novikov symbols $Q^{\beta }$ and insertion coordinates, define the cohomology-valued $J$-function by its pairing with every $b\in H$: $(b,J_Y(\tau ,z){)}_Y={\int }_Y{e}^{\tau \mathrm{/}z}b+{\sum }_{\beta \mathrm{\ne }0}{\sum }_{m\ge 0}\frac{Q^{\beta }}{m!}{\int }_{[{\bar{\mathcal{M}}}_{0,m+2}(Y,\beta ){]}^{\mathrm{v}\mathrm{i}\mathrm{r}}}\frac{{\mathrm{ev}}_1^{\ast }(b){\mathrm{ev}}_2^{\ast }(1){\prod }_{j=3}^{m+2}{\mathrm{ev}}_j^{\ast }(\tau )}{z-{\psi }_1}\mathrm{.}$ The empty product for $m=0$ is $1$, and $(z-{\psi }_1{)}^{-1}={\sum }_{r\ge 0}{\psi }_1^r{z}^{-r-1}$. This is a formal series in $z^{-1}$ with coefficients in $H\otimes R_Y$: at a fixed insertion degree only one $m$ contributes, and curve sums use the ample-degree completion. The curve-class-zero contribution is prescribed as $e^{\tau \mathrm{/}z}$, without an unstable two-point degree-zero integral. The test insertion and descendant are at marking 1; marking 2 carries the unit. Positive-degree two-point cohomological virtual data are required separately from the algebraic supplier's stable-curve-range statement.

For the unit/divisor restriction write $\tau =t_{0}1+D(t)$ with $D(t)={\sum }_{i=1}^r{t}_i{p}_i$ and fixed divisor classes $p_i\in H^2(Y;\mathbb{Q})$. The compatibility data needed to identify this restriction with Givental's source convention are the descendant string and divisor identities: if $C_{\beta }(t;b)$ denotes the coefficient of $Q^{\beta }$ in the positive-degree pairing just defined, they read ${\mathrm{\partial }}_{t_0}{C}_{\beta }(t;b)=z^{-1}{C}_{\beta }(t;b)$ and ${\mathrm{\partial }}_{t_i}{C}_{\beta }(t;b)=(p_i\cdot \beta )C_{\beta }(t;b)+z^{-1}{C}_{\beta }(t;p_i\cup b)$. These identities are additional hypotheses on the supplied descendants, not conclusions of the algebraic splitting supplier. Their formal solution with the displayed two-point initial value at $t=0$ gives the source-normalized restriction $(b,J_Y(t_{0}1+D(t),z){)}_Y={\int }_Y{e}^{(t_{0}1+D(t))\mathrm{/}z}b+{\sum }_{\beta \mathrm{\ne }0}{Q}^{\beta }{e}^{⟨D(t),\beta ⟩}{\int }_{[{\bar{\mathcal{M}}}_{0,2}(Y,\beta ){]}^{\mathrm{v}\mathrm{i}\mathrm{r}}}\frac{{\mathrm{ev}}_1^{\ast }(e^{(t_{0}1+D(t))\mathrm{/}z}b){\mathrm{ev}}_2^{\ast }(1)}{z-{\psi }_1}\mathrm{.}$ Indeed differentiation of the right-hand coefficient gives the two stated identities, and coefficient recursion in the $t_i$ makes their solution unique over $\mathbb{Q}$. In particular the factor $e^{⟨D(t),\beta ⟩}$ is required while $Q$ and $t$ are independent. Givental's logarithmic presentation is defined instead by the classical term $e^{(t_0+{\sum }_i{p}_i\log q_i)\mathrm{/}z}$ and the same two-point coefficients with weights $q^{\beta }$; it absorbs the divisor weight into the curve monomial. The symbols $\log q_i$ are adjoined logarithmic coordinates, not power series at $q=0$. When $J_Y$ is compared with the following $I_Y$, use this logarithmic presentation, with the same curve monomials $q^{\beta }$ in both series. This is a separate formal presentation, not a convergence claim for substituting $Q=1$ into an arbitrary Novikov series, and the divisor-only formula is not used for general $\tau$.

