Blowup Relations in Mathematics & Physics

• Blowup relations are constructions that express identities between blow-ups and blow-downs in diverse fields, including geometric group theory and supersymmetric gauge theory.
• They establish equivalences via peripheral structures, factorization of partition functions, and bilinear tau-function frameworks, clarifying complex relationships.
• This framework unifies methods across algebraic geometry, analysis, and integrable systems, offering practical insights into finite-time blowup and quantization phenomena.

Blowup relations are a family of constructions and identities attached to blowups, blow-downs, or finite-time blowup phenomena, and the expression is used in several technically distinct senses across current mathematics and mathematical physics. In geometric group theory, a blow-up of a compactum is defined by the existence of a $G$-equivariant continuous surjection between compacta carrying minimal non-elementary convergence actions, and the relation is controlled by peripheral structures in the geometrically finite case (Matsuda et al., 2012). In supersymmetric gauge theory and integrable systems, blowup relations reconstruct partition functions on $\mathbb C^2\times S^1$ or on orbifolds from partition functions on blown-up patches, and these identities generate bilinear tau-function equations for Painlevé systems (Bershtein et al., 2018, Stoyan, 13 Sep 2025). In algebraic and symplectic geometry, the term refers to equivalences among constructions of the blowup, birational correspondences between simultaneous and iterated blow-ups, and closed formulae describing how enumerative invariants change under blow-up (Hauser, 2014, Huang, 14 Jun 2026, He et al., 2014). In analysis, related terminology governs comparison principles, quantization laws, and PDE–ODE correspondences for finite-time blowup (Nie, 16 Mar 2025, Bauer et al., 2023).

1. Blow-up and blow-down for convergence actions

For a countable group $G$ acting by homeomorphisms on a compact metrizable space $X$, a convergence action is defined by the requirement that whenever $\{g_i\}\subset G$ is an infinite sequence of distinct elements, there is a subsequence $\{g_{i_n}\}$ and points $r,a\in X$ such that

$$g_{i_n}|_{X\setminus\{r\}}\to a \quad\text{and}\quad g_{i_n}^{-1}|_{X\setminus\{a\}}\to r$$

uniformly on compact subsets. The limit set $\Lambda(G,X)$ has cardinality $0,1,2$ or $\mathbb C^2\times S^1$0, and if $\mathbb C^2\times S^1$1 the action is non-elementary. Parabolic subgroups, bounded parabolic points, and conical limit points then furnish the usual geometric finiteness dichotomy. The action is geometrically finite when every point of $\mathbb C^2\times S^1$2 is either a conical limit point or a bounded parabolic point (Matsuda et al., 2012).

Given such an action, the peripheral structure is

$\mathbb C^2\times S^1$3

If $\mathbb C^2\times S^1$4 and $\mathbb C^2\times S^1$5 carry minimal non-elementary convergence actions of $\mathbb C^2\times S^1$6, a $\mathbb C^2\times S^1$7-equivariant continuous surjection

$\mathbb C^2\times S^1$8

makes $\mathbb C^2\times S^1$9 a blow-up of $G$0 and $G$1 a blow-down of $G$2. The map forces an inclusion relation on peripherals,

$G$3

meaning that every member of $G$4 is conjugate into some member of $G$5.

The central characterization due to Matsuda–Oguni–Yamagata states that if both $G$6 and $G$7 support geometrically finite convergence actions of $G$8, then

$G$9

In particular, $X$0 implies that $X$1 and $X$2 are equivariantly homeomorphic. The proof realizes both boundaries as ideal boundaries of augmented spaces

$X$3

uses the inclusion of peripheral data to obtain a natural coarse embedding, and then shows that the identity on $X$4 extends continuously to the boundaries by a comparison criterion for geodesics, together with typical lifts of relative geodesics and quasi-geodesic stability.

