Tropological Sigma Models in TQFT

• Tropological sigma models are BRST-localized topological field theories derived via Maslov dequantization that replace standard complex structures with nilpotent Jordan structures.
• They employ tropicalization of worldsheet and target geometries, leading to localization on tropical maps and reproducing enumerative data from the relativistic A-model.
• The models exhibit nonrelativistic, anisotropic features with foliation-preserving symmetries and extend to higher-dimensional filtered geometries with nilpotent extensions.

Tropological sigma models are nonrelativistic, BRST-type topological field theories obtained by Maslov dequantization of both worldsheet and target-space complex structures. They were introduced as a path-integral tropical analog of the topological A-model, with the aim of producing a field-theoretic framework whose correlators localize on tropical maps and, in explicit cases, reproduce the same enumerative data as the relativistic theory (Albrychiewicz et al., 2023). Subsequent work generalized the construction to higher-dimensional targets, where nested Maslov dequantizations yield filtered rather than merely foliated geometries, together with nilpotent global symmetries and nil-equivariant extensions (Albrychiewicz et al., 30 Jul 2025).

1. Terminology and theoretical setting

The term has acquired two nearby uses in the literature. In a review of twisted $R$-Poisson sigma models, “Tropological Sigma Models” is used simply to mean “Topological Sigma Models” (Chatzistavrakidis, 2022). In the more specific sense developed for tropical geometry, however, tropological sigma models form a distinguished class of topological sigma theories obtained from the tropical limit of the topological sigma model before coupling to topological gravity (Albrychiewicz et al., 2023).

This placement is significant because standard topological sigma models are cohomological field theories whose correlators depend only on topology, and AKSZ provides a canonical construction from a dg source $T[1]\Sigma$ and a QP target (Ikeda, 2012). Tropological sigma models retain the BRST-localized, cohomological character of that broader family, but replace ordinary complex structures by nilpotent “Jordan structures,” and replace the usual relativistic worldsheet geometry by a foliated or, in higher dimensions, filtered geometry (Albrychiewicz et al., 2023).

A central motivation comes from tropical geometry. Mikhalkin’s program showed that certain Gromov–Witten invariants of pseudoholomorphic-map theories can be computed by passing to the tropical limit. The tropological construction translates that statement into the language of topological quantum field theory and path integrals, seeking a first-principles field-theoretic realization of tropical localization (Albrychiewicz et al., 2023).

2. Tropicalization of worldsheet and target geometry

The basic tropicalization is implemented by writing local complex coordinates in exponential form,

$$z=\exp\left\{ \frac{r}{\hbar}+i\theta\right\},\qquad Z=\exp\left\{\frac{X}{\hbar} +i\Theta\right\},$$

and taking the singular limit $\hbar\to 0$ (Albrychiewicz et al., 2023). On the target, one keeps the same differentiable topology as the original complex manifold, but the almost complex structure and metric degenerate to tropical objects.

The relativistic complex structures $\widehat{\varepsilon}$ on the worldsheet and $\widehat{J}$ on the target tropicalize to nilpotent endomorphisms

$$\varepsilon_\alpha{}^\beta d\sigma^\alpha\frac{\partial}{\partial\sigma^\beta}=dr\frac{\partial}{\partial\theta},\qquad J_i{}^jdY^i\frac{\partial}{\partial Y^j}=dX\frac{\partial}{\partial \Theta},$$

with

$\varepsilon^2=0,\qquad J^2=0.$

These tensors take Jordan form in adapted coordinates. Correspondingly, a relativistic metric and its inverse tropicalize to the rank-one pair

$$g_{\alpha\beta}d\sigma^\alpha d\sigma^\beta=dr^2,\qquad h^{\alpha\beta}\frac{\partial}{\partial\sigma^\alpha}\frac{\partial}{\partial\sigma^\beta}=\left(\frac{\partial}{\partial\theta}\right)^2,$$

satisfying

$$g_{\alpha\beta}h^{\beta\gamma}=0,\qquad h^{\alpha\beta}g_{\beta\gamma}=0.$$

On the worldsheet, the nilpotent $T[1]\Sigma$0 defines a one-dimensional distribution $T[1]\Sigma$1, integrable to a foliation by compact $T[1]\Sigma$2 leaves. Locally the worldsheet is built from sleeves $T[1]\Sigma$3, and the complex structure is replaced by a foliation-preserving Jordan structure. The local infinitesimal symmetries preserving $T[1]\Sigma$4 are

$T[1]\Sigma$5

This is the geometric origin of the model’s anisotropic, nonrelativistic character.

