T-Manifolds in Geometry and Physics

• T-Manifolds are a family of objects characterized by torus actions, T-duality, and doubled geometry across various mathematical and physical contexts.
• They encompass smooth manifolds with effective torus actions in topology, Hamiltonian and Kähler structures in geometry, and non-geometric fibrations in string theory.
• These diverse constructions enable advances in quantization, symmetry analysis, and special holonomy, linking classical methods with modern theoretical frameworks.

In the literature represented here, the expression T-manifold is context-dependent rather than uniform. In equivariant topology it often means a smooth manifold equipped with an effective torus action; in Hamiltonian geometry it typically means a symplectic or Kähler manifold with an effective Hamiltonian torus action and moment map; in Hamiltonian dynamics it can denote local manifolds of $T$-relative equilibria; and in string theory it may refer to non-geometric torus fibrations, often called T-folds when T-duality is the only duality used. A further doubled-geometric usage identifies the total space of $TM\oplus T^*M$ as a big-tangent manifold endowed with canonical tensorial structures. These usages are related by their reliance on torus symmetry, torus fibrations, or $T$-duality, but they are not equivalent notions (Sarkar et al., 2019, Leung et al., 2023, Achmed-Zade et al., 2018, Vaisman, 2013).

1. Core meanings and terminological range

In equivariant topology, a $T$-manifold means a smooth manifold $M$ equipped with an effective smooth action of a torus $T$ (Sarkar et al., 2019). In the closely related language of locally standard $T^k$-manifolds, a smooth $T^k$-manifold is again a smooth manifold with a smooth effective $T^k$-action (Wiemeler, 2020). In Hamiltonian Kähler geometry, the basic setting is a compact Kähler manifold $(M,\omega,J)$ with an effective Hamiltonian $TM\oplus T^*M$0-action by isometries and a moment map $TM\oplus T^*M$1 (Leung et al., 2023). In string theory, by contrast, T-manifolds are often called T-folds and are defined by allowing transition functions in the T-duality group $TM\oplus T^*M$2 rather than only geometric transition functions (Achmed-Zade et al., 2018).

Context	Meaning of “T-manifold”	Hallmark data
Equivariant topology	Smooth manifold with effective torus action	$TM\oplus T^*M$3 (Sarkar et al., 2019)
Hamiltonian/Kähler geometry	Compact Kähler or symplectic manifold with Hamiltonian torus action	$TM\oplus T^*M$4 (Leung et al., 2023)
Hamiltonian dynamics	Local manifold of $TM\oplus T^*M$5-relative equilibria	$TM\oplus T^*M$6 (Sommerfeld, 2020)
String theory	T-fold / non-geometric torus-fibered background	$TM\oplus T^*M$7-valued monodromy (Achmed-Zade et al., 2018)
Generalized geometry	Big-tangent manifold $TM\oplus T^*M$8	canonical $TM\oplus T^*M$9 (Vaisman, 2013)

This range already indicates that the letter $T$0 may denote a torus, a maximal torus, the T-duality group, or the doubled tangent object $T$1. Accordingly, the term is best read relative to its research program.

2. Hamiltonian and Kähler $T$2-manifolds

A Hamiltonian $T$3-manifold in the Kähler setting consists of a compact Kähler manifold $T$4, an effective Hamiltonian torus action

$T$5

and a moment map $T$6 satisfying

$T$7

If $T$8 is a $T$9-invariant pre-quantum line bundle with $T$0, the induced infinitesimal action on sections is

$T$1

This is the basic Hamiltonian $T$2-manifold framework used by Leung–Wang (Leung et al., 2023).

A central construction in that setting is the mixed polarization

$T$3

where $T$4, $T$5, and $T$6. On toric manifolds, where $T$7, this reduces to the singular real polarization generated by the torus fibers of the moment map; for $T$8, it is genuinely mixed (Leung et al., 2023). The associated quantum space is defined distributionally by

$T$9

and for $M$0 one obtains a decomposition

$M$1

with supports inside $M$2. For regular integral $M$3,

$M$4

so geometric quantization commutes with symplectic reduction in this mixed-polarization picture (Leung et al., 2023).

A second major Hamiltonian usage appears in complexity one geometry. A Hamiltonian $M$5-manifold $M$6 has complexity one if

$M$7

For compact, connected, Kähler complexity one Hamiltonian $M$8-manifolds, every painting is trivial; as a corollary, two tall compact, connected Kähler complexity one Hamiltonian $M$9-manifolds are symplectomorphic exactly if they have the same genus, Duistermaat–Heckman measure, and skeleton (Charton et al., 12 Mar 2026). This places a strong rigidity constraint on the Kähler subclass inside the broader Karshon–Tolman framework.

