N=4, N=2-star, and the adjoint hypermultiplet mass

Before this lesson: Add fundamental matter: masses, flavors, and decoupling.

The fundamental-matter example left an important alternative: put the hypermultiplet in the adjoint representation, so it has the same gauge transformation law as the gauge field. With the required interactions and zero hypermultiplet mass, the symmetry grows from N=2 to N=4. Giving this particular multiplet a mass removes the additional supersymmetry while preserving N=2. The resulting theory is called N=2-star, written $\mathcal N=2^*$ .

We work in flat four-dimensional spacetime with global gauge group $SU(2)$ , of rank one. The massless theory has sixteen real Poincaré supercharges; the massive theory has eight. Four are manifest in the N=1 notation below. Neither the gauge rank nor the number of manifest charges changes when the mass is switched on. No preparation from the N=4 volume is needed.

Find the mass before naming the deformation

An N=2 vector multiplet contains a gauge field, a complex adjoint scalar $\phi$ , and two Weyl fermions. Its N=1 description is a vector superfield together with an adjoint chiral $\Phi$ . A full adjoint hypermultiplet adds two adjoint chirals $Q,\widetilde Q$ : two complex scalars and two Weyl fermions. Thus the combined massless fields are one vector, six real scalars and four Weyl fermions, all adjoint. Per Lie-algebra generator, they have eight propagating bosonic and eight fermionic states. This counts fields before forming gauge-invariant multiparticle states.

Take Hermitian generators $T^a=\sigma^a/2$ with ordinary fundamental trace $\operatorname{Tr}(T^aT^b)=\delta^{ab}/2$ . Normalize the hypermultiplet components by the Kähler potential $2\operatorname{Tr}(Q^\dagger Q+\widetilde Q^\dagger\widetilde Q)$ , with the usual gauge-covariant completion. Retain $\Phi_{\rm SW}=g\Phi_{\rm can}$ from the preceding lesson. The superpotential is

W=2\operatorname{Tr}\!\left[ \widetilde Q\left(\sqrt2[\Phi_{\rm SW},Q]+mQ\right) \right]. \tag{SW15.1}

Here $m$ is a complex coefficient of mass dimension one. The kinetic normalization makes its magnitude the hypermultiplet mass at $\phi=0$ . The cubic coefficient is fixed by the N=2 gauge interaction; it is not an independently adjustable coupling. At $m=0$ , the complete gauge, kinetic and interaction package has N=4 symmetry. A field count or the cubic polynomial alone would not establish that symmetry. Tachikawa (2.1.11)–(2.1.18) constructs this organization and explains the enhancement.

Now set $Q=\widetilde Q=0$ and $\phi_{\rm SW}=a_{\rm cl}T^3$ . The commutators $[T^3,T^1\pm iT^2]=\pm(T^1\pm iT^2)$ separate the adjoint into two charged directions and one neutral direction. Writing the canonical component vectors as $q^a,\widetilde q^a$ , the quadratic superpotential is $\widetilde q^T\mathcal M q$ , with

\mathcal M= \begin{pmatrix} m&-i\sqrt2\,a_{\rm cl}&0\\ i\sqrt2\,a_{\rm cl}&m&0\\ 0&0&m \end{pmatrix}. \tag{SW15.2}

A constant unitary change to the charged basis diagonalizes this matrix. Eliminating the hypermultiplet auxiliaries gives a scalar mass-squared matrix $\mathcal M^\dagger\mathcal M$ for $q$ and the corresponding transpose for $\widetilde q$ . The fermionic Dirac masses have the same singular values:

M_+=|m+\sqrt2\,a_{\rm cl}|,\qquad M_-=|m-\sqrt2\,a_{\rm cl}|,\qquad M_0=|m|. \tag{SW15.3}

For example, $\det\mathcal M=m(m^2-2a_{\rm cl}^2)$ . A charged hyper becomes massless at $a_{\rm cl}=m/\sqrt2$ , or $u_{\rm cl}=\operatorname{Tr}\phi_{\rm SW}^2=m^2/4$ . This differs from the fundamental threshold $u_{\rm cl}=m^2$ because the adjoint root has twice the fundamental weight.

