Twisted Indices and Elliptic Genera

Twisted indices and elliptic genera are protected traces whose path integrals retain topological sectors explicitly. Magnetic fluxes are summed, flat holonomies are integrated, and one-loop determinants become meromorphic forms. A residue formula is meaningful only after the spin and background bundles, flux lattice, counterterm scheme, singular hyperplanes, Jeffrey–Kirwan chamber, and treatment of infinity have been fixed.

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

The three-dimensional A-twisted trace

Put a three-dimensional $N=2$ theory on $\Sigma_g\times S^1$ and use periodic fermions along $S^1$. In the convention of Closset and Kim, the R-symmetry background along $\Sigma_g$ is

$$A^{(R)}=\frac12\omega_{\Sigma}, \qquad \mathfrak n_R=\frac1{2\pi}\int_{\Sigma_g}F_R=g-1.$$

It cancels the spin connection for two supercharges of R-charge $\pm1$. The protected observable is

$$I_g(\mathbf y;\boldsymbol{\mathfrak n}) =\operatorname{Tr}_{\mathcal H(\Sigma_g;\boldsymbol{\mathfrak n})} (-1)^F\prod_A y_A^{Q_A},$$

where $Q_A$ commute with the preserved supercharges, and $\mathfrak n_A$ are quantized background flavor fluxes. Reversing the sign of $A^{(R)}$ defines the opposite twist and changes every formula containing $(g-1)r$; it is not a harmless relabeling Closset and Kim 2016, eqs. (1.1) and (2.3)–(2.5).

The R-background is a line bundle $L_R$ with $L_R^2\cong K_{\Sigma}$. Thus a choice of square root, equivalently a spin structure, is part of the background when odd powers of $L_R$ occur. More generally, every field must couple to an honest line bundle. For a component of gauge weight $\rho$, R-charge $r$, and total flavor flux $\mathfrak n_i$, the integer

$$d_i(\rho,\mathfrak m) =\rho(\mathfrak m)+\mathfrak n_i+(g-1)(r-1)$$

is the relevant Dolbeault index. Fractional trial R-charges are allowed only when mixing with background bundles makes all physical powers and fluxes integral.

Flux sum and the localized integrand

Let $G$ be the compact global gauge group, $T$ a maximal torus, and

$$\mathfrak m\in\Gamma_{G^\vee} =\operatorname{Hom}(U(1),T)$$

its cocharacter lattice. This lattice changes when the global form of $G$ changes. Write $x_a=e^{2\pi i u_a}$ for the complexified gauge holonomies. In one complete convention, localization gives

$$Z_{\Sigma_g\times S^1} =\frac1{|W_G|} \sum_{\mathfrak m\in\Gamma_{G^\vee}} \sum_{u_*} \operatorname{JK-Res}_{u=u_*}[Q(u_*),\eta]\, \mathcal I_{\mathfrak m}(u),$$

with

$$\mathcal I_{\mathfrak m} =(-2\pi i)^{\operatorname{rk}G}Z_{\mathrm{cl}} \left(\prod_i Z_{\Phi_i}\right) Z_{\mathrm{vec}}\,H(u)^g\, du_1\wedge\cdots\wedge du_{\operatorname{rk}G}.$$

For a chiral $\Phi_i$ in representation $\mathcal R_i$, define $y_i=\prod_Ay_A^{Q_i^A}$ and $\mathfrak n_i=\sum_AQ_i^A\mathfrak n_A$. Its determinant is

$$Z_{\Phi_i} =\prod_{\rho\in\mathcal R_i} \left( \frac{x^{\rho/2}y_i^{1/2}} {1-x^\rho y_i} \right)^{\rho(\mathfrak m)+\mathfrak n_i+(g-1)(r_i-1)},$$

while the vector multiplet contributes

$$Z_{\mathrm{vec}} =(-1)^{\sum_{\alpha>0}\alpha(\mathfrak m)} \prod_{\alpha\in\Delta} (1-x^\alpha)^{1-g}.$$

Here $\Delta$ is the root system. The Hessian $H(u)=\det(\partial_{u_a}\partial_{u_b}\widetilde W_{\mathrm{eff}})$ absorbs the one-form gaugino zero modes, up to the fixed $2\pi i$ convention for $\widetilde W_{\mathrm{eff}}$. These formulas are a matched set: moving square-root signs or powers between factors without also changing classical Chern–Simons terms changes the answer Closset and Kim 2016, eqs. (2.59)–(2.63).

