Twisted Indices and Elliptic Genera

Twisted indices and elliptic genera are protected traces whose localized formulas retain topological sectors explicitly. Magnetic fluxes are summed, flat holonomies are integrated, and one-loop determinants become meromorphic forms. A Jeffrey–Kirwan prescription selects residues only after the gauge anomaly, flux lattice, auxiliary covector, boundary poles, and convergence chamber are fixed.

Required background. Start from supersymmetric indices, use the two-dimensional elliptic-genus and anomaly conventions, and import the complete Jeffrey–Kirwan residue prescription.

Fluxes, holonomies, and the topological twist

On $\Sigma_g\times S^1$, a three-dimensional $N=2$ theory can preserve scalar supercharges by turning on an R-symmetry flux that cancels the spin connection on $\Sigma_g$. The BPS locus contains a gauge magnetic flux

$$\mathfrak m\in\Gamma_{G^\vee}$$

and a complexified holonomy $u$ around $S^1$, often exponentiated as $x=e^{iu}$. The localized topologically twisted index has the form

$$Z_{\Sigma_g\times S^1} =\frac1{|W|} \sum_{\mathfrak m\in\Gamma_{G^\vee}} \oint_{\mathrm{JK}} \prod_{a=1}^{r}\frac{dx_a}{2\pi i x_a}\, Z_{\mathrm{cl}}(x,\mathfrak m) Z_{\mathrm{1-loop}}(x,\mathfrak m) \mathcal H_g(x,\mathfrak m).$$

The factor $\mathcal H_g$ contains gaugino zero modes and the Hessian associated with the genus-$g$ handles. Authors redistribute powers of this Hessian between $Z_{\mathrm{1-loop}}$ and $\mathcal H_g$, so a comparison must use one complete convention rather than comparing isolated factors.

The cocharacter lattice is determined by the global gauge group. Classical Chern–Simons, FI, and topological terms contribute powers of $x$ and phases depending on $\mathfrak m$. Matter determinants have hyperplane poles where charged modes become massless. The JK covector $\eta$ selects a set of intersecting charge hyperplanes; possible residues at $x=0$ and $x=\infty$ encode asymptotic Coulomb-branch data and cannot be discarded automatically Benini and Zaffaroni 2015, §§2–3.

When the flux sum can be resummed, the same answer may be written as a sum over Bethe vacua:

$$Z_{\Sigma_g\times S^1} =\sum_{\widehat u\in\mathcal S_{\mathrm{BE}}} \mathcal H(\widehat u)^{g-1} \prod_A\Pi_A(\widehat u)^{\mathfrak n_A}.$$

Here $\exp(\partial\widetilde W_{\mathrm{eff}}/\partial u_a)=1$ defines $\mathcal S_{\mathrm{BE}}$, $\mathcal H$ is the handle-gluing operator, and $\Pi_A$ is a flavor-flux operator. This form assumes isolated Bethe solutions and excludes roots at infinity unless they are treated separately.

The two-dimensional elliptic genus

For a two-dimensional theory on a torus of complex structure $\tau$, the elliptic genus is a Ramond–Ramond trace

$$Z_{T^2}(\tau,z,\xi) =\operatorname{Tr}_{\mathrm{RR}} (-1)^F q^{L_0-c/24}\bar q^{\bar L_0-\bar c/24} y^{J}\prod_A x_A^{F_A},$$

where $q=e^{2\pi i\tau}$ and the chosen right-moving supercharge forces cancellation of states with positive $\{\bar Q,\bar Q^\dagger\}$. In a compact theory the result is holomorphic in $q$; a continuum can leave nonholomorphic dependence through spectral asymmetry.