The tropicalized localization equations are the degenerate counterparts of the Cauchy–Riemann equations,

$T[1]\Sigma$6

In adapted coordinates they reduce to

$T[1]\Sigma$7

Locally the solutions are

$T[1]\Sigma$8

and on compact $T[1]\Sigma$9-circles global consistency imposes integer slope,

$$z=\exp\left\{ \frac{r}{\hbar}+i\theta\right\},\qquad Z=\exp\left\{\frac{X}{\hbar} +i\Theta\right\},$$0

Forgetting $$z=\exp\left\{ \frac{r}{\hbar}+i\theta\right\},\qquad Z=\exp\left\{\frac{X}{\hbar} +i\Theta\right\},$$1 recovers the piecewise-linear tropical curve with integral slope. At singular leaves where sleeves meet, continuity imposes tropical balancing, for example

$$z=\exp\left\{ \frac{r}{\hbar}+i\theta\right\},\qquad Z=\exp\left\{\frac{X}{\hbar} +i\Theta\right\},$$2

3. BRST formulation and localization

The field content consists of maps $$z=\exp\left\{ \frac{r}{\hbar}+i\theta\right\},\qquad Z=\exp\left\{\frac{X}{\hbar} +i\Theta\right\},$$3, represented locally by adapted target coordinates $$z=\exp\left\{ \frac{r}{\hbar}+i\theta\right\},\qquad Z=\exp\left\{\frac{X}{\hbar} +i\Theta\right\},$$4, together with BRST multiplets for the map and for gauge fixing (Albrychiewicz et al., 2023). The basic BRST algebra is

$$z=\exp\left\{ \frac{r}{\hbar}+i\theta\right\},\qquad Z=\exp\left\{\frac{X}{\hbar} +i\Theta\right\},$$5

and for the antighost and auxiliary fields,

$$z=\exp\left\{ \frac{r}{\hbar}+i\theta\right\},\qquad Z=\exp\left\{\frac{X}{\hbar} +i\Theta\right\},$$6

After fixing the redundant components in a conformally compatible way, one introduces

$$z=\exp\left\{ \frac{r}{\hbar}+i\theta\right\},\qquad Z=\exp\left\{\frac{X}{\hbar} +i\Theta\right\},$$7

The BRST-exact action is

$$z=\exp\left\{ \frac{r}{\hbar}+i\theta\right\},\qquad Z=\exp\left\{\frac{X}{\hbar} +i\Theta\right\},$$8

which expands to

$$z=\exp\left\{ \frac{r}{\hbar}+i\theta\right\},\qquad Z=\exp\left\{\frac{X}{\hbar} +i\Theta\right\},$$9

Integrating out $\hbar\to 0$0 gives

$\hbar\to 0$1

Because the action is BRST-exact, the path integral localizes to the tropical equations $\hbar\to 0$2, and the semiclassical one-loop approximation in the topological Planck constant $\hbar\to 0$3 is exact. The residual bosonic and fermionic gauge symmetries are

$\hbar\to 0$4

$\hbar\to 0$5

with $\hbar\to 0$6 and $\hbar\to 0$7. These gauge symmetries remove sleeve zero-modes and encode the hybrid local/global structure characteristic of the foliated worldsheet.

Near punctures the Jordan structure becomes singular but remains admissible. In non-adapted coordinates $\hbar\to 0$8, the localization equations admit smooth monomial solutions

$\hbar\to 0$9

which represent sleeves of winding number $\widehat{\varepsilon}$0. This furnishes the local field-theoretic realization of tropical ends.

4. Observables and enumerative content

Observables are constructed by the standard cohomological mechanism: a closed form $\widehat{\varepsilon}$1 on the target defines

$\widehat{\varepsilon}$2

with Stora–Zumino descent

$\widehat{\varepsilon}$3

(Albrychiewicz et al., 2023). In this respect the theory remains squarely within the cohomological sigma-model paradigm.