3. Topological and toric-topological frameworks

In equivariant topology, a broad class of $T$0-manifolds is provided by locally $T$1-standard $T$2-manifolds. These are $T$3-dimensional smooth manifolds with effective $T$4-action that are locally modeled on $T$5 with the $T$6-standard action. Their orbit spaces are manifolds with corners of dimension $T$7, and when the orbit space is a simple polytope $T$8, the action is encoded by a hyper characteristic function

$T$9

satisfying the direct-summand condition at each vertex (Sarkar et al., 2019). The associated model space is

$T^k$0

and every locally $T^k$1-standard $T^k$2-manifold with simple polytope orbit is equivariantly homeomorphic to some $T^k$3 (Sarkar et al., 2019).

This framework includes quasitoric manifolds ($T^k$4), moment-angle manifolds, good contact toric manifolds ($T^k$5), and hyperplane cuts of quasitoric manifolds (Sarkar et al., 2019). Under maximal-rank hypotheses, the equivariant cohomology ring is

$T^k$6

and the $T^k$7-algebra structure recovers the hyper characteristic vectors $T^k$8. Sarkar and Song prove an equivariant cohomological rigidity theorem: under the stated direct-summand condition, weak equivariant homeomorphism is equivalent to weak isomorphism of equivariant cohomology algebras as $T^k$9-algebras (Sarkar et al., 2019).

A related smooth-classification result is due to Wiemeler for locally standard $T^k$0-manifolds with a section of the orbit map $T^k$1. In this setting, $T^k$2 is a smooth manifold with corners, the isotropy data define a characteristic function $T^k$3 on faces, and the pair $T^k$4 determines the $T^k$5-manifold up to equivariant diffeomorphism once a section exists (Wiemeler, 2020).

Buchstaber–Terzić introduce a further axiomatic package for $T^k$6-manifolds: a smooth compact $T^k$7-manifold with effective $T^k$8-action, an open almost moment map $T^k$9 whose image is a convex polytope $T^k$0, and six axioms governing strata, admissible polytopes, parameter spaces, and gluing maps. The complexity is

$T^k$1

From these data they construct a model space $T^k$2 with a $T^k$3-action and a $T^k$4-equivariant homeomorphism $T^k$5, inducing $T^k$6. Their theory contains toric geometry and toric topology when $T^k$7, and also covers positive-complexity examples such as complex Grassmann manifolds $T^k$8 (Buchstaber et al., 2018).

4. $T^k$9-manifolds of relative equilibria

In equivariant Hamiltonian dynamics, the phrase denotes a different object. Let $(M,\omega,J)$0 be a compact connected Lie group, $(M,\omega,J)$1 a maximal torus, and $(M,\omega,J)$2 a symplectic $(M,\omega,J)$3-representation modeling a neighborhood of a completely symmetric equilibrium. For $(M,\omega,J)$4, define

$(M,\omega,J)$5

Under the generic conditions denoted (GC) and (NR′), and when the weights in $(M,\omega,J)$6 are linearly independent and satisfy the maximality condition of Theorem 2.15, there exists a local $(M,\omega,J)$7-invariant manifold of $(M,\omega,J)$8-relative equilibria tangent to $(M,\omega,J)$9 at the origin (Sommerfeld, 2020).

The main result of that paper is an isotropy-preserving normal form: for sufficiently small $TM\oplus T^*M$00, the map

$TM\oplus T^*M$01

defines a manifold $TM\oplus T^*M$02 of $TM\oplus T^*M$03-relative equilibria with $TM\oplus T^*M$04 and

$TM\oplus T^*M$05

Thus the local $TM\oplus T^*M$06-isotropy structure of the nonlinear manifold is modeled exactly by the linear $TM\oplus T^*M$07-action on $TM\oplus T^*M$08 (Sommerfeld, 2020).

The $TM\oplus T^*M$09-orbit of the union of these $TM\oplus T^*M$10-manifolds is Whitney-stratified by isotropy type. For isotropy type $TM\oplus T^*M$11, the corresponding stratum has dimension

$TM\oplus T^*M$12

where $TM\oplus T^*M$13 is the orthogonal complement of $TM\oplus T^*M$14 and $TM\oplus T^*M$15 is the minimal adjoint isotropy subgroup of an element of $TM\oplus T^*M$16 containing $TM\oplus T^*M$17 (Sommerfeld, 2020). This usage is local, representation-theoretic, and bifurcation-theoretic rather than global and topological.

