Equivariant Tropological Sigma Models

• Equivariant tropological sigma models are cohomological field theories that incorporate nilpotent Jordan structures and equivariant nilmanifold regularization to define filtered target geometries.
• They replace standard complex structures with nested Maslov dequantizations, leading to refined localization equations and an exotic BRST symmetry modeled on the step-3 Engel algebra.
• This framework conjecturally produces filtered Gromov–Witten invariants, capturing advanced enumerative geometry features on nonholonomic, filtered manifolds.

Searching arXiv for the cited paper and closely related background on equivariant sigma models. Equivariant tropological sigma models are a class of cohomological sigma models in which the complex structures of both worldsheet and target are replaced, after Maslov dequantization, by nilpotent Jordan structures, and in which the resulting theory is further extended equivariantly with respect to a nilpotent symmetry associated to the target filtration structure. In the formulation developed for four-dimensional targets, the central example is the nil-equivariant extension of tropological sigma models on filtered manifolds induced by the $J_{3,1}$ Jordan type, where the target acquires a nonholonomic filtration with Engel symbol algebra rather than an ordinary foliation (Albrychiewicz et al., 30 Jul 2025). The theory is a BRST-type topological field theory whose localization equations tropicalize the pseudoholomorphic map equations of the A-model, while the equivariant extension replaces compact-group equivariance by a nilmanifold regularization of a noncompact nilpotent symmetry group and is conjectured to define filtered Gromov–Witten invariants on filtered manifolds (Albrychiewicz et al., 30 Jul 2025).

1. Tropological sigma models and Maslov dequantization

Tropological sigma models arise from tropicalizing the A-model localization equations while retaining both logarithmic moduli and phases of complex coordinates. Locally, on a complex curve, one writes

$z = e^{\frac{r}{\hbar} + i\theta},$

and on the target

$Z = \exp\Big(\frac{X}{\hbar} + i\Theta\Big).$

Instead of quotienting by phase, the construction keeps both $r,\theta$ and $X,\Theta$, so the tropical limit $\hbar\to 0$ produces smooth manifolds equipped not with honest complex structures but with nilpotent endomorphisms (Albrychiewicz et al., 30 Jul 2025).

In the two-dimensional worldsheet case, the tropical limit of the worldsheet complex structure gives a nilpotent endomorphism

$$\epsilon : T\Sigma \to T\Sigma,\qquad \epsilon(\partial_r) = \partial_\theta,\quad \epsilon(\partial_\theta)=0,\quad \epsilon^2 =0.$$

This is a second-order Jordan structure. It induces the filtration

$$0 \subset \ker\epsilon = \operatorname{im}\epsilon \subset T\Sigma,$$

which in two dimensions is equivalent to a foliation by curves tangent to $\ker\epsilon$ (Albrychiewicz et al., 30 Jul 2025).

The tropological localization equations are obtained by tropicalizing the A-model pseudoholomorphicity equations,

$$E_\alpha{}^i = \varepsilon_\alpha{}^\beta\,\partial_\beta Y^i - J_j{}^i\,\partial_\alpha Y^j = 0,$$

with $z = e^{\frac{r}{\hbar} + i\theta},$0 the worldsheet Jordan structure and $z = e^{\frac{r}{\hbar} + i\theta},$1 the tropical target structure. In the simplest two-dimensional case this gives

$z = e^{\frac{r}{\hbar} + i\theta},$2

The cohomological action is

$z = e^{\frac{r}{\hbar} + i\theta},$3

with BRST multiplet

$z = e^{\frac{r}{\hbar} + i\theta},$4

and matter fields $z = e^{\frac{r}{\hbar} + i\theta},$5 with fermionic partners $z = e^{\frac{r}{\hbar} + i\theta},$6 (Albrychiewicz et al., 30 Jul 2025).

For two-dimensional targets such as $z = e^{\frac{r}{\hbar} + i\theta},$7, the tropological sigma model is described as an anisotropic or nonrelativistic limit of the A-model, its BRST observables correspond to the same de Rham classes as in the A-model, and its correlators reproduce the usual Gromov–Witten invariants of the original complex target, in agreement with Mikhalkin’s tropical curve counts (Albrychiewicz et al., 2023). This establishes the baseline meaning of “tropological”: topological field theory formulated with nilpotent Jordan structures and tropical pseudoholomorphicity rather than ordinary complex geometry (Albrychiewicz et al., 2023).

