A- and B-Twists and Cohomological Observables

A topological twist changes the Lorentz generator by an R-symmetry generator so that one supercharge becomes a scalar. Observables are then classes of that scalar supercharge, and metric dependence becomes exact 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 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

Let a field have Lorentz spin $s$ and R charge $q_R$. With the 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. The names are therefore

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

Central extensions can make $Q_A^2$ or $Q_B^2$ a symmetry transformation rather than zero. On gauge-invariant local operators in the appropriate topological sector one requires nilpotence, possibly only modulo a gauge transformation. Boundary conditions must also make the surface terms vanish or be canceled by boundary degrees of freedom.

Quantum anomaly freedom is essential. In an ordinary Kähler sigma model $U(1)_V$ is non-anomalous, so the A-twist exists generally. The axial anomaly is proportional to $c_1(TX)$, so the B-twist requires its cancellation, for example on a Calabi–Yau target.

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 turns one combination of fermions into a worldsheet scalar valued in $\phi^*TX$ and the complementary combinations into one-forms. The localization equations are holomorphic-map equations,

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

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

$$\omega=\frac1{p!}\omega_{i_1\cdots i_p}(\phi) d\phi^{i_1}\wedge\cdots\wedge d\phi^{i_p}$$

maps to an operator built from scalar fermion zero modes, and $Q_A$ acts as 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:

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

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 $c_1(TX)=0$, the B-twist localizes on constant maps. Its 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 $\bar\partial$. Correlators depend on complex structure but not on the Kähler class. On a compact Calabi–Yau, the holomorphic volume form converts polyvectors into differential forms and supplies the trace pairing.

For a Landau–Ginzburg B-model, the differential and residue correlators take the form developed in Vafa 1991:

$$Q_B\sim\bar\partial+\iota_{dW}.$$

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 residues. 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

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

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.

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.

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

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

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

References

• Vafa, C. “Topological Landau–Ginzburg Models.” Modern Physics Letters A 6 (1991): 337–346. doi:10.1142/S0217732391000356.
• 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.