Follow a deformation through SQED-XYZ duality

A useful duality should survive controlled changes of the theory. A mass that is elementary in one description may become a linear superpotential term in another; an FI parameter may become a real mass for monopole fields. We will follow both changes and compare their vacuum equations.

Start with the SQED–XYZ dictionary and sources and currents. Both descriptions remain three-dimensional $\mathcal N=2$ theories with four real Lorentzian supercharges. We work in flat space, with all rigid supergravity sources zero. SQED has compact gauge group $U(1)$, one pair $Q,\widetilde Q$ of charges $+1,-1$, gauge Chern–Simons level $k=0$, and the preceding paired regulator. Gauge parameters have period $2\pi$, and global checks use spin manifolds. Background counterterms will be specified below. No gauge rank or number of supercharges changes when we turn on the parameters.

A complex mass joins two branches

Retain the holomorphic normalization and phase choice $X=M=Q\widetilde Q$, $Y=V_+$, $Z=V_-$, with $W=XYZ$. Adding a source $m$ for the meson gives

$$W_{\rm SQED}=mM \quad\longleftrightarrow\quad W_{\rm XYZ}=XYZ+mX. \tag{S16.1}$$

For now these are matched holomorphic operator coordinates and sources. We distinguish canonical ultraviolet parameters shortly.

The three F-term equations on the XYZ side are

$$yz+m=0,\qquad xz=0,\qquad xy=0. \tag{S16.2}$$

If $m\ne0$, then $y$ and $z$ are both nonzero. Either of the last two equations therefore sets $x=0$. The complete solution is

$$x=0,\qquad y\in\mathbb C^*,\qquad z=-\frac m y, \tag{S16.3}$$

where $\mathbb C^*$ means the nonzero complex numbers. The chiral ring becomes

$$\frac{\mathbb C[X,Y,Z]}{(YZ+m,XZ,XY)} \simeq\mathbb C[Y,Y^{-1}], \qquad Y^{-1}=-Z/m. \tag{S16.4}$$

This is one smooth complex cylinder. Choosing a nonzero reference $y_0$ of the same units as $y$, write

$$y=y_0e^u,\qquad z=-\frac m{y_0}e^{-u}, \qquad u\sim u+2\pi i. \tag{S16.5}$$

The noncompact real part and periodic imaginary part exhibit the cylinder. The three branches of the undeformed theory have changed: the Higgs branch is removed and the two Coulomb coordinates are joined by a nonzero product.

The sign is part of the previous lesson’s convention. AHISS uses $W=-MV_+V_-+mM$ and obtains $V_+V_-=m$. Our replacement $V_-^{\rm here}=-V_-^{\rm AHISS}$ gives $YZ=-m$. A deformation test must carry this phase choice along with the operator map. AHISS v1, section 3.1, following (3.2)

Compare with the massive gauge theory

For canonically normalized ultraviolet SQED fields, denote the superpotential coefficient by $m_{\rm e}$: $W=m_{\rm e}Q\widetilde Q$. It has engineering dimension one. Eliminating the matter auxiliaries adds

$$U_F=|m_{\rm e}|^2 \left(|\phi_+|^2+|\phi_-|^2\right).$$

At zero FI parameter the full classical potential is therefore

$$U_{\rm SQED} =(\sigma^2+|m_{\rm e}|^2) (|\phi_+|^2+|\phi_-|^2) +\frac{g_{\rm YM}^2}{2} (|\phi_+|^2-|\phi_-|^2)^2. \tag{S16.6}$$

For nonzero $m_{\rm e}$, zero energy requires $\phi_+=\phi_-=0$. The real vector scalar $\sigma$ remains free to vary, and the unbroken gauge field supplies a periodic dual photon. The resulting low-energy Coulomb cylinder agrees with the complex space (S16.3). The quantum holomorphic relation is $V_+V_-=-m$ in our normalization; it does not follow by multiplying the two undeformed semiclassical exponentials.

There is a useful weak-coupling check. If $|m_{\rm e}|\gg g_{\rm YM}^2$, charged matter can be integrated out at energies $E\ll |m_{\rm e}|$. This complex mass removes the charge-opposite pair without leaving an uncancelled gauge Chern–Simons level in the paired convention. A massless Abelian vector remains. Its dual-photon kinetic coefficient can receive threshold corrections. The relation (S16.3) determines a complex manifold, not that coefficient or a globally flat metric.

For deformations of the common interacting infrared fixed point, instead take the deformation scales small compared with the respective ultraviolet crossover scales. These two regimes serve different purposes: the first controls a semiclassical calculation; the second defines a flow away from the shared fixed point. The vacuum-space comparison is evidence consistent with the duality, not a proof that the two bare actions agree.

