Monodromy

Monodromy records how a germ of a holomorphic function changes after analytic continuation around a closed path. The source point returns to itself, but the continued germ of a holomorphic function need not. For $z(t)=e^{it}$, $0\le t\le 2\pi$, starting with holomorphic square root value $1$ gives $w(t)=e^{it/2}$ and hence $w(2\pi )=-1$; starting with complex logarithm value $0$ gives $L(t)=it$ and hence $L(2\pi )=2\pi i$. After $m$ counterclockwise turns the values are $(-1{)}^m$ and $2\pi im$. Thus monodromy exchanges the two square-root branches of a multivalued function, whereas it shifts through infinitely many logarithm branches of a multivalued function. ^[ref]

The three panels use the same parameter $t$. They show analytic continuation through local holomorphic patches, including across a chosen branch cut; repeatedly evaluating the principal logarithm would instead reset its imaginary part at the cut. Monodromy makes the obstruction to a global branch of a multivalued function visible and motivates the use of Riemann surfaces as domains on which continued values stay distinct.

A parameter deformation preserves generalized monodromy data, including Stokes matrices and formal monodromy at irregular points. If normalized sectorial matrices have parameter-independent jumps, $B=Y_z{Y}^{-1}$ agrees across overlaps. Its finite singularities and growth at infinity determine it by complex analytic arguments. Compatibility of $Y_{\lambda }=AY$ and $Y_z=BY$ is precisely the displayed zero-curvature identity, obtained by differentiating both equations and subtracting.

An unramified formal fundamental matrix $\hat{Y}=\hat{G}(\lambda ){\lambda }^{\Theta }{e}^{Q(\lambda )}$ acquires the right factor $e^{2\pi i\Theta }$ on a positive turn of the logarithm branch. This is formal monodromy. It can be nontrivial even when the true equation has an entire fundamental matrix and trivial actual monodromy, because Stokes matrices compensate for it.

When Stokes factors act on solution coefficient vectors, a full positive turn gives their ordered product equal to the inverse of formal monodromy for a polynomial system with no finite singularities. The true fundamental matrix is entire, so its actual monodromy is trivial, but its normalized formal branch changes by $M_f$. Indeed, write $Y_{j+1}=Y_j{S}_j^{-1}$; then $Y_{N+1}=Y_1(S_N\cdots S_1{)}^{-1}$, whereas normalized continuation after the positive turn gives $Y_{N+1}=Y_1{M}_f$. Cancelling the invertible fundamental matrix proves the identity. The uncorrected product is identity only when $M_f=I$. For a cubic-exponential two-by-two system with $U=1,V=-1,P=1,R=2,d=3/2$, the formal exponent is $\theta =1/2$, so $M_f=-I$ and the Stokes product is $-I$.

For a linear ordinary differential equation with holomorphic coefficients on a punctured disc, analytic continuation of a basis gives an invertible monodromy matrix. A nonzero eigenvector with eigenvalue $\lambda =e^{2\pi i\sigma }$ selects a solution $w$ for which $z^{-\sigma }w$ is single-valued and holomorphic. Its Laurent series gives $w=z^{\sigma }{\sum }_{n\in \mathbb{Z}}{c}_n{z}^n$. No regular-singular hypothesis is needed; infinitely many negative powers may occur.