For the tropical target $\widehat{\varepsilon}$4, the BRST cohomology of point operators is two-dimensional, spanned by the identity and the top-degree class. To avoid divergence in $\widehat{\varepsilon}$5, one uses the $\widehat{\varepsilon}$6-translation-invariant representative

$\widehat{\varepsilon}$7

The instanton number becomes the sleeve winding number,

$\widehat{\varepsilon}$8

The genus-zero correlators with instanton weight $\widehat{\varepsilon}$9 are

$\widehat{J}$0

These values match the relativistic A-model on $\widehat{J}$1. A torus handle with one finite sleeve carries zero winding and collapses to the local operator

$\widehat{J}$2

so that the all-genus partition function becomes

$\widehat{J}$3

This agreement provides the principal explicit check of the construction. At the same time, the initial paper does not introduce Mikhalkin multiplicities or general tropical counts for higher-dimensional toric targets; its concrete enumerative verification is confined to $\widehat{J}$4.

5. Symmetry, nonrelativistic structure, and relation to standard topological sigma models

The worldsheet theory exhibits a nonrelativistic structure “similar to theories of the Lifshitz type” and is governed by foliation-preserving rather than full relativistic conformal symmetry (Albrychiewicz et al., 2023). Before integrating out $\widehat{J}$5, the action is invariant under the transformations preserving the Jordan structure, and the degenerate metric pair $\widehat{J}$6 supplies a compatible notion of anisotropic conformal symmetry. The intersection of the conformal symmetries of $\widehat{J}$7 and $\widehat{J}$8 coincides with the symmetries of the Jordan structure.

Besides the infinite-dimensional foliation-preserving algebra, the model admits finite-dimensional subalgebras. The paper identifies a Heisenberg–Weyl algebra generated by rigid translations and nonrelativistic boosts, as well as the indecomposable solvable algebra $\widehat{J}$9. An anisotropic deformation parameterized by $$\varepsilon_\alpha{}^\beta d\sigma^\alpha\frac{\partial}{\partial\sigma^\beta}=dr\frac{\partial}{\partial\theta},\qquad J_i{}^jdY^i\frac{\partial}{\partial Y^j}=dX\frac{\partial}{\partial \Theta},$$0 yields dynamical exponent

$$\varepsilon_\alpha{}^\beta d\sigma^\alpha\frac{\partial}{\partial\sigma^\beta}=dr\frac{\partial}{\partial\theta},\qquad J_i{}^jdY^i\frac{\partial}{\partial Y^j}=dX\frac{\partial}{\partial \Theta},$$1

although the original tropological model focuses on the isotropic case $$\varepsilon_\alpha{}^\beta d\sigma^\alpha\frac{\partial}{\partial\sigma^\beta}=dr\frac{\partial}{\partial\theta},\qquad J_i{}^jdY^i\frac{\partial}{\partial Y^j}=dX\frac{\partial}{\partial \Theta},$$2, hence $$\varepsilon_\alpha{}^\beta d\sigma^\alpha\frac{\partial}{\partial\sigma^\beta}=dr\frac{\partial}{\partial\theta},\qquad J_i{}^jdY^i\frac{\partial}{\partial Y^j}=dX\frac{\partial}{\partial \Theta},$$3. In the cross-channel, Fourier modes generate a BMS/GCA-type algebra with possible central extensions,

$$\varepsilon_\alpha{}^\beta d\sigma^\alpha\frac{\partial}{\partial\sigma^\beta}=dr\frac{\partial}{\partial\theta},\qquad J_i{}^jdY^i\frac{\partial}{\partial Y^j}=dX\frac{\partial}{\partial \Theta},$$4

$$\varepsilon_\alpha{}^\beta d\sigma^\alpha\frac{\partial}{\partial\sigma^\beta}=dr\frac{\partial}{\partial\theta},\qquad J_i{}^jdY^i\frac{\partial}{\partial Y^j}=dX\frac{\partial}{\partial \Theta},$$5

$$\varepsilon_\alpha{}^\beta d\sigma^\alpha\frac{\partial}{\partial\sigma^\beta}=dr\frac{\partial}{\partial\theta},\qquad J_i{}^jdY^i\frac{\partial}{\partial Y^j}=dX\frac{\partial}{\partial \Theta},$$6

This nonrelativistic structure sharply distinguishes tropological models from standard A- and B-models. In the latter, the worldsheet requires a nondegenerate complex structure and metric, and localization is onto holomorphic or pseudo-holomorphic maps. By contrast, the tropological construction does not require a worldsheet complex structure; it is based on a worldsheet foliation structure (Albrychiewicz et al., 2023). The comparison is nonetheless structurally close: standard topological sigma models are cohomological and metric-independent, and AKSZ furnishes a general construction for many such theories (Ikeda, 2012). Tropological sigma models preserve the BRST-localized and topological features of that setting while replacing complex geometry by tropical Jordan geometry.

