Extended Topological Field Theory

• Extended Topological Field Theory is a framework that assigns higher-categorical structures to manifolds of various dimensions, refining the classic TQFT axioms.
• It provides concrete models such as the Rozansky–Witten theory, where invariants, vector spaces, and categories emerge through rigorous gluing operations.
• The theory interconnects higher category theory, algebraic geometry, and quantum field theory, advancing areas like geometric Langlands duality and derived algebraic geometry.

Extended Topological Field Theory (Extended TFT) generalizes the Atiyah–Segal axiomatization of topological quantum field theory, assigning not only numbers or vector spaces to closed manifolds but also higher-category structures (categories, 2-categories, ... n-categories) to manifolds of lower dimension—reflecting data such as boundary conditions and defect operators. This extension naturally encodes the local, multi-layered structure necessary for describing both physical theories and intricate topological invariants in mathematics. The modern realization of Extended TFT tightly intertwines developments in higher category theory, geometric representation theory, and quantum field theory, culminating in deep structural results and powerful classification theorems (Kapustin, 2010, Feshbach et al., 2011, Schommer-Pries, 2015).

1. Higher Categories and the Axiomatic Framework

Conventional n-dimensional TQFTs are defined as symmetric monoidal functors

$Z : \mathrm{Bord}_n \to \mathrm{Vect}$

where $\mathrm{Bord}_n$ is the category of closed oriented n-manifolds and their bordisms. This setting suffices for associating numbers to n-manifolds and vector spaces to (n–1)-manifolds.

The extended formalism refines this structure: to a closed oriented (n–k–1)-manifold, it assigns a k-category, organizing local or codimension-k data (such as boundary conditions, surface/line/point operators). For instance, in 3d TFT:

• Closed 3-manifolds $\longmapsto$ numbers
• Closed 2-manifolds $\longmapsto$ vector spaces
• Closed 1-manifolds $\longmapsto$ categories (e.g., branes)
• Points $\longmapsto$ 2-categories

Such an Extended TFT is a symmetric monoidal functor from a higher-categorical bordism $(\infty,n)$-category to an $(\infty,n)$-category of algebraic structures, often constructed as pseudo n-fold categories with explicit associators and interchangers for rigorous gluing operations (Feshbach et al., 2011).

2. Concrete Examples and Monoidal Structures

A prototypical example is the three-dimensional Rozansky–Witten (RW) model. The RW construction assigns:

• To a closed 3-manifold $M$, a finite-type invariant (perturbatively finite partition function).
• To a closed oriented surface $\Sigma_g$ of genus $g$, a vector space

$$\bigoplus_p H^p\left(X, (\wedge T^{(1,0)}X)^{\otimes g}\right)$$

where $X$ is a holomorphic symplectic manifold (the target).

• To a point, a 2-category of boundary conditions, with objects given by Lagrangian submanifolds and morphisms often described in terms of matrix factorizations.

Upon reduction from $S^1 \times \Sigma$ to $\Sigma$, the effective 2d theory is the B-model with target $X$. The category of branes is a 2-periodic version of the derived category $\mathrm{D}_{\mathbb{Z}_2}(\mathrm{Coh}(X))$, but quantum corrections deform the obvious symmetric monoidal tensor product to a braided monoidal structure. Corrections to the associator in this category scale according to the symplectic form $\Omega$ as $\Omega \to \lambda \Omega$ with scaling $\lambda^{1-p}$ at p-th order (Kapustin, 2010).

3. Higher-Categorical Formulations: Pseudo n-Fold Categories

Extended TQFTs are rigorously described as symmetric monoidal functors from a $(\infty,n)$-category of bordisms with corners to a target category of higher algebraic data (n-fold spans, cospans, categories of sheaves, etc.) (Feshbach et al., 2011). In these structures:

• Composition in multiple directions is encoded via associator (pentagon and hexagon) diagrams.
• Explicit decomposition and gluing of bordisms are governed by coherence isomorphisms and natural transformations.

This approach gives computational control over compositions: for instance, explicit pushouts (or fiber products) model the geometric operations and capture locality at every categorical level. Such frameworks contrast with the more abstract quasi-category approaches (e.g., Lurie's models), offering more concrete tools