A notable application is that a compactum with a geometrically finite convergence action admits blow-downs whose convergence actions are not geometrically finite. The construction introduces a descending chain of virtually free, almost-malnormal, quasiconvex subgroups

$X$5

builds an ascending chain of relative structures $X$6, and then forms inverse limits. An alternate refinement produces uncountably many blow-downs that are pairwise non-homeomorphic and pairwise incomparable under blow-up.

2. Gauge-theoretic blowup relations

In five-dimensional $X$7 $X$8 gauge theory with four fundamental flavors, the basic blowup relation states that the partition function on $X$9 can be reconstructed from partition functions on the blown-up $\{g_i\}\subset G$0 with an exceptional $\{g_i\}\subset G$1 inserted. Writing the full partition function as $\{g_i\}\subset G$2, one has, for each integer shift $\{g_i\}\subset G$3,

$\{g_i\}\subset G$4

In exponential variables $\{g_i\}\subset G$5, $\{g_i\}\subset G$6, the same factorization is written with shifted $\{g_i\}\subset G$7 and instanton parameter $\{g_i\}\subset G$8. In refined topological-string language one passes to the Weyl-invariant combination $\{g_i\}\subset G$9, where the coefficient $\{g_{i_n}\}$0 is built out of $\{g_{i_n}\}$1-loop pieces so that each factor is Weyl-invariant (Stoyan, 13 Sep 2025).

The integer data of these relations admit a representation-theoretic organization. For the $\{g_{i_n}\}$2 quiver one obtains integer tuples $\{g_{i_n}\}$3 together with an extra integer $\{g_{i_n}\}$4, and the associated vector

$\{g_{i_n}\}$5

transforms as a weight under the simple reflections of $\{g_{i_n}\}$6. The $\{g_{i_n}\}$7 nontrivial blowup relations are grouped into $\{g_{i_n}\}$8-orbits of size $\{g_{i_n}\}$9, matching the weights of the four fundamental representations of $r,a\in X$0. Simultaneous Weyl action on Coulomb–mass parameters $r,a\in X$1 and on the lattice variables $r,a\in X$2 makes the blowup relation covariant.

For five-dimensional pure $r,a\in X$3 SYM with Chern–Simons invariant $r,a\in X$4, Nakajima–Yoshioka blowup equations introduce a blow-up partition function $r,a\in X$5 on the one-point blowup of $r,a\in X$6, with discrete labels $r,a\in X$7 and $r,a\in X$8. The vanishing conditions

$r,a\in X$9

combine with the factorization of $$g_{i_n}|_{X\setminus\{r\}}\to a \quad\text{and}\quad g_{i_n}^{-1}|_{X\setminus\{a\}}\to r$$0 into two shifted flat-space partition functions to give a master K-theoretic blowup equation. In the Nekrasov–Shatashvili limit, this becomes a compatibility formula that expands into infinitely many algebraic-linear relations among refined BPS invariants of the toric Calabi–Yau $$g_{i_n}|_{X\setminus\{r\}}\to a \quad\text{and}\quad g_{i_n}^{-1}|_{X\setminus\{a\}}\to r$$1, and the resulting system can be solved recursively by degree (Grassi et al., 2016).

3. Painlevé equations and bilinear tau forms

A major development is the extraction of Painlevé tau-functions from Nakajima–Yoshioka blowup relations. For four-dimensional $$g_{i_n}|_{X\setminus\{r\}}\to a \quad\text{and}\quad g_{i_n}^{-1}|_{X\setminus\{a\}}\to r$$2 Nekrasov partition functions, the specialization $$g_{i_n}|_{X\setminus\{r\}}\to a \quad\text{and}\quad g_{i_n}^{-1}|_{X\setminus\{a\}}\to r$$3 identifies the full partition function with a $$g_{i_n}|_{X\setminus\{r\}}\to a \quad\text{and}\quad g_{i_n}^{-1}|_{X\setminus\{a\}}\to r$$4 Virasoro conformal block. The Painlevé VI tau-function is then written as a Fourier series

$$g_{i_n}|_{X\setminus\{r\}}\to a \quad\text{and}\quad g_{i_n}^{-1}|_{X\setminus\{a\}}\to r$$5

