Boundaries, Gluing, and Jeffrey–Kirwan Residues

Supersymmetric cutting and gluing require more than multiplying two localized answers: one must choose a boundary polarization, match the preserved supercharge and integration cycle, cancel boundary anomalies, and quotient boundary gauge transformations exactly once. In Coulomb-branch formulas, the Jeffrey–Kirwan (JK) residue packages a related chamber choice determined by charge covectors and an auxiliary vector $\eta$; it is not an ordinary residue rule independent of asymptotic data.

Required background. Complex contours, Stokes chambers, and regularization supplies the cycle and chamber data. Boundaries, interfaces, and domain walls supplies boundary conditions, anomaly inflow, and interface degrees of freedom.

Helpful background. $Q$-cohomology and path-integral deformation supplies the boundary term in the Ward identity.

Supersymmetry at a boundary

Let $M$ have boundary $Y$. A supersymmetry variation of the bulk action generally has the form

$$\delta_QS_{\rm bulk}=\int_Y\mathcal B_Q.$$

A valid boundary problem must cancel $\mathcal B_Q$ using a boundary condition, a boundary action, or additional boundary degrees of freedom. It must also make the variational principle well posed and preserve the even symmetry $Q^2$. In particular, a vector appearing in $Q^2$ should be tangent to $Y$ for a fixed boundary, and gauge transformations allowed at $Y$ must be classified as either redundancies or physical boundary symmetries.

Quantum mechanically, the boundary measure can have a gauge, R, or gravitational anomaly. A bulk inflow term may cancel it; otherwise the claimed symmetry and the localization Ward identity fail. Boundary counterterms can also shift phases and contact terms. These conditions are independent of whether the bulk BPS equations have a smooth solution.

Cutting and gluing as a pairing

Suppose $M=M_1\cup_YM_2$. Choosing a polarization of the boundary phase space identifies boundary data $a$ and produces wavefunctionals

$$\Psi_1(a)=Z_{M_1}[a], \qquad \Psi_2(a)=Z_{M_2}[a].$$

The closed-manifold answer is a pairing

$$Z_M=\langle\Psi_2,\Psi_1\rangle_Y =\sum_\nu\int_{\Gamma_Y^\nu} d\mu_Y^\nu(a)\; \Psi_2^\nu(a)\Psi_1^\nu(a).$$

The sector label $\nu$ includes boundary flux or bundle topology. The measure $d\mu_Y$ contains the boundary gauge quotient, possible one-loop factors, and orientation. If both halves divide by the same boundary gauge group and the pairing divides again, gauge zero modes are removed more than once; if none does, an infinite gauge volume remains.

Localization commutes with gluing only when the two halves use compatible $Q$, polarization, regulator, and contour. Boundary zero modes that are nonnormalizable on a half-space may become ordinary modes after gluing, so determinants do not always multiply without an interface factor. Holomorphic blocks in three-dimensional $\mathcal N=2$ theories provide a concrete realization: closed partition functions are sums of products of block wavefunctions, with Stokes transformations acting on the block basis Beem, Dimofte, and Pasquetti 2014, §§4–6.

The Jeffrey–Kirwan residue

Let $u=(u_1,\ldots,u_r)$ be complexified Cartan variables. Near an isolated intersection $u_*$ of hyperplanes, a localized meromorphic form has the model

$$\omega(u)= \frac{g(u)\,du_1\wedge\cdots\wedge du_r} {Q_1(u-u_*)\cdots Q_n(u-u_*)},$$

where the $Q_i\in(\mathbb R^r)^*$ are charge covectors. For exactly $r$ linearly independent charges and a generic covector $\eta$, fix an orientation and define

$$\operatorname{JKRes}_{u_*}(Q_1,\ldots,Q_r;\eta) \frac{du_1\wedge\cdots\wedge du_r} {Q_1(u-u_*)\cdots Q_r(u-u_*)} = \begin{cases} \displaystyle\frac{1}{\det(Q_1,\ldots,Q_r)}, & \eta\in\operatorname{Cone}(Q_1,\ldots,Q_r),\\[6pt] 0,&\text{otherwise}. \end{cases}$$

If more than $r$ hyperplanes meet, one must use a flag prescription or a consistent decomposition; choosing arbitrary $r$-subsets can double count. The arrangement is projective near $u_*$ when its charges lie in an open half-space. Nonprojective intersections require a deformation or an additional prescription, and poles at infinity must be analyzed separately. The underlying residue construction for symplectic quotients is due to Jeffrey and Kirwan 1995, §§3 and 8.