Keep the units of the source

Canonical XYZ fields $X_c,Y_c,Z_c$ have ultraviolet engineering dimension $1/2$. Their action uses

$$W_{\rm can}=\lambda X_cY_cZ_c+fX_c,\qquad \lambda\ne0,\qquad [\lambda]=\frac12,\qquad [f]=\frac32.$$

One convenient holomorphic rescaling is $X=\lambda X_c$, $Y=Y_c$, $Z=Z_c$. It gives (S16.1) with $m=f/\lambda$. This normalization has $[X]=1$, $[Y]=[Z]=1/2$, and $[m]=1$, but its kinetic terms are no longer all canonical.

For example, if a matching convention at scale $\mu_0$ is $X_c=c_M\mu_0^{-1/2}M_{\rm ren}(\mu_0)$, then the electric source maps to $f=m_{\rm e}\mu_0^{1/2}/c_M$. If $M_{\rm ren}$ is precisely the normalized $M$ in (S16.1), compatibility with $X=M$ requires $\lambda c_M\mu_0^{-1/2}=1$, giving $m=m_{\rm e}$ at that matching scale. Those normalization choices are not independent. A different normalization requires carrying its matching factor through the source map. In a canonical weakly coupled electric description the tree-level matter mass at $\sigma=0$ is $|m_{\rm e}|$; wavefunction normalization matters beyond that approximation. The holomorphic cylinder parameter is not automatically a pole mass.

At the undeformed interacting fixed point, $\Delta_X=2/3$, so the source for $X$ has scaling dimension $2-2/3=4/3$. This infrared scaling dimension is distinct from either ultraviolet engineering convention. A fixed nonzero complex source preserves supersymmetry but breaks the undeformed axial symmetry and conformal invariance; assigning compensating charges to the source is bookkeeping, not an unbroken symmetry at a fixed source value.

Real masses are scalar sources

A supersymmetric flavor source contains more than the connection used to differentiate a current. In flat three-dimensional $\mathcal N=2$ language it is a vector multiplet

$$(B_\mu,\nu,\lambda_B,\widetilde\lambda_B,D_B).$$

Here $B_\mu$ is a nondynamical connection, $\nu$ a scalar, and $D_B$ an auxiliary source. The constant configuration $B_\mu=0$, $\nu\in\mathbb R$, $\lambda_B=\widetilde\lambda_B=D_B=0$ preserves all four supercharges: the flat gaugino variations contain only the field strength, the derivative of $\nu$, and $D_B$, all zero.

For a chiral field of flavor charge $q_F$, this source gives the physical real fermion mass $q_F\nu$ and scalar potential $(q_F\nu)^2|\phi|^2$ in the component convention already fixed for SQED. A real mass is thus different from a holomorphic coefficient in $W$. It uses an actual conserved flavor symmetry. AHISS v1, section 2.3, (2.8)–(2.9)

For reference, let $\mu$ source the axial symmetry and $\nu$ the topological symmetry. At zero complex mass the charge table gives

Fields	Real mass contribution
SQED $Q,\widetilde Q$	$\sigma+\mu,\ -\sigma+\mu$
XYZ $X$	$2\mu$
XYZ $Y,Z$	$-\mu+\nu,\ -\mu-\nu$

The two SQED entries also contain the dynamical scalar $\sigma$. On that side $\nu$ acts through an FI coupling, which we now normalize. We set $\mu=0$ in the worked deformation. This keeps the calculation within the topological symmetry, which is also preserved by the complex mass. The table matches the flavor/FI assignments in Willett–Yaakov v2, section 3.1.

Normalize the FI coupling

Use orientation $\epsilon^{123}=+1$, field strength $f=da$, and unit magnetic flux $\int_{S^2}f=2\pi n$. Couple a compact topological background $B_J$ at level one. The bosonic part of its supersymmetric mixed term is

$$\mathcal L_{\rm BF} =\frac1{2\pi} \left(i\epsilon^{\mu\nu\rho}B_{J\mu}\partial_\nu a_\rho -\nu D-\sigma D_J\right). \tag{S16.7}$$

“BF” denotes the connection–field-strength term $B_J\wedge f$; it is not a kinetic term for a new photon. The supersymmetric fermion partners vanish on the bosonic frozen source used here.

The factor and signs can be checked without a curved-space construction. Polarize the Abelian supersymmetric Chern–Simons density $(i\,a\wedge da-2\sigma D)/(4\pi)$ between the two vector multiplets. The connection cross terms are $i(B_J\wedge da+a\wedge dB_J)/(4\pi)$. On a closed manifold, or with vanishing boundary terms, integration by parts makes their integrals equal, giving the first term of (S16.7). The auxiliary cross terms are $-(\nu D+\sigma D_J)/(2\pi)$. This uses the normalization of CT v2, (4.5).

