Real masses and flavor backgrounds

A background connection probes a conserved current. Supersymmetry places that connection in a multiplet with a scalar and an auxiliary field. Giving the scalar a constant value produces a real mass. On a curved space, keeping the source supersymmetric also fixes its auxiliary value.

Use the charged matter action and the definition of source responses. We consider three-dimensional $\mathcal N=2$ theories with four real Lorentzian supercharges, using their Euclidean continuation. Our flat and round-sphere backgrounds preserve the full continued algebra. On a general rigid background, only the supercharges admitted by its Killing spinors can survive.

Take one chiral multiplet $\Phi=(\phi,\psi,F)$ with a canonical kinetic term and a compact flavor symmetry $U(1)_F$. Its flavor charge $q_F$ is an integer in a unit-charge probe convention; its scalar R-charge is $r$, and its central-charge parameter is $z$. There is initially no superpotential. The Euclidean antichiral fields are independent partners, not additional chiral flavors. No dynamical gauge field is needed for this example.

Freeze the whole flavor multiplet

Introduce the nondynamical vector multiplet

$$\mathcal V_F=(a_{F\mu},\sigma_F,\lambda_F,\widetilde\lambda_F,D_F).$$

Here $[\sigma_F]=[a_F]=1$ and $[D_F]=2$ in mass units. These fields are external sources: we do not integrate over them or solve a Yang–Mills equation for them.

Set the source gauginos to zero. Their variations must vanish for every retained supersymmetry. With commuting Killing-spinor labels $\zeta,\widetilde\zeta$, define

$$K_F=D_F+H\sigma_F,\qquad \mathcal B_{F\mu}=(*f_F)_\mu+i\sigma_F V_\mu, \qquad f_F=da_F.$$

The vector transformations give

$$\begin{aligned} \delta_\zeta\lambda_F &=i\left[K_F-\gamma^\mu(\mathcal B_{F\mu}+\partial_\mu\sigma_F)\right]\zeta,\\ \delta_{\widetilde\zeta}\widetilde\lambda_F &=-i\left[K_F+\gamma^\mu(\mathcal B_{F\mu}-\partial_\mu\sigma_F)\right]\widetilde\zeta. \end{aligned}$$

For two independent labels in each family, both matrices must vanish. Taking their traces isolates $K_F$; taking traces after multiplying by $\gamma^\nu$ isolates their vector coefficients. Adding and subtracting those coefficients gives

$$\partial_\mu\sigma_F=0,\qquad D_F=-H\sigma_F,\qquad *f_F=-i\sigma_F V.$$

Since the rigid source satisfies $V=-i*dC$, write the resulting background as

$$\boxed{ \sigma_F=\mu,\qquad D_F=-H\mu,\qquad a_F=-\mu C+a_{\rm flat},\qquad da_{\rm flat}=0.} \tag{A05.1}$$

This derives the constant-source family in P v3, (6.20) and (6.26). It is sufficient for any subset of these supercharges; fewer retained labels can allow more general sources. The local equations must still be compatible with the bundles and boundary conditions.

For a real mass deformation take $\mu$ real. Euclidean supersymmetric backgrounds need not have every component real: in particular, $D_F$ will be imaginary on our sphere. This is a fixed source condition, not a choice of integration contour for a dynamical auxiliary.

Extract the mass and fix its sign

The matter derivative before freezing is

$$D_\mu\phi= \left[\nabla_\mu-ir\left(A_\mu-\frac12V_\mu\right) -izC_\mu-iq_Fa_{F\mu}\right]\phi.$$

Substituting (A05.1) combines two terms:

$$-izC_\mu-iq_Fa_{F\mu} =-iM C_\mu-iq_Fa_{{\rm flat}\mu}, \qquad M=z-q_F\mu. \tag{A05.2}$$

The same $M$ occurs in the chiral transformations. A constant flavor scalar therefore shifts the effective central-charge parameter; it does not change the R-charge. Do not count a mass once in $z$ and again in $\mu$ unless two distinct contributions are intended.

Set $\rho=\widetilde\phi\phi$ and $S=\widetilde\psi\psi$. In the charged action, the source-dependent algebraic terms before imposing $D_F=-H\mu$ are

$$\begin{aligned} \mathcal L_{\rm alg}={}& \left[ q_F(D_F+H\mu)+M^2-2(r-1)HM+\mathcal U_{\rm geom} \right]\rho\\ &+i\left[M-\left(r-\frac12\right)H\right]S-\widetilde F F,\\ \mathcal U_{\rm geom}={}& -\frac r4R+\frac12\left(r-\frac12\right)V^2 +r\left(r-\frac12\right)H^2. \end{aligned} \tag{A05.3}$$

This is the matter action’s algebraic part, not a potential obtained by conjugating every Euclidean field. The kinetic derivatives are the $\mathscr D_\mu=D_\mu+ir_0V_\mu$ of the prerequisite, with $r_{0\phi}=1/2$, $r_{0\psi}=-1/2$ and opposite partner values.

