Follow the fields around a singularity

Before this lesson: What becomes massless at strong coupling?.

After walking around a closed path, a coordinate need not return to its original value. The familiar example is a continuously followed polar angle: a full turn changes it by $2\pi$ although the point returns to its starting position. A logarithm remembers that angle. In Seiberg–Witten theory, this memory mixes the electric and magnetic variables.

Read the light-monopole effective theory first. Its logarithm will let us calculate the mixing, rather than memorize three matrices. We continue to study pure four-dimensional N=2 $SU(2)$ Yang–Mills, with eight real supercharges preserved by Coulomb-branch vacua and a generic rank-one Abelian low-energy theory. There is no supersymmetry-breaking deformation.

Keep the first-paper convention

\Pi=\begin{pmatrix}a_D\\a\end{pmatrix},\qquad \gamma=(n_m,n_e),\qquad Z_\gamma=\gamma\Pi,\qquad M_{\rm BPS}=\sqrt2\,|Z_\gamma|. \tag{SW08.1}

The last equality presupposes an existing BPS particle. The periods have mass dimension one; the vacuum coordinate $u$ has dimension two. Monodromy concerns continuation in the complex space of vacua, not motion around a magnetic source in physical space.

Follow the logarithm before the fields

Start at $b=\rho>0$ and traverse $b(t)=\rho e^{2\pi i t}$ , $0\leq t\leq1$ . With a positive reference mass $\mu_0$ , the logarithm followed along this path is

L(t)=\log(\rho/\mu_0)+2\pi i t,\qquad L(1)-L(0)=\int_0^1\frac{\dot b(t)}{b(t)}\,dt=2\pi i. \tag{SW08.2}

Evaluating the principal logarithm afresh at every point would insert a jump at the negative real axis and erase this information at the endpoint. A branch cut is a bookkeeping choice for a single chart. It is not a physical wall forbidding continuation.

Near the monopole point, use $b=a_D$ and the dual electric description. The preceding lesson gave

\tau_D=h(b)-\frac{i}{\pi}\log(b/\mu_0),\qquad a=a_0-\int_0^b h(w)\,dw+ \frac{i}{\pi}\big[b\log(b/\mu_0)-b\big]. \tag{SW08.3}

Here $h$ is holomorphic and single-valued in a disk containing $b=0$ ; $a_0\ne0$ . Differentiation checks the sign: $\tau_D=-da/db$ . The logarithmic coefficient comes from one full unit-charge hypermultiplet, not from a choice of branch.

Assume $b=c_0(u-u_+)+O((u-u_+)^2)$ with $c_0\ne0$ . A sufficiently small counterclockwise loop around $u_+$ then winds once around $b=0$ . All analytic terms return to themselves. The logarithm alone gives

a_D\longmapsto a_D,\qquad a\longmapsto a-2a_D, \qquad M_+=\begin{pmatrix}1&0\\-2&1\end{pmatrix}, \qquad \Pi\longmapsto M_+\Pi. \tag{SW08.4}

The dual coupling changes by $\tau_D\mapsto\tau_D+2$ ; its imaginary part returns unchanged. Reversing the loop gives $M_+^{-1}$ , and winding $w$ times gives $a\mapsto a-2w a_D$ . These are consequences of actual analytic continuation of the light-field threshold. The corresponding primary calculation is Seiberg–Witten, (5.8)–(5.12).

There is a useful check that does not require a circular path. Along any closed path avoiding zero, $da=[-h(b)+(i/\pi)L]\,db$ . Integration by parts gives $\oint L\,db=[bL]_{\rm start}^{\rm end}-\oint db=2\pi i w b_{\rm start}$ . The integral of $h\,db$ vanishes within the disk. Thus $\Delta a=-2w b_{\rm start}$ even when the loop is distorted. This statement needs the continuous $L$ , including its endpoint jump.

Fix the paths before multiplying matrices

A local calculation determines a matrix in the period frame carried to that neighborhood. Comparing different singularities also requires paths from a common base point.

