Supersymmetric B-Twisted Landau–Ginzburg Model

• The supersymmetric B-twisted Landau–Ginzburg model is a topological sector of 2D N=(2,2) theories defined by a holomorphic superpotential and characterized by Jacobian and twisted-Dolbeault data.
• It features a bulk construction using polyvector-valued forms, Koszul complexes, and Frobenius manifold structures that underpin mirror symmetry and singularity theory.
• Its open sector is realized via holomorphic and matrix factorizations, enabling the analysis of D-brane categories, defect fusion, and interfaces in the theory.

The supersymmetric B-twisted Landau–Ginzburg model is the B-type topological sector of a two-dimensional $\mathcal N=(2,2)$ Landau–Ginzburg theory with holomorphic superpotential $W$. In the broad geometric formulation, it is associated to a Landau–Ginzburg pair $(X,W)$, where $X$ is a non-compact Kählerian manifold with holomorphically trivial canonical line bundle and $W:X\to \mathbb C$ is a non-constant holomorphic function. Across the physical and mathematical literature, its bulk sector is described by Jacobian or twisted-Dolbeault data, its open sector by holomorphic factorizations or matrix factorizations, and its mirror-theoretic incarnations range from singularity theory to orbifold and Weyl-orbifolded Landau–Ginzburg models (Babalic et al., 2017, He et al., 2015).

1. Supersymmetric input and the twisted Landau–Ginzburg datum

The untwisted starting point is a two-dimensional $\mathcal N=2$ Wess–Zumino/Landau–Ginzburg model with chiral superfields $\{\Phi_I\}$, complex scalars $A_I$, fermions, and auxiliary fields $F_I$. In the momentum-space formulation with UV cutoff, the Euclidean action contains the bosonic combination

$$S_B=\frac{1}{L_0L_1}\sum_p \sum_I N_I^*(-p)N_I(p),\qquad N_I(p)=2p_zA_I(p)+\frac{\partial W(\{A\})^*}{\partial A_I^*}(p),$$

while the superpotential determines the Yukawa couplings through $W$0 and the scalar potential through $W$1. The same holomorphic superpotential is the central datum in the B-twisted theory, especially through the critical locus $W$2 and the Jacobi-ring structure (Morikawa, 2018).

A standard condition underlying the twist is quasi-homogeneity. For a quasi-homogeneous superpotential one assigns weights $W$3 so that

$W$4

In standard Landau–Ginzburg/SCFT correspondence, the infrared central charge is

$W$5

This structure is explicit in ADE examples such as

$W$6

which are used as concrete quasi-homogeneous Landau–Ginzburg models with the required $W$7 data for topological twisting (Morikawa, 2018).

In the singularity-theoretic formulation of the B-model, the basic bulk observables are expected to be the Jacobi ring of the superpotential,

$W$8

with $W$9 in the mirror-symmetry setting of Berglund–Hübsch–Krawitz. This identifies the B-twisted model as a theory governed by the singularity of a holomorphic potential rather than by Kähler data (He et al., 2015).

2. Bulk topological sector, Jacobian structures, and primitive forms

For a Landau–Ginzburg pair $(X,W)$0 of complex dimension $(X,W)$1, the differential-geometric bulk model is built from polyvector-valued $(X,W)$2-forms: $(X,W)$3 The twisted differential is

$(X,W)$4

and the on-shell bulk algebra is

$(X,W)$5

This cohomology is naturally identified with the hypercohomology of the Koszul complex

$(X,W)$6

When $(X,W)$7 is Stein and the critical set is finite, one has

$(X,W)$8

so the familiar Jacobi algebra is recovered as the bulk state space (Babalic et al., 2017).

The singularity-theoretic enhancement of this picture is K. Saito’s theory. For a quasi-homogeneous singularity $(X,W)$9, the completed Brieskorn lattice is

$X$0

equipped with the higher residue pairing

$X$1

A primitive form is encoded by a good section of $X$2, equivalently by a homogeneous good basis satisfying

$X$3

With a primitive form fixed, the deformation

$X$4

defines a Frobenius-manifold structure whose flat coordinates arise from

$X$5

In this formulation, $X$6 is the primitive form and $X$7 is the B-model $X$8-function (He et al., 2015).

