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 contours, cancel boundary anomalies, and quotient boundary gauge transformations exactly once. In effective Cartan integrals, the Jeffrey–Kirwan (JK) residue packages a related oriented chamber choice. It is not an instruction to take ordinary residues of every pole with a favored charge sign.

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 $Y=\partial M$. A supersymmetry variation of the bulk action generally reduces on shell to

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

A valid boundary problem cancels this term through boundary conditions, a boundary action, or boundary degrees of freedom. It must simultaneously:

• make the ordinary variational principle well posed;
• preserve $Q$ and the even transformation $Q^2$;
• specify which gauge transformations at $Y$ remain redundancies and which become physical boundary symmetries;
• give the bulk fluctuation complex a compatible domain, or add edge modes that restore it.

For a fixed geometric boundary, a vector field appearing in $Q^2$ must be tangent to $Y$. A gauge parameter in $Q^2$ must preserve the boundary condition. These closure tests are independent of whether the bulk localization equations admit a smooth solution.

Quantum mechanically, the boundary measure can carry gauge, R-symmetry, or gravitational anomalies. A bulk inflow term or additional boundary matter must cancel any anomaly in a symmetry that is to be gauged. Otherwise the $Q$ Ward identity and the boundary gauge quotient fail. The half-space path integral is then best regarded as a section of an anomaly line, and gluing pairs dual anomaly lines rather than multiplying ordinary numbers.

Cutting and gluing as a dual pairing

Suppose $M=M_1\cup_YM_2$. A polarization is, locally, a Lagrangian foliation of the boundary phase space. It chooses commuting boundary data $a$ that can label states. The two path integrals define states in dual boundary state spaces because the induced orientations of $Y$ are opposite:

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

Orientation reversal gives a dual vector; it gives complex conjugation only when a compatible unitary real structure has also been chosen. A general localized pairing has the schematic form

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

The sector $\nu$ contains the boundary bundle and flux data that can extend over both halves. The kernel $\mathcal K_Y$ can contain an interface or boundary-localization theory; in a simple polarization it may be absorbed into $d\mu_Y$. The measure carries orientations, one-loop factors, and the boundary gauge quotient. That quotient is performed once. Dividing on both halves and in the pairing over-removes gauge zero modes; omitting it everywhere leaves an infinite gauge volume.

The polarization-space integral and the symmetry conditions behind it are explained in Dedushenko 2018a, §§2–4. Supersymmetric boundary localization requires a polarization preserved by $Q$ and produces model-dependent finite-dimensional gluing formulas Dedushenko 2018b, §§2–5. Consequently, “localization commutes with gluing” is a result to establish after matching $Q$, the regulator, polarization, contours, zero modes, and anomaly trivializations on both halves.

Holomorphic blocks in three-dimensional $\mathcal N=2$ theories give an important realization: a block is a boundary wavefunction labeled by a massive vacuum or thimble, and closed partition functions are bilinear combinations of blocks. The block basis itself undergoes Stokes transformations Beem, Dimofte, and Pasquetti 2014, §§4–6.

The nondegenerate JK residue

Let $u=(u_1,\ldots,u_r)$ be complexified Cartan coordinates with a chosen orientation. Near an isolated intersection $u_*$, let $Q_i\in(\mathbb R^r)^*$ be the real charge covectors of the vanishing hyperplanes. For exactly $r$ linearly independent charges, define

$$\operatorname{Cone}(Q_1,\ldots,Q_r) =\left\{\sum_{i=1}^r a_iQ_i:\ a_i>0\right\}.$$

For a generic covector $\eta$ that lies on no cone generated by $r-1$ charges, the nondegenerate convention used in supersymmetric residue formulas is

$$\operatorname{JKRes}_{u_*}(Q_1,\ldots,Q_r;\eta) \frac{du_1\wedge\cdots\wedge du_r} {Q_1(u-u_*)\cdots Q_r(u-u_*)} = \begin{cases} \displaystyle\frac1{|\det(Q_1,\ldots,Q_r)|}, & \eta\in\operatorname{Cone}(Q_1,\ldots,Q_r),\\[7pt] 0,&\text{otherwise}. \end{cases}$$

The absolute value is essential. Orientation has not disappeared: for the logarithmic form,

$$\operatorname{JKRes}_{u_*}(Q;\eta) \left( \frac{dQ_1}{Q_1}\wedge\cdots\wedge \frac{dQ_r}{Q_r} \right) = \begin{cases} \operatorname{sgn}\det(Q_1,\ldots,Q_r), &\eta\in\operatorname{Cone}(Q_1,\ldots,Q_r),\\ 0,&\text{otherwise}. \end{cases}$$

These two formulas are equivalent because wedging the $dQ_i$ contributes $\det Q$. Confusing them is the usual source of a missing sign. The gauge-theory convention and its relation to oriented local cycles are stated in Benini et al. 2015, §2.4.1, Eqs. (2.27)–(2.30); the residue construction originates in Jeffrey and Kirwan 1995, §§3 and 8.

Degenerate intersections and projectivity

If more than $r$ hyperplanes meet, the residue is degenerate. It requires a flag or iterated-residue prescription compatible with the full charge arrangement. Summing arbitrarily over $r$-element subsets can double count and need not reproduce the JK functional.

