Sphere Partition Functions and Matrix Models

Localization reduces a supersymmetric sphere path integral to an integral over its BPS zero modes, multiplied by classical, one-loop, and nonperturbative factors. There is no dimension-independent “sphere matrix model”: the locus, measure, determinant, flux sum, contour, and counterterms depend on the theory and background. This page gives a reusable assembly procedure and three benchmark formulas.

Required background. Use the derivation of localization loci and one-loop determinants together with complex contours, zero modes, and regularization.

Helpful background. Boundary gluing and Jeffrey–Kirwan residues control sphere formulas built from hemispheres or contour residues.

From the BPS locus to a finite-dimensional integral

After adding a positive $QV$ deformation, expand about each connected component $\mathcal M_\alpha$ of the BPS locus. Schematically,

$$Z_{S^d} =\sum_\alpha\frac1{|W_\alpha|} \int_{\Gamma_\alpha}d\mu_\alpha\, e^{-S_{\mathrm{cl}}(\alpha)} Z_{\mathrm{1-loop}}(\alpha) Z_{\mathrm{np}}(\alpha).$$

The Weyl factor $|W_\alpha|$, induced measure $d\mu_\alpha$, and nonperturbative term are part of the formula, not optional decorations. A Vandermonde determinant may be placed in the measure or absorbed into $Z_{\mathrm{1-loop}}$, but never counted twice.

A dependable derivation follows this order:

1. solve $Q\Psi=0$ for every fermion and classify smooth and singular sectors;
2. divide by gauge transformations and identify the residual Weyl group;
3. normalize bosonic and fermionic zero modes separately;
4. regularize the determinant with a symmetry-preserving phase convention;
5. include flux, vortex, or instanton contributions at fixed points;
6. determine the physical cycle from the original real fields, then track any contour deformation;
7. test a free or weak-coupling limit.

Three-dimensional N=2 theories on the round sphere

For a three-dimensional $N=2$ gauge theory on a unit round $S^3$, with compact gauge group $G$, a standard convention gives

$$\begin{aligned} Z_{S^3} ={}&\frac1{|W|}\int_{\mathfrak h}d^r\sigma\, e^{i\pi k\operatorname{Tr}\sigma^2+2\pi i\zeta\cdot\sigma} \prod_{\alpha>0}4\sinh^2\!\bigl(\pi\alpha(\sigma)\bigr) \\ &\times \prod_I\prod_{\rho\in R_I} \exp\!\left[\ell\!\left(1-\Delta_I+i\rho(\sigma)+i m_I\right)\right]. \end{aligned}$$

Here $\sigma$ is the constant vector-multiplet scalar, $k$ and $\zeta$ are Chern–Simons and FI parameters in the displayed normalization, and $m_I$ and $\Delta_I$ are dimensionless real masses and trial R-charges. The special function can be fixed by

$$\ell'(z)=-\pi z\cot(\pi z), \qquad \ell(0)=0.$$

For one free chiral multiplet with $\Delta=1/2$ and zero mass,

$$Z_{\mathrm{chiral}}=e^{\ell(1/2)}=2^{-1/2}, \qquad -\log|Z_{\mathrm{chiral}}|=\frac12\log2.$$

This tiny benchmark catches sign errors in $1-\Delta$, missing square roots, and incompatible definitions of $\ell$. The general matrix model was derived directly from the localized path integral by Kapustin, Willett, and Yaakov 2010, §§3–4.