The genus-zero theory is encoded by the prepotential $X$9, while higher genus is reconstructed by Givental quantization and Teleman’s classification of semisimple CohFTs. For invertible quasi-homogeneous singularities, this yields the Saito–Givental realization of the Landau–Ginzburg B-model at all genera, with the all-genus ancestor potential $W:X\to \mathbb C$0 mirror to the FJRW A-model (He et al., 2015).

3. Open sector, B-branes, and interfaces

The open-string sector is described globally by holomorphic factorizations. A holomorphic factorization of $W:X\to \mathbb C$1 is a pair

$W:X\to \mathbb C$2

where $W:X\to \mathbb C$3 is a holomorphic vector superbundle and

$W:X\to \mathbb C$4

The off-shell brane category $W:X\to \mathbb C$5 has morphism spaces

$W:X\to \mathbb C$6

with differential $W:X\to \mathbb C$7, and its total cohomology

$W:X\to \mathbb C$8

is the on-shell category of topological D-branes. On Stein targets, $W:X\to \mathbb C$9 simplifies to categories of holomorphic or projective analytic factorizations, providing a direct analytic analogue of matrix-factorization categories (Babalic et al., 2017).

For interfaces between Landau–Ginzburg theories with superpotentials $\mathcal N=2$0 and $\mathcal N=2$1, B-type supersymmetry leads to matrix factorizations of the difference: $\mathcal N=2$2 Thus a B-type defect from $\mathcal N=2$3 to $\mathcal N=2$4 is a $\mathcal N=2$5-graded free module over $\mathcal N=2$6 with odd differential $\mathcal N=2$7 satisfying the above equation. Boundary conditions are the special case $\mathcal N=2$8, so B-type branes and B-type defects are unified by the same factorization formalism (0707.0922).

Fusion of defects is represented algebraically by tensor product of matrix factorizations. If $\mathcal N=2$9 is a $\{\Phi_I\}$0-factorization and $\{\Phi_I\}$1 a $\{\Phi_I\}$2-factorization, then

$\{\Phi_I\}$3

This is the algebraic realization of physical interface fusion. The same framework describes the action of a defect on a B-type boundary condition by fusion with a factorization of $\{\Phi_I\}$4 (Fredenhagen, 2022).

A more recent reformulation replaces explicit tensor-product reductions by fusion functors. A $\{\Phi_I\}$5-fusion functor is a $\{\Phi_I\}$6-linear functor

$\{\Phi_I\}$7

satisfying

$\{\Phi_I\}$8

for every module map $\{\Phi_I\}$9. Such a functor sends a matrix factorization of $A_I$0 to one of $A_I$1, and the corresponding operator-like interface factorization is recovered from the identity defect as

$A_I$2

In this picture, horizontal composition of interfaces becomes ordinary composition of functors, and cone constructions can be lifted from matrix factorizations to the functorial level (Fredenhagen, 2022).

4. Global analytic models and the one-dimensional case

In complex dimension one, the B-type Landau–Ginzburg model admits a particularly explicit classification. A one-dimensional Landau–Ginzburg pair is

$A_I$3

where $A_I$4 is an open Riemann surface and $A_I$5 is a non-constant holomorphic function. The allowed targets are very broad: $A_I$6 need not be affine algebraic and may have infinite genus or infinitely many Freudenthal ends. Every open Riemann surface is Stein, holomorphically parallelizable, and holomorphically Calabi–Yau, so the one-dimensional theory lies automatically inside the non-anomalous B-type open-closed Landau–Ginzburg framework (Lazaroiu et al., 2018).

The critical set is

$A_I$7

Since $A_I$8 is Stein, compact analytic subsets are finite, and the bulk state space decomposes as

$A_I$9

where $F_I$0 is the analytic Milnor algebra and $F_I$1 its Milnor number. The bulk sector therefore localizes at the critical points, exactly as in standard B-twisted Landau–Ginzburg physics (Lazaroiu et al., 2018).