$Z_{\mathrm{cl}}$ includes gauge, mixed gauge–flavor, gauge–R, flavor, and gravitational Chern–Simons counterterms. Three dimensions has a parity anomaly rather than a perturbative chiral gauge anomaly: effective Chern–Simons levels must obey the quantization rules of the declared global gauge group. Different allowed background contact terms can multiply the result by quantized fugacity, flux, or framing phases, so a duality comparison must use a common counterterm scheme.

Matter poles occur at $x^\rho y_i=1$. The compactification of each $\mathbb C^*$ holonomy also has loci $x_a=0,\infty$, whose charges are determined by the asymptotic effective Chern–Simons levels and monopole operators. The auxiliary covector $\eta$ selects projective intersections of these charge hyperplanes. It must not be used to erase a physical pole at infinity Benini and Zaffaroni 2015, §§2–3.

Bethe-vacuum form and its hypotheses

When the flux sum converges in a chosen chamber and can be resummed, the answer becomes

$$Z_{\Sigma_g\times S^1} =\sum_{\widehat u\in\mathcal S_{\mathrm{BE}}} \mathcal U(\widehat u;\boldsymbol{\mathfrak n}) \mathcal H(\widehat u)^{g-1}.$$

The Bethe set consists of Weyl-inequivalent solutions away from root hyperplanes,

$$\exp\!\left(2\pi i\frac{\partial\widetilde W_{\mathrm{eff}}} {\partial u_a}\right)=1.$$

$\mathcal U$ contains background-flux operators and contact terms, while $\mathcal H$ is the full handle-gluing operator, not just a bare determinant. This expression assumes that the circle-reduced two-dimensional theory is massive, the Bethe roots are isolated and nondegenerate, and no missing root or continuum lies at infinity. For $g=1$ and zero background flux, each simple massive Bethe vacuum contributes one; if the hypotheses fail, “count the roots” is not valid Closset and Kim 2016, eqs. (2.71)–(2.77).

The two-dimensional elliptic genus

For a two-dimensional theory on a torus of complex structure $\tau$, choose Ramond boundary conditions around the spatial circle and insert $(-1)^F$, which makes fermions periodic around Euclidean time as well. A $(2,2)$ elliptic genus can be written

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

where $q=e^{2\pi i\tau}$, $y=e^{2\pi iz}$, and $x_A=e^{2\pi i\xi_A}$. The chosen right-moving supercharge pairs states with $\bar L_0-\bar c/24>0$. For a compact theory with discrete spectrum, the surviving trace is holomorphic in $q$; noncompact continua are discussed below Benini et al. 2015, §2.

For a gauge theory of rank $r_G$, localization gives

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

with additional disconnected flat-bundle sectors when the global gauge group requires them. In an additive $(2,2)$ convention, a chiral 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 the product over roots. Every theta-function argument is a point on $\mathbb C/(\mathbb Z+\tau\mathbb Z)$, so the complete meromorphic form must be single valued under large gauge transformations 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 $\xi_i$, vector R-charge zero, and positive FI chamber. The positive JK covector selects the denominator poles

$$u_i=-\xi_i.$$

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

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

At $z=0$ and generic $\xi_i$, every fixed-point contribution is one, so $Z=N$, the Witten index and Euler characteristic of $\mathbb{CP}^{N-1}$. When flavor holonomies coincide, individual summands develop apparent poles; first form the symmetric sum, then take the regulated limit Benini et al. 2014, §§3–4.

