Canonical Local B Model Framework

• Canonical local B model is a framework that uses mirror curves of non-compact Calabi–Yau threefolds to extract Picard–Fuchs systems for 5d gauge theories.
• It employs a canonical algebra on the ramification locus and, in simply-laced cases, Frobenius manifold structures to compute unique structure constants.
• The method is validated through precision tests against gauge theory prepotentials, matching both perturbative and instanton corrections.

The expression “Canonical Local B Model” is not introduced as a formal standalone term in the cited literature. The closest precise usage is a canonical effective framework for the local topological B-model on non-compact Calabi–Yau threefolds that engineer $5d\; \mathcal N=1$ pure super Yang–Mills theory compactified on a circle. In that setting, the genus-zero B-model is encoded by Seiberg–Witten or mirror curves, and its Picard–Fuchs system is extracted canonically from the curve rather than guessed case by case. In the simply-laced case this construction is governed by Frobenius manifolds of extended affine Weyl groups; in the general case it is formulated algebraically through a canonical subring of regular functions on the ramification locus of the reduced spectral curve (Brini et al., 2021).

1. Terminology, scope, and physical setting

In the relevant usage, the local B-model is the B-model of a non-compact Calabi–Yau threefold whose effective geometry is encoded by a family of affine spectral curves

$$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$

These geometries engineer pure $5d$ gauge theory with simple gauge group $G$ on

$\mathbb R^4\times S^1_{R_5}.$

The Coulomb branch is described by the complexified adjoint scalar

$\varphi=\phi+iA_5=\sum_{i=1}^r a_i h_i,$

with exponentiated Cartan coordinates

$q_i=e^{2\pi iR_5 a_i},$

Weyl-invariant coordinates $u_i$ defined from holonomies of $(A_5-i\phi)_{\rm cl}$ around $S^1_{R_5}$, and the scale/radius modulus

$$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$0

The low-energy effective theory is governed by a prepotential $$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$1. In the decompactified $$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$2 limit it is the IMS cubic,

$$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$3

while on $$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$4 it receives KK and instanton corrections,

$$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$5

$$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$6

The B-model data are extracted from periods of the Seiberg–Witten differential

$$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$7

with special geometry relations

$$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$8

This is the operational meaning of the local B-model in this framework: the non-compact Calabi–Yau is represented effectively by its mirror curve together with these period integrals. The candidate spectral polynomials include the M-theory/brane-engineering curve $$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$9, the untwisted relativistic Toda curve $5d$0, and the twisted relativistic Toda curve $5d$1 (Brini et al., 2021).

2. Canonical extraction of the Picard–Fuchs system

The adjective canonical refers to a curve-intrinsic construction of the differential system. A key simplification is that, in the cases of interest, the curve can be written in reduced form

$5d$2

with

$5d$3

The reduced polynomial describes the perturbative geometry and is the input for the canonical algebraic construction.

The starting point is the ideal

$5d$4

which cuts out the ramification locus of the projection to the $5d$5-line. Given distinguished coordinates $5d$6, one defines

$5d$7

and then the $5d$8-dimensional vector space

$5d$9

The central claim is closure under multiplication: $G$0 for unique structure constants $G$1. This yields a commutative associative algebra canonically attached to the ramification scheme of the curve. Practically, the ring is computed by taking a reduced Gröbner basis $G$2 of $G$3 and reducing products modulo that basis; vanishing of the remainder gives an overdetermined linear system for the $G$4, which has a unique solution for the Seiberg–Witten curves considered.

Once $G$5 is known, the Picard–Fuchs ideal is generated by the second-order operators

$G$6

This universal relation,

$G$7

is the core Picard–Fuchs system of the canonical local B-model. The construction is canonical precisely because the multiplication tensor $G$8 is extracted functorially from the curve and its ramification algebra, rather than being reverse-engineered from low-order periods or toric GKZ data (Brini et al., 2021).

3. Simply-laced formulation via Frobenius manifolds

For simply-laced gauge groups, the construction has a geometric realization in terms of Frobenius manifolds associated to extended affine Weyl groups of type $G$9. For a simple ADE Lie algebra $\mathbb R^4\times S^1_{R_5}.$0, one considers the extended affine Weyl action

$\mathbb R^4\times S^1_{R_5}.$1

on $\mathbb R^4\times S^1_{R_5}.$2, and the quotient

$\mathbb R^4\times S^1_{R_5}.$3

This space is identified with the classical Coulomb branch times the circle modulus.