2. Four-dimensional Jordan structures and filtered targets

For four real target dimensions, the nilpotent endomorphisms are classified by Jordan type. The relevant nilpotent forms are

$z = e^{\frac{r}{\hbar} + i\theta},$8

with

$z = e^{\frac{r}{\hbar} + i\theta},$9

A single Maslov dequantization of both complex directions of $Z = \exp\Big(\frac{X}{\hbar} + i\Theta\Big).$0 gives

$Z = \exp\Big(\frac{X}{\hbar} + i\Theta\Big).$1

but the $Z = \exp\Big(\frac{X}{\hbar} + i\Theta\Big).$2 type cannot be reached by a single Maslov dequantization. Instead, it requires a nested Maslov dequantization: first obtaining something equivalent to $Z = \exp\Big(\frac{X}{\hbar} + i\Theta\Big).$3, then rotating coordinates, then applying a second Maslov limit to remove an additional entry (Albrychiewicz et al., 30 Jul 2025).

This distinction is geometrically decisive. The types $Z = \exp\Big(\frac{X}{\hbar} + i\Theta\Big).$4 and $Z = \exp\Big(\frac{X}{\hbar} + i\Theta\Big).$5 lead to integrable distributions and hence foliated geometries. By contrast, $Z = \exp\Big(\frac{X}{\hbar} + i\Theta\Big).$6 produces a genuine filtration

$Z = \exp\Big(\frac{X}{\hbar} + i\Theta\Big).$7

with

$Z = \exp\Big(\frac{X}{\hbar} + i\Theta\Big).$8

Equivalently,

$Z = \exp\Big(\frac{X}{\hbar} + i\Theta\Big).$9

and the associated graded is

$r,\theta$0

The graded Lie algebra is nonabelian; it is precisely the four-dimensional step-3 Engel algebra (Albrychiewicz et al., 30 Jul 2025).

A filtered manifold in this sense is a manifold $r,\theta$1 equipped with a flag of subbundles

$r,\theta$2

such that Lie brackets respect the filtration and define a symbol algebra

$r,\theta$3

For $r,\theta$4, the target is therefore not foliated in the same sense as the two-dimensional case but filtered and nonholonomic, analogous to the Engel distribution of four-dimensional sub-Riemannian geometry (Albrychiewicz et al., 30 Jul 2025).

This shift from foliation to filtration is the core technical distinction behind equivariant tropological sigma models in four dimensions. A plausible implication is that the relevant enumerative geometry is no longer standard tropical GW theory on foliated spaces but a refinement sensitive to the symbol algebra of the filtration.

3. Localization equations on filtered geometries

For the doubled foliation type $r,\theta$5, the theory decomposes into two independent two-dimensional sectors $r,\theta$6 and $r,\theta$7, and the localization equations are just two copies of tropical pseudoholomorphicity: $r,\theta$8 The target geometry is then effectively $r,\theta$9 as a foliated manifold (Albrychiewicz et al., 30 Jul 2025).

For the exotic $X,\Theta$0 case, the localization equations become hierarchical: $X,\Theta$1 The dependence of $X,\Theta$2 on $X,\Theta$3 is not a direct product structure: $X,\Theta$4 couples to $X,\Theta$5, reflecting the filtration tower (Albrychiewicz et al., 30 Jul 2025).

The fields remain ordinary smooth maps $X,\Theta$6 into the underlying $X,\Theta$7 manifold, but the target filtration enters through the localization equations and the induced global symmetry algebra on field space. This is structurally reminiscent of AKSZ-type sigma models in which target geometry is encoded by a dg or algebroid structure rather than by an ordinary Riemannian metric (Kotov et al., 2010). The paper on algebroids and sigma models emphasizes that topological sigma models are naturally organized by target-side algebraic structures such as Lie algebroids, Courant algebroids, and Dirac structures, with field equations expressing morphism conditions (Kotov et al., 2010). This suggests, by analogy, that filtered tropological targets play an analogous organizing role for the nilpotent Jordan framework.