The gaugino Yukawa terms vanish because both source gauginos vanish. The explicit source-auxiliary coupling vanishes because $D_F+H\mu=0$. Relative to $\mu=0$, the remaining algebraic change is

$$\Delta\mathcal L_{\rm alg} =\left[q_F^2\mu^2-2q_Fz\mu+2q_F(r-1)H\mu\right]\rho -iq_F\mu S. \tag{A05.4}$$

The derivative also changes through (A05.2) when $C\ne0$. Thus the mass-squared term, the curvature-dependent linear term, and the fermion term all follow from the same frozen multiplet. P v3, (6.31)–(6.32)

In flat space take $A=V=H=C=0$ and a gauge-trivial $a_{\rm flat}=0$. Then

$$\mathcal L_{\rm flat} =\partial_\mu\widetilde\phi\,\partial^\mu\phi -i\widetilde\psi\gamma^\mu\partial_\mu\psi -\widetilde F F+M^2\rho+iMS.$$

To identify a familiar Dirac mass, define the independent row $\chi=-i\widetilde\psi^{\mathsf T}C_{\rm sp}$, where $C_{\rm sp}=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ contracts spinor indices. This matrix is unrelated to the one-form $C_\mu$. The fermion density becomes

$$\chi(\gamma^\mu\partial_\mu+m_{\rm phys})\psi, \qquad \boxed{m_{\rm phys}=-M=q_F\mu-z.} \tag{A05.5}$$

For real $\mu,z$, the scalar mass-squared is $m_{\rm phys}^2$. The free particle mass is $|m_{\rm phys}|$; its signed fermion coefficient remains meaningful. This agrees with P’s explicit real-mass example $z=-m$, $q=0$ following (7.11).

A direct propagator check is useful. In Euclidean momentum space, $\{\gamma^\mu,\gamma^\nu\}=2\delta^{\mu\nu}$ implies

$$(i\gamma\cdot p+m_{\rm phys})^{-1} =\frac{-i\gamma\cdot p+m_{\rm phys}}{p^2+m_{\rm phys}^2}.$$

The scalar and fermion have the same denominator. Squaring a mass without first fixing the fermion convention would have hidden the sign in (A05.5).

Worked example on the round sphere

Supply a round sphere of radius $\ell$ with

$$A=V=C=0,\qquad H=-\frac i\ell,\qquad R=-\frac6{\ell^2}, \qquad \nabla_\mu\zeta=\frac{i}{2\ell}\gamma_\mu\zeta,\quad \nabla_\mu\widetilde\zeta=\frac{i}{2\ell}\gamma_\mu\widetilde\zeta.$$

There are two global solutions in each family. These complexified solutions represent the continued four-supercharge algebra, not eight real Lorentzian charges. The smooth compact $U(1)$ flat connection on $S^3$ is gauge trivial. Choose $a_F=0$ and

$$\sigma_F=\mu,\qquad D_F=\frac{i\mu}{\ell}.$$

Setting $D_F=0$ instead would leave $\delta_\zeta\lambda_F=(\mu/\ell)\zeta$: a constant scalar alone is insufficient.

Inserting the actual invariant source into (A05.3) gives

$$\begin{aligned} \mathcal L_{S^3}={}& \nabla_\mu\widetilde\phi\,\nabla^\mu\phi -i\widetilde\psi\gamma^\mu\nabla_\mu\psi-\widetilde F F\\ &+\left[M^2+\frac{2i(r-1)M}{\ell} +\frac{r(2-r)}{\ell^2}\right]\rho +\left[iM-\frac{r-\frac12}{\ell}\right]S. \end{aligned} \tag{A05.6}$$

For example, the mass-independent scalar terms combine as $3r/(2\ell^2)-r(r-1/2)/\ell^2=r(2-r)/\ell^2$. This checks the negative-sphere-curvature convention. The large-radius limit at fixed $M$ recovers the flat action.