The charges at an intersection are projective if they all lie in an open half-space: equivalently, there is a vector $\delta\in\mathbb R^r$ with $Q_i(\delta)>0$ for every incident charge. Projectivity makes a local oriented-cycle construction possible. A nonprojective arrangement requires a deformation or additional global contour data. Neither prescription removes the separate obligation to analyze singularities at infinity.

In a particular localization derivation, $\eta$ may be related to an FI parameter or to the direction in which a contour is closed. The sign of that map depends on conventions and on asymptotic charges; it is not part of the abstract JK definition. Crossing a cone wall changes the finite-pole prescription, but a physical wall-crossing formula also includes Coulomb directions and residues at infinity. In one-dimensional gauged quantum mechanics this asymptotic contribution is derived explicitly in Hori, Kim, and Yi 2015, §§4–5.

Rank-one orientation and infinity

For $r=1$, write a vanishing denominator locally as $Qv$ with $Q\ne0$. Then

$$\operatorname{JKRes}_{v=0}(Q;\eta) \frac{dv}{Qv} = \begin{cases} 1/|Q|,&\eta Q>0,\\ 0,&\eta Q<0. \end{cases}$$

Consequently $\operatorname{JKRes}(Q;\eta),dv/v=\operatorname{sgn}(Q)$ when selected. This charge orientation distinguishes a JK residue from an ordinary complex residue.

Consider

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

At $u=a$ the local charge is $+1$; at $u=b$ it is $-1$. The ordinary residues are

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

The JK values, however, are

$$\operatorname{JKRes}_{u=a,\,\eta>0}\omega =\frac1{b-a}, \qquad \operatorname{JKRes}_{u=b,\,\eta<0}\omega =\frac1{b-a}.$$

For the negative charge, the charge-oriented local circle reverses the ordinary-residue sign. The form falls as $du/u^2$, so its ordinary residue at infinity vanishes; the two JK chambers therefore agree once orientation is treated correctly.

By contrast, take

$$\widetilde\omega=\frac{du}{u-a}.$$

Its finite pole has charge $+1$. The finite JK sum is $1$ for $\eta>0$ and $0$ for $\eta<0$, while $\operatorname{Res}_\infty\widetilde\omega=-1$. Thus the chamber difference is supplied by the boundary at infinity. This toy example captures why a finite-pole JK rule alone is not a physical wall-crossing theorem.

Residues and boundary blocks are related, not identical

A JK prescription selects charge-oriented linking cycles around hyperplane poles in an effective Cartan integral. A holomorphic block is a boundary wavefunction, usually with a thimble or vacuum label. To identify a residue sum with a block gluing formula, one must establish:

1. the same global charge and flux lattices;
2. the map between $\eta$, FI parameters, and thimble asymptotics;
3. the boundary polarization and the status of boundary gauge transformations;
4. anomaly inflow and any required edge multiplets;
5. determinant phases, local counterterms, and the gluing kernel;
6. the residual Weyl quotient and every contribution at infinity.

Without these checks, agreement of formal integrands can still miss a phase, a wall-crossing term, or an entire gauge-volume factor.

Exercises

1. Charge orientation in rank one. At the negative-charge pole $u=b$ in the two-pole example, rewrite $\omega$ using $v=u-b$ and evaluate the JK residue for $\eta<0$.

Solution

Near $v=0$,

$$\omega =\frac{dv}{(b-a+v)(-v)} =\frac{g(v)\,dv}{Qv}, \qquad g(0)=\frac1{b-a}, \quad Q=-1.$$

Because $\eta Q>0$, the pole is selected, and the nondegenerate formula gives $g(0)/|Q|=1/(b-a)$. The ordinary residue is negative because it uses the coordinate orientation rather than the charge-oriented JK cycle.

2. A wall at infinity. Verify the chamber difference for $\widetilde\omega=du/(u-a)$ and relate it to the residue at infinity.

Solution

The only finite charge is $+1$, so the finite JK sums are $Z_{\eta>0}=1$ and $Z_{\eta<0}=0$. On the Riemann sphere the sum of all residues vanishes, hence $\operatorname{Res}_\infty\widetilde\omega=-1$. Therefore

$$Z_{\eta>0}-Z_{\eta<0} =1=-\operatorname{Res}_\infty\widetilde\omega.$$

The finite chamber jump is exactly an asymptotic contribution in this convention.

3. Anomaly under gluing. Suppose the boundary theory on $M_1$ has anomaly polynomial $I$ and the orientation-reversed boundary contribution from $M_2$ also has $I$, rather than $-I$. Can the boundary symmetry be gauged in the gluing integral without extra data?

Solution

No. The anomalous variations add to $2I$ instead of cancelling. One needs bulk inflow, interface degrees of freedom with anomaly $-2I$, or different boundary data. Otherwise the boundary gauge quotient and hence the gluing pairing are 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.
• Dedushenko, Mykola. “Gluing I: Integrals and Symmetries.” arXiv:1807.04274 [hep-th] (2018). Open preprint.
• Dedushenko, Mykola. “Gluing II: Boundary Localization and Gluing Formulas.” arXiv:1807.04278 [hep-th] (2018). Open preprint.
• Hori, Kentaro, Heeyeon Kim, and Piljin Yi. “Witten Index and Wall Crossing.” Journal of High Energy Physics 2015, no. 1 (2015): 124. doi:10.1007/JHEP01(2015)124. Open preprint.
• Jeffrey, Lisa C., and Frances C. Kirwan. “Localization for Nonabelian Group Actions.” Topology 34, no. 2 (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.