For the following reconstruction, assume the two finite singularities are $u_\pm=\pm u_0$ , with $u_0>0$ , and there are no other finite singularities. This is the minimal singularity pattern used in the original solution; the local logarithm alone does not prove it or fix $u_0$ . Choose a real base point $P>u_0$ far in the weak-coupling region, with the branch $a(P)\sim+\sqrt{2P}$ .

Use the following rooted paths:

$\ell_+$ goes along the real axis toward $u_+$ , circles it once counterclockwise, and retraces its approach.
$\ell_-$ approaches $u_-$ by passing below $u_+$ , circles $u_-$ once counterclockwise, and retraces that approach.
$\ell_\infty$ is a large counterclockwise circle enclosing both finite points, based at $P$ .

The small circles and approach paths avoid both punctures. Shrinking the large loop without crossing a puncture gives $\ell_+$ followed by $\ell_-$ . The lower approach to $u_-$ matters: taking the upper approach instead changes the rooted loop by conjugation.

We specify multiplication using continued functions. Write $\mathcal C_\ell\Pi=M_\ell\Pi$ for the pair of local functions near $P$ after a loop. Following $\ell_1$ and then $\ell_2$ means

\mathcal C_{\ell_2}(\mathcal C_{\ell_1}\Pi) =\mathcal C_{\ell_2}(M_1\Pi) =M_1M_2\Pi. \tag{SW08.5}

The first matrix consists of constant coefficients, so continuation acts on the functions to its right. In this convention the chronological word “first $+$ , then $-$ ” therefore gives $M_+M_-$ . This is not the rule for applying two fixed linear maps successively to an ordinary numerical column. Keeping those two operations distinct prevents an order error.

Accordingly, our rooted paths obey $M_\infty=M_+M_-$ . The numerical relation agrees with the original paper’s (5.13) and its based-loop discussion. “Monodromy at infinity” here means this large counterclockwise loop in the $u$ plane; in the local coordinate $1/u$ it is clockwise.

The weak-coupling loop fixes the other matrix

At large $u$ , the asymptotic periods are

a\sim\sqrt{2u},\qquad a_D\sim\frac{2ia}{\pi}\log(a/\Lambda)+\frac{ia}{\pi}. \tag{SW08.6}

On a counterclockwise $u$ loop, the continuously followed square root gives $a\mapsto-a$ and $\log a\mapsto\log a+\pi i$ . Substituting both changes, including the minus sign multiplying the logarithm, gives

a_D\longmapsto-a_D+2a,\qquad M_\infty=\begin{pmatrix}-1&2\\0&-1\end{pmatrix}. \tag{SW08.7}

The overall minus sign already reflects the classical Weyl identification. The shear by $2a$ reflects the quantum logarithm. The asymptotically smaller instanton terms are odd in $a$ and single-valued in $(\Lambda/a)^4$ , so they do not change this matrix. Equivalently, an integral monodromy cannot acquire an arbitrarily small continuous correction as the loop radius is increased without crossing a singularity. See Seiberg–Witten, (5.1)–(5.4).

Under the stated two-finite-singularity assumption, ordinary multiplication now determines

M_-=M_+^{-1}M_\infty =\begin{pmatrix}-1&2\\-2&3\end{pmatrix}, \qquad M_+M_-=\begin{pmatrix}-1&2\\0&-1\end{pmatrix}. \tag{SW08.8}

Both finite matrices have determinant one and trace two; their difference from the identity squares to zero. They do not commute: $M_-M_+=\begin{pmatrix}-5&2\\-8&3\end{pmatrix}$ . Changing the path convention requires changing the corresponding matrices, not merely swapping factors in (SW08.8).

The primitive invariant charge rows are $(1,0)M_+=(1,0)$ and $(1,-1)M_-=(1,-1)$ . Thus the respective vanishing central-charge coordinates are $a_D$ and $a_D-a$ . “Primitive” means the two integer entries have no common divisor. The monodromy test identifies the charge direction of the hypothesized light hypermultiplet. It does not prove from an eigenvector alone that a particle exists.

What changes, and what stays physical

Two related uses of a monodromy matrix must be separated.