The mass term preserves the entire N=2 algebra: it can be introduced as a constant scalar of a background vector multiplet for the flavor rotation $Q\mapsto e^{i\alpha}Q$ , $\widetilde Q\mapsto e^{-i\alpha}\widetilde Q$ . The Coulomb configuration has zero auxiliary fields and preserves all eight charges. The reduction from sixteen to eight is explicit breaking of the action’s extra supersymmetry, not spontaneous breaking in this vacuum. A term proportional to $\operatorname{Tr}\Phi_{\rm SW}^2$ would instead mass the chiral inside the vector multiplet and generally leave only N=1.

Separate physical masses from parameters of a regulator

The mass $m$ in (SW15.1) and the curve below is a flat-space complex mass; $|m|$ is a positive mass at the origin. It is not automatically the symbol called “mass” in an instanton formula.

For a concrete translation, rotations in two Euclidean planes can be assigned parameters $\epsilon_1,\epsilon_2$ of dimension one. In the particular convention of Pestun’s corrected (5.11), the equivariant mass is

m_{\rm eq}=m_{\rm P}+\frac{\epsilon_1+\epsilon_2}{2}. \tag{SW15.4}

The shift incorporates the R-symmetry action used with those rotations. In that paper’s four-sphere integration convention, $m_{\rm P}=i m_E$ with real $m_E$ , as specified before (4.47); this is its analytic continuation and mass contour, not a universal equation between a Lorentzian positive mass and a Euclidean parameter. Continuing its complex mass variable to the flat-space convention identifies $m_{\rm P}$ with our $m$ , while both rotation parameters go to zero.

At finite deformation, physical zero mass means $m_{\rm eq}=(\epsilon_1+\epsilon_2)/2$ in this convention, not $m_{\rm eq}=0$ . On its round sphere, $\epsilon_1=\epsilon_2=1/R$ , so the former value is $1/R$ . The special choice $\epsilon_1+\epsilon_2=0$ hides the distinction. None of these finite-background statements changes the eight-charge count of our massive flat-space theory, or asserts that a generic rotation background preserves those eight charges. The actual instanton calculation keeps its equivariant conventions separately.

Fix the curve and the period normalization together

For this adjoint theory, Seiberg and Witten return to their original normalization:

\tau=\frac{\theta}{2\pi}+\frac{4\pi i}{g^2},\qquad q=e^{2\pi i\tau},\qquad a\sim\sqrt{2u},\qquad Z=n_m a_D+n_e a+\frac{S m}{\sqrt2}. \tag{SW15.5}

The bound is $M\geq\sqrt2|Z|$ , a W boson has $n_e=1$ , and $S$ is the flavor charge. Thus the weak-coupling root states with $n_e=\pm1,S=1$ reproduce (SW15.3). The functions $a,a_D$ belong to the massive adjoint theory. The preceding lesson’s fundamental normalization $A=a/2$ , $B=a_D$ must not be substituted into this adjoint curve unchanged. Seiberg–Witten (16.1)–(16.3) states the two packages separately. Our global gauge group remains $SU(2)$ ; modular properties of a curve alone do not identify its line operators with those of an $SO(3)$ theory.

We will use the source’s coordinate prescription for $u$ , which agrees with $\operatorname{Tr}\phi_{\rm SW}^2$ at the weak cusp. In a massive quantum theory, specifying the additive mixing of this composite with $m^2$ matters. The convenient modular coordinate is a different variable, $\widetilde u=u-e_1m^2/8$ .

Here are explicit definitions for the dimensionless coefficients, requiring no prior theta-function theory. Put $t=e^{\pi i\tau}$ , with $|t|<1$ , and define the convergent series

\vartheta_2(t)=\sum_{n\in\mathbb Z}t^{(n+1/2)^2},\quad \vartheta_3(t)=\sum_{n\in\mathbb Z}t^{n^2},\quad \vartheta_4(t)=\sum_{n\in\mathbb Z}(-1)^n t^{n^2}. \tag{SW15.6}

The powers use the branch $\log t=\pi i\tau$ . Use $e_1=(\vartheta_3^4+\vartheta_4^4)/3$ , $e_2=-(\vartheta_2^4+\vartheta_3^4)/3$ , $e_3=(\vartheta_2^4-\vartheta_4^4)/3$ . They sum to zero; $\vartheta_3^4=\vartheta_2^4+\vartheta_4^4$ . These are modern theta labels: the source’s $\theta_1,\theta_2,\theta_3$ correspond to our $\vartheta_2,\vartheta_4,\vartheta_3$ .