The same blowup identity also yields a bilinear factorization in terms of $$g_{i_n}|_{X\setminus\{r\}}\to a \quad\text{and}\quad g_{i_n}^{-1}|_{X\setminus\{a\}}\to r$$6 conformal blocks through the “short” and “long” tau-functions

$$g_{i_n}|_{X\setminus\{r\}}\to a \quad\text{and}\quad g_{i_n}^{-1}|_{X\setminus\{a\}}\to r$$7

with

$$g_{i_n}|_{X\setminus\{r\}}\to a \quad\text{and}\quad g_{i_n}^{-1}|_{X\setminus\{a\}}\to r$$8

Eliminating $$g_{i_n}|_{X\setminus\{r\}}\to a \quad\text{and}\quad g_{i_n}^{-1}|_{X\setminus\{a\}}\to r$$9 between differential blowup relations produces Hirota identities and the Toda-like system

$\Lambda(G,X)$0

which is equivalent to the $\Lambda(G,X)$1-form of Painlevé VI (Bershtein et al., 2018).

The five-dimensional $\Lambda(G,X)$2-difference analogue gives $\Lambda(G,X)$3-Painlevé equations. In the self-dual specialization $\Lambda(G,X)$4, the blowup identities collapse to a single relation and define a short $\Lambda(G,X)$5-deformed tau-function

$\Lambda(G,X)$6

This function satisfies the $\Lambda(G,X)$7-Hirota bilinear equations recorded as (4.8) and (4.9) in the source, and an elementary elimination yields the standard q-Toda-like form

$\Lambda(G,X)$8

equivalent to the tau-form of the $\Lambda(G,X)$9 $0,1,2$0-Painlevé III equation (Bershtein et al., 2018).

For $0,1,2$1-Painlevé VI, a further sequence of operations is used: one performs “Higgsing,” then takes the self-dual limit $0,1,2$2 with $0,1,2$3. The resulting limit of the blowup relations produces linear relations among shifted tau-functions; multiplying appropriate relations eliminates the auxiliary action functional and yields pure bilinear identities. One representative system is

$0,1,2$4

together with three companion equations. Eliminating the ratio $0,1,2$5 recovers the standard bilinear form

$0,1,2$6

for $0,1,2$7-Painlevé VI tau-functions (Stoyan, 13 Sep 2025).

Nekrasov’s BPS/CFT treatment identifies the bulk partition function at $0,1,2$8 with the $0,1,2$9 Liouville four-point conformal block and the defect partition function with a level-$\mathbb C^2\times S^1$00 $\mathbb C^2\times S^1$01 four-point WZW block; in the NS limit $\mathbb C^2\times S^1$02, the defect expectation yields a Hamilton–Jacobi form of Painlevé VI and reproduces the Gamayun–Iorgov–Lysovyy expansion (Nekrasov, 2020). A later refinement formulates quantum Painlevé tau-functions by noncommutative Zak transforms, derives bilinear tau forms for canonically quantized Painlevé equations, and identifies the $\mathbb C^2\times S^1$03 blowup relations in the nontrivial holonomy sector as odd or translation-type Hirota equations (Bonelli et al., 31 Dec 2025).

4. Algebraic and symplectic geometric relations of blowups

In algebraic geometry, blowups themselves are related by several equivalent constructions. If $\mathbb C^2\times S^1$04 and $\mathbb C^2\times S^1$05, the blowup is defined by the universal property that $\mathbb C^2\times S^1$06 makes $\mathbb C^2\times S^1$07 a Cartier divisor and is final among such morphisms. The same object is constructed as

$\mathbb C^2\times S^1$08

and also as the closure of the graph of the rational map to projective space defined by generators of $\mathbb C^2\times S^1$09. The affine charts

$\mathbb C^2\times S^1$10

show that the Proj-of-Rees and closure-of-graph constructions agree. The