At a fixed vacuum, comparing period frames $\Pi'=M\Pi$ requires the passive change $\gamma'=\gamma M^{-1}$ . It preserves the same central charge:

\gamma'\Pi'=\gamma\Pi. \tag{SW08.9}

Alternatively, continue the function $Z_\gamma(u)=\gamma\Pi(u)$ with its numerical row $\gamma$ held fixed. At the endpoint it is $\gamma M\Pi$ in the initial branch, so its expression in that branch has row $\gamma M$ . This is the sense of Bilal’s (5.14). It is not the passive inverse rule. Nor does formal continuation guarantee that an associated particle stays stable all along the path.

For a real matrix $M=\begin{pmatrix}A&B\\C&D\end{pmatrix}$ of determinant one, differentiating the periods gives

\tau'=\frac{A\tau+B}{C\tau+D},\qquad da'=(C\tau+D)\,da,\qquad \operatorname{Im}\tau'=\frac{\operatorname{Im}\tau}{|C\tau+D|^2}. \tag{SW08.10}

The last identity follows by subtracting the conjugate fraction and using $AD-BC=1$ . It proves $\operatorname{Im}\tau'|da'|^2=\operatorname{Im}\tau|da|^2$ : the positive scalar kinetic metric is unchanged. The on-shell electric and magnetic field strengths change with the same duality frame, as derived in Abelian dualization. A globally preferred electric potential is not required.

For a concrete numerical check, take illustrative local data $\Pi=m_0(i,2)^T$ , $\tau=i$ , with $m_0>0$ . These are test data, not a claimed evaluation of the exact periods at a specified $u$ . Under $M_+$ , $\Pi'=m_0(i,2-2i)^T$ , $\tau'=(-2+i)/5$ , and $da'=(1-2i)da$ . The metric coefficient changes from $1$ to $1/5$ , while the squared coordinate differential gains a factor five. The charge $(0,1)$ becomes $(2,1)$ passively, giving $Z'=m_0[2i+(2-2i)]=2m_0$ . Holding $(0,1)$ fixed during continuation instead gives $(2-2i)m_0$ , represented in the old branch by $(-2,1)$ . Confusing these operations would falsely suggest a failure of (SW08.9).

These integral matrices preserve the integer particle-charge lattice and its antisymmetric pairing. That is not a classification of line probes or a proof of microscopic S-duality. In the inherited $SU(2)$ normalization, fundamental Wilson probes carry electric half-units relative to the charged vector particle; the full global data have not been replaced by a list of particle labels.

Exercises

1. Wind twice, or change the local coordinate

Let $h(b)=i t_0+\alpha b/\mu_0$ in (SW08.3), where $t_0>0$ and $\alpha$ is dimensionless. Calculate the change in $a$ , $\tau_D$ , and the dual prepotential $\mathcal F_D$ after $w$ windings, using $\mathcal F_D'=-a$ . Fix its additive continuation constant by the local singular expression. Then consider $b=c(u-u_+)^k$ with integer $k>1$ : what matrix does one $u$ loop produce, and which earlier assumption fails?

Hint 1

The polynomial terms return to themselves. Integrate the logarithmic term once more to find a local prepotential.

Hint 2

A full turn of $u-u_+$ changes the argument of $b$ by $2\pi k$ . A valid local coordinate must have nonzero first derivative at the center.

Solution

The dual prepotential can be taken as

\mathcal F_D=-a_0b+\frac{i t_0}{2}b^2+ \frac{\alpha}{6\mu_0}b^3 -\frac{i}{2\pi}b^2\log(b/\mu_0)+\frac{3i}{4\pi}b^2.

Differentiating twice gives $\tau_D=i t_0+\alpha b/\mu_0-(i/\pi)\log(b/\mu_0)$ . The continued logarithm changes by $2\pi i w$ , so $\Delta\mathcal F_D=w b^2$ , $\Delta(-a)=2w b$ , and $\Delta\tau_D=2w$ . The metric is unchanged because this coupling shift is real. Every prepotential term has dimension two.

