Factorization, Holomorphic Blocks, and Gluing

A supersymmetric partition function factorizes into holomorphic blocks only when its BPS vacua, integration cycles, anomaly factors, and gluing operation have been specified. A block is a wavefunction associated with a boundary torus and a massive vacuum, not an invariant number by itself. Stokes jumps change the block basis while the correctly glued observable remains unchanged.

Required background. Use a concrete sphere matrix model and import its cycles and kernels from boundary gluing and residues.

Helpful background. Twisted indices and elliptic genera exhibit the same chamber-dependent residue structures.

Blocks as vacuum wavefunctions

For a three-dimensional $N=2$ theory with isolated massive vacua $\alpha$, a holomorphic block is the supersymmetric path integral on a solid torus $D^2\times_q S^1$ with vacuum $\alpha$ selected asymptotically. Localization produces a contour integral

$$B^\alpha(x;q) =\int_{\Gamma_\alpha} \prod_{a=1}^{r}\frac{ds_a}{2\pi i s_a}\, \Upsilon(s,x;q).$$

The variables $x$ are exponentiated complex masses or FI parameters, $q$ is the rotation fugacity of the cigar, $\Upsilon$ is a product of $q$-special functions, and $\Gamma_\alpha$ is a middle-dimensional cycle associated with the vacuum. The same meromorphic integrand on a different cycle is a different block.

Supersymmetric line operators act as $q$-difference operators. If $\widehat x$ multiplies by $x$ and $\widehat p$ shifts $x\mapsto qx$, then

$$\widehat p\,\widehat x=q\,\widehat x\,\widehat p,$$

and the blocks solve a common system

$$\widehat f_i(\widehat x,\widehat p;q)B^\alpha(x;q)=0.$$

Different vacua give independent solutions of the same difference equations Beem, Dimofte, and Pasquetti 2014, §§2–3.

Free-chiral block

For $|q|<1$, define the $q$-Pochhammer symbol

$$(z;q)_\infty=\prod_{n=0}^{\infty}(1-zq^n).$$

For $|q|>1$, use its analytic continuation

$$(z;q)_\infty =\frac{1}{(q^{-1}z;q^{-1})_\infty}.$$

In one standard contact-term convention, a free chiral block is

$$B_\Delta(x;q)=(q x^{-1};q)_\infty.$$

It obeys

$$B_\Delta(qx;q) =(1-x^{-1})B_\Delta(x;q),$$

which is its elementary line-operator difference equation. Multiplying the block by an elliptic function $c(x;q)$ with $c(qx;q)=c(x;q)$ preserves this equation. The difference equation alone therefore does not fix the normalization or contact terms.

Gluing two solid tori

A closed three-manifold obtained by gluing two solid tori pairs the corresponding wavefunctions. Schematically,

$$Z_M(x) =e^{P_M(x)} \sum_{\alpha,\beta} B^\alpha(x;q)\, K^M_{\alpha\beta}\, \widetilde B^\beta(\widetilde x;\widetilde q).$$

The matrix $K^M$ implements the mapping-class-group element used for gluing, while $P_M$ is a local anomaly or contact-term polynomial. For squashed-sphere fusion, a frequently used analytic-continuation convention has

$$q=e^{2\pi i b^2}, \qquad \widetilde q=e^{2\pi i/b^2},$$

and pairs blocks associated with the same massive vacuum. Identity fusion instead produces an $S^2\times S^1$-type index with a different relation between $(x,q)$ and $(\widetilde x,\widetilde q)$. The word “gluing” is not enough to choose between them.

For the free chiral there is one massive vacuum. With the $S$-fusion convention above, set

$$x=e^{2\pi b\mu}, \qquad \widetilde x=e^{2\pi\mu/b},$$

where $\mu$ is the complexified mass including the chosen R-charge shift, and interpret each Pochhammer by the convergent product on its side of the unit circle, followed by analytic continuation. Its explicitly glued one-loop determinant is

$$\begin{aligned} Z^{S}_{\chi,\mathrm{1-loop}} (\mu;b) &=e^{P_\chi(\mu;b)} \,B_\Delta(x;q)B_\Delta(\widetilde x;\widetilde q) \\ &=e^{P_\chi(\mu;b)} \,(qx^{-1};q)_\infty \,(\widetilde q\widetilde x^{-1};\widetilde q)_\infty. \end{aligned}$$

The right-hand side is the $q$-Pochhammer representation of the corresponding double-sine determinant. The fixed quadratic polynomial $P_\chi$ records the chosen parity-anomaly and background-contact counterterms; changing it changes the scheme, not the block difference equation. Analytic continuation then reaches real positive $b$. This equality is the elementary factorization benchmark, and it makes the second block, the gluing map, and the local prefactor explicit Pasquetti 2012, §§3–4.

Stokes transformations

As masses or FI parameters cross a Stokes wall, steepest-descent cycles jump by an integral matrix:

$$\Gamma'_\alpha=M_\alpha{}^\beta\Gamma_\beta, \qquad B'^\alpha=M_\alpha{}^\beta B^\beta.$$

The complementary cycles and gluing kernel transform contragrediently, so the closed-manifold partition function stays fixed. A single block is therefore chamber dependent even when the glued answer is analytic.

This leads to three distinct notions of equality:

• the same formal $q$-series in one chamber;
• analytic continuations of the same block solution;
• two bases related by a Stokes matrix that give the same glued observable.

Only the last statement is basis independent. Publishing a block without its chamber and cycle leaves these possibilities unresolved.

When factorization is valid

A finite vacuum sum is justified when:

1. the compactified theory has isolated massive supersymmetric vacua;
2. the associated cycles span the relevant relative homology;
3. the block integrals converge or have a specified resummation;
4. gauge and background anomalies are cancelled or retained as $P_M$;
5. no continuum or noncompact branch contributes an extra integral;
6. the gluing measure and global gauge sectors are complete.

If vacua collide, cycles can become linearly dependent and the block basis can develop logarithmic solutions. If a Coulomb branch remains noncompact, factorization may require a continuous spectral integral rather than a discrete sum. These are changes of structure, not merely numerical complications.

Reconstruction limits

Knowing $Z_M$ does not uniquely recover the individual blocks. A basis change

$$B\longmapsto C B, \qquad K^M\longmapsto(C^{-1})^T K^M C^{-1}$$

leaves the bilinear pairing invariant. Elliptic prefactors and contact terms add further ambiguity. Independent difference equations, asymptotics, and cycle data are needed to reconstruct a meaningful block basis.

Exercises

Verify the free-chiral difference equation.

Solution

Using $(z;q)_\infty=(1-z)(qz;q)_\infty$, one has $B(qx;q)=(x^{-1};q)_\infty=(1-x^{-1})(qx^{-1};q)_\infty=(1-x^{-1})B(x;q)$.

References

• Beem, C., T. Dimofte, and S. Pasquetti. “Holomorphic Blocks in Three Dimensions.” Journal of High Energy Physics 2014, no. 12 (2014): 177. DOI; Open PDF.
• Pasquetti, S. “Factorisation of $N=2$ Theories on the Squashed 3-Sphere.” Journal of High Energy Physics 2012, no. 4 (2012): 120. DOI; Open PDF.