The open sector is modeled by matrix factorizations over $F_I$2: $F_I$3 When $F_I$4 is critically finite, the category

$F_I$5

is Krull–Schmidt, and its non-zero indecomposable objects are precisely the nontrivial primary elementary factorizations. Writing

$F_I$6

one has the explicit decomposition

$F_I$7

Thus the global D-brane category splits into local singularity categories, one for each multiple zero of $F_I$8. In dimension one, the full topological D-brane sector is controlled by the finite critical divisor of the superpotential (Lazaroiu et al., 2018).

5. Orbifolds, mirrors, and global B-model geometry

Mirror symmetry supplies several large-scale realizations of B-twisted Landau–Ginzburg models. For invertible quasi-homogeneous singularities, the Landau–Ginzburg mirror symmetry conjecture identifies the FJRW A-model of $F_I$9 with the Saito–Givental B-model of the transpose singularity $$S_B=\frac{1}{L_0L_1}\sum_p \sum_I N_I^*(-p)N_I(p),\qquad N_I(p)=2p_zA_I(p)+\frac{\partial W(\{A\})^*}{\partial A_I^*}(p),$$0. At genus zero this is an identification of Frobenius structures; at all genera it is an equality of ancestor potentials after the correct primitive form is chosen on the B-side (He et al., 2015).

Orbifold B-models admit a noncommutative formulation in terms of curved algebras. The basic datum is a triple

$$S_B=\frac{1}{L_0L_1}\sum_p \sum_I N_I^*(-p)N_I(p),\qquad N_I(p)=2p_zA_I(p)+\frac{\partial W(\{A\})^*}{\partial A_I^*}(p),$$1

with $$S_B=\frac{1}{L_0L_1}\sum_p \sum_I N_I^*(-p)N_I(p),\qquad N_I(p)=2p_zA_I(p)+\frac{\partial W(\{A\})^*}{\partial A_I^*}(p),$$2, $$S_B=\frac{1}{L_0L_1}\sum_p \sum_I N_I^*(-p)N_I(p),\qquad N_I(p)=2p_zA_I(p)+\frac{\partial W(\{A\})^*}{\partial A_I^*}(p),$$3 a $$S_B=\frac{1}{L_0L_1}\sum_p \sum_I N_I^*(-p)N_I(p),\qquad N_I(p)=2p_zA_I(p)+\frac{\partial W(\{A\})^*}{\partial A_I^*}(p),$$4-invariant central element, and $$S_B=\frac{1}{L_0L_1}\sum_p \sum_I N_I^*(-p)N_I(p),\qquad N_I(p)=2p_zA_I(p)+\frac{\partial W(\{A\})^*}{\partial A_I^*}(p),$$5 a finite group acting compatibly on $$S_B=\frac{1}{L_0L_1}\sum_p \sum_I N_I^*(-p)N_I(p),\qquad N_I(p)=2p_zA_I(p)+\frac{\partial W(\{A\})^*}{\partial A_I^*}(p),$$6. The orbifold Landau–Ginzburg B-model is modeled by the curved crossed-product algebra

$$S_B=\frac{1}{L_0L_1}\sum_p \sum_I N_I^*(-p)N_I(p),\qquad N_I(p)=2p_zA_I(p)+\frac{\partial W(\{A\})^*}{\partial A_I^*}(p),$$7

Its closed sector is governed by orbifold Hochschild/Jacobian data,

$$S_B=\frac{1}{L_0L_1}\sum_p \sum_I N_I^*(-p)N_I(p),\qquad N_I(p)=2p_zA_I(p)+\frac{\partial W(\{A\})^*}{\partial A_I^*}(p),$$8

and the deformation theory carries a Getzler–Gauss–Manin connection on periodic cyclic homology. In the $$S_B=\frac{1}{L_0L_1}\sum_p \sum_I N_I^*(-p)N_I(p),\qquad N_I(p)=2p_zA_I(p)+\frac{\partial W(\{A\})^*}{\partial A_I^*}(p),$$9-orbifold of $W$00-type singularities, the resulting variation of semi-infinite Hodge structures matches that of the $W$01-type singularity under deformation, giving a deformed McKay-type correspondence for Landau–Ginzburg B-models (He et al., 2019).

