Topological and Holomorphic Twists

A twist changes the action of rotations by composing them with R symmetry. If a supercharge becomes a scalar under the new rotation group, it can be defined on a broad class of curved manifolds and used as a cohomological differential. A topological twist makes all translations $Q$-exact; a holomorphic twist makes only anti-holomorphic translations $Q$-exact, so its observables retain holomorphic position dependence.

Required background. Generalized Killing spinors and global spin-R bundles supplies the global bundle condition. $Q$-cohomology and Hodge complexes supplies the passage from a nilpotent supercharge to cohomology.

Helpful background. Two-dimensional A- and B-twists develops the dimension-specific sigma-model constructions that are only used here as examples.

Twisting the rotation group

Let $H\subset\operatorname{Spin}(d)$ be a rotation subgroup and let

$$\rho:H\longrightarrow G_R$$

be a homomorphism into the global R-symmetry group. The twisted rotation group is the diagonal copy

$$H'=\operatorname{diag}\bigl(H\times\rho(H)\bigr).$$

A field carrying Lorentz representation $R_L$ and R representation $R_R$ is re-decomposed under $H'$ using $R_L\otimes R_R|_{\rho(H)}$. No dynamics has yet changed: the twist is initially a reassignment of geometric spin. It becomes a curved-space construction when an R connection is chosen so that the $\rho(H)$ part compensates the spin connection. Globally, the associated spin-R bundles must exist and the R symmetry used by $\rho$ must be nonanomalous.

The supercharges are decomposed the same way. A singlet $Q$ under $H'$ is a scalar differential. In a gauge theory it usually obeys

$$Q^2=\delta_{\rm gauge}(\phi)+\mathcal H,$$

where $\mathcal H$ is a residual isometry, R rotation, or flavor transformation. Thus $Q$ is strictly nilpotent on gauge-invariant $\mathcal H$-invariant operators, and equivariantly nilpotent on the full field complex.

Worked example: the Donaldson–Witten twist

For four-dimensional Euclidean $\mathcal N=2$ supersymmetry,

$$\operatorname{Spin}(4)=SU(2)_+\times SU(2)_-, \qquad G_R=SU(2)_R\times U(1)_r.$$

The left-handed supercharges transform as $(\mathbf2,\mathbf1;\mathbf2)$. Identifying $SU(2)_+$ with $SU(2)_R$ gives

$$\mathbf2\otimes\mathbf2=\mathbf1\oplus\mathbf3,$$

so the twisted algebra contains a scalar $Q$ and a self-dual two-form supercharge. The right-handed supercharges become a one-form. This is the twist underlying Witten’s cohomological formulation of Donaldson theory Witten 1988, §§2–3.

For the vector multiplet, the fermions reorganize into a scalar $\eta$, a one-form $\psi$, and a self-dual two-form $\chi^+$. With an auxiliary self-dual two-form $H^+$, one convenient off-shell convention is

$$\begin{aligned} QA&=\psi, & Q\psi&=D_A\phi, & Q\phi&=0,\\ Q\bar\phi&=\eta, & Q\eta&=[\bar\phi,\phi],\\ Q\chi^+&=H^+, & QH^+&=[\chi^+,\phi]. \end{aligned}$$

These equations give $Q^2=\delta_{\rm gauge}(\phi)$ with the convention $\delta_\phi A=D_A\phi$. Factors of $i$ move if Lie-algebra fields are chosen Hermitian instead of anti-Hermitian. The auxiliary $H^+$ is essential for off-shell closure.

Gauge-invariant polynomials $\mathcal O^{(0)}_k=\operatorname{Tr}\phi^k$ are $Q$ closed. Their descendants satisfy

$$Q\mathcal O^{(p)}_k+d\mathcal O^{(p-1)}_k=0,$$

so integrating $\mathcal O^{(p)}_k$ over a closed $p$-cycle gives a $Q$-closed observable. Moving the cycle through a homologous family changes the insertion by a $Q$-exact term.

Topological versus holomorphic protection

In a topological twist, the stress tensor is expected to have the form

$$T_{\mu\nu}=\{Q,G_{\mu\nu}\}+T^{\rm anom}_{\mu\nu}.$$

If the anomaly term vanishes, the measure and boundary conditions are $Q$ invariant, and insertions do not collide, then metric variations of $Q$-closed correlators vanish. The qualification is important: a classical $Q$-exact stress tensor does not remove quantum anomalies, contact terms, wall crossing, or boundary dependence.

A holomorphic twist is weaker. On a complex manifold with local coordinates $(z^i,\bar z^{\bar i})$, one seeks a supercharge for which

$$P_{\bar i}=\{Q,G_{\bar i}\},$$

while $P_i$ need not be $Q$ exact. Correlators in $Q$ cohomology are then independent of $\bar z$ but can vary holomorphically with $z$. Four-dimensional $\mathcal N=1$ theories with a nonanomalous $U(1)_R$ on suitable complex manifolds admit a twisted description of this kind Closset et al. 2014, §§2–4.

What must be checked

A valid twist requires all of the following:

• a genuine global R symmetry and a well-defined homomorphism $\rho$;
• compatible spin-R transition functions for every field;
• a globally defined scalar or holomorphic $Q$;
• off-shell closure, or a stated replacement such as BV closure;
• $Q$-invariance of the action, measure, contour, insertions, and boundary conditions;
• control of R-symmetry and gravitational anomalies;
• proof of the claimed stress-tensor exactness in the protected directions.

Twisting does not automatically make a theory topological. A scalar supercharge can coexist with metric dependence if only part of the stress tensor is $Q$ exact, if the partition function has local counterterm ambiguities, or if the selected boundary condition introduces scale or polarization data.

Exercises

1. Twisted fermions. Verify that the $\mathcal N=2$ left-handed gaugino representation produces a scalar and a self-dual two-form after the Donaldson–Witten twist.

Solution

The left-handed spin index and the $SU(2)_R$ index are both doublets. Under the diagonal group their tensor product decomposes as $\mathbf2\otimes\mathbf2=\wedge^2\mathbf2\oplus\operatorname{Sym}^2\mathbf2=\mathbf1\oplus\mathbf3$. The singlet is $\eta$ and the triplet is identified with a self-dual two-form $\chi^+$.

2. Descent. Show that $\int_C\mathcal O^{(1)}$ is $Q$ closed when $C$ is a closed curve and $Q\mathcal O^{(1)}+d\mathcal O^{(0)}=0$.

Solution

$Q\int_C\mathcal O^{(1)}=-\int_Cd\mathcal O^{(0)}=0$ by Stokes’ theorem. If $C$ had endpoints, boundary insertions or boundary conditions would be needed to cancel the endpoint term.

References

• Closset, Cyril, Thomas T. Dumitrescu, Guido Festuccia, and Zohar Komargodski. “From Rigid Supersymmetry to Twisted Holomorphic Theories.” Physical Review D 90 (2014): 085006. doi:10.1103/PhysRevD.90.085006. Open preprint.
• Witten, Edward. “Topological Quantum Field Theory.” Communications in Mathematical Physics 117 (1988): 353–386. doi:10.1007/BF01223371. Open PDF.

Next step

The finite-dimensional model behind cohomological localization is the Atiyah–Bott–Berline–Vergne fixed-point formula.