4. BRST structure and Engel symmetry

For the four-dimensional target, the BRST action on matter is

$X,\Theta$8

with

$X,\Theta$9

Antighosts and auxiliaries satisfy

$\hbar\to 0$0

After gauge fixing, only

$\hbar\to 0$1

and

$\hbar\to 0$2

remain (Albrychiewicz et al., 30 Jul 2025).

For $\hbar\to 0$3, the action before integrating out auxiliaries is

$\hbar\to 0$4

Beyond the BRST charge $\hbar\to 0$5, the theory has four fermionic symmetry generators $\hbar\to 0$6 whose associated charges $\hbar\to 0$7 obey

$\hbar\to 0$8

with all other commutators vanishing. This is exactly the four-dimensional step-3 Engel algebra, isomorphic to $\hbar\to 0$9 in Mubarakzyanov’s classification (Albrychiewicz et al., 30 Jul 2025).

The paper identifies this field-space symmetry algebra with the symbol algebra of the filtered tangent bundle induced by $$\epsilon : T\Sigma \to T\Sigma,\qquad \epsilon(\partial_r) = \partial_\theta,\quad \epsilon(\partial_\theta)=0,\quad \epsilon^2 =0.$$0. This is the central reason nil-equivariant extension becomes natural: the additional fermionic symmetries are not accidental but encode the filtered geometry itself. In that respect, the construction is close in spirit to the role of equivariant cohomology in organizing sigma-model currents and target-space symmetries when those symmetries project to target vector fields (Bernardes et al., 2023). There, the transformation of currents is encoded in a target-space bicomplex $$\epsilon : T\Sigma \to T\Sigma,\qquad \epsilon(\partial_r) = \partial_\theta,\quad \epsilon(\partial_\theta)=0,\quad \epsilon^2 =0.$$1 with differential $$\epsilon : T\Sigma \to T\Sigma,\qquad \epsilon(\partial_r) = \partial_\theta,\quad \epsilon(\partial_\theta)=0,\quad \epsilon^2 =0.$$2, and obstructions or descendants are packaged cohomologically (Bernardes et al., 2023). The nil-equivariant tropological construction can be viewed as a specialized realization of this general paradigm, with the Lie algebra replaced by the Engel algebra arising from filtration.

5. Nilmanifold regularization and the nil-equivariant extension

The global Engel symmetry integrates to a simply connected nilpotent Lie group $$\epsilon : T\Sigma \to T\Sigma,\qquad \epsilon(\partial_r) = \partial_\theta,\quad \epsilon(\partial_\theta)=0,\quad \epsilon^2 =0.$$3, but this group is noncompact. The construction therefore introduces a discrete cocompact subgroup $$\epsilon : T\Sigma \to T\Sigma,\qquad \epsilon(\partial_r) = \partial_\theta,\quad \epsilon(\partial_\theta)=0,\quad \epsilon^2 =0.$$4, yielding a compact nilmanifold

$$\epsilon : T\Sigma \to T\Sigma,\qquad \epsilon(\partial_r) = \partial_\theta,\quad \epsilon(\partial_\theta)=0,\quad \epsilon^2 =0.$$5

This step is necessary because there are no nontrivial compact Lie subalgebras of a nilpotent Lie algebra with a positive definite invariant inner product; any compact subgroup of $$\epsilon : T\Sigma \to T\Sigma,\qquad \epsilon(\partial_r) = \partial_\theta,\quad \epsilon(\partial_\theta)=0,\quad \epsilon^2 =0.$$6 lies in the center and must be discrete (Albrychiewicz et al., 30 Jul 2025).

