Construction, existence, and wall-crossing of generalized multi-monopole Seiberg–Witten invariants $SW_N(c_1,\dots,c_N;J)$ at the $E_1$ point of 5d $SU(2)$ super Yang–Mills

The path-integral derivation of K-theoretic Donaldson invariants by Kim, Manschot, Moore, Tao and Zhang (arXiv:2509.23042) predicts, from the wall-crossing of the U-plane integral at the superconformal $E_1$ locus ($R^4=1$), the existence of a new family of smooth four-manifold invariants: generalized Seiberg–Witten invariants $SW_N(c_1,\dots,c_N;J)$ associated with multi-monopole equations in which $N$ monopole spinors couple to $N$ distinct Spin^c classes. For $N=2$ the invariants $SW_2(c,c-2n;J)$ would jump across walls $B(c-n,J)=0$ that, when $n$ is not even, are not associated with any Spin^c structure, and could be nonzero, for example on $\mathbb{CP}^2$ with nonvanishing 't Hooft flux, where ordinary Seiberg–Witten invariants vanish. No mathematical definition is known: for $N\ge 2$ the moduli spaces are noncompact, so the standard Seiberg–Witten compactness argument fails. The open problem is to construct these invariants (or prove that they cannot exist), establish their chamber structure and wall-crossing formula, and decide whether the Higgs-branch contribution to the twisted 5d partition function is given by the conjectural formula of the source in terms of $SW_2$ with universal couplings.

Background

Donaldson invariants of a closed oriented smooth four-manifold $X$ are counts obtained by integrating cohomology classes over the moduli space of anti-self-dual instantons (solutions of the Yang–Mills self-duality equations). Their abelian gauge-theoretic counterpart, the Seiberg–Witten (SW) invariants, are defined as follows. A Spin^c structure is a lift of the $SO(4)$ frame bundle of $X$ to the group $\mathrm{Spin}^c(4)=(\mathrm{Spin}(4)\times U(1))/\mathbb{Z}_2$; it has a determinant line bundle with first Chern class $c$. For a Spin^c connection $A$ one solves the monopole equations $F^+_A=\bar M M$, $\not{D}_A M=0$, where $M$ is a positive spinor, $F^+_A$ is the self-dual part of the curvature, and $\not{D}_A$ is the Spin^c Dirac operator. Solutions modulo $U(1)$ gauge form a compact moduli space, and its signed count (with cohomological insertions) defines the invariant $SW(c)$. When $b_2^+(X)=1$ (dimension of the space of self-dual harmonic 2-forms), the invariant depends on the chamber of the period point $J$, a class in the positive cone of $H^2(X,\mathbb{R})$ recording the self-dual harmonic form, and jumps across walls $B(c,J)=0$, where $B$ is the intersection form.

K-theoretic Donaldson invariants refine these counts by an extra circle action: physically they are BPS indices of instanton particles in 5d $\mathcal{N}=1$ $SU(2)$ super Yang–Mills on $X\times S^1$, partially topologically twisted along $X$ so that the partition function is formally metric-independent and its $R$-expansion (with $R$ the circle circumference) has Witten indices as coefficients. Kim–Manschot–Moore–Tao–Zhang derive this partition function for $b_2^+(X)>0$ by integration over the Coulomb branch of the effective abelian theory (the U-plane integral, an integral over the complexified coupling), and independently for toric $X$ by equivariant localization in the Ω-background, a deformation by two equivariant parameters $\epsilon_1,\epsilon_2$ for a $\mathbb{C}^*\times\mathbb{C}^*$ symmetry.

At the special value $R^4=1$ the 5d theory sits at the superconformal $E_1$ point. Near a Coulomb-branch singularity two monopoles become massless simultaneously, and the U-plane wall-crossing involves walls $B(c-n,J)=0$, with $n$ the 't Hooft flux (the second Stiefel–Whitney class of the $SO(3)$ gauge bundle obstructing the lift to $SU(2)$), which, when $n$ is not even, are not the walls of any Spin^c structure. Ordinary monopole invariants cannot absorb this wall-crossing, so the authors introduce generalized multi-monopole equations: for a Spin^c connection $A$ of class $c$ and background Spin^c connections $A+2A_j$ of classes $c_j=c+2c_1(L_j)$, $N$ monopole fields $M_j$ solve

