Effective Twisted Superpotentials and Quantum Vacua

On a two-dimensional Coulomb branch, charged multiplets acquire complex masses and can be integrated out. Holomorphy packages their effect into an effective twisted superpotential $\widetilde W_{\mathrm{eff}}(\Sigma)$. Its derivative is multivalued, but its exponential is single-valued; the resulting equations determine isolated quantum Coulomb vacua and, after compactification in related settings, Bethe-type equations. The derivation is valid only away from massless matter, excluded gauge roots, and uncontrolled continua.

Required background. We use the GLSM charge, FI, and global-group data and the distinction between Wilsonian and 1PI effective actions. Helpful background. Instanton zero modes and selection rules help assess possible nonperturbative terms.

Wilsonian setup and its domain

Take an Abelian rank-$k$ gauge theory with Coulomb scalars $\sigma_a$, chiral multiplets of charges $Q_i^{\ a}$, and twisted masses $m_i$. The complex mass of the $i$th chiral is

$$M_i(\sigma)=\sum_aQ_i^{\ a}\sigma_a+m_i.$$

Choose a Wilsonian scale $E$ satisfying

$$E\ll |M_i(\sigma)|$$

for every field being removed. Then the low-energy twisted F-terms are encoded in a holomorphic $\widetilde W_{\mathrm{eff}}(\Sigma)$. This construction must not be evaluated at $M_i=0$: the logarithm is warning that a supposedly heavy field has become light.

For one $U(1)$ and our coordinate $q=e^{-2\pi r+i\theta}$, a convenient scheme is

$$\widetilde W_{\mathrm{eff}}(\sigma)= \frac1{2\pi}\left[ \sum_i M_i(\sigma) \left(\log\frac{M_i(\sigma)}{\mu}-1\right) -(\log q)\sigma \right].$$

A finite local counterterm linear in $\sigma$ rescales $q$ by a constant, so the numerical coordinate of a discriminant is scheme-dependent. Vacuum multiplicities and invariant monodromy data are not.

Why the logarithm appears

Differentiate with respect to a Coulomb scalar:

$$2\pi\frac{\partial\widetilde W_{\mathrm{eff}}}{\partial\sigma_a} =\sum_iQ_i^{\ a}\log\frac{M_i(\sigma)}{\mu}-\log q_a.$$

The derivative of $M(\log M-1)$ is $Q\log M$; the constant $-1$ removes an unwanted linear contribution. Equivalently, the one-loop fermion and boson determinants produce the logarithmic running of the FI-theta coupling. Supersymmetric holomorphy promotes that running to the twisted superpotential.

Each logarithm is multivalued:

$$\log M_i\longmapsto\log M_i+2\pi in_i.$$

Consequently $\partial\widetilde W_{\mathrm{eff}}$ is defined modulo $i$ times the integral cocharacter lattice. The physical vacuum equation is not “the derivative equals zero on a chosen branch,” but

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

Therefore

$$\prod_i \left(\frac{M_i(\sigma)}{\mu}\right)^{Q_i^{\ a}} =q_a, \qquad a=1,\ldots,k.$$

Negative charges create denominators. This is harmless on the domain $M_i\ne0$ but makes the excluded divisor explicit.

Worked example: $\mathbb{CP}^{N-1}$

The GLSM has one $U(1)$ and $N$ chiral fields of charge $+1$. With vanishing twisted masses,

$$\widetilde W_{\mathrm{eff}}(\sigma)= \frac1{2\pi}\left[ N\sigma\left(\log\frac{\sigma}{\mu}-1\right) -(\log q)\sigma \right].$$

The exponentiated equation is

$$\left(\frac{\sigma}{\mu}\right)^N=q, \qquad \sigma^N=q\mu^N.$$

For $q\ne0$ there are $N$ isolated vacua,

$$\sigma_\ell=\mu q^{1/N}e^{2\pi i\ell/N}, \qquad \ell=0,\ldots,N-1.$$

This is also the small quantum-cohomology relation $H^N=q$ after identifying $H$ with the twisted-chiral operator $\sigma$ and absorbing $\mu$ into the definition of $q$. The axial anomaly manifests itself as the generated scale and the cyclic permutation of vacua when the theta angle winds.

The Hessian is

$$\mathcal H(\sigma) =\frac{\partial^2\widetilde W_{\mathrm{eff}}}{\partial\sigma^2} =\frac{N}{2\pi\sigma}.$$

It is nonzero at every vacuum for $q\ne0$, confirming isolation within the Coulomb description.

With twisted masses $m_i$, the relation becomes

$$\prod_{i=1}^{N}\frac{\sigma+m_i}{\mu}=q.$$

Vacua can collide where this polynomial and its derivative vanish simultaneously. At a collision the Hessian vanishes; the local massive-vacuum approximation and ordinary residue formulas fail together.

