A- and B-Twists and Cohomological Observables

A topological twist changes the Lorentz generator by an R-symmetry generator so that a supercharge becomes a scalar on an oriented Euclidean worldsheet. Observables are classes of that scalar supercharge, and metric variations vanish in cohomology when the stress tensor is a supercharge commutator. In a $(2,2)$ theory the vector and axial R symmetries give inequivalent A- and B-twists: the A-model probes symplectic and complexified Kähler data, while the B-model probes complex structure and holomorphic superpotential data.

Required background. We use the $(2,2)$ algebra and R-charge convention and the relation between supercharge cohomology and Hodge complexes. Helpful background. The axiomatic meaning of a topological field theory clarifies what the twist does and does not establish.

Making scalar supercharges

Continue the Lorentzian charges on the algebra page to a Euclidean rotation charge $s$. Let a field have R charge $q_R$. With the local convention

$$s'=s+\frac12q_R,$$

the vector twist makes

$$Q_A=\bar Q_++Q_-$$

a scalar, while the axial twist makes

$$Q_B=\bar Q_++\bar Q_-$$

a scalar. Indeed, $\bar Q_+$ has $(s,q_V,q_A)=(\tfrac12,-1,-1)$, $Q_-$ has $(-\tfrac12,+1,-1)$, and $\bar Q_-$ has $(-\tfrac12,-1,+1)$. Applying the displayed shift gives

Twist	R symmetry	Scalar supercharge	Basic dependence
A	$U(1)_V$	$Q_A=\bar Q_++Q_-$	symplectic/complexified Kähler data
B	$U(1)_A$	$Q_B=\bar Q_++\bar Q_-$	complex and holomorphic data

This scalar-charge assignment is the translation of Hori et al. 2003, §16.2.1, pp. 400–403 into the R-charge signs fixed on the prerequisite page. Literature using $s'=s-q_R/2$ selects the conjugate scalar charges instead, so the sign of the spin shift and the charge table must always be translated together.

Central extensions give $Q_A^2=2\widetilde Z$ and $Q_B^2=2Z$ in the algebra fixed on the prerequisite page. The relevant central charge must vanish on the local-operator sector, or $Q^2$ must act only by a gauge or flavor transformation that is quotiented in equivariant cohomology. Boundary conditions must also cancel the surface terms.

Quantum anomaly freedom is essential, and integral R charges are needed for the shifted fields to define honest tensor bundles on a general Riemann surface. In an ordinary Kähler sigma model without a superpotential, $U(1)_V$ is non-anomalous, so the A-twist exists for a generic Kähler target. The axial anomaly is measured by $c_1(TX)$, so the B-twist requires its cancellation for every allowed map; a Calabi–Yau target supplies the standard case. In an Abelian GLSM, the corresponding perturbative B-twist condition is $\sum_iQ_i^{\ a}=0$ for each gauge factor. With a superpotential, the chosen R symmetry must also preserve the action—for example, an A-twist of an LG model requires an appropriate quasi-homogeneous vector R assignment. These conditions are summarized in Hori et al. 2003, §16.4, pp. 408–428.

Twisted sigma-model fields

For an ordinary chiral multiplet whose scalar has zero vector and axial R charge, the fermion bundles reorganize as follows. Bars label the fields obtained by Lorentzian conjugation before Euclidean complexification.

Twist	Scalar fermions	One-form fermions	Semiclassical local complex
A	$\psi_-^i$, $\bar\psi_+^{\bar i}$	$\psi_+^i$, $\bar\psi_-^{\bar i}$	$\Omega^\bullet(X)$ with differential $\mathrm d$
B	$\bar\psi_+^{\bar i}$, $\bar\psi_-^{\bar i}$	$\psi_+^i$, $\psi_-^i$	$\Omega^{0,\bullet}(X,\wedge^\bullet T^{1,0}X)$ with differential $\bar\partial$

In the B-twist the two scalar fermions are conventionally recombined into $\eta^{\bar i}$ and a covector $\vartheta_i=g_{i\bar j}(\bar\psi_+^{\bar j}-\bar\psi_-^{\bar j})$, while the unbarred fermions combine into a one-form $\rho^i$. This table is the field-level reason the two scalar charges compute different cohomologies; it is not merely a mnemonic about which moduli each model remembers.

Schematically, and suppressing convention-dependent constants,

$$Q_A\phi^i\sim\psi_-^i, \qquad Q_A\bar\phi^{\bar i}\sim\bar\psi_+^{\bar i}, \qquad Q_B\phi^i=0, \qquad Q_B\bar\phi^{\bar i}\sim\eta^{\bar i}.$$

The remaining transformations pair the one-form fermions with derivatives and auxiliary fields. Eliminating those auxiliaries gives the de Rham complex in the A-model and the Dolbeault–polyvector complex in the B-model. In a gauged theory these become gauge-covariant, equivariant complexes and must be supplemented by the twisted vector multiplet.