Using generators $$\epsilon : T\Sigma \to T\Sigma,\qquad \epsilon(\partial_r) = \partial_\theta,\quad \epsilon(\partial_\theta)=0,\quad \epsilon^2 =0.$$7 of the Engel algebra,

$$\epsilon : T\Sigma \to T\Sigma,\qquad \epsilon(\partial_r) = \partial_\theta,\quad \epsilon(\partial_\theta)=0,\quad \epsilon^2 =0.$$8

one writes exponential coordinates of the second kind and obtains the group law by BCH. After rescaling $$\epsilon : T\Sigma \to T\Sigma,\qquad \epsilon(\partial_r) = \partial_\theta,\quad \epsilon(\partial_\theta)=0,\quad \epsilon^2 =0.$$9 by $$0 \subset \ker\epsilon = \operatorname{im}\epsilon \subset T\Sigma,$$0, the lattice

$$0 \subset \ker\epsilon = \operatorname{im}\epsilon \subset T\Sigma,$$1

is closed under the group law, producing a compact nilmanifold $$0 \subset \ker\epsilon = \operatorname{im}\epsilon \subset T\Sigma,$$2 (Albrychiewicz et al., 30 Jul 2025).

The left-invariant one-forms are

$$0 \subset \ker\epsilon = \operatorname{im}\epsilon \subset T\Sigma,$$3

with

$$0 \subset \ker\epsilon = \operatorname{im}\epsilon \subset T\Sigma,$$4

The dual Killing vector fields are

$$0 \subset \ker\epsilon = \operatorname{im}\epsilon \subset T\Sigma,$$5

satisfying

$$0 \subset \ker\epsilon = \operatorname{im}\epsilon \subset T\Sigma,$$6

up to normalization (Albrychiewicz et al., 30 Jul 2025).

The equivariant BRST differential $$0 \subset \ker\epsilon = \operatorname{im}\epsilon \subset T\Sigma,$$7 is then defined by adding Grassmann-even Nil ghosts $$0 \subset \ker\epsilon = \operatorname{im}\epsilon \subset T\Sigma,$$8 of ghost number $$0 \subset \ker\epsilon = \operatorname{im}\epsilon \subset T\Sigma,$$9. On matter fields,

$\ker\epsilon$0

On the antighost multiplet,

$\ker\epsilon$1

As usual in the Cartan model,

$\ker\epsilon$2

on all fields (Albrychiewicz et al., 30 Jul 2025).

With the same gauge-fixing fermion as in the non-equivariant $\ker\epsilon$3 model,

$\ker\epsilon$4

one defines

$\ker\epsilon$5

To ensure off-shell $\ker\epsilon$6-invariance, one adds

$\ker\epsilon$7

so the total action

$\ker\epsilon$8

is $\ker\epsilon$9-invariant (Albrychiewicz et al., 30 Jul 2025).

The localization locus is the intersection of the tropical localization equations $$E_\alpha{}^i = \varepsilon_\alpha{}^\beta\,\partial_\beta Y^i - J_j{}^i\,\partial_\alpha Y^j = 0,$$0 with the fixed-point conditions of the nilpotent symmetry generated by $$E_\alpha{}^i = \varepsilon_\alpha{}^\beta\,\partial_\beta Y^i - J_j{}^i\,\partial_\alpha Y^j = 0,$$1. This is the precise sense in which the theory becomes nil-equivariant.

6. Observables, equivariant cohomology, and filtered Gromov–Witten invariants

For the nonequivariant theory on $$E_\alpha{}^i = \varepsilon_\alpha{}^\beta\,\partial_\beta Y^i - J_j{}^i\,\partial_\alpha Y^j = 0,$$2, the observables are inherited from the usual A-model. The cohomology ring is

$$E_\alpha{}^i = \varepsilon_\alpha{}^\beta\,\partial_\beta Y^i - J_j{}^i\,\partial_\alpha Y^j = 0,$$3

with primary observables corresponding to degree $$E_\alpha{}^i = \varepsilon_\alpha{}^\beta\,\partial_\beta Y^i - J_j{}^i\,\partial_\alpha Y^j = 0,$$4, $$E_\alpha{}^i = \varepsilon_\alpha{}^\beta\,\partial_\beta Y^i - J_j{}^i\,\partial_\alpha Y^j = 0,$$5, and $$E_\alpha{}^i = \varepsilon_\alpha{}^\beta\,\partial_\beta Y^i - J_j{}^i\,\partial_\alpha Y^j = 0,$$6 classes. For the degree-2 observable,

$$E_\alpha{}^i = \varepsilon_\alpha{}^\beta\,\partial_\beta Y^i - J_j{}^i\,\partial_\alpha Y^j = 0,$$7

and the descent tower is

$$E_\alpha{}^i = \varepsilon_\alpha{}^\beta\,\partial_\beta Y^i - J_j{}^i\,\partial_\alpha Y^j = 0,$$8

The paper states that in the tropological limit the same form of observables survives, while the target symplectic form becomes a tropical distributional object; the ring of observables remains identified with $$E_\alpha{}^i = \varepsilon_\alpha{}^\beta\,\partial_\beta Y^i - J_j{}^i\,\partial_\alpha Y^j = 0,$$9 (Albrychiewicz et al., 30 Jul 2025).