5. T-duality, T-folds, and doubled geometry

In string theory, T-manifolds are often called T-folds when T-duality is the only duality used. They are torus-fibered backgrounds whose transition functions are allowed to lie in the T-duality group

$TM\oplus T^*M$18

rather than only in the mapping class group of the torus fiber (Achmed-Zade et al., 2018). For $TM\oplus T^*M$19, the relevant group is $TM\oplus T^*M$20, and the paper studies $TM\oplus T^*M$21-fibered T-folds together with the map

$TM\oplus T^*M$22

This realizes certain $TM\oplus T^*M$23-fibered T-folds as geometric $TM\oplus T^*M$24-fibrations in a dual picture. The monodromies around duality defects split into geometric diffeomorphisms, $TM\oplus T^*M$25-shifts, and genuinely non-geometric $TM\oplus T^*M$26-transformations (Achmed-Zade et al., 2018).

A dg-geometric formulation of T-duality is developed by Lupercio–Rengifo–Uribe. They consider dg-manifolds that are $TM\oplus T^*M$27-bundles over $TM\oplus T^*M$28-bundles over manifolds, define T-dual dg-manifolds, construct the T-duality map as a degree $TM\oplus T^*M$29 map between the cohomologies of T-dual dg-manifolds, and prove an explicit isomorphism between the differential graded algebras of their symmetries (Lupercio et al., 2012). In the self T-dual case, the derived dg-Leibniz algebra of the fixed points of the T-dual automorphism recovers the algebraic structure underlying $TM\oplus T^*M$30 generalized geometry, and higher dg-manifolds similarly yield structures associated with exceptional generalized geometry (Lupercio et al., 2012).

A related doubled-geometric construction is Vaisman’s big-tangent manifold

$TM\oplus T^*M$31

the total space of the bundle $TM\oplus T^*M$32 over $TM\oplus T^*M$33 (Vaisman, 2013). Its vertical leaves are para-Hermitian vector spaces, and $TM\oplus T^*M$34 carries canonical tensor fields

$TM\oplus T^*M$35

together with a canonical presymplectic form. From the viewpoint of $TM\oplus T^*M$36-structures, big-tangent geometry is equivalent to a suitable triple $TM\oplus T^*M$37 satisfying $TM\oplus T^*M$38, $TM\oplus T^*M$39, and compatibility conditions linking the three tensors. Vaisman then defines horizontal bundles, compatible vertical metrics, canonical double-metric connections, and an action functional for such double fields (Vaisman, 2013). In this sense, the term points not to torus actions but to doubled tangent–cotangent geometry.

6. Torus-bundle special holonomy manifolds

A broader torus-geometric usage arises in special holonomy. The paper on complete non-compact $TM\oplus T^*M$40-manifolds constructs torsion-free $TM\oplus T^*M$41-structures on the total spaces of principal $TM\oplus T^*M$42-bundles over asymptotically conical Calabi–Yau 3-folds (Cavalleri, 2024). The central ansatz is a $TM\oplus T^*M$43-invariant 4-form

$TM\oplus T^*M$44

where $TM\oplus T^*M$45 is a principal $TM\oplus T^*M$46-connection and $TM\oplus T^*M$47 is an $TM\oplus T^*M$48-structure on the base. The resulting metrics are complete, Ricci-flat, and asymptotically $TM\oplus T^*M$49-fibred conical, abbreviated $TM\oplus T^*M$50 (Cavalleri, 2024).

This construction produces infinitely many diffeomorphism types of $TM\oplus T^*M$51 $TM\oplus T^*M$52-manifolds and, notably, the first known examples of complete toric $TM\oplus T^*M$53-manifolds (Cavalleri, 2024). Here the torus is not merely an acting symmetry group; it is built into the geometry both as a principal bundle and, in the toric examples, as an effective multi-Hamiltonian $TM\oplus T^*M$54-action.

Taken together, these usages show that T-manifold is a family resemblance term rather than a single definition. Its common thread is the structural role of a torus or of $TM\oplus T^*M$55-duality: as an acting compact group, as a reduction datum in Hamiltonian geometry, as a source of local bifurcation manifolds, as a non-geometric transition group in string theory, or as the doubled $TM\oplus T^*M$56 object of generalized geometry.