Period Calculations

In this note we review how the period vector \(\Pi\) of a Calabi-Yau threefold \(X\) is computed in practice. This extends the conceptual introduction in Type IIB Flux Compactifications, where the period vector and prepotential were defined, to the concrete computational procedure used in JAXVacua. The discussion follows [hep-th/9308122], [hep-th/9406055], and the review in [2512.17095].

Mirror symmetry and period computation

Consider a mirror pair of Calabi-Yau threefolds \((X, \widetilde{X})\), where we compactify Type IIB string theory on \(X\). The complex structure moduli space \(\mathcal{M}_{\text{cs}}(X)\) is classically exact and is identified, via mirror symmetry, with the complexified Kähler moduli space \(\mathcal{K}_{\widetilde{X}}\) of the mirror \(\widetilde{X}\):

(1)\[ \vec{\Pi}_{\text{IIB}} = \vec{\Pi}_{\text{IIA}} \equiv \vec{\Pi}\;. \]

In the symplectic basis \(\{\alpha_A, \beta^A\}\), \(A = 0, \ldots, h^{2,1}\), the periods of the holomorphic \((3,0)\)-form \(\Omega\) are

(2)\[\begin{split} \vec{\Pi}_{\text{IIB}} \coloneqq \begin{pmatrix} \int_X \Omega \wedge \beta_A \\ \int_X \Omega \wedge \alpha^A \end{pmatrix}\;. \end{split}\]

Computing these periods directly is difficult in general. However, when \(X\) is a hypersurface in a toric variety, one can obtain the periods around the large complex structure (LCS) point following a procedure due to Hosono, Klemm, Theisen, and Yau [hep-th/9308122].

The fundamental period

Let \((X, \widetilde{X})\) be a mirror pair of Calabi-Yau hypersurfaces in toric varieties \((\widetilde{V}, V)\), defined by fine, regular, star triangulations \((\widetilde{\mathcal{T}}, \mathcal{T})\) of a dual pair of reflexive polytopes \((\Delta, \Delta^\circ)\). We take \(X\) to be the vanishing locus of a generic anticanonical polynomial

(3)\[ f(\mathbbm{t}) = \Psi^0 S_0(\mathbbm{t}) - \sum_{I=1}^m \Psi^I S_I(\mathbbm{t})\;, \]

specified in terms of \(m+1\) complex parameters \(\Psi^I\) and monomials \(S_I(\mathbbm{t})\) of the toric coordinates \((\mathbbm{t}^1, \ldots, \mathbbm{t}^4)\).

We introduce the gauged linear sigma model (GLSM) matrix with entries \(Q^a_{~I}\), \(a = 1, \ldots, h^{1,1}\), \(I = 1, \ldots, h^{1,1} + 4\), recording the divisor class decomposition \([\widehat{D}_I] = \sum_a Q^a_{~I} \widehat{H}_a\). The anticanonical charge is \(Q^a_{~0} \coloneqq \sum_I Q^a_{~I}\).

Writing the holomorphic \(3\)-form as

\[ \Omega = \oint_{f=0} \frac{d\mathbbm{t}^1 \wedge d\mathbbm{t}^2 \wedge d\mathbbm{t}^3 \wedge d\mathbbm{t}^4}{(2\pi i)^4 \cdot f(\mathbbm{t})}\;, \]

the fundamental period \(\varpi_0(\psi)\) is defined by integrating \(\Omega\) over the SYZ \(T^3\) fiber:

(4)\[ \varpi_0(\psi) \coloneqq \Psi^0 \int_{T^3} \Omega = \Psi^0 \oint_{|\mathbbm{t}^1|=\epsilon} \frac{d\mathbbm{t}^1}{2\pi i} \cdots \oint_{|\mathbbm{t}^4|=\epsilon} \frac{d\mathbbm{t}^4}{2\pi i} \frac{1}{f(\mathbbm{t})}\;. \]

Applying the residue theorem and multinomial expansion, one arrives at the expression

(5)\[ \varpi_0(\psi) = \sum_{\tilde{\mathbf{q}} \in \mathcal{M}_{\widetilde{V}} \cap H^2(\widetilde{V}, \mathbb{Z})} \frac{\Gamma(1 + \tilde{q}_a Q^a_{~0})}{\prod_I \Gamma(1 + \tilde{q}_a Q^a_{~I})} \, \psi^{\tilde{\mathbf{q}}} =: \sum_{\tilde{\mathbf{q}}} c_{\tilde{\mathbf{q}}} \, \psi^{\tilde{\mathbf{q}}}\;, \]

where the sum runs over the Mori cone \(\mathcal{M}_{\widetilde{V}}\) of the mirror toric variety \(\widetilde{V}\), and \(\psi^{\tilde{\mathbf{q}}} \coloneqq \prod_a (\psi^a)^{\tilde{q}_a}\).