The quotient carries a canonical semisimple Frobenius manifold structure with flat coordinates $\mathbb R^4\times S^1_{R_5}.$4, prepotential $\mathbb R^4\times S^1_{R_5}.$5, metric

$\mathbb R^4\times S^1_{R_5}.$6

and multiplication

$\mathbb R^4\times S^1_{R_5}.$7

The Euler vector field is

$\mathbb R^4\times S^1_{R_5}.$8

and, in the normalization used in the introduction,

$\mathbb R^4\times S^1_{R_5}.$9

The proposal is that Seiberg–Witten periods are odd periods of this Frobenius manifold and therefore satisfy

$\varphi=\phi+iA_5=\sum_{i=1}^r a_i h_i,$0

This is the ADE canonical Picard–Fuchs system. For $\varphi=\phi+iA_5=\sum_{i=1}^r a_i h_i,$1, it reduces to the standard GKZ system of the toric local Calabi–Yau mirror. The relation to the reduced spectral polynomial is expressed by the Landau–Ginzburg/Jacobi-ring identity

$\varphi=\phi+iA_5=\sum_{i=1}^r a_i h_i,$2

so the algebraic construction in the general case is the direct shadow of Frobenius-manifold multiplication in the simply-laced case (Brini et al., 2021).

4. Non-simply-laced extension and the role of privileged coordinates

The non-simply-laced case is the most distinctively canonical part of the framework. The paper states that there is no suitable twisted affine Frobenius manifold underlying the actual $\varphi=\phi+iA_5=\sum_{i=1}^r a_i h_i,$3 gauge-theory curves: the naive twisted analogue fails because the natural metric is degenerate or curved, and the twisted relativistic Toda chain does not reproduce the correct $\varphi=\phi+iA_5=\sum_{i=1}^r a_i h_i,$4 prepotential. The replacement is therefore purely algebraic.

The main claim is that there exists, up to affine-linear transformations, a unique coordinate chart $\varphi=\phi+iA_5=\sum_{i=1}^r a_i h_i,$5 such that:

1. the span $\varphi=\phi+iA_5=\sum_{i=1}^r a_i h_i,$6 closes with structure constants $\varphi=\phi+iA_5=\sum_{i=1}^r a_i h_i,$7;
2. the Seiberg–Witten periods satisfy

$\varphi=\phi+iA_5=\sum_{i=1}^r a_i h_i,$8

1. the inverse map is quasi-polynomial,

$\varphi=\phi+iA_5=\sum_{i=1}^r a_i h_i,$9

and the large-complex-structure system admits single-logarithmic solutions.

For $q_i=e^{2\pi iR_5 a_i},$0, the hyperelliptic form allows a direct homogeneity relation. In general, the periods can be written as

$q_i=e^{2\pi iR_5 a_i},$1

and one finds equations of the form

$q_i=e^{2\pi iR_5 a_i},$2

The $q_i=e^{2\pi iR_5 a_i},$3 example is representative. For the M-theory curve

$q_i=e^{2\pi iR_5 a_i},$4

the canonical coordinates are

$q_i=e^{2\pi iR_5 a_i},$5

with inverse

$q_i=e^{2\pi iR_5 a_i},$6

The resulting homogeneity equation is

$q_i=e^{2\pi iR_5 a_i},$7

A useful caution appears already in rank one. For $q_i=e^{2\pi iR_5 a_i},$8, the algebraic Picard–Fuchs system is still canonical, but rank-one geometry leaves residual ambiguities not fixed by the PF equations alone; these must be resolved by direct period expansions from the full curve. This suggests that canonicity at the level of the PF ideal does not always determine the full physical geometry without additional special-geometry input (Brini et al., 2021).