For a smooth projective toric ambient variety $X$ over $\mathbb{C}$ with torus-invariant prime divisors $D_1,\dots ,D_N$ and nef line bundles $L_1,\dots ,L_l$ cutting out a smooth complete intersection $Y$, use $\mathrm{NE}(X)$ for the additive monoid of integral homology classes represented by effective algebraic $1$-cycles on $X$. Each $D_j$ is Cartier: on a smooth toric chart it is a coordinate hyperplane or is absent, so it is locally cut out by one nonzero divisor. Consequently ${\mathcal{O}}_X(D_j)$ is a line bundle. Set $u_j=c_1({\mathcal{O}}_X(D_j)){\mathrm{\mid }}_Y$, $v_a=c_1(L_a){\mathrm{\mid }}_Y$, $D_j(\beta )=⟨c_1({\mathcal{O}}_X(D_j)),\beta ⟩\in \mathbb{Z}$, and $L_a(\beta )=⟨c_1(L_a),\beta ⟩\in {\mathbb{Z}}_{\ge 0}$ for $\beta \in \mathrm{NE}(X)$; the latter inequality is the nef hypothesis. These degree pairings are taken on $X$ before restriction to $Y$. The source hypergeometric function is $I_Y(q,z)=e^{(t_0+{\sum }_i{p}_i\log q_i)\mathrm{/}z}{\sum }_{\beta \in \mathrm{NE}(X)}{q}^{\beta }{\prod }_{a=1}^l\frac{{\prod }_{m=-\mathrm{\infty }}^{L_a(\beta )}(v_a+mz)}{{\prod }_{m=-\mathrm{\infty }}^0(v_a+mz)}{\prod }_{j=1}^N\frac{{\prod }_{m=-\mathrm{\infty }}^0(u_j+mz)}{{\prod }_{m=-\mathrm{\infty }}^{D_j(\beta )}(u_j+mz)}\mathrm{.}$ Cancel each half-infinite ratio before interpreting any inverse. The $a$th bundle factor is ${\prod }_{m=1}^{L_a(\beta )}(v_a+mz)$. For $r=D_j(\beta )$ the $j$th toric factor is ${\prod }_{m=1}^r(u_j+mz{)}^{-1}$ if $r\ge 0$, and ${\prod }_{m=r+1}^0(u_j+mz)$ if $r<0$; empty products equal $1$. Thus the factor with $m=0$ is never inverted. For $m\mathrm{\ne }0$ the inverse is the finite expansion $(u_j+mz{)}^{-1}=(mz{)}^{-1}{\sum }_{k=0}^{{\dim }_{\mathbb{C}}Y}(-u_j\mathrm{/}(mz){)}^k$, since $u_j^{{\dim }_{\mathbb{C}}Y+1}=0$. Interpret the curve sum in the ample-degree completion of the $\mathrm{NE}(X)$ monoid algebra with coefficients in $H^{\ast }(Y;\mathbb{Q})((z^{-1}))$, keeping the displayed logarithmic exponential as a separate prefactor. Here the $p_i$ are a basis of divisor classes on $X$, restricted to $Y$ in the exponential; $q^{\beta }$ are monoid symbols with $q^{\beta }{q}^{\gamma }=q^{\beta +\gamma }$, and $\log q_i$ are the associated adjoined logarithmic coordinates. The ambient cohomology component is the image of $H^{\ast }(X;\mathbb{Q})$ in $H^{\ast }(Y;\mathbb{Q})$; no surjectivity onto all cohomology is asserted.

For each effective curve class $\beta$, put $d_{\beta }=⟨c_1(TY),\beta ⟩=(-K_Y)\cdot \beta$. If a divisor $D$ representing $-K_Y$ is specified, this equals $D\cdot \beta$; no arbitrary divisor is used for this exponent. For $d_{\beta }\ge 2$, define the one-point descendant $n_{\beta }={\int }_{[{\bar{\mathcal{M}}}_{0,1}(Y,\beta ){]}^{\mathrm{v}\mathrm{i}\mathrm{r}}}{\psi }_1^{d_{\beta }-2}{\mathrm{ev}}_1^{\ast }[\mathrm{p}\mathrm{t}],$ and set $n_{\beta }=0$ when $d_{\beta }<2$. Here $[\mathrm{p}\mathrm{t}]$ is the cohomology class of a point, of complex degree ${\dim }_{\mathbb{C}}Y$, and ${\psi }_1=c_1(s_1^{\ast }{\omega }_{\pi })$ has complex degree one. The one-point genus-zero virtual dimension is ${\dim }_{\mathbb{C}}Y-2+c_1(TY)\cdot \beta$; hence the insertion has the required degree exactly with the stated $d_{\beta }$. The regularized quantum period is the formal series ${\widehat{G}}_Y=1+{\sum }_{\begin{array}{c}\beta \mathrm{\ne }0\\ d_{\beta }\ge 2\end{array}}{d}_{\beta }!n_{\beta }{Q}^{\beta }\mathrm{.}$ When $-K_Y$ is ample, grouping terms by $d_{\beta }=-K_Y\cdot \beta$ gives the one-variable Fano period.