With $f=m^2/4$ , the exact curve is $y^2=\prod_i(X-e_i\widetilde u-e_i^2f)$ . To take a weak-coupling limit without a diverging coordinate shift, substitute $X=x-e_1u/2+e_1^2f/2$ . Define $c_1=3e_1/2$ and $c_2=(e_3-e_2)/2=\vartheta_2^4/2$ . The result is

\begin{aligned} y^2&=(x-r_1)(x-r_2)(x-r_3),\\ r_1&=c_1u,\\ r_2&=-c_2u+c_2(c_1+c_2)f,\\ r_3&=c_2u-c_2(c_1-c_2)f. \end{aligned} \tag{SW15.7}

The curve, including $f=m^2/4$ , is the nonperturbative input from Seiberg–Witten (16.17)–(16.25). We are checking its limits, not deriving the full solution from a classical mass matrix.

There are two useful checks on its mass normalization. The root collisions occur at $u=c_1f$ and $u=\pm c_2f$ . Since $c_1\to1$ at weak coupling, the first collision agrees with the charged massless point $u\simeq m^2/4$ . Independently, the source’s residue construction gives

2\pi i\,\operatorname{Res}\lambda =-i\sqrt2\sqrt{\frac{\Theta}{(e_1-e_2)(e_1-e_3)}}, \qquad \Theta=-(e_1-e_2)(e_1-e_3)f. \tag{SW15.8}

Choosing the pole with square root $+im/2$ gives $m/\sqrt2$ , exactly the unit-flavor contribution in (SW15.5); the opposite pole reverses the sign. This is the short normalization check extracted from the source’s (17.63), (17.67)–(17.68), whose full residue construction is additional geometry.

The derivative differential has coefficient $\sqrt2/(4\pi)$ in the source, twice its fundamental-matter coefficient. Choose its sheet and electric cycle so that $da/du>0$ on the real weak branch. In the earlier pure-curve sheet convention this is $\partial_u\lambda=-\sqrt2\,dx/(4\pi y_{\rm site})$ : $y_{\rm source}=-y_{\rm site}$ reconciles the signs. A sheet reversal changes neither the polynomial nor the physical periods when the convention is translated consistently.

Two limits with different low-energy content

First take $m\to0$ at fixed $\tau$ and nonzero $u$ . The curve becomes $y^2=\prod_i(X-e_i u)$ . The rescaling $X=u\xi$ , $y=u^{3/2}\eta$ makes its shape independent of $u$ . The normalized derivative periods scale as $u^{-1/2}$ ; their ratio remains $\tau$ . With the weak-branch normalization and vanishing periods at $u=0$ ,

a=\sqrt{2u},\qquad a_D=\tau a,\qquad \mathcal F_{\rm vector}(a)=\frac12\tau a^2 \quad\text{up to linear and constant terms}. \tag{SW15.9}

This is the N=4 vector-sector prepotential, as in source (16.10). At generic $a$ , the neutral hypermultiplet is also massless when $m=0$ . Thus the low-energy theory is not a pure N=2 photon theory with all hypermultiplets removed. A Coulomb expectation value breaks conformal symmetry spontaneously; it does not make the parent coupling run.

Now take the opposite limit. In the trace normalization above, the one-loop N=2 coefficient is $2T(\mathrm{adj})-2T(R)$ , with $T(\mathrm{adj})=2$ for $SU(2)$ . It is zero with one full adjoint hyper and four after that hyper is removed. Holomorphic threshold matching therefore has the form $\Lambda^4\propto m^4q$ . The coefficient depends on the specified scale convention, and the curve fixes it. Tachikawa (3.1.3)–(3.1.10)

Expand the defining series: $c_1=1+24q+O(q^2)$ and $c_2=8\sqrt q+32q^{3/2}+O(q^{5/2})$ . Take

m\longrightarrow\infty,\qquad q\longrightarrow0,\qquad \Lambda^4=4m^4q\ \text{fixed}. \tag{SW15.10}