5. Period reconstruction, weak-coupling expansion, and prepotential recovery

The weak-coupling or large-complex-structure variables are

$q_i=e^{2\pi iR_5 a_i},$9

where $u_i$0 is the Cartan matrix. Near $u_i$1, periods are sought in the form

$u_i$2

with at most double logarithms. The single-logarithmic solutions are identified with the special coordinates $u_i$3, while the double-logarithmic solutions are identified with the dual periods $u_i$4. The prepotential is then reconstructed from

$u_i$5

For non-simply-laced groups, the remaining ambiguities are fixed by imposing the existence of a prepotential with weak-coupling analytic structure

$u_i$6

Concrete examples illustrate the method. For $u_i$7,

$u_i$8

and the Picard–Fuchs system yields the $u_i$9 prepotential. For $(A_5-i\phi)_{\rm cl}$0 with $(A_5-i\phi)_{\rm cl}$1,

$(A_5-i\phi)_{\rm cl}$2

and the reconstructed prepotential matches the $(A_5-i\phi)_{\rm cl}$3 instanton expansion. The $(A_5-i\phi)_{\rm cl}$4 prepotential recovered from the canonical system begins

$(A_5-i\phi)_{\rm cl}$5

This matches gauge theory. By contrast, in rank-one symplectic theory the modern $(A_5-i\phi)_{\rm cl}$6 curve yields

$(A_5-i\phi)_{\rm cl}$7

in agreement with the local A-model on $(A_5-i\phi)_{\rm cl}$8 (Brini et al., 2021).

6. Precision tests, diagnostics, and related local B-model literature

The framework is not merely constructive; it is also diagnostic. Whenever a candidate curve is available from brane engineering or M-theory, the canonically extracted Picard–Fuchs system is solved near weak coupling and the resulting prepotential is compared with gauge-theory prepotentials computed from K-theoretic blow-up equations. In successful cases these comparisons include perturbative, one-instanton, and higher-instanton terms and show perfect agreement. This applies to simply-laced examples such as $(A_5-i\phi)_{\rm cl}$9, $S^1_{R_5}$0, $S^1_{R_5}$1, and $S^1_{R_5}$2, and to non-simply-laced M-theory curves such as $S^1_{R_5}$3 and $S^1_{R_5}$4 (Brini et al., 2021).

The same method rules out incorrect geometries. For $S^1_{R_5}$5, both the preferred modern curve and an older Brandhuber et al. curve produce the same discriminant algebra and the same $S^1_{R_5}$6,

$S^1_{R_5}$7

with

$S^1_{R_5}$8

but the full periods differ once integration constants are fixed; only the modern $S^1_{R_5}$9 curve reproduces the instanton expansion. Similarly, the canonical PF machinery applies to twisted relativistic Toda curves, but for twisted $$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$00 and $$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$01 the resulting prepotentials disagree with gauge theory already at one loop. In the twisted $$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$02 case the perturbative term contains

$$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$03

instead of the gauge-theory sum with unit weights. The construction itself remains mathematically consistent; the failure lies in the physical identification of the twisted Toda curve as the Seiberg–Witten geometry.

Mirror symmetry provides the broader interpretation. For $$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$04, the Seiberg–Witten curve is the Hori–Iqbal–Vafa mirror of a toric local Calabi–Yau, and the Picard–Fuchs system reduces to GKZ. A related literature computes local B-model Yukawa couplings directly from A-twisted GLSM correlators for local toric Calabi–Yau targets; in that setting, ambiguities of classical intersection numbers are interpreted as degrees of freedom of twisted-mass deformations (Honma et al., 2018). This suggests a useful distinction. The canonical local B-model of the $$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$05 gauge-theory framework is primarily a curve-intrinsic construction of Picard–Fuchs systems and genus-zero prepotentials, whereas the GLSM approach is a localization-based computation of Yukawa couplings for toric local mirrors. The two are compatible at the level of local mirror symmetry, but they are organized around different canonical data.

In this sense, the canonical local B-model is best understood not as a universally standardized term, but as a precise program: the extraction of special-geometry data from local Calabi–Yau mirror curves by a canonical algebra on the ramification locus, supplemented in ADE by Frobenius-manifold geometry and validated by non-perturbative comparison with $$C_{G;u}=\{(\mu,\lambda)\in(\mathbb C^\ast)^2\mid \mathsf P_{G;u}(\mu,\lambda)=0\}.$$06 gauge theory (Brini et al., 2021).