The constant-term period of a specified mirror algebra element $W$ uses the trace-coefficient construction in the earlier intrinsic-mirror-algebra definition. An oscillatory period additionally requires the explicit space, holomorphic volume form, nonzero parameter, relative rapid-decay cycle, orientation, normalization, and a proof of convergence or a declared formal completion; its value is not identified with ${\widehat{G}}_Y$ by definition.

A mirror-coordinate change is an explicitly specified invertible formal substitution between named coefficient rings and parameter spaces. In the Givental convention it has the form $t_0\mapsto t_0+f_0(q)z+h(q)$ and $\log q_i\mapsto \log q_i+f_i(q)$, with support and weighted degrees stated for each theorem. In the Johnston convention it is the stated base change of effective-curve monomials. For the Berglund–Gräfnitz–Lathwood coordinate convention, let $z=(z_1,\dots ,z_r)$ be formal curve parameters and let $W(x_1,\dots ,x_n;z)\in \mathbb{Q}\mathrm{⟦}z\mathrm{⟧}[x_1^{\pm 1},\dots ,x_n^{\pm 1}]$ be the specified Laurent potential in the torus variables $x_1,\dots ,x_n$. Assume the sum $a_W(z)={\sum }_{m\ge 1}\frac{1}{m}{\mathrm{CT}}_{x_1,\dots ,x_n}(W(x_1,\dots ,x_n;z{)}^m)$ is coefficientwise locally finite and belongs to $(z_1,\dots ,z_r)\mathbb{Q}\mathrm{⟦}z\mathrm{⟧}$. The curve parameters are already in $W$; there is no additional factor $z^m$ in this sum. Write $-z=(-z_1,\dots ,-z_r)$. For a chosen curve basis ${\beta }_i$ set $d_i={\beta }_i\cdot D$; the closed substitutions are $Q_i=z_i\exp (d_i{a}_W(-z))$, and the open mirror map is $U=x\exp (-a_W(-z))$, where here $x$ denotes the chosen mirror-curve coordinate and $U$ its open parameter. Since $Q_i=z_i+O((z{)}^2)$, coefficient recursion gives a unique formal inverse $z=z(Q)$. Solving the displayed open substitution gives $x=U{M}_W^{\mathrm{i}\mathrm{n}\mathrm{v}}(Q)$ with $M_W^{\mathrm{i}\mathrm{n}\mathrm{v}}(Q)=\exp (a_W(-z(Q)))$. Indeed substituting this expression in $x\exp (-a_W(-z(Q)))$ gives $U$ exactly. These are separate source coordinate conventions; no mirror map is inferred from equality of units or from a numerical period match.

Stable n-pointed algebraic curves and the moduli functor

BF perfect obstruction theory and intrinsic-normal-cone class

Algebraic stable-map descendants and deformation/splitting identities

Associativity from genus-zero WDVV under a splitting axiom

Specified mathematical mirror datum

Polarized variation of Hodge structure and period map

Gauss–Manin connection and Griffiths transversality

Intrinsic mirror algebra data and classical period

Singular cohomology ring

Berglund–Gräfnitz–Lathwood Definition 4.6 prints an unsigned expression $M_W(Q)=\exp (a_W(z(Q)))$ while calling it the inverse of its signed open map. With $z(Q)$ defined here as the literal inverse of the displayed closed substitution, that expression does not in general invert the open map. The explicitly signed $M_W^{\mathrm{i}\mathrm{n}\mathrm{v}}$ above resolves this convention conflict; no equality with the source's unsigned expression is asserted.

This is a mathematical interface, not an assertion that the required stable-map, logarithmic, or mirror-algebra constructions have been proved in this item. In the current run the R4 stable-map descendant/splitting supplier and the conditional quantum-product associativity item are escalated, and the specified mirror-datum supplier has no current readiness record. The exact supplier statements remain the hypotheses for any consuming theorem.