The original work treats worldsheet gravity as a fixed, nondynamical background and postpones coupling to tropicalized topological gravity to a sequel. It also raises several open problems: extension beyond $$\varepsilon_\alpha{}^\beta d\sigma^\alpha\frac{\partial}{\partial\sigma^\beta}=dr\frac{\partial}{\partial\theta},\qquad J_i{}^jdY^i\frac{\partial}{\partial Y^j}=dX\frac{\partial}{\partial \Theta},$$7, incorporation of tropical multiplicities, classification of boundary sectors and D-branes, and a refinement of Atiyah–Segal-type axioms appropriate to foliated, nonrelativistic topological quantum field theories (Albrychiewicz et al., 2023).

6. Higher-dimensional targets, filtered manifolds, and nil-equivariant extensions

The higher-dimensional generalization shows that tropological sigma models are not confined to foliated targets. For 4D targets, nested Maslov dequantizations lead generically to filtered manifolds with nonholonomic tangent structure, and the corresponding sigma models acquire enhanced global symmetries described by nilpotent Lie algebras (Albrychiewicz et al., 30 Jul 2025).

The 4D target-space classification is organized by Jordan type:

Jordan structure	Nilpotency/class	Geometric interpretation
$$\varepsilon_\alpha{}^\beta d\sigma^\alpha\frac{\partial}{\partial\sigma^\beta}=dr\frac{\partial}{\partial\theta},\qquad J_i{}^jdY^i\frac{\partial}{\partial Y^j}=dX\frac{\partial}{\partial \Theta},$$8	degree-2 nilpotent	foliation
$$\varepsilon_\alpha{}^\beta d\sigma^\alpha\frac{\partial}{\partial\sigma^\beta}=dr\frac{\partial}{\partial\theta},\qquad J_i{}^jdY^i\frac{\partial}{\partial Y^j}=dX\frac{\partial}{\partial \Theta},$$9	degree-2 nilpotent	foliation
$\varepsilon^2=0,\qquad J^2=0.$0	degree-3 nilpotent	nontrivial filtration
$\varepsilon^2=0,\qquad J^2=0.$1	degree-4 nilpotent	chain structure with constrained winding

A single dequantization of $\varepsilon^2=0,\qquad J^2=0.$2 yields

$\varepsilon^2=0,\qquad J^2=0.$3

while a nested double dequantization, with an intermediate basis rotation, can produce

$\varepsilon^2=0,\qquad J^2=0.$4

For $\varepsilon^2=0,\qquad J^2=0.$5, the target carries the filtration

$\varepsilon^2=0,\qquad J^2=0.$6

with $\varepsilon^2=0,\qquad J^2=0.$7, $\varepsilon^2=0,\qquad J^2=0.$8, and $\varepsilon^2=0,\qquad J^2=0.$9. The associated graded symbol algebra is the step-3 Engel algebra.

In the $$g_{\alpha\beta}d\sigma^\alpha d\sigma^\beta=dr^2,\qquad h^{\alpha\beta}\frac{\partial}{\partial\sigma^\alpha}\frac{\partial}{\partial\sigma^\beta}=\left(\frac{\partial}{\partial\theta}\right)^2,$$0 model, the BRST-exact action takes the form

$$g_{\alpha\beta}d\sigma^\alpha d\sigma^\beta=dr^2,\qquad h^{\alpha\beta}\frac{\partial}{\partial\sigma^\alpha}\frac{\partial}{\partial\sigma^\beta}=\left(\frac{\partial}{\partial\theta}\right)^2,$$1

and after integrating out $$g_{\alpha\beta}d\sigma^\alpha d\sigma^\beta=dr^2,\qquad h^{\alpha\beta}\frac{\partial}{\partial\sigma^\alpha}\frac{\partial}{\partial\sigma^\beta}=\left(\frac{\partial}{\partial\theta}\right)^2,$$2,

$$g_{\alpha\beta}d\sigma^\alpha d\sigma^\beta=dr^2,\qquad h^{\alpha\beta}\frac{\partial}{\partial\sigma^\alpha}\frac{\partial}{\partial\sigma^\beta}=\left(\frac{\partial}{\partial\theta}\right)^2,$$3