In the nil-equivariant model, physical observables lie in the equivariant BRST cohomology

$z = e^{\frac{r}{\hbar} + i\theta},$00

computed in the Cartan model with Nil ghosts $z = e^{\frac{r}{\hbar} + i\theta},$01. Assuming the $z = e^{\frac{r}{\hbar} + i\theta},$02-action is Hamiltonian with moment maps $z = e^{\frac{r}{\hbar} + i\theta},$03 satisfying

$z = e^{\frac{r}{\hbar} + i\theta},$04

the equivariant symplectic form is

$z = e^{\frac{r}{\hbar} + i\theta},$05

The basic observable becomes

$z = e^{\frac{r}{\hbar} + i\theta},$06

and higher components are obtained by equivariant descent (Albrychiewicz et al., 30 Jul 2025).

The paper argues that the resulting equivariant ring is

$z = e^{\frac{r}{\hbar} + i\theta},$07

where $z = e^{\frac{r}{\hbar} + i\theta},$08 is the equivariant extension of the Kähler class (Albrychiewicz et al., 30 Jul 2025). This parallels standard equivariant A-model constructions, where local observables are equivariant cohomology classes and localization computes equivariant GW invariants. The general framework of equivariant A-twists on $z = e^{\frac{r}{\hbar} + i\theta},$09 shows that equivariant parameters deform quantum cohomology relations into difference or Picard–Fuchs-type equations, with correlation functions computed by localization residues (Closset et al., 2015). A plausible implication is that nil-equivariant tropological sigma models should admit an analogous localization formalism once the moduli problem of filtered tropical maps is made explicit.

The central conjecture is that the nil-equivariant correlators define filtered Gromov–Witten invariants: GW-type intersection numbers refined by the nilpotent Engel symmetry and the target filtration (Albrychiewicz et al., 30 Jul 2025). No explicit closed formulas are given. The proposal is instead structural: filtered manifolds equipped with a Jordan structure and associated nilpotent symmetry should admit invariants generalizing equivariant GW invariants for compact Lie group actions, but now with nilpotent noncompact groups regularized through nilmanifold quotients (Albrychiewicz et al., 30 Jul 2025).

7. Conceptual position and related frameworks

The term “equivariant tropological sigma model” as used here is highly specific: it denotes the nil-equivariant extension of a tropological sigma model whose target carries a filtration induced by an exotic Jordan structure such as $z = e^{\frac{r}{\hbar} + i\theta},$10, with equivariance implemented via the associated nilpotent group and compactified nilmanifold quotient (Albrychiewicz et al., 30 Jul 2025). It is not merely an A-model with an $z = e^{\frac{r}{\hbar} + i\theta},$11-background, nor simply a tropicalized version of an ordinary equivariant sigma model.

Nonetheless, several adjacent frameworks clarify its meaning.

The AKSZ and algebroid perspective emphasizes that topological sigma models are organized by target-side dg or algebroid data, with classical solutions described as morphisms of geometric-algebraic structures (Kotov et al., 2010). This is relevant because the filtered manifold plus symbol algebra in the tropological setting plays a comparable structural role, even though the paper does not phrase the model explicitly as an AKSZ theory. This suggests that a future intrinsic formulation of filtered GW theory may require algebroidal or higher-geometric language.

The literature on equivariant cohomology of sigma-model currents shows how target symmetries project to vector fields, produce descent towers, and are encoded by bicomplexes combining de Rham and Lie algebra differentials (Bernardes et al., 2023). The nil-equivariant BRST operator $z = e^{\frac{r}{\hbar} + i\theta},$12 fits naturally into this pattern, with the crucial modification that the symmetry algebra is the Engel algebra and the equivariant parameter space is nilpotent rather than reductive.