Hessians and topological correlators

For rank $k$, define

$$\mathcal H_{ab} =\frac{\partial^2\widetilde W_{\mathrm{eff}}} {\partial\sigma_a\partial\sigma_b} =\frac1{2\pi}\sum_i \frac{Q_i^{\ a}Q_i^{\ b}}{M_i(\sigma)}.$$

An isolated solution requires $\det\mathcal H\ne0$, after quotienting by any residual Weyl action. In an A-twisted Coulomb-branch localization formula, correlators take the schematic form

$$\langle f(\sigma)\rangle =\sum_{\sigma_*} \frac{f(\sigma_*)\,\mathcal U(\sigma_*)} {\det\mathcal H(\sigma_*)},$$

where $\mathcal U$ includes one-loop measure and flux-dependent factors. The bare Hessian alone is not a universal correlator formula; genus, gauge group, R charges, and operator normalization fix the remaining factors.

Non-Abelian gauge groups

For compact $G$, restrict $\sigma$ to a Cartan subalgebra and divide solutions by the Weyl group. Matter weights $\rho$ replace charges:

$$M_{i,\rho}=\rho(\sigma)+m_i.$$

Several extra checks are required:

• remove root hyperplanes $\alpha(\sigma)=0$, where $W$-bosons become massless;
• include the vector-multiplet one-loop contribution with a consistent root convention;
• sum over the flux lattice of the actual global group, not merely the Lie algebra;
• identify Weyl-related roots and handle fixed points separately;
• include discrete theta angles and disconnected gauge sectors when present.

A formal polynomial root lying on a root hyperplane is not a valid Abelianized vacuum. Nor may one count all Weyl images as different physical vacua.

From vacuum equations to Bethe equations

Write

$$\Pi_a(\sigma)= \exp\left(2\pi\frac{\partial\widetilde W_{\mathrm{eff}}} {\partial\sigma_a}\right).$$

The equations $\Pi_a=1$ have the same multiplicative structure as Bethe ansatz equations: ratios of shifted masses play the role of scattering phases, and $q_a$ are twist parameters. In gauge/Bethe correspondences, $\widetilde W_{\mathrm{eff}}$ is identified with a Yang–Yang function. The identification is model-specific: the precise spin chain, boundary conditions, inhomogeneities, and admissible Bethe roots must all be given, as emphasized in Nekrasov and Shatashvili 2009.

Exactness and possible corrections

The displayed matter logarithm is a one-loop Wilsonian twisted F-term. In many $(2,2)$ GLSMs it is perturbatively one-loop exact because higher loops cannot generate the required holomorphic dependence. This does not license a universal “one loop is exact” claim:

• vortices can generate twisted-superpotential terms in dual variables;
• compactification from higher dimensions produces Kaluza–Klein sums and trigonometric or elliptic functions;
• noncompact Coulomb directions can make the 1PI action nonlocal;
• fields that become massless must be restored;
• finite counterterms change the coordinate called $q$;
• boundary conditions can add effective degrees of freedom.

An exact vacuum equation is exact only for a specified theory, parameter chamber, and Wilsonian domain.

Quintic discriminant as a check

For charges $(1,1,1,1,1,-5)$ and zero masses,

$$\left(\frac{\sigma}{\mu}\right)^5 \left(\frac{-5\sigma}{\mu}\right)^{-5}=q,$$

so the $\sigma$ dependence cancels and

$$q=(-5)^{-5}.$$

At generic $q$ there is no finite isolated Coulomb vacuum of this type; at the displayed value an unlifted Coulomb direction appears. This reproduces the Coulomb singularity in the original GLSM phase analysis Witten 1993, §3.2. It is a useful sign and normalization check.

Exercises

1. Solve the $\mathbb{CP}^1$ equation with masses $m_1=-m_2=m$.

Solution

The equation is $(\sigma+m)(\sigma-m)=q\mu^2$, hence

$$\sigma_\pm=\pm\sqrt{m^2+q\mu^2}.$$

The vacua collide when $m^2+q\mu^2=0$. At that point the Hessian $\frac1{2\pi}[(\sigma+m)^{-1}+(\sigma-m)^{-1}]$ vanishes after imposing the collision condition, and the massive description degenerates.

2. Show that changing a logarithm branch does not change the exponentiated vacuum equation when all charges are integral.

Solution

A branch shift adds $2\pi in_iQ_i^{\ a}$ to $2\pi\partial_a\widetilde W_{\mathrm{eff}}$. Its exponential is multiplied by $e^{2\pi i\sum_i n_iQ_i^{\ a}}=1$ because the charges and $n_i$ are integers.

3. Why must $\sigma=0$ be excluded from the massless $\mathbb{CP}^{N-1}$ derivation even though one can formally write $\sigma^N=q\mu^N$ there when $q=0$?

Solution

At $\sigma=0$ every charged chiral has zero mass, violating the Wilsonian inequality used to integrate them out. The $q=0$ limit is a boundary of parameter space and must be analyzed with the original light fields, not by extending the logarithmic action through its singularity.

References

• Nekrasov, N. A., and Shatashvili, S. L. “Supersymmetric Vacua and Bethe Ansatz.” Nuclear Physics B Proceedings Supplements 192–193 (2009): 91–112. doi:10.1016/j.nuclphysbps.2009.07.047; arXiv:0901.4744.
• Witten, E. “Phases of $N=2$ Theories in Two Dimensions.” Nuclear Physics B 403 (1993): 159–222. doi:10.1016/0550-3213(93)90033-L; arXiv:hep-th/9301042.