Anomalies, modularity, and continua

For two-dimensional Weyl fermions, define the anomaly matrix

$$\mathcal A^{AB}=\operatorname{Tr}_{\mathrm{fermions}} \gamma^3Q_AQ_B.$$

Every entry with a gauged index must vanish; otherwise the holonomy integrand fails to descend to the gauge-holonomy torus. Entries involving only background symmetries determine the Jacobi Gaussian under modular transformations. The gravitational anomaly $\operatorname{Tr}\gamma^3=c_R-c_L$ controls an additional modular multiplier. Thus three phenomena must remain separate:

• a gauge anomaly makes the proposed two-dimensional gauge theory inconsistent;
• a background ‘t Hooft anomaly gives controlled quasi-periodicity and is physical data;
• a noncompact continuum can produce a nonholomorphic completion through boson–fermion spectral asymmetry.

The last effect is not a gauge anomaly. In the supersymmetric cigar, for example, the holomorphic discrete-state contribution is mock modular, while a regulator-dependent continuum remainder restores modular covariance Troost 2010, §§3–4.

A reproducible contour checklist

Before evaluating either observable, record:

1. the spatial and temporal spin structures and the R-background convention;
2. the global gauge group, cocharacter lattice, and any disconnected bundle sectors;
3. background fluxes, holonomies, and their integrality conditions;
4. classical Chern–Simons/contact terms and one-loop sign convention;
5. all finite and asymptotic singular hyperplanes with their charge covectors;
6. the JK covector and chamber, plus the wall-crossing term if that chamber changes;
7. the convergence or regulated meaning of every flux sum;
8. gauge-anomaly cancellation, large-gauge invariance, and the predicted modular multiplier;
9. any continuum contribution omitted by a holomorphic residue sum.

Changing one item can define a different observable. Equality after silently dropping a boundary residue or changing a contact term is not a valid protected-sector comparison.

Exercises

1. Derive the chiral exponent

A component of a three-dimensional chiral is valued in

$$L_\rho\otimes L_i\otimes K_\Sigma^{r/2},$$

with degrees $\rho(\mathfrak m)$, $\mathfrak n_i$, and $r(g-1)$. Use Riemann–Roch to derive the exponent in $Z_{\Phi_i}$.

Solution

For a line bundle $L$ on $\Sigma_g$, Riemann–Roch gives $\chi(L)=\deg L+1-g$. Here

$$\deg L=\rho(\mathfrak m)+\mathfrak n_i+r(g-1),$$

so

$$\chi(L)=\rho(\mathfrak m)+\mathfrak n_i+(g-1)(r-1),$$

which is precisely the determinant exponent. This also shows why it must be integral.

2. The $\mathbb{CP}^{N-1}$ Witten limit

Evaluate the residue formula at $z=0$ with distinct $\xi_i$.

Solution

For every ordered pair $j\ne i$, numerator and denominator become identical. Each product is therefore one and the sum has $N$ terms:

$$Z_{\mathbb{CP}^{N-1}}(z=0)=N.$$

The result agrees with $\chi(\mathbb{CP}^{N-1})=N$.

3. Genus one and Bethe vacua

Assume zero background flux, $g=1$, and $n$ isolated nondegenerate Bethe vacua. Compute the twisted index.

Solution

At $g=1$, the handle factor is $\mathcal H^{g-1}=1$. With zero background flux, $\mathcal U=1$ in the stated convention, so every simple vacuum contributes one and

$$Z_{T^2\times S^1}=n.$$

If a root is degenerate or runs to infinity, the hypotheses fail and this count must be replaced by the regulated contour formula.

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.
• Closset, C., and H. Kim. “Comments on Twisted Indices in 3d Supersymmetric Gauge Theories.” Journal of High Energy Physics 2016, no. 8 (2016): 059. DOI; Open PDF.
• Troost, J. “The Non-Compact Elliptic Genus: Mock or Modular.” Journal of High Energy Physics 2010, no. 6 (2010): 104. DOI; Open PDF.