Why correlation functions become topological

Suppose the twisted stress tensor obeys

$$T_{\mu\nu}=\{Q,G_{\mu\nu}\}$$

up to equations of motion and improvement terms. For $Q$-closed operators $\mathcal O_a$ and a $Q$-invariant measure,

$$\frac{\delta}{\delta g^{\mu\nu}} \left\langle\prod_a\mathcal O_a\right\rangle =-\left\langle\{Q,G_{\mu\nu}\} \prod_a\mathcal O_a\right\rangle=0.$$

The last equality is a Ward identity. It can fail if the measure is anomalous, the integration region has a boundary at infinity, operator collisions generate contact terms, or the boundary conditions are not $Q$-invariant. “$T$ is $Q$-exact” is therefore a local algebraic mechanism, not a substitute for compactness and anomaly checks.

The action often decomposes as

$$S=\{Q,V\}+S_{\mathrm{top}}.$$

Rescaling the coefficient of the $Q$-exact term does not change protected correlators, permitting localization onto the zero locus of its bosonic part.

The A-model

For a Kähler sigma model, the A-twist uses the scalar fields $\psi_-^i$ and $\bar\psi_+^{\bar i}$ and the complementary one-form fields shown above. The bosonic localization equations are the pseudoholomorphic-map equations,

$$\bar\partial\phi=0$$

for a chosen orientation. Local $Q_A$-cohomology is represented semiclassically by differential forms on $X$. A $(p,q)$-form

$$\begin{aligned} \omega&=\frac{1}{p!q!}\, \omega_{I\bar J}(\phi,\bar\phi)\, \mathrm d\phi^I\wedge\mathrm d\bar\phi^{\bar J},\\ \mathcal O_\omega&=\frac{1}{p!q!}\, \omega_{I\bar J}(\phi,\bar\phi)\, \psi_-^I\bar\psi_+^{\bar J}, \end{aligned}$$

where $I=(i_1,\ldots,i_p)$ and $\bar J=(\bar j_1,\ldots,\bar j_q)$ are antisymmetrized multi-indices. These are the corresponding differential form and scalar-fermion operator. Acting with $Q_A$ reproduces the de Rham differential.

At zero instanton degree, operator multiplication is the cup product. Holomorphic maps deform it to quantum cohomology, as in the topological sigma-model construction of Witten 1988, §3.2, pp. 430–436:

$$\mathcal O_a*\mathcal O_b =\sum_{\beta\in H_2(X,\mathbb Z)}\sum_c q^\beta C_{ab}^{\ \ c}(\beta)\mathcal O_c.$$

Here $q^\beta$ is the instanton weight and $C_{ab}^{\ \ c}(\beta)$ denotes the appropriate Gromov–Witten coefficient when the stable-map problem and its virtual cycle are defined. For a noncompact target or an uncontrolled boundary of map space, the formal sum is not by itself a correlator.

For $X=\mathbb{CP}^{N-1}$ with hyperplane class $H$,

$$QH^*(\mathbb{CP}^{N-1}) =\mathbb C[H,q]/(H^N-q).$$

The equality is understood with a convention-dependent normalization of $q$. It matches the GLSM Coulomb relation $\sigma^N=q$ under $H\leftrightarrow\sigma$.

The A-model depends on the complexified Kähler class $B+i\omega$ but not on complex-structure deformations. It remains meaningful even when the untwisted sigma model is not conformal, provided the twist and path integral are well defined.

The B-model

For a sigma model with a nonanomalous axial R symmetry, the B-twist combines the unbarred fermions into $\rho^i$ and the barred fermions into the scalars $\eta^{\bar i}$ and $\vartheta_i$. Its bosonic localization locus consists of constant maps. Local operators are represented by Dolbeault cohomology of holomorphic polyvector fields,

$$\bigoplus_{p,q}H^q(X,\wedge^pT^{1,0}X),$$

with $Q_B$ represented by $\bar\partial$. Fixed-worldsheet correlators depend on complex structure but not on the Kähler class. On a compact Calabi–Yau, a holomorphic volume form converts polyvectors into differential forms and supplies the trace pairing. This statement concerns the topological sigma model; coupling to topological gravity introduces additional moduli-space boundary effects.

For a Landau–Ginzburg B-model, the cohomological differential becomes

$$Q_B\longleftrightarrow\bar\partial+\iota_{\mathrm d W}.$$

On affine space with isolated critical points, its degree-zero cohomology is the Jacobi ring

$$\operatorname{Jac}(W)= \mathbb C[X_1,\ldots,X_n]/(\partial_1W,\ldots,\partial_nW).$$

Genus-zero correlators reduce to the residues developed in Vafa 1991, pp. 337–346. For a single field with simple critical points,

$$\langle f(X)\rangle =\sum_{W'(X_*)=0}\frac{f(X_*)}{W''(X_*)},$$

up to normalization. The B-model does not sum over nonconstant worldsheet instantons, but singular critical loci and noncompact integration cycles can still cause divergences.