For the ramified map, $c$ has mass dimension $1-2k$ , and one $u$ loop gives $M_+^k=\begin{pmatrix}1&0\\-2k&1\end{pmatrix}$ . Its coefficient must not be mistaken automatically for $k$ independent light hypermultiplets: here it arose from the coordinate map. But $db/du$ vanishes at $u_+$ , so this map violates the regular-coordinate assumption used in (SW08.4). It is a changed local model, not a new result about the pure- $SU(2)$ solution.

2. Recover the matrix of a charged hypermultiplet

Suppose a single hypermultiplet with primitive row $\gamma=(p,q)$ becomes massless and admits a local unit-electric description with coordinate $b=p a_D+q a$ . Choose integers $\alpha,\beta$ such that $\alpha q-\beta p=1$ , and use $C=\begin{pmatrix}\alpha&\beta\\p&q\end{pmatrix}$ to define $(b_D,b)^T=C\Pi$ . Derive the old-frame monodromy from $b_D\mapsto b_D+2b$ . Show that it does not depend on $\alpha,\beta$ , and test $(p,q)=(1,-1)$ .

Hint 1

Conjugate the electric shear by the change of frame: the old variables are obtained with $C^{-1}$ .

Hint 2

Write the shear as the identity plus a rank-one matrix. This makes the independence of the first row of $C$ visible.

Solution

The local matrix is $E=\begin{pmatrix}1&2\\0&1\end{pmatrix}$ , so

M_\gamma=C^{-1}EC =I+2\begin{pmatrix}q\\-p\end{pmatrix}(p,q) =\begin{pmatrix}1+2pq&2q^2\\-2p^2&1-2pq\end{pmatrix}. \tag{SW08.11}

Indeed, $C^{-1}(1,0)^T=(q,-p)^T$ and $(0,1)C=(p,q)$ ; the other row drops out. The scalar $(p,q)(q,-p)^T$ vanishes. Consequently $(M_\gamma-I)^2=0$ , $\det M_\gamma=1$ , and $\gamma M_\gamma=\gamma$ . For $(1,-1)$ this reproduces $M_-$ , including both diagonal signs.

Primitivity permits the integer choice of $\alpha,\beta$ by Bezout’s identity. The result assumes the specified single hypermultiplet and simple local zero. An invariant eigenvector by itself does not determine multiplicity or a stable spectrum. This formula agrees with Bilal, (5.15).

3. Change the approach path without creating a new singularity

Using the chronological convention (SW08.5), consider the loop that first goes around $\ell_\infty$ once in reverse, then around $\ell_+$ , then around $\ell_\infty$ forward. Find its matrix $N$ , a primitive invariant row, and its relation to $M_\gamma$ in (SW08.11). Does it describe a third finite puncture or establish a new stable particle?

Hint 1

The word is $M_\infty^{-1}M_+M_\infty$ . It is a conjugate loop with a different approach to the same puncture.

Hint 2

If $\gamma_+M_+=\gamma_+$ , multiply this identity by $M_\infty$ to find an invariant row for $N$ .

Solution

Direct multiplication gives $N=\begin{pmatrix}-3&8\\-2&5\end{pmatrix}$ . The invariant row is $\gamma_+M_\infty=(-1,2)$ , or its opposite $(1,-2)$ ; substituting $(p,q)=(1,-2)$ in (SW08.11) gives exactly $N$ . The opposite row represents the CPT-conjugate charge direction and leaves this matrix unchanged.

The loop still encircles the same monopole puncture after changing its approach. It has not introduced another zero or extra matter. The changed row records how that local vanishing charge is expressed after the chosen continuation. Whether any fixed charge is represented by a stable particle throughout a particular path needs spectral and stability information in addition to this calculation. Conjugating an integral matrix cannot supply that information.

The next question

We now know how the two local electric–magnetic descriptions must join, but we have not constructed the functions $a_D(u)$ and $a(u)$ between the singularities. Riemann surfaces from square roots shows how surfaces and their closed cycles encode these continuations. Before that construction, check that you can reproduce the sign in $a\mapsto a-2a_D$ , explain the chosen loop order, and distinguish $\gamma M$ from $\gamma M^{-1}$ without appealing to a memorized table.