Four-dimensional N=2 theories on S⁴

For a Lagrangian four-dimensional $N=2$ gauge theory on a round sphere of radius $r$, the Coulomb-locus variable $a$ lies on a real Cartan cycle. In a common convention,

$$Z_{S^4} =\frac1{|W|}\int_{\mathfrak h}d^r a\, e^{-8\pi^2r^2(a,a)/g^2} Z_{\mathrm{1-loop}}(a,m;r) \left|Z_{\mathrm{inst}}(a,m,q;\epsilon_1,\epsilon_2)\right|^2,$$

with

$$\epsilon_1=\epsilon_2=\frac1r.$$

The north-pole contribution is an Omega-background instanton sum and the south-pole contribution is its conjugate on the physical contour. The theta angle is contained in $q=e^{2\pi i\tau}$. Depending on convention, the Cartan Vandermonde is included in $d^r a$ or in the vector one-loop determinant; this must be stated explicitly. Pestun’s construction also fixes which curvature couplings are required for the chosen supercharge Pestun 2012, §§3–4.

Two-dimensional N=(2,2) theories on S²

For gauge group $G$, the Coulomb-branch representation contains a sum over cocharacters $\mathfrak m$ and an integral over the constant scalar $\sigma$:

$$Z_{S^2} =\frac1{|W|} \sum_{\mathfrak m\in\Gamma_{G^\vee}} \int_{\Gamma}d^r\sigma\, Z_{\mathrm{cl}}(\sigma,\mathfrak m) Z_{\mathrm{vec}}(\sigma,\mathfrak m) Z_{\mathrm{chiral}}(\sigma,\mathfrak m).$$

The chiral factors are ratios of Gamma functions whose arguments depend on gauge weights, twisted masses, R-charges, and $\mathfrak m/2$. Flux quantization depends on the global gauge group and matter charges. Higgs-branch formulas obtained by closing the contour are equivalent only after all residues at infinity and vortex sectors are included Benini and Cremonesi 2015, §§3–4; Doroud et al. 2013, §§3–5.

Convergence, contours, and numerical evaluation

The original localization derivation determines a real integration cycle, but analytic continuation of masses or R-charges can force a deformation in the complexified Cartan. Poles crossing the cycle generate residues. A numerical integral is meaningful only after this chamber information is frozen.

Check the following before trusting a value:

• large-$|\sigma|$ asymptotics and absolute versus oscillatory convergence;
• singular hyperplanes and the prescription for poles on the cycle;
• Weyl quotient and the normalization of the Cartan metric;
• determinant phases and background Chern–Simons terms;
• decoupled $U(1)$ or center-of-mass factors;
• flux or instanton truncation error;
• stability under working precision and contour deformation;
• agreement with a free determinant or weak-coupling expansion.

Sphere matrix models compute the defined supersymmetric background observable. Interpreting it as a free energy, Kähler potential, R-symmetry functional, or duality test requires the additional counterterm and normalization analysis appropriate to that use.

Exercises

Evaluate the round-$S^3$ formula for a free chiral multiplet with $\Delta=1/2$.

Solution

There is no Cartan integral or vector determinant. The one-loop factor is $e^{\ell(1-\Delta)}=e^{\ell(1/2)}$. Integrating $\ell'(z)=-\pi z\cot(\pi z)$ with $\ell(0)=0$ gives $\ell(1/2)=-(\log2)/2$, so $Z=2^{-1/2}$ and $-\log|Z|=(\log2)/2$.

References

• Benini, F., and S. Cremonesi. “Partition Functions of $N=(2,2)$ Gauge Theories on $S^2$ and Vortices.” Communications in Mathematical Physics 334 (2015): 1483–1527. DOI; Open PDF.
• Doroud, N., J. Gomis, B. Le Floch, and S. Lee. “Exact Results in $D=2$ Supersymmetric Gauge Theories.” Journal of High Energy Physics 2013, no. 5 (2013): 093. DOI; Open PDF.
• Kapustin, A., B. Willett, and I. Yaakov. “Exact Results for Wilson Loops in Superconformal Chern–Simons Theories with Matter.” Journal of High Energy Physics 2010, no. 3 (2010): 089. DOI; Open PDF.
• Pestun, V. “Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops.” Communications in Mathematical Physics 313 (2012): 71–129. DOI; Open PDF.