The theory possesses additional fermionic symmetries with charges $$g_{\alpha\beta}d\sigma^\alpha d\sigma^\beta=dr^2,\qquad h^{\alpha\beta}\frac{\partial}{\partial\sigma^\alpha}\frac{\partial}{\partial\sigma^\beta}=\left(\frac{\partial}{\partial\theta}\right)^2,$$4 satisfying

$$g_{\alpha\beta}d\sigma^\alpha d\sigma^\beta=dr^2,\qquad h^{\alpha\beta}\frac{\partial}{\partial\sigma^\alpha}\frac{\partial}{\partial\sigma^\beta}=\left(\frac{\partial}{\partial\theta}\right)^2,$$5

with all other commutators vanishing. This realizes the 4D step-3 Engel algebra on field space.

To regularize the noncompact symmetry, the construction passes to the simply connected Engel group $$g_{\alpha\beta}d\sigma^\alpha d\sigma^\beta=dr^2,\qquad h^{\alpha\beta}\frac{\partial}{\partial\sigma^\alpha}\frac{\partial}{\partial\sigma^\beta}=\left(\frac{\partial}{\partial\theta}\right)^2,$$6 and a lattice $$g_{\alpha\beta}d\sigma^\alpha d\sigma^\beta=dr^2,\qquad h^{\alpha\beta}\frac{\partial}{\partial\sigma^\alpha}\frac{\partial}{\partial\sigma^\beta}=\left(\frac{\partial}{\partial\theta}\right)^2,$$7, producing the compact nilmanifold $$g_{\alpha\beta}d\sigma^\alpha d\sigma^\beta=dr^2,\qquad h^{\alpha\beta}\frac{\partial}{\partial\sigma^\alpha}\frac{\partial}{\partial\sigma^\beta}=\left(\frac{\partial}{\partial\theta}\right)^2,$$8. The left-invariant Maurer–Cartan forms are

$$g_{\alpha\beta}d\sigma^\alpha d\sigma^\beta=dr^2,\qquad h^{\alpha\beta}\frac{\partial}{\partial\sigma^\alpha}\frac{\partial}{\partial\sigma^\beta}=\left(\frac{\partial}{\partial\theta}\right)^2,$$9

with

$$g_{\alpha\beta}h^{\beta\gamma}=0,\qquad h^{\alpha\beta}g_{\beta\gamma}=0.$$0

This nilmanifold symmetry is then used to define a nil-equivariant tropological sigma model. Introducing Grassmann-even Nil ghosts $$g_{\alpha\beta}h^{\beta\gamma}=0,\qquad h^{\alpha\beta}g_{\beta\gamma}=0.$$1, the equivariant BRST differential acts by

$$g_{\alpha\beta}h^{\beta\gamma}=0,\qquad h^{\alpha\beta}g_{\beta\gamma}=0.$$2

$$g_{\alpha\beta}h^{\beta\gamma}=0,\qquad h^{\alpha\beta}g_{\beta\gamma}=0.$$3

so that

$$g_{\alpha\beta}h^{\beta\gamma}=0,\qquad h^{\alpha\beta}g_{\beta\gamma}=0.$$4

The equivariant symplectic form is

$$g_{\alpha\beta}h^{\beta\gamma}=0,\qquad h^{\alpha\beta}g_{\beta\gamma}=0.$$5

and the basic equivariant observable is

$$g_{\alpha\beta}h^{\beta\gamma}=0,\qquad h^{\alpha\beta}g_{\beta\gamma}=0.$$6

This framework leads to the conjectural notion of filtered Gromov–Witten invariants. The proposal is that nil-equivariant correlators in the $$g_{\alpha\beta}h^{\beta\gamma}=0,\qquad h^{\alpha\beta}g_{\beta\gamma}=0.$$7-cohomology define invariants of filtered manifolds, refining ordinary Gromov–Witten theory by incorporating the filtration and its symbol algebra. The higher-dimensional theory therefore extends the original tropological program from foliated tropical worldsheets to anisotropic filtered geometries with explicit nilpotent symmetry, while preserving the BRST-localized and metric-independent character of the construction (Albrychiewicz et al., 30 Jul 2025).