For a unit-flux configuration, changing a background holonomy by $2\pi$ changes the BF action by an integer multiple of $2\pi i$. The path-integral weight is therefore well defined. Its Euclidean source derivative is $\delta S_E/\delta B_{J\mu}=i(*f)^\mu/(2\pi)$. This states the Euclidean source-current normalization explicitly; the integer monopole label remains $J(V_n)=n$.

Freeze $\nu$ and set $B_J=D_J=0$. Define the FI parameter used in the moment-map equation by

$$\zeta=\frac{\nu}{2\pi},\qquad \mathcal L_{\rm FI}=-\zeta D. \tag{S16.8}$$

Thus the previously constructed auxiliary density becomes

$$-\frac{D^2}{2g_{\rm YM}^2} +D\bigl(|\phi_+|^2-|\phi_-|^2-\zeta\bigr).$$

Completing its square gives the positive reduced potential $g_{\rm YM}^2(|\phi_+|^2-|\phi_-|^2-\zeta)^2/2$. As in the preceding component action, the Euclidean auxiliary follows a shifted imaginary contour; this is stationary auxiliary elimination, not a convergent integral over real $D$. A convention writing the FI term as $+\xi D$ instead has $\xi=-\zeta$. We have fixed the sign through the source coupling rather than through the name of the parameter.

State the contact-term scheme

Keep the paired gauge regulator, $k=0$, and the level-one BF coupling fixed. Set the additional integer $B_J$ background Chern–Simons counterterm to zero in the matched source convention on both sides, and keep that choice as $\nu$ changes. Other flavor and R connections are zero in this calculation.

A permitted integer change would add

$$S_{\rm ct} =\frac{k_{JJ}}{4\pi}\int \left(iB_J\wedge dB_J-2\nu D_J\,d^3x\right), \qquad k_{JJ}\in\mathbb Z.$$

It changes background contact terms and, on suitable nonzero backgrounds, partition-function phases. It contains no dynamical field and does not change the vacuum equations derived here. This choice does not assert that every quantum contact coefficient, particularly its fractional part, vanishes. Those fractional parts cannot in general be removed by integer counterterms. CT v2, (1.3)–(1.7) and (4.5)

Worked FI deformation: retain one complex branch

Set the complex mass to zero and take $\nu\ne0$. On XYZ the real masses are $(0,\nu,-\nu)$. Write $x_c,y_c,z_c$ for the scalar components of the canonical fields $X_c,Y_c,Z_c$. Their potential is

$$U_{\rm XYZ} =|\lambda|^2(|y_cz_c|^2+|x_cz_c|^2+|x_cy_c|^2) +\nu^2(|y_c|^2+|z_c|^2).$$

Zero energy requires $y_c=z_c=0$, while $x_c$ is arbitrary. Only the Higgs coordinate $X=\lambda X_c$ survives.

On SQED the zero-energy equations are

$$\sigma\phi_+=\sigma\phi_-=0,\qquad |\phi_+|^2-|\phi_-|^2=\zeta. \tag{S16.9}$$

Since $\zeta\ne0$, at least one charged field is nonzero and hence $\sigma=0$. Divide by the gauge transformation $(\phi_+,\phi_-)\mapsto(e^{i\alpha}\phi_+,e^{-i\alpha}\phi_-)$. The remaining complex coordinate is again $M=\phi_+\phi_-$.

The quotient also lets us calculate its classical metric. Set $\rho=|M|$, $u=|\phi_+|^2$, $v=|\phi_-|^2$. Then

$$u-v=\zeta,\quad uv=\rho^2,\qquad u=\frac{\sqrt{\zeta^2+4\rho^2}+\zeta}{2},\quad v=\frac{\sqrt{\zeta^2+4\rho^2}-\zeta}{2}.$$

Let $\theta=\arg M$ away from the origin. Eliminating the gauge phase from the canonical scalar kinetic terms gives

$$ds^2=\frac{du^2}{4u}+\frac{dv^2}{4v} +\frac{uv}{u+v}\,d\theta^2 =\frac{d\rho^2+\rho^2d\theta^2}{\sqrt{\zeta^2+4\rho^2}} =\frac{dM\,d\overline M}{\sqrt{\zeta^2+4|M|^2}}. \tag{S16.10}$$

For $\zeta\ne0$ the coefficient is finite at $M=0$, equal to $1/|\zeta|$. The FI deformation smooths the classical cone. Unit electric charges leave no discrete gauge stabilizer there.

This quotient metric is a tree-level low-energy result: massive gauge fluctuations have been eliminated. It is controlled when $u+v\gg g_{\rm YM}^2$ and $E\ll g_{\rm YM}\sqrt{2(u+v)}$. In particular, $|\zeta|\gg g_{\rm YM}^2$ controls the whole branch. At small FI parameter, quantum corrections matter near the origin. The XYZ $X$ axis and SQED Higgs branch agree as complex spaces under the dictionary; their canonical ultraviolet metrics are not asserted to agree. AHISS describes this FI-to-monopole-real-mass deformation in section 3.1.

