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 branch-dependent, but an exponentiated derivative is single-valued; its unit locus gives candidate quantum Coulomb vacua. In specified gauge/Bethe correspondences the same equations become Bethe equations. The calculation is valid only away from massless matter, excluded gauge roots, degenerate solutions, 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 primitive $U(1)^k$ charge normalization, 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, with the masses also large compared with gradients and any other infrared scale. 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 warns that a supposedly heavy field has become light.

Use the FI–theta convention of the phase page,

$$t_a(\mu)=2\pi r_a(\mu)-i\theta_a, \qquad q_a(\mu)=e^{-t_a(\mu)}.$$

Below, $q_a$ without an explicit argument means $q_a(\mu)$ at the same Wilsonian scale.

In a convenient subtraction scheme, the Abelian one-loop result is

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

A branch of every logarithm is chosen on a simply connected patch with $M_i\ne0$. The formula is defined up to allowed linear terms and corresponding finite redefinitions of $q_a$, so the numerical coordinate of a discriminant is scheme-dependent. Once the theory, global gauge group, and scheme are fixed, vacuum multiplicities and monodromy of the solution set are invariant.

Why the logarithm appears

Differentiate and define the dimensionless effective FI–theta function

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

The derivative of $M(\log M-1)$ is $Q\log M$; the constant $-1$ removes an unwanted linear contribution. The $\mu$ dependence cancels between the logarithms and $q_a(\mu)$ because $d\log q_a/d\log\mu=-\sum_iQ_i^{\ a}$. The determinant derivation, its sign, and the resulting vacuum equation appear in Hori et al. 2003, §15.5.1, pp. 384–387.

Each logarithm is multivalued:

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

Consequently $\tau_a$ changes by $2\pi i\sum_iQ_i^{\ a}n_i$. A branch change of $\log q_a$ changes it by another element of $2\pi i\mathbb Z$. The physical equation is therefore not “the derivative vanishes on one preferred branch,” but

$$\Pi_a(\sigma)\equiv e^{\tau_a(\sigma)}=1.$$

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. For a nonprimitive charge basis or another global form, the cocharacter lattice replaces the displayed componentwise $2\pi i\mathbb Z$ condition.

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

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

The running of $q(\mu)$ makes

$$\Lambda^N\equiv\mu^Nq(\mu)$$

RG invariant. For $\Lambda\ne0$ there are $N$ candidate Coulomb vacua,

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

This is also the small quantum-cohomology relation $H^N=\Lambda^N$ after identifying $H$ with the twisted-chiral operator $\sigma$; a dimensionless convention instead writes $H^N=q$. The axial anomaly manifests itself as the generated scale and as the cyclic permutation of the $N$ roots when the theta angle winds once. The vacuum count and its domain are analyzed in Hori et al. 2003, §15.5.2, pp. 387–391.

The ordinary Hessian in the sign convention fixed above 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. If no $M_i$ vanishes there, the Hessian vanishes and the local isolated-vacuum approximation and ordinary residue formulas fail together. If some $M_i=0$, the Wilsonian action has already failed and that matter field must first be restored.

Hessians and topological correlators

For rank $k$, it is useful to keep both the Hessian and the Jacobian of the exponentiated equations:

$$\begin{aligned} \mathcal J_{ab} &\equiv\frac{\partial\tau_a}{\partial\sigma_b} =\sum_i\frac{Q_i^{\ a}Q_i^{\ b}}{M_i(\sigma)},\\ \mathcal H_{ab} &\equiv\frac{\partial^2\widetilde W_{\mathrm{eff}}} {\partial\sigma_a\partial\sigma_b} =-\frac1{2\pi}\mathcal J_{ab}. \end{aligned}$$

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

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

where $\mathcal U$ includes the remaining one-loop measure, flux, and sign factors. This is deliberately schematic: the exact handle-gluing operator, genus dependence, gauge group, R charges, and operator normalization must be specified before using a residue formula.

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

The equations $\Pi_a=e^{\tau_a}=1$ have the same multiplicative structure as Bethe ansatz equations: ratios of shifted masses can play the role of scattering phases, and FI parameters can become twists. In a gauge/Bethe correspondence, a conventionally normalized $\widetilde W_{\mathrm{eff}}$ is identified with a Yang–Yang function. This is an additional, model-specific dictionary—not a property of every GLSM. The precise integrable system, boundary conditions, inhomogeneities, and admissible roots must be supplied, as emphasized in Nekrasov and Shatashvili 2009, §1, pp. 1–4.

Exactness and possible corrections

For the specified two-dimensional $(2,2)$ Coulomb-branch theory, the displayed matter logarithm is a Wilsonian twisted F-term and is one-loop exact under the holomorphy and decoupling assumptions used in its derivation. This does not license a universal “one loop is exact” claim for every effective action or every dimensional lift:

• vortex sectors can generate twisted-superpotential terms after dualization, where the variables and weakly coupled description have changed;
• 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 twisted F-term equation is exact only for a specified theory, parameter chamber, global gauge group, and Wilsonian domain. It does not by itself prove that the listed Coulomb solutions exhaust Higgs, mixed, or strongly coupled vacua.

The large-$|\sigma|$ determinant calculation and the reason it must not be continued through $\sigma=0$ are already explicit in Witten 1993, §3.2, pp. 20–24.

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 agrees with the quantum phase analysis in Hori et al. 2003, §15.5.3, pp. 391–393 and is a sensitive sign and normalization check. A finite linear counterterm rescales the coordinate called $q$, so comparisons must use the same scheme.

The chapter’s phase–mirror–tt* map places this discriminant beside a separate exact $\mathbb{CP}^1$ ring–mirror–tt* chain. The comparison shares conventions; it does not identify the two fixtures as one flow.

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 $\sigma=0$. At that point

$$\mathcal J=(\sigma+m)^{-1}+(\sigma-m)^{-1}=0,$$

and $\mathcal H=-\mathcal J/(2\pi)$ also vanishes. For $m\ne0$ the integrated-out fields remain massive at the collision, so it is the isolated-vacuum approximation—not the matter threshold—that 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 $\tau_a$. 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

• Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., and Zaslow, E. Mirror Symmetry. Clay Mathematics Monographs 1. Providence, RI: American Mathematical Society and Clay Mathematics Institute, 2003, ch. 15. Clay Mathematics Institute PDF.
• 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.