Work on equivariant gerbes and WZ sigma models shows that full gauging of topological terms requires higher equivariant structures, often encoded simplicially or as equivariant gerbes (Murray et al., 2015, Gawedzki et al., 2010). Although nil-equivariant tropological sigma models are not presented as gerbe-gauged WZ models, the need to regularize a noncompact symmetry by a lattice and descend to a nilmanifold strongly echoes the geometric logic of equivariant gerbes on quotient stacks and descent groupoids (Murray et al., 2015).

The original tropological sigma model on two-dimensional targets provides the non-equivariant baseline: it is a BRST TFT with worldsheet foliation rather than worldsheet complex structure, reproducing usual GW invariants of $z = e^{\frac{r}{\hbar} + i\theta},$13 while localizing onto tropical maps (Albrychiewicz et al., 2023). The 2025 extension shows that once higher-dimensional nested Maslov dequantizations are allowed, the geometry ceases generically to be foliated and becomes filtered; the corresponding field theory acquires Engel symmetry and therefore admits a new kind of equivariance (Albrychiewicz et al., 30 Jul 2025).

A common misconception would be to assume that all tropological sigma models live on foliated targets by direct analogy with the two-dimensional case. The four-dimensional classification shows this is false: only $z = e^{\frac{r}{\hbar} + i\theta},$14 and related degenerate cases remain foliated, whereas $z = e^{\frac{r}{\hbar} + i\theta},$15 is intrinsically filtered and nonholonomic (Albrychiewicz et al., 30 Jul 2025). Another misconception would be to think the equivariant extension uses an ordinary compact group action. Instead, the relevant symmetry is noncompact and nilpotent, and compactness is restored only after passing to a cocompact lattice and nilmanifold quotient (Albrychiewicz et al., 30 Jul 2025).

Within the four-dimensional classification, all inequivalent tropological sigma models are indexed by nilpotent Jordan forms:

1. $z = e^{\frac{r}{\hbar} + i\theta},$16, effectively the $z = e^{\frac{r}{\hbar} + i\theta},$17 model with spectators.
2. $z = e^{\frac{r}{\hbar} + i\theta},$18, the doubled foliated model.
3. $z = e^{\frac{r}{\hbar} + i\theta},$19, a more degenerate foliated case.
4. $z = e^{\frac{r}{\hbar} + i\theta},$20, the filtered exotic case admitting nil-equivariant extension (Albrychiewicz et al., 30 Jul 2025).

This isolates the $z = e^{\frac{r}{\hbar} + i\theta},$21 theory as the canonical instance of an equivariant tropological sigma model in the strict sense.

8. Outlook

The present state of the subject is structurally rich but still conjectural at the enumerative level. The nil-equivariant extension is fully specified as a BRST theory with compact nilmanifold symmetry and Cartan-type differential $z = e^{\frac{r}{\hbar} + i\theta},$22, but explicit evaluations of filtered GW invariants are not yet available (Albrychiewicz et al., 30 Jul 2025). The paper instead formulates the conjecture that these correlators refine standard GW invariants by recording dependence on the filtered symbol algebra and nilpotent equivariant parameters.

The most immediate open directions stated are rigorous construction and localization of filtered GW invariants, understanding the moduli space of Jordan structures and its relation to enhanced global symmetries, and extending the framework to higher-dimensional targets where nested Maslov dequantizations should yield higher-step filtrations and more complicated nilpotent symmetry algebras (Albrychiewicz et al., 30 Jul 2025). This suggests that the four-dimensional Engel case is likely only the first nontrivial member of a broader hierarchy of filtered enumerative theories.

In that sense, equivariant tropological sigma models sit at an intersection of tropical geometry, cohomological TFT, filtered and sub-Riemannian geometry, nilpotent Lie theory, and equivariant enumerative geometry. Their defining move is the replacement of target complex structure by a nested nilpotent Jordan structure whose symbol algebra survives in field space as a nilpotent symmetry, and whose equivariant refinement is expected to generate a new class of filtered Gromov–Witten invariants (Albrychiewicz et al., 30 Jul 2025).