Exercises

1. Check the massless tangent direction

In canonical XYZ variables choose $W=\lambda X_c(Y_cZ_c-v^2)$, with $v>0$ real and $\lambda\ne0$. At $X_c=0$, $Y_c=Z_c=v$, find the massless chiral fluctuation and both nonzero fermion mass magnitudes. Check the corresponding quadratic scalar potential.

Hint 1

Use $\delta Y_\pm=(\delta Y_c\pm\delta Z_c)/\sqrt2$.

Hint 2

The quadratic superpotential is $\sqrt2\lambda v\,\delta X_c\delta Y_+$. Physical masses are singular values of its complex Hessian.

Solution

The linearized constraint is $\delta Y_c+\delta Z_c=0$, so $\delta Y_-$ is tangent to the cylinder and massless. The quadratic superpotential pairs $\delta X_c$ and $\delta Y_+$ with mass magnitude $\sqrt2|\lambda|v$. Its two nonzero singular values are equal. Directly expanding $\sum_i|W_i|^2$ gives

$$U^{(2)}=2|\lambda|^2v^2 \left(|\delta X_c|^2+|\delta Y_+|^2\right).$$

There are four massive real scalar components and two massless tangent components, agreeing with the fermionic count. This canonical tree calculation is perturbatively controlled for $v\gg|\lambda|$. It checks local smoothness away from the undeformed junction, not an exact quantum metric.

2. Reverse the FI parameter

At $M=0$, find the gauge-equivalence class of vacua for each sign of nonzero $\zeta$. Does reversing its sign create a second Higgs branch or leave a residual $\mathbb Z_2$ gauge symmetry? What happens to the classical metric as $\zeta\to0$?

Hint 1

Combine $uv=0$ with $u-v=\zeta$.

Hint 2

A unit-charge nonzero field fixes the gauge phase completely modulo $2\pi$. Examine the metric both at fixed nonzero $M$ and near $M=0$.

Solution

For $\zeta>0$, representatives are $\phi_+=\sqrt\zeta$, $\phi_-=0$; for $\zeta<0$, they are $\phi_+=0$, $\phi_-=\sqrt{-\zeta}$. In each case the phase of the nonzero field is gauge, and a transformation fixing it has $e^{i\alpha}=1$. There is no residual $\mathbb Z_2$.

Each sign gives one complex $M$ plane, with a different charged field nonzero at its origin. At fixed $M\ne0$ the metric approaches $dM\,d\overline M/(2|M|)$, the original classical cone. At $M=0$ its coefficient $1/|\zeta|$ diverges as $\zeta\to0$, signaling the return of the singular quotient. The low-energy elimination of the massive vector also ceases to be uniform there. On XYZ either sign of $\nu=2\pi\zeta$ gives the same positive term $\nu^2(|y|^2+|z|^2)$, correctly retaining one branch.

3. Turn on both deformations

Keep a nonzero complex mass and a nonzero topological real mass, with $\mu=0$. Can either description have a finite homogeneous supersymmetric vacuum? Distinguish explicit preservation of supersymmetry by the action from invariance of a vacuum.

Hint 1

On XYZ, the real-mass terms constrain $y,z$ before the F-term $yz+m$ is imposed.

Hint 2

On SQED, the complex mass constrains both charged scalars before the FI equation is imposed.

Solution

On XYZ, zero energy from the real-mass terms requires $y=z=0$. The remaining F-term is then $yz+m=m\ne0$, so the supersymmetry conditions are incompatible. Equivalently, the nonzero-product cylinder contains no point fixed by the topological mass deformation.

On SQED, $m_{\rm e}\ne0$ forces $\phi_+=\phi_-=0$. The FI equation would then require $0=\zeta$, contrary to the assumption. In the classical canonical action the auxiliary stationary value is $D=-g_{\rm YM}^2\zeta$, whereas an invariant homogeneous vector configuration requires $D=0$.

Both deformations preserve the four-supercharge action: the complex mass is a superpotential source and the topological vector source is supersymmetric. Their simultaneous vacuum equations have no finite solution. This is a statement about supersymmetric vacua, not a proof about the full nonsupersymmetric spectrum or a calculation of its quantum vacuum energy.

These tests distinguish holomorphic constraints, real source couplings, and kinetic approximations. A comparison of partition functions or their phases must additionally keep the background counterterm convention fixed; agreement of branch equations alone cannot supply that information.

Reproduce the full complex sphere duality with both axial mass and topological source, then distinguish a quantized contact phase from a mistaken source dictionary.