For a rank-$r$ gauge theory, localization gives

$$Z_{T^2} =\frac1{|W|} \sum_{u_*} \operatorname{JK-Res}_{u=u_*} \bigl(Q(u_*),\eta\bigr) Z_{\mathrm{1-loop}}(u;\tau,z,\xi)\,d^r u,$$

with additional disconnected bundle sectors when the global gauge group requires them. In one standard $(2,2)$ convention, a chiral multiplet of vector R-charge $R$ contributes

$$Z_\Phi =\prod_{\rho\in\mathcal R} \frac{\theta_1\!\left(\tau,\rho(u)+\xi+(R/2-1)z\right)} {\theta_1\!\left(\tau,\rho(u)+\xi+(R/2)z\right)}.$$

The vector determinant supplies the Cartan zero-mode normalization and a product over roots. All theta-function arguments are defined modulo the torus lattice; gauge-anomaly cancellation is exactly what makes the complete meromorphic form elliptic in each gauge holonomy Benini et al. 2015, §§2–3.

A U(1) residue benchmark

Consider a $(2,2)$ $U(1)$ GLSM with $N$ charge-$+1$ chirals, generic flavor holonomies $\xi_i$, and positive FI chamber. Set the vector R-charges to zero in the displayed convention. The positive JK covector selects the $N$ denominator poles

$$u_i=-\xi_i.$$

After the vector prefactor cancels the simple-pole normalization, the equivariant elliptic genus of the geometric $\mathbb{CP}^{N-1}$ phase is

$$Z_{\mathbb{CP}^{N-1}} =\sum_{i=1}^{N} \prod_{j\ne i} \frac{\theta_1(\tau,\xi_j-\xi_i-z)} {\theta_1(\tau,\xi_j-\xi_i)}.$$

For $N=1$ the empty product gives one, as required for a massive theory with one vacuum. When flavor holonomies coincide, individual terms develop apparent singularities; the symmetric sum has the appropriate regulated limit. Taking that limit term by term is a common error Benini et al. 2014, §§3–4.

Anomalies and modular behavior

Under modular transformations, an elliptic genus behaves as a Jacobi form or a vector-valued generalization. Its index and multiplier are fixed by flavor and gravitational anomalies. Gauge anomalies must vanish for the holonomy integral to be well defined. A nonzero gravitational anomaly controls the modular weight or phase rather than invalidating the observable.

Three different obstructions must be separated:

• gauge anomaly: a failure of ellipticity in a gauge holonomy, which makes the gauge theory inconsistent;
• ‘t Hooft anomaly: controlled quasi-periodicity in a background holonomy, which is physical data;
• noncompact continuum: a nonholomorphic completion or mock-modular behavior, which reflects infrared spectral density.

Contour and convergence checks

Before evaluating either observable, state the flux lattice, spin structure, R-flux integrality, one-loop normalization, JK covector, chamber, and poles at infinity. Then check:

1. cancellation of local gauge and mixed gauge anomalies;
2. invariance under large gauge transformations of every holonomy;
3. convergence or regulated meaning of the magnetic-flux sum;
4. stability when $\eta$ moves inside one chamber;
5. the wall-crossing contribution when $\eta$ crosses a wall;
6. modular covariance and the anomaly-predicted multiplier;
7. any continuum contribution omitted by a purely holomorphic residue sum.

An equality obtained only after changing $\eta$ or dropping a boundary residue compares different definitions unless an independent wall-crossing argument supplies the missing term.

Exercises

Evaluate the $N=1$ specialization of the $\mathbb{CP}^{N-1}$ residue formula.

Solution

There is one selected pole and no index $j\ne i$. The product over an empty set equals one, so the elliptic genus is $Z=1$, consistent with one supersymmetric ground state and no nontrivial compact target directions.

References

• Benini, F., R. Eager, K. Hori, and Y. Tachikawa. “Elliptic Genera of Two-Dimensional $N=2$ Gauge Theories with Rank-One Gauge Groups.” Letters in Mathematical Physics 104 (2014): 465–493. DOI; Open PDF.
• Benini, F., R. Eager, K. Hori, and Y. Tachikawa. “Elliptic Genera of 2d $N=2$ Gauge Theories.” Communications in Mathematical Physics 333 (2015): 1241–1286. DOI; Open PDF.
• Benini, F., and A. Zaffaroni. “A Topologically Twisted Index for Three-Dimensional Supersymmetric Theories.” Journal of High Energy Physics 2015, no. 7 (2015): 127. DOI; Open PDF.