For a unit-charge chiral with $z=0$ and $r=1/2$, the result is particularly informative:

$$\mathcal L_{S^3} =\nabla\widetilde\phi\cdot\nabla\phi +\left(\mu^2+\frac{i\mu}{\ell}+\frac{3}{4\ell^2}\right)\rho +\chi(\gamma\cdot\nabla+\mu)\psi-\widetilde F F. \tag{A05.7}$$

At $\mu=0$ this is the conformally coupled free scalar and massless Dirac field. The extra $i\mu/\ell$ is required by the chosen supersymmetric source; it is not part of the flat pole mass. For real $\mu$ the Euclidean density is complex, although its scalar quadratic form has positive real part on $\widetilde\phi=\phi^*$. Curved Euclidean coefficients should not be read as Lorentzian particle masses.

Differentiate along the supersymmetric family

Changing a real mass changes more than one source. Define responses with the positive source-variation convention of the prerequisite:

$$\delta S_E=\int\sqrt g\, \left(j_F^\mu\delta a_{F\mu} +\mathcal O_\sigma\delta\sigma_F +\mathcal O_D\delta D_F\right).$$

With the geometry and $a_{\rm flat}$ fixed, (A05.1) gives

$$\frac{dS_E}{d\mu} =\int\sqrt g\, \left(\mathcal O_\sigma-H\mathcal O_D-C_\mu j_F^\mu\right). \tag{A05.8}$$

For the sphere, $C=0$. Differentiating (A05.3) before freezing gives

$$\mathcal O_D=q_F\rho,\qquad \mathcal O_\sigma= q_F\left[(2r-1)H-2M\right]\rho-iq_FS.$$

Subtracting $H\mathcal O_D$ therefore yields the insertion

$$\frac{d\mathcal L_{S^3}}{d\mu} =2q_F\left[(r-1)H-M\right]\rho-iq_FS. \tag{A05.9}$$

Direct differentiation of (A05.6) agrees. Holding $D_F$ fixed would miss $-q_FH\rho$ and follow a different, generally nonsupersymmetric path in source space.

For a regulated generating functional $\mathcal W=-\log Z$, this implies

$$\mathcal W'=\langle S_E'\rangle,\qquad \mathcal W''=\langle S_E''\rangle-\langle S_E'S_E'\rangle_c, \qquad S_E''=2q_F^2\int\sqrt g\,\rho$$

for this canonical matter action along the sphere family. The last term is an explicit second source derivative; it must not be discarded.

The current multiplet and background vector coupling are discussed in CT v2, (4.3)–(4.5). CT writes the linear source density as $-j_{\rm CT}a_F-K_{\rm CT}\sigma_F-J_{\rm CT}D_F$. With the same source variables, our response definitions have the opposite operator signs. Quantum comparisons also require a fixed background-counterterm convention. We keep any regulator-fixed fractional contact data and set an additional integer flavor Chern–Simons counterterm to zero unless explicitly changing it; this does not set every contact coefficient to zero.

Separate a real mass from other deformations

A complex superpotential mass is a different coupling. For two independent chirals, $W=m_c\Phi_1\Phi_2$ contributes F terms and a fermion pairing. A flavor scalar instead couples through the kinetic D-term. A fixed $m_c$ preserves the proposed flavor symmetry only if $q_1+q_2=0$. Assigning a compensating charge to $m_c$ as a formal source does not make that flavor symmetry unbroken at a fixed nonzero value.

A flat connection is different again. Locally it can be removed from the derivative, but around a circle a charged field measures $\exp(iq_F\oint a_F)$. With compact $U(1)_F$ and a unit-charge probe, $\oint a_F$ is defined modulo $2\pi$. Its value is not determined by $\mu$. Global spin structure also matters: the local gaugino equations alone cannot guarantee global supersymmetry. The next exercises keep these distinctions explicit.

Exercises

1. Add a complex mass without losing the flavor symmetry

In flat space take two canonically normalized chirals of charges $+1,-1$, with $z_1=z_2=0$, a constant real flavor scalar $\mu$, and $W=m_c\Phi_1\Phi_2$. Show that the fixed complex mass is compatible with this flavor background. Find the scalar masses and the two signed eigenvalues of the fermion mass matrix after charge conjugating the negative-charge fermion. Would the same fixed superpotential preserve the symmetry assigning charge $+1$ to both chirals?

Hint 1

First check the superpotential’s total charge. Eliminating the two matter auxiliaries adds $|m_c|^2(|\phi_1|^2+|\phi_2|^2)$.

Hint 2

In a same-charge fermion basis, a constant phase choice gives the Hermitian mass matrix $\begin{pmatrix}\mu&\overline m_c\\m_c&-\mu\end{pmatrix}$. Square it before finding its eigenvalues.

Solution

The charges in $\Phi_1\Phi_2$ sum to zero, so a fixed $m_c$ preserves the flavor symmetry. The real-mass scalar contribution is $\mu^2(|\phi_1|^2+|\phi_2|^2)$. Adding the eliminated F terms gives two equal scalar mass-squared values $\mu^2+|m_c|^2$.