For toric orbifolds, the B-side is described by Laurent-polynomial Landau–Ginzburg mirrors on

$W$02

with superpotential

$W$03

Near the large-volume point in the compactified moduli space, the Gauss–Manin and Brieskorn data extend logarithmically across the boundary divisor

$W$04

and the special fiber satisfies

$W$05

This produces a logarithmic Frobenius manifold mirror to orbifold quantum cohomology, together with a variation of pure and polarized TERP structures and hence $W$06-geometry. When a crepant resolution exists, these structures glue globally across birational phases (Mann et al., 2016).

A different mirror-theoretic direction constructs mirrors of nonabelian A-twisted gauge theories as B-twisted Landau–Ginzburg orbifolds. For a gauge group $W$07 of rank $W$08, dimension $W$09, and matter representation of dimension $W$10, the proposed mirror has twisted chiral fields $W$11, matter mirrors $W$12, and root/W-boson fields $W$13, with superpotential

$W$14

Its critical loci reproduce Coulomb-branch and quantum-cohomology relations, and its Hessian-weighted B-model correlators reproduce A-twisted gauge-theory correlators. The full theory is a Weyl-group orbifold, so the nonabelian Lie-theoretic data appear explicitly in the B-twisted Landau–Ginzburg model (Gu et al., 2018).

6. Non-compact theories, related constructions, and important distinctions

The B-twisted Landau–Ginzburg framework extends beyond compact polynomial superpotentials. A non-compact class arises from $W$15 Liouville/cigar theories and is described by a negative-power superpotential

$W$16

In these models the strict chiral ring is

$W$17

and the identity operator is not normalizable. The paper describes the ring as naturally without unit, introduces an extended ring including the almost-normalizable operator $W$18, computes the topological metric

$W$19

and develops deformed operators using

$W$20

The corresponding $W$21 equations reduce to affine Toda systems, and for the negative cubic case the metric is governed by the Painlevé III equation (Li et al., 2018).

Several nearby constructions are often conflated with the B-twist but are distinct. In three dimensions, a $W$22 Landau–Ginzburg theory on a half-space admits B-type boundary conditions preserving a two-dimensional $W$23 supersymmetry algebra. The resulting B-branes are holomorphic submanifolds on which the superpotential is constant. This is the three-dimensional analogue of B-type brane geometry, but it is not a topological B-twist and does not introduce BRST operators (Okazaki et al., 2013).

Likewise, the numerical study of the two-dimensional massless $W$24 Wess–Zumino model with two superfields provides explicit component-field realizations, quasi-homogeneous superpotentials, exact supersymmetry-preserving nonperturbative formulations, the Witten index, and infrared central charges for $W$25, $W$26, and $W$27 models, but it does not construct the B-twist itself. Its relevance is as structural background for the untwisted Landau–Ginzburg theory on which the twist is performed (Morikawa, 2018).

Another frequent source of confusion is mirror-symmetric indirect evidence. The computation of the Landau–Ginzburg $W$28-model for

$W$29

via microlocal sheaves on the skeleton $W$30 yields

$W$31

which identifies the mirror $W$32-model of the pair-of-pants. This is a precise homological mirror symmetry statement, but it is not a direct construction of the B-twisted Landau–Ginzburg model of $W$33 itself (Nadler, 2015).

Finally, in heterotic $W$34 Landau–Ginzburg models, the literature summarized here centers on the A-twist, not the B-twist. Its main direct relevance to B-twisted heterotic theories is the supersymmetry-imposed curvature constraint for nonholomorphic superpotentials,

$W$35

together with

$W$36

These conditions are stated there to govern Mathai–Quillen-type structures arising in correlation functions of both A-twisted and B-twisted heterotic Landau–Ginzburg models, but the paper does not construct the B-twist explicitly (Garavuso, 2022).

The supersymmetric B-twisted Landau–Ginzburg model is therefore best understood not as a single formalism but as a tightly connected family of equivalent descriptions: a twisted $W$37 field theory controlled by a holomorphic superpotential; a bulk algebra given by twisted Dolbeault or Jacobian data; an open sector given by holomorphic or matrix factorizations; a singularity-theoretic Frobenius-manifold and primitive-form package; and, in mirror-symmetric contexts, a broad class of orbifolded, logarithmic, and gauge-theoretic Landau–Ginzburg realizations.