Higher periods

All remaining periods are determined by the fundamental period \(\varpi_0(\psi) = \sum_{\tilde{\mathbf{q}}} c_{\tilde{\mathbf{q}}} \, \psi^{\tilde{\mathbf{q}}}\) via \(\rho\)-derivatives [hep-th/9308122]:

(6)\[ \varpi^a(\psi) = \sum_{\tilde{\mathbf{q}}} \left. \frac{\partial_{\rho_a}}{2\pi i} \left( c_{\tilde{\mathbf{q}}+\vec{\rho}} \, \psi^{\tilde{\mathbf{q}}+\vec{\rho}} \right) \right|_{\vec{\rho}=0}\;, \quad \varpi^{ab}(\psi) = \sum_{\tilde{\mathbf{q}}} \left. \frac{\partial_{\rho_a} \partial_{\rho_b}}{(2\pi i)^2} \left( c_{\tilde{\mathbf{q}}+\vec{\rho}} \, \psi^{\tilde{\mathbf{q}}+\vec{\rho}} \right) \right|_{\vec{\rho}=0}\;, \]

(7)\[ \varpi^{abc}(\psi) = \sum_{\tilde{\mathbf{q}}} \left. \frac{\partial_{\rho_a} \partial_{\rho_b} \partial_{\rho_c}}{(2\pi i)^3} \left( c_{\tilde{\mathbf{q}}+\vec{\rho}} \, \psi^{\tilde{\mathbf{q}}+\vec{\rho}} \right) \right|_{\vec{\rho}=0}\;. \]

At zeroth order in \(\psi\), these reduce to

\[ \varpi_0 \simeq 1\;, \quad \varpi^a \simeq \frac{\log(\psi^a)}{2\pi i}\;, \quad \frac{1}{2} \widetilde{\kappa}_{abc} \varpi^{ab} \simeq \frac{1}{2} \widetilde{\kappa}_{abc} \varpi^a \varpi^b - \frac{1}{24} \tilde{c}_a\;, \]

\[ \frac{1}{3!} \widetilde{\kappa}_{abc} \varpi^{abc} \simeq \frac{1}{3!} \widetilde{\kappa}_{abc} \varpi^a \varpi^b \varpi^c - \frac{1}{24} \tilde{c}_a \varpi^a + \frac{\zeta(3)}{(2\pi i)^3} \chi(\widetilde{X})\;, \]

in terms of the triple intersection numbers \(\widetilde{\kappa}_{abc}\) of the mirror \(\widetilde{X}\), the second Chern class integrals \(\tilde{c}_a = \int_{\widetilde{X}} c_2(\widetilde{X}) \wedge \tilde{\beta}_a\), and the Euler characteristic \(\chi(\widetilde{X})\).

Integral symplectic basis

Mirror symmetry implies that the LCS monodromies in Type IIB on \(X\) equal the large-volume monodromies of Type IIA on \(\widetilde{X}\), which are determined by the intersection form of \(\widetilde{X}\). Using these monodromies to fix all integration constants and adopting a suitable normalisation, one obtains the periods in an integral symplectic basis:

(8)\[\begin{split} \Pi(\psi) = \frac{1}{\varpi_0} \begin{pmatrix} \frac{1}{3!} \widetilde{\kappa}_{abc} \varpi^{abc} + \frac{1}{12} \tilde{c}_a \varpi^a \\ -\frac{1}{2} \widetilde{\kappa}_{abc} \varpi^{bc} + \tilde{a}_{ab} \varpi^b \\ \varpi_0 \\ \varpi^a \end{pmatrix}\;, \end{split}\]

where the matrix \(\tilde{a}_{ab}\) is defined as

\[\begin{split} \tilde{a}_{ab} \equiv \frac{1}{2} \begin{cases} \widetilde{\kappa}_{aab} & a \geq b \\ \widetilde{\kappa}_{abb} & a < b \end{cases}\;. \end{split}\]

Gopakumar-Vafa invariants

The flat coordinates \(z^a\) are related to the algebraic coordinates \(\psi^a\) via

(9)\[ z^a = \frac{\varpi^a}{\varpi_0} = \frac{\log(\psi^a)}{2\pi i} + \frac{1}{2\pi i} \frac{c^a(\psi)}{\varpi_0}\;, \]

with correction terms \(c^a(\psi) = \sum_{\tilde{\mathbf{q}}} c^a_{\tilde{\mathbf{q}}} \psi^{\tilde{\mathbf{q}}}\) where \(c^a_{\tilde{\mathbf{q}}} = \partial_{\rho_a} c_{\tilde{\mathbf{q}}+\vec{\rho}} |_{\vec{\rho}=0}\). The LCS limit corresponds to \(e^{2\pi i z^a} \ll 1\).