In supersymmetric gauge theory, $\eta$ is related to the direction used to close the contour and often to an FI chamber. Changing $\eta$ across a cone wall changes the selected residues. This wall crossing can be physical, but only after contributions at infinity and changing asymptotic vacua are included. The two-dimensional elliptic-genus formula gives a precise gauge-theory implementation Benini et al. 2015, §§2–3.

Rank-one example

For $r=1$, a pole whose vanishing denominator has charge $Q_i\ne0$ is selected when $\eta/Q_i>0$. Consider

$$\omega=\frac{du}{(u-a)(-u+b)}, \qquad a\ne b.$$

The pole at $u=a$ has charge $+1$, while the pole at $u=b$ has charge $-1$. Their ordinary residues are

$$\operatorname{Res}_{u=a}\omega=\frac1{b-a}, \qquad \operatorname{Res}_{u=b}\omega=-\frac1{b-a}.$$

Thus $\eta>0$ selects the first value and $\eta<0$ selects the second. The sign change is chamber data, not an algebraic ambiguity. In a physical integral the difference must be matched to the contour at infinity or to states that enter or leave the spectrum at the wall.

Relating residues to boundary blocks

Both JK residues and block decompositions encode integration-cycle data, but they operate at different stages. A JK prescription selects cycles around hyperplane poles in an effective Cartan integral. A holomorphic block is a wavefunction obtained from a manifold with boundary and normally carries a thimble label or massive-vacuum label. To compare them, establish:

1. the same effective variables and charge lattice;
2. the map between $\eta$, FI parameters, and thimble asymptotics;
3. boundary anomaly cancellation and any edge multiplets;
4. matching determinant phases and local counterterms;
5. the gluing measure and residual Weyl quotient;
6. treatment of poles and Coulomb directions at infinity.

Without these checks, agreement of a formal residue sum with a product of blocks can miss an overall phase, a chamber-dependent term, or a duplicated gauge volume.

Exercises

1. Rank-one wall crossing. Recompute the two residues in the example and verify their sum is zero. What does this say about a contour enclosing both poles?

Solution

At $u=a$, the second factor equals $b-a$, giving $1/(b-a)$. At $u=b$, $-u+b=-(u-b)$, giving $-1/(b-a)$. A contour enclosing both finite poles has zero residue sum, consistent with no residue at infinity for this form. Each JK chamber instead selects one oriented local cycle.

2. Anomaly under gluing. Suppose the boundary theory on $M_1$ has anomaly polynomial $I$ and the orientation-reversed boundary of $M_2$ has the same, rather than opposite, anomaly. Can the two wavefunctionals be paired gauge invariantly without extra data?

Solution

No. Their anomalous gauge variations add instead of cancelling. One needs an inflow term or interface degrees of freedom with anomaly $-2I$, or a different boundary condition. Otherwise the boundary gauge quotient in the pairing is not defined.

References

• Beem, Christopher, Tudor Dimofte, and Sara Pasquetti. “Holomorphic Blocks in Three Dimensions.” Journal of High Energy Physics 2014, no. 12 (2014): 177. doi:10.1007/JHEP12(2014)177. Open preprint.
• Benini, Francesco, Richard Eager, Kentaro Hori, and Yuji Tachikawa. “Elliptic Genera of Two-Dimensional $\mathcal N=2$ Gauge Theories.” Communications in Mathematical Physics 333 (2015): 1241–1286. doi:10.1007/s00220-014-2210-y. Open preprint.
• Jeffrey, Lisa C., and Frances C. Kirwan. “Localization for Nonabelian Group Actions.” Topology 34 (1995): 291–327. doi:10.1016/0040-9383(94)00028-J. Open preprint.

Where this leads

The chapter has now supplied the validity conditions needed before interpreting exact partition functions and protected observables. Continue to the next chapter only after the background, global supercharge, deformation complex, saddle sectors, contour, regulator, counterterms, and boundary data are all fixed.