The displayed Hermitian fermion matrix squares to $(\mu^2+|m_c|^2)I$ and has trace zero. Its signed eigenvalues are therefore $\pm\sqrt{\mu^2+|m_c|^2}$; the two physical masses agree with the scalar masses. The relative off-diagonal phase is a basis choice, whereas the gap is invariant. This is a free quadratic calculation, not a statement about an interacting pole mass.

If both chiral charges were $+1$, the superpotential would have charge two. A nonzero fixed neutral coefficient would break that proposed flavor symmetry. A spurion charge for $m_c$ describes how to compare different sources; it does not authorize an independent constant real mass for an unbroken symmetry that is absent.

2. Test a flat connection globally

Put the free chiral on $\mathbb R^2\times S^1$, with circle coordinate $y\sim y+L$, periodic spin structure and zero rigid sources. Let $a_{Fy}=\alpha/L$, $\sigma_F=\mu$, and $D_F=0$. Derive the scalar mode eigenvalues and check invariance under $\alpha\mapsto\alpha+2\pi$. Can this source preserve four supercharges? What changes if the fermions and supersymmetry parameters are antiperiodic, with no R-symmetry twist?

Hint 1

Apply $\partial_y-iq_F\alpha/L$ to $\exp(2\pi iny/L)$. The source scalar still contributes $(q_F\mu-z)^2$.

Hint 2

A large flavor gauge transformation relabels an integer mode. Supersymmetry parameters carry R-charge but no flavor charge.

Solution

For transverse momentum $k_\perp$, the scalar eigenvalues are

$$\Lambda_n=k_\perp^2+(q_F\mu-z)^2+ \frac{(2\pi n-q_F\alpha)^2}{L^2},\qquad n\in\mathbb Z.$$

Changing $\alpha$ by $2\pi$ gives $\Lambda_n(\alpha+2\pi)=\Lambda_{n-q_F}(\alpha)$, so the full spectrum is unchanged. A single labeled mode need not be unchanged. The unit-charge probe fixes the global period even if a particular field has $|q_F|>1$.

The flat gaugino variations vanish, and constant flavor-neutral spinors exist with the periodic spin structure. Thus all four supercharges survive, for every flavor holonomy. With antiperiodic spin structure and no compensating R background, the only covariantly constant globally defined spinor is zero: a constant component cannot also change sign around the circle. The same local source equations then preserve no global supercharge.

3. Follow the auxiliary and the contact term

For the unit-charge sphere example (A05.7), calculate $d\mathcal L/d\mu$ and compare it with varying $\sigma_F$ at fixed $D_F$. Then change the additional background counterterm by integer $k$:

$$\Delta S_{\rm ct}=\frac{k}{4\pi}\int \left(i a_F\wedge da_F-2\sigma_FD_F\,d{\rm vol}\right).$$

On the sphere family, find its contribution to $\mathcal W''$. Does it change the matter propagating masses?

Hint 1

Use $D_F=i\mu/\ell$ and $\mathcal O_D=\rho$. Keep the source auxiliary dependent on $\mu$ in the total derivative.

Hint 2

The sphere volume is $2\pi^2\ell^3$. A background-only term multiplies the partition function by its exponential and adds directly to $\mathcal W=-\log Z$.

Solution

Direct differentiation gives

$$\frac{d\mathcal L}{d\mu} =\left(2\mu+\frac i\ell\right)\rho-iS.$$

At fixed $D_F$, (A05.3) instead gives $\mathcal O_\sigma=2\mu\rho-iS$. The missing contribution is $(dD_F/d\mu)\mathcal O_D=i\rho/\ell$. It is required for the derivative along the supersymmetric family.

The additional term has density $-ik\mu^2/(2\pi\ell)$, hence

$$\Delta\mathcal W=-i\pi k\ell^2\mu^2,\qquad \Delta\mathcal W''=-2i\pi k\ell^2.$$

It changes the local source response and partition-function phase. It contains no dynamical matter field, so it does not change the quadratic matter operators. The integer ambiguity is consistent with compact flavor gauge invariance on spin manifolds; it cannot remove arbitrary fractional contact data. CT v2, (1.3)–(1.7), (4.5)

A real mass is now a specified family of supersymmetric sources, with a fixed charge normalization and global data. That specification is what makes comparisons of massive theories and their source-dependent observables meaningful.

Paper guides

Connect this lesson to the original papers and their notation.

• Paper guide: Section 6, curved multiplets and actions