$\sum_{j=1}^{N}F^+(A+2A_j)+\sum_{j=1}^{N}\bar M_j M_j=0,\qquad \not{D}_j M_j=0,$

where $\not{D}_j$ is the Spin^c Dirac operator coupled to $A+2A_j$. The diagonal $U(1)$ acting on all $M_j$ is gauged; a complementary $U(1)^{N-1}$ is a global symmetry. These generalize the equal-class equations studied on Kähler surfaces by Bryan–Wentworth and the $U(1)$, $N_f$-flavor equations analyzed by Losev–Nekrasov–Shatashvili and by Dedushenko–Gukov–Putrov. For $N\ge 2$ the moduli space is noncompact: in the analogous three-dimensional equations, sequences of solutions degenerate along so-called Fueter sections (Haydys–Walpuski), and the resulting counts are chamber-dependent rather than topological (Doan; Wilson). The paper's conjecture (its Sec. 8, Eq. 8.31 and App. H) is that generalized invariants $SW_N(c_1,\dots,c_N;J)$ nevertheless exist, with simple wall-crossing at the walls $B\big(\sum_j c_j,J\big)=0$, and that for $N=2$ the Higgs-branch contribution to the 5d partition function at the $E_1$ locus is a sum over $SW_2(c,c-2n;J)$ with universal $c$- and $n$-dependent couplings. An analogous conjecture for massive 4d SQCD was made earlier by Aspman–Furrer–Manschot. The localization formulae of Losev–Nekrasov–Shatashvili and Dedushenko–Gukov–Putrov do not apply directly, because their walls of marginal stability differ.

Problem Statement

Let $X$ be a closed oriented smooth four-manifold with $b_2^+(X)\ge 1$, let $J$ be a period point in the positive cone of $H^2(X,\mathbb{R})$, and let $c,c_1,\dots,c_N$ be characteristic classes of Spin^c structures on $X$. Consider the generalized multi-monopole equations of Kim–Manschot–Moore–Tao–Zhang (their Eq. H.8): for a Spin^c connection $A$ of class $c$ and background Spin^c connections $A+2A_j$ of classes $c_j=c+2c_1(L_j)$, monopole fields $M_j\in\Gamma(W_j^+)$ solve

$\sum_{j=1}^{N}F^+(A+2A_j)+\sum_{j=1}^{N}\bar M_j M_j=0,\qquad \not{D}_j M_j=0,$

with $\not{D}_j$ the Spin^c Dirac operator coupled to $A+2A_j$; solutions are taken modulo the diagonal (gauged) $U(1)$. Denote the moduli space by $\mathcal{M}_N(\{c_j\})$; its expected complex dimension is $m(\{c_j\})=(-2\chi(X)-2\sigma(X)+\sum_j(c_j^2-\sigma(X)))/8$, where $\chi(X)$ and $\sigma(X)$ denote the Euler characteristic and the signature of $X$.

The problem, in three linked parts with one objective — establish the multi-monopole invariants predicted by the 5d $E_1$ path integral — is:

(1) Existence and construction: define invariants $SW_N(c_1,\dots,c_N;J)$ of these equations — handling the noncompactness of the moduli space — reducing to the ordinary Seiberg–Witten invariants $SW(c;J)$ at $N=1$; or prove that no invariants with these properties exist.

(2) Chamber structure and wall-crossing: prove that for $b_2^+(X)>1$ the invariants are independent of $J$, and that for $b_2^+(X)=1$ they jump only across the walls $B(\sum_j c_j,J)=0$; determine the wall-crossing formula.

(3) The $N=2$ case and the physical prediction: for $SW_2(c,c-2n;J)$, with $n$ the 't Hooft flux and expected dimension $\frac{1}{4}(2n^2-2B(n,c)+c^2-\chi(X)-2\sigma(X))$, decide whether the invariant can be nonzero when the ordinary SW invariants vanish (for example on $\mathbb{CP}^2$ with $n\ne 0$), and whether the Higgs-branch contribution to the twisted 5d partition function at the $E_1$ locus is given by the source's conjectural formula (8.32) with the universal couplings (8.33).

Parts (2) and (3) are conditional on the affirmative resolution of part (1): they presuppose that the invariants $SW_N(c_1,\dots,c_N;J)$ exist. A negative resolution of part (1) — a proof that no invariants with these properties exist — therefore resolves the task, with parts (2) and (3) becoming moot.

References

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