Descent and integrated observables

Start with a local scalar operator $\mathcal O^{(0)}$ satisfying $Q\mathcal O^{(0)}=0$. Translational generators are $Q$-exact in a topological theory, so one can construct descendants obeying

$$\mathrm d\mathcal O^{(0)}=\{Q,\mathcal O^{(1)}\}, \qquad \mathrm d\mathcal O^{(1)}=\{Q,\mathcal O^{(2)}\}.$$

Then

$$\oint_\gamma\mathcal O^{(1)}, \qquad \int_{\Sigma_2}\mathcal O^{(2)}$$

are $Q$-closed on closed cycles. The two-form descendant deforms the action by the coupling associated with $\mathcal O^{(0)}$. Boundary terms in Stokes’ theorem explain why open worldsheets require compatible branes and boundary descendants.

A- and B-branes in one paragraph

On a worldsheet with boundary, preserving $Q_A$ leads in the simplest geometric limit to A-branes supported on Lagrangian or more general coisotropic submanifolds with suitable bundles. Preserving $Q_B$ leads to B-branes described semiclassically by holomorphic submanifolds and holomorphic bundles or complexes. These are entry points, not complete definitions: disk anomalies, gradings, stability, curvature, and instanton corrections are essential. Mirror symmetry is expected to exchange the two categories.

What the twist forgets

Protected feature	Retained	Not determined
Scalar-supercharge cohomology	rings and topological correlators	ordinary operator norms
Metric independence	deformations by $Q$-exact terms	physical stress-tensor correlators
Localization locus	exact protected integral when compact	generic real-time dynamics
Brane category	protected open-string sector	all massive boundary excitations

A matching A-model and B-model can be compelling evidence for mirror symmetry, but it does not alone prove equality of every unprotected observable. Their mirror exchange was formulated directly in topological field theory in Witten 1992, pp. 120–158.

The mirror-symmetry dictionary owns the object-level exchange between the two protected sectors. The general framework for topological and holomorphic twists owns twist homomorphisms, global background bundles, and anomaly obstructions beyond this two-dimensional construction.

Common pitfalls

Reversing vector and axial twists. In the convention fixed here, the A-twist uses $U(1)_V$ and the B-twist uses $U(1)_A$. Recheck the scalar-supercharge spins rather than memorizing a sign.

Calling every $Q$-closed integral topological. Metric independence also requires a $Q$-invariant measure, control of moduli-space boundaries, and a suitable stress-tensor improvement.

Identifying the classical and quantum rings. A-model instantons deform cup product. In contrast, the affine LG B-model ring is the Jacobi quotient, subject to orbifold and noncompact refinements.

Ignoring global twist data. A scalar charge in flat space is not enough. Its R symmetry must be exact, its charges must define the required bundles on the chosen worldsheet, and boundaries must preserve the same $Q$ or supply compensating boundary data.

Exercises

1. Derive the scalar nature of $Q_A$ and $Q_B$ from the R-charge table on the algebra page.

Solution

With $s'=s+q_R/2$, $\bar Q_+$ has $s=+1/2$ and charge $-1$ under both R symmetries. $Q_-$ has $s=-1/2$ and vector charge $+1$, so it is scalar in the vector twist. $\bar Q_-$ has $s=-1/2$ and axial charge $+1$, so it is scalar in the axial twist.

2. For $W=X^{k+2}/(k+2)$, compute the B-model ring and its dimension.

Solution

$W'=X^{k+1}$, hence $\operatorname{Jac}(W)=\mathbb C[X]/(X^{k+1})$ with basis $1,X,\ldots,X^k$. Its dimension is $k+1$.

3. Show that the integral of a one-form descendant around a homologously deformed closed contour changes by a $Q$-exact term.

Solution

If $\gamma'-\gamma=\partial S$, Stokes’ theorem and descent give

$$\oint_{\gamma'}\mathcal O^{(1)}-\oint_\gamma\mathcal O^{(1)} =\int_S\mathrm d\mathcal O^{(1)} =\left\{Q,\int_S\mathcal O^{(2)}\right\}.$$

Thus the cohomology class depends only on the homology class of the contour.

References

• Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., and Zaslow, E. Mirror Symmetry. Clay Mathematics Monographs 1. Providence, RI: American Mathematical Society and Clay Mathematics Institute, 2003, ch. 16. Clay Mathematics Institute PDF.
• Vafa, C. “Topological Landau–Ginzburg Models.” Modern Physics Letters A 6 (1991): 337–346. doi:10.1142/S0217732391000324.
• Witten, E. “Mirror Manifolds and Topological Field Theory.” In Essays on Mirror Manifolds, edited by S.-T. Yau, 120–158. Hong Kong: International Press, 1992. arXiv:hep-th/9112056.
• Witten, E. “Topological Sigma Models.” Communications in Mathematical Physics 118 (1988): 411–449. doi:10.1007/BF01466725.