The non-perturbative part of the Type IIA prepotential can be written in terms of genus-zero Gopakumar-Vafa (GV) invariants \(\mathscr{N}_{\tilde{\mathbf{q}}} \in \mathbb{Z}\) [hep-th/9809187], [hep-th/9812127]:

(10)\[ \mathcal{F}_{\text{inst}}(z) = -\frac{1}{(2\pi i)^3} \sum_{\tilde{\mathbf{q}} \in \mathcal{M}(\widetilde{X})} \mathscr{N}_{\tilde{\mathbf{q}}} \, \text{Li}_3\!\left(e^{2\pi i \, \tilde{\mathbf{q}} \cdot \mathbf{z}}\right)\;, \]

where the sum runs over effective curve classes in the Mori cone \(\mathcal{M}_{\widetilde{X}}\). By comparing the instanton expansion (10) with the period expressions, one can extract the GV invariants order by order in \(\psi\). Specialized algorithms for this extraction in models with many moduli were implemented in [2303.00757].

Summary of LCS formulas

The prepotential \(\mathcal{F}\) in the LCS patch decomposes as

(11)\[ \mathcal{F}(z) = \mathcal{F}_{\text{poly}}(z) + \mathcal{F}_{\text{inst}}(z)\;, \]

with the polynomial part

(12)\[ \mathcal{F}_{\text{poly}}(z) = -\frac{1}{3!} \widetilde{\kappa}_{abc} z^a z^b z^c + \frac{1}{2} \tilde{a}_{ab} z^a z^b + \frac{1}{24} \tilde{c}_a z^a + \frac{\zeta(3) \chi(\widetilde{X})}{2(2\pi i)^3}\;, \]

and the instanton part \(\mathcal{F}_{\text{inst}}(z)\) given in (10), in terms of the polylogarithm \(\text{Li}_k(x) = \sum_{n=1}^\infty x^n / n^k\).

The polynomial terms are computed from the triple intersection numbers \(\widetilde{\kappa}_{abc}\) of the mirror threefold \(\widetilde{X}\), together with

\[\begin{split} \tilde{c}_a = \int_{\widetilde{X}} c_2(\widetilde{X}) \wedge \tilde{\beta}_a\;, \quad \tilde{a}_{ab} \equiv \frac{1}{2} \begin{cases} \widetilde{\kappa}_{aab} & a \geq b \\ \widetilde{\kappa}_{abb} & a < b \end{cases}\;, \quad \chi(\widetilde{X}) = \int_{\widetilde{X}} c_3(\widetilde{X})\;, \end{split}\]

for a basis \(\{\tilde{\beta}_a\}_{a=1}^{h^{1,1}(\widetilde{X})}\) of \(H^2(\widetilde{X}, \mathbb{Z})\). The instanton contribution \(\mathcal{F}_{\text{inst}}(z)\) accounts for Type IIA worldsheet instanton corrections, encoded in the genus-zero Gopakumar-Vafa invariants \(\mathscr{N}_{\tilde{\mathbf{q}}}\).

The period vector (9) can then be computed from \(\mathcal{F}\) as reviewed in Type IIB Flux Compactifications.

Implementation in jaxvacua

The period computation infrastructure is spread across several modules:

periods.period_vector_per(XPer[, conj])	Computes the period vector \(\Pi\) in terms of the periods \(X^I\).
periods.prepot_per(XPer[, conj])	Computes the prepotential \(F\) in terms of the periods \(X^I\).
periods.kahler_potential_per(XPer, cXPer)	Computes the value of the Kähler potential \(K\) as a function of the periods \(X^I\).

css.F_LCS_poly(moduli[, conj])	Computes the polynomial contribution \(F_{\mathrm{poly}}\) to the LCS prepotential \(F_{\mathrm{LCS}}\) in terms of the complex structure moduli \(z^i\).
css.F_inst(moduli[, conj])	Returns the instanton part \(F_{\mathrm{inst}}\) of the LCS prepotential \(F_{\mathrm{LCS}}\) in terms of the complex structure moduli \(z^i\).
css.F_LCS(moduli[, conj])	Calculates the value of the LCS prepotential in terms of the complex structure moduli \(z^{i}\).

The topological input data (intersection numbers \(\widetilde{\kappa}_{abc}\), second Chern class integrals \(\tilde{c}_a\), GV invariants \(\mathscr{N}_{\tilde{\mathbf{q}}}\)) is provided through the jaxvacua.lcs module, which constructs the data tree from CYTools or from pre-computed model files.