Choose the phase branch $\Lambda^2=2m^2\sqrt q$ and keep $u$ fixed. Then $c_1\to1$ , $c_2\to0$ but $c_2f\to\Lambda^2$ . Equation (SW15.7) therefore gives

y^2\longrightarrow(x-u)(x-\Lambda^2)(x+\Lambda^2). \tag{SW15.11}

Both the polynomial and the normalized derivative differential return directly to the original pure-theory package. This is the adjoint decoupling limit of source (16.22), not the fundamental-matter isogeny used in the preceding lesson. In a unit-threshold convention $\Lambda_{\rm hol}^4=m^4q$ , the curve’s scale is $\Lambda^4=4\Lambda_{\rm hol}^4$ ; dropping this finite factor moves the singularities.

The physical low-energy regime requires external energies and the Coulomb scale small compared with $|m|$ , and avoids a charged-hyper threshold. Fixing $m$ and merely sending $q\to0$ gives a free weak-coupling limit, whereas fixing $q$ and sending $m\to\infty$ sends the pure scale to infinity. Neither is (SW15.10).

Worked check: a period approaches the pure answer

Use units $\Lambda=1$ , real $u=3$ , positive $m$ and $t=1/(2m^2)$ . The roots in (SW15.7) obey $r_3<r_2<r_1$ . Substituting $x=r_3+(r_2-r_3)\sin^2\theta$ removes the endpoint square roots in the electric integral and gives

\frac{da}{du} =\frac{\sqrt2}{\pi}\int_0^{\pi/2} \frac{d\theta}{\sqrt{r_1-r_3-(r_2-r_3)\sin^2\theta}}. \tag{SW15.12}

At infinite mass this is $0.4173134208370366$ . Evaluating the full theta-series coefficients, rather than their leading terms, gives:

$m$	$da/du$	$m^2[(da/du)-(da/du)_{\rm pure}]$
8	0.4184446407985145	0.0723980775
16	0.4175946615550715	0.0719976238
32	0.4173836352688717	0.0718995782
64	0.4173309684919868	0.0718751947

The approach is proportional to $m^{-2}$ . To predict its coefficient, expand the roots with $s=\Lambda^2$ : $r_1=u+O(m^{-4})$ , $r_2=s+(4s^2-4su)/m^2+O(m^{-4})$ and $r_3=-s+(4s^2+4su)/m^2+O(m^{-4})$ . At $s=1,u=3$ , differentiating the integrand in (SW15.12) gives

C=\frac{\sqrt2}{\pi}\int_0^{\pi/2} \frac{2(4-6\sin^2\theta)} {(4-2\sin^2\theta)^{3/2}}\,d\theta =0.0718670781722600. \tag{SW15.13}

The last column approaches $C$ . Independent complex-contour quadrature and the complete elliptic-integral expression agree with the real integral at 70-digit working precision, with scaled discrepancies below $10^{-60}$ . This checks period normalization and the controlled limit; it does not construct the whole massive BPS spectrum. The expansion is not uniform when branch points collide, so one cannot use this error estimate at a singular vacuum.

Exercises

1. A neutral field is not absent

For $a_{\rm cl}=1+i$ and $m=2$ , find all three hypermultiplet masses. Determine what happens to the neutral hyper when $m$ is subsequently set to zero, and compare this with adding a mass to $\Phi_{\rm SW}$ instead.

Hint 1

The physical masses are singular values of the quadratic superpotential matrix, not its complex eigenvalues themselves.

Hint 2

The zero adjoint weight has no Coulomb contribution. Changing the mass term changes which N=2 multiplet is removed.

Solution

The charged coefficients are $2\pm\sqrt2(1+i)$ . Hence $M_\pm^2=8\pm4\sqrt2$ , while $M_0^2=4$ . All are nonnegative, and the two charged values are interchanged by reversing the Coulomb direction.

At $m=0$ , the charged masses are both $\sqrt2|1+i|=2$ , but the neutral hyper has zero mass and remains in the low-energy Abelian N=4 multiplet. It cannot be integrated out by the same heavy-mass expansion. The action restores sixteen real Poincaré charges, all preserved by this commuting Coulomb vacuum.

Adding $\mu\operatorname{Tr}\Phi_{\rm SW}^2$ instead gives a mass to the chiral inside the N=2 vector. It generically preserves only the four charges of N=1. An “adjoint mass” is therefore insufficient information: both the vector chiral and the hypermultiplet are adjoint, but their mass deformations have different supersymmetry.

2. Keep the first correction to decoupling

Using $h=m^{-2}$ and $s=\Lambda^2$ , derive the first correction to the cubic in (SW15.11). Also locate the two finite singularities and the heavy one to their first nonzero corrections. Explain why a dimensionally correct threshold rule without its finite coefficient would fail.

Hint 1

Insert $t=sh/2$ into $c_1=1+24t^2+\cdots$ and $c_2=8t+32t^3+\cdots$ . For singularities, use $u=c_1f,\pm c_2f$ rather than solving a cubic anew.

Hint 2

To order $h$ , the sum of the two finite $x$ roots is $8s^2h$ and their product is $-s^2+8s^2uh$ .

Solution

The root expansions in the worked example give

y^2=(x-u)(x^2-s^2)-8s^2h(x-u)^2+O(h^2),

at fixed $x,u,s$ . Every term has dimension six: $[x]=[u]=[s]=2$ and $[h]=-2$ . The singularities require one further term:

u_\pm=\pm\left(s+s^3h^2+O(h^4)\right),\qquad u_{\rm heavy}=\frac1{4h}+\frac32s^2h+O(h^3).

Thus the two pure singularities stay finite, while the elementary charged-hyper threshold runs to infinity. The polynomial has an order- $h$ correction even though the finite singularity locations first shift at order $h^2$ ; locating singularities alone does not check the whole effective coupling.

If one called $m^4q$ the same $\Lambda^4$ as in the pure benchmark, then $c_2f\to2\Lambda^2$ and the finite singularities would be $\pm2\Lambda^2$ . Dimensions would pass, but the normalized curve and period comparison would fail. One must translate the definition of the scale.

3. Diagnose a shifted zero mass

In the convention (SW15.4), write the symmetric combination $m_{\rm eq}(\epsilon_1+\epsilon_2-m_{\rm eq})$ in terms of $m_{\rm P}$ . A calculation on a sphere of radius $R$ sets $m_{\rm eq}=0$ and calls this the undeformed N=4 mass. Diagnose the error, including what is meant by the Euclidean mass convention.

Hint 1

Write $E=\epsilon_1+\epsilon_2$ and complete the square around $m_{\rm eq}=E/2$ .

Hint 2

The flat-space limit removes both $\epsilon_i$ . It is not the same operation as setting one parameter named “mass” to zero at fixed $R$ .

Solution

Substitution gives

m_{\rm eq}(E-m_{\rm eq})=\frac{E^2}{4}-m_{\rm P}^2.

The physical massless value $m_{\rm P}=0$ is therefore $m_{\rm eq}=E/2$ . For the cited round sphere it is $1/R$ . Instead, setting $m_{\rm eq}=0$ gives $m_{\rm P}=-1/R$ , which is a different complex mass value. On the source’s contour $m_{\rm P}=i m_E$ , this would require imaginary $m_E$ rather than a real Euclidean integration parameter.

When both $\epsilon_i\to0$ at fixed complex $m_{\rm P}$ , the shift disappears and the continued flat-space mass is $m=m_{\rm P}$ . The fact that the shift also vanishes on $\epsilon_1=-\epsilon_2$ does not make it absent in other backgrounds. This parameter check does not infer a generic finite-background supersymmetry enhancement from the polynomial alone.

The two limits now have separate meanings: restore the adjoint hypermultiplet to recover N=4, or remove it while holding a matched low-energy scale fixed to recover pure N=2. The instanton-counting calculation asks how to reproduce a prepotential coefficient microscopically. For the broader massless theory, the N=4 field-content lesson develops its other descriptions; the calculations here do not depend on that additional route.

The adjoint section of the original-paper guide follows these conventions into Section 16. Its exit calculations change the heavy-mass phase and the modulus held fixed, testing which low-energy comparison survives.