---
title: Supersymmetric B-Twisted Landau–Ginzburg Model
url: https://www.emergentmind.com/topics/supersymmetric-b-twisted-landau-ginzburg-model
type: topic
---

# Supersymmetric B-Twisted Landau–Ginzburg Model

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 [1709.00684][1503.01757].

## 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 \(\partial_I\partial_JW\) and the scalar potential through \(|\partial_IW|^2\). The same holomorphic superpotential is the central datum in the B-twisted theory, especially through the critical locus \(\partial_iW=0\) and the Jacobi-ring structure [1810.02519].

A standard condition underlying the twist is quasi-homogeneity. For a quasi-homogeneous superpotential one assigns weights \(q_i\) so that
\[
W(\lambda^{q_i}\Phi_i)=\lambda\,W(\Phi_i).
\]
In standard Landau–Ginzburg/SCFT correspondence, the infrared central charge is
\[
c=3\sum_i(1-2q_i).
\]
This structure is explicit in ADE examples such as
\[
D_n:\quad W(\{\Phi\})=\frac{\lambda_1}{n}\Phi_1^n+\frac{\lambda_2}{2}\Phi_1\Phi_2^2,
\qquad
E_7:\quad W(\{\Phi\})=\frac{\lambda_1}{3}\Phi_1^3+\frac{\lambda_2}{3}\Phi_1\Phi_2^3,
\]
which are used as concrete quasi-homogeneous Landau–Ginzburg models with the required \(U(1)\) data for topological twisting [1810.02519].

In the singularity-theoretic formulation of the B-model, the basic bulk observables are expected to be the Jacobi ring of the superpotential,
\[
\mathrm{Jac}(f)=\mathbb C[x_1,\dots,x_N]\Big/\left(\frac{\partial f}{\partial x_1},\dots,\frac{\partial f}{\partial x_N}\right),
\]
with \(f=W^T\) 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 [1503.01757].

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

For a Landau–Ginzburg pair \((X,W)\) of complex dimension \(d\), the differential-geometric bulk model is built from polyvector-valued \((0,j)\)-forms:
\[
\PV(X)=\bigoplus_{i=-d}^{0}\bigoplus_{j=0}^{d}\PV^{i,j}(X)
=\bigoplus_{i=-d}^{0}\bigoplus_{j=0}^{d}\Omega^{0,j}\!\left(X,\wedge^{|i|}TX\right).
\]
The twisted differential is
\[
\delta_W=\bar\partial+\iota_W,\qquad \iota_W=-\mathbf{i}\,(\partial W)\lrcorner,
\]
and the on-shell bulk algebra is
\[
\HPV(X,W)=H(\PV(X),\delta_W).
\]
This cohomology is naturally identified with the hypercohomology of the Koszul complex
\[
(Q_W):\quad 0\to \wedge^dTX\xrightarrow{\iota_W}\wedge^{d-1}TX\xrightarrow{\iota_W}\cdots\xrightarrow{\iota_W}O_X\to 0.
\]
When \(X\) is Stein and the critical set is finite, one has
\[
\HPV^k(X,W)=0\quad(k\neq 0),\qquad \HPV^0(X,W)\simeq \operatorname{Jac}(X,W),
\]
so the familiar Jacobi algebra is recovered as the bulk state space [1709.00684].

The singularity-theoretic enhancement of this picture is K. Saito’s theory. For a quasi-homogeneous singularity \(f\), the completed Brieskorn lattice is
\[
\mathcal H_f^{(0)}=
\Omega^N_{\mathbb C^N,0}[z]\Big/(df\wedge + z\,d)\Omega^{N-1}_{\mathbb C^N,0},
\]
equipped with the higher residue pairing
\[
K_f:\mathcal H_f^{(0)}\otimes \mathcal H_f^{(0)}\to z^N\mathbb C[z].
\]
A primitive form is encoded by a good section of \(\mathcal H_f^{(0)}\to \Omega_f\), equivalently by a homogeneous good basis satisfying
\[
K_f(\phi_\alpha d^Nx,\phi_\beta d^Nx)\in z^N\mathbb C.
\]
With a primitive form fixed, the deformation
\[
F=f+\sum_\alpha s_\alpha\phi_\alpha
\]
defines a Frobenius-manifold structure whose flat coordinates arise from
\[
e^{(F-f)/z}\,\zeta(z,s)=\mathcal J.
\]
In this formulation, \(\zeta\) is the primitive form and \(\mathcal J\) is the B-model \(J\)-function [1503.01757].

The genus-zero theory is encoded by the prepotential \(\mathcal F_{0,f,\zeta}^{\mathrm{SG}}\), 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 \(\mathscr A^{\mathrm{SG}}_{W^T,\zeta}\) mirror to the FJRW A-model [1503.01757].

## 3. Open sector, B-branes, and interfaces

The open-string sector is described globally by holomorphic factorizations. A holomorphic factorization of \(W\) is a pair
\[
a=(E,D),
\]
where \(E=E^{\hat 0}\oplus E^{\hat 1}\) is a holomorphic vector superbundle and
\[
D\in \Gamma(X,\End^{\hat 1}(E)),\qquad D^2=W\,\id_E.
\]
The off-shell brane category \(DF(X,W)\) has morphism spaces
\[
\Hom_{DF(X,W)}(a_1,a_2)=\mathcal A\bigl(X,\Hom(E_1,E_2)\bigr)
\]
with differential \(\bar\partial+\mathfrak d\), and its total cohomology
\[
\HDF(X,W)=H(DF(X,W))
\]
is the on-shell category of topological D-branes. On Stein targets, \(\HDF(X,W)\) simplifies to categories of holomorphic or projective analytic factorizations, providing a direct analytic analogue of matrix-factorization categories [1709.00684].

For interfaces between Landau–Ginzburg theories with superpotentials \(W_1(X)\) and \(W_2(Y)\), B-type supersymmetry leads to matrix factorizations of the difference:
\[
Q^2=\bigl(W_1(X)-W_2(Y)\bigr)\mathbf 1.
\]
Thus a B-type defect from \(W_1\) to \(W_2\) is a \(\mathbb Z_2\)-graded free module over \(\mathbb C[X_i,Y_i]\) with odd differential \(Q\) satisfying the above equation. Boundary conditions are the special case \(W_2=0\), 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 \(Q_1\) is a \((W',W)\)-factorization and \(Q_2\) a \((W,W'')\)-factorization, then
\[
Q_1\boxtimes Q_2=Q_1\otimes \mathbf 1+\sigma_1\otimes Q_2,
\qquad
(Q_1\boxtimes Q_2)^2=(W'-W'')\,\mathrm{id}.
\]
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 \(W\) [2212.11573].

A more recent reformulation replaces explicit tensor-product reductions by fusion functors. A \((W',W)\)-fusion functor is a \(\mathbb C\)-linear functor
\[
U:R\text{-mod}_f^{\mathbb Z_2}\to R'\text{-mod}_f^{\mathbb Z_2}
\]
satisfying
\[
U(Wf)=W'U(f)
\]
for every module map \(f\). Such a functor sends a matrix factorization of \(W\) to one of \(W'\), and the corresponding operator-like interface factorization is recovered from the identity defect as
\[
\Pi^{I_W}(U)=U(I_W).
\]
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 [2212.11573].

## 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
\[
(\Sigma,W),
\]
where \(\Sigma\) is an open Riemann surface and \(W\in \mathrm O(\Sigma)\) is a non-constant holomorphic function. The allowed targets are very broad: \(\Sigma\) 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 [1811.05650].

The critical set is
\[
Z_W=\{x\in \Sigma\mid (\partial W)(x)=0\}.
\]
Since \(\Sigma\) is Stein, compact analytic subsets are finite, and the bulk state space decomposes as
\[
H\simeq_{\mathrm O(X)}\bigoplus_{p\in Z_W}\mathrm M(\widehat W_p),
\qquad
\dim_{\mathbb C}H=\sum_{p\in Z_W}\nu_p,
\]
where \(\mathrm M(\widehat W_p)\) is the analytic Milnor algebra and \(\nu_p\) its Milnor number. The bulk sector therefore localizes at the critical points, exactly as in standard B-twisted Landau–Ginzburg physics [1811.05650].

The open sector is modeled by matrix factorizations over \(\mathrm O(\Sigma)\):
\[
T\simeq_{\mathrm O(\Sigma)} HMF(\mathrm O(\Sigma),W).
\]
When \(W\) is critically finite, the category
\[
hmf(\mathrm O(\Sigma),W)
\]
is Krull–Schmidt, and its non-zero indecomposable objects are precisely the nontrivial primary elementary factorizations. Writing
\[
W=W_0\prod_{i=1}^N t_{x_i}^{n_i},
\qquad
A_i=\mathrm O(\Sigma)/\langle t_{x_i}^{n_i}\rangle,
\]
one has the explicit decomposition
\[
hmf(\mathrm O(\Sigma),W)\simeq \bigvee_{i=1}^N D_{\mathrm{Sing}(A_i)}
\simeq \bigvee_{i=1}^N \underline{\mathrm{mod}_{A_i}}.
\]
Thus the global D-brane category splits into local singularity categories, one for each multiple zero of \(W\). In dimension one, the full topological D-brane sector is controlled by the finite critical divisor of the superpotential [1811.05650].

## 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 \(W\) with the Saito–Givental B-model of the transpose singularity \(W^T\). 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 [1503.01757].

Orbifold B-models admit a noncommutative formulation in terms of curved algebras. The basic datum is a triple
\[
(A,W,G),
\]
with \(A=\mathbb C[x,y]\), \(W\) a \(G\)-invariant central element, and \(G\) a finite group acting compatibly on \(A\). The orbifold Landau–Ginzburg B-model is modeled by the curved crossed-product algebra
\[
A_W[G]=A\rtimes \mathbb C[G].
\]
Its closed sector is governed by orbifold Hochschild/Jacobian data,
\[
\operatorname{Jac}(W,G)=\bigoplus_{g\in G}\operatorname{Jac}(W_g)^G[-\iota_g],
\]
and the deformation theory carries a Getzler–Gauss–Manin connection on periodic cyclic homology. In the \(\mathbb Z_2\)-orbifold of \(A\)-type singularities, the resulting variation of semi-infinite Hodge structures matches that of the \(D\)-type singularity under deformation, giving a deformed McKay-type correspondence for Landau–Ginzburg B-models [1904.09426].

For toric orbifolds, the B-side is described by Laurent-polynomial Landau–Ginzburg mirrors on
\[
Y=(\mathbb C^*)^d
\]
with superpotential
\[
F(\underline y,\underline\chi)= -\sum_{i=1}^{n}\underline{\chi}^{\,\underline n_i}\,\underline y^{a_i}.
\]
Near the large-volume point in the compactified moduli space, the Gauss–Manin and Brieskorn data extend logarithmically across the boundary divisor
\[
D_X=\{\chi_1\cdots \chi_r=0\},
\]
and the special fiber satisfies
\[
{}_0\mathcal Q\mathcal M^{\log,X}_{\,|\, z=\underline\chi=0}
\simeq H^*_{\mathrm{orb}(X,\mathbb C)}.
\]
This produces a logarithmic Frobenius manifold mirror to orbifold quantum cohomology, together with a variation of pure and polarized TERP structures and hence \(tt^*\)-geometry. When a crepant resolution exists, these structures glue globally across birational phases [1605.08937].

A different mirror-theoretic direction constructs mirrors of nonabelian A-twisted gauge theories as B-twisted Landau–Ginzburg orbifolds. For a gauge group \(G\) of rank \(r\), dimension \(n\), and matter representation of dimension \(N\), the proposed mirror has twisted chiral fields \(\sigma_a\), matter mirrors \(Y_i\), and root/W-boson fields \(X_{\tilde\mu}\), with superpotential
\[
W = \sum_{a=1}^r \sigma_a \Big( \sum_{i=1}^N \rho_i^a Y_i - \sum_{\tilde\mu=1}^{n-r} \alpha_{\tilde\mu}^a \ln X_{\tilde\mu} - t_a \Big) + \sum_{i=1}^N \exp(-Y_i) + \sum_{\tilde\mu=1}^{n-r} X_{\tilde\mu}.
\]
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 [1806.04678].

## 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 \(N=2\) Liouville/cigar theories and is described by a negative-power superpotential
\[
W=\frac{Y^{-k}}{k}.
\]
In these models the strict chiral ring is
\[
R=\mathrm{span}\{Y^{-2},Y^{-3},\dots,Y^{-k}\},
\]
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 \(Y^{-1}\), computes the topological metric
\[
\eta_{ij}=\delta_{i+j,k+2},
\]
and develops deformed operators using
\[
L=(kW)^{1/k},
\qquad
\phi_i=\big[-L^{\,i-2}\partial_YL\big]_{\le -2}.
\]
The corresponding \(tt^*\) equations reduce to affine Toda systems, and for the negative cubic case the metric is governed by the Painlevé III equation [1803.08710].

Several nearby constructions are often conflated with the B-twist but are distinct. In three dimensions, a \(\mathcal N=2\) Landau–Ginzburg theory on a half-space admits B-type boundary conditions preserving a two-dimensional \(N=(2,0)\) 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 [1302.6593].

Likewise, the numerical study of the two-dimensional massless \(\mathcal N=2\) 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 \(D_3\), \(D_4\), and \(E_7\) 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 [1810.02519].

Another frequent source of confusion is mirror-symmetric indirect evidence. The computation of the Landau–Ginzburg \(A\)-model for
\[
M=\mathbb C^3,\qquad W=z_1z_2z_3
\]
via microlocal sheaves on the skeleton \(L=\Cone(T^2)\) yields
\[
\mu Sh_\Lambda(X)\simeq Coh_{\torsion}(\mathbb P^1\setminus\{0,1,\infty\}),
\]
which identifies the mirror \(B\)-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 \((\mathbb C^3,W)\) itself [1507.08735].

Finally, in heterotic \((0,2)\) 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,
\[
\partial_i \overline{D}_{\overline{\imath} \overline{F}_{\overline{a}}
+ F_{i \overline{\imath} a \overline{a}}\, h^{a \overline{b}} \overline{F}_{\overline{b}}=0,
\]
together with
\[
\overline{F}_{\overline{a}, k}\,\partial_z\phi^k\lambda_-^{\overline a}=0.
\]
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 [2205.15533].

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 \(\mathcal N=(2,2)\) 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.

Source: https://www.emergentmind.com/topics/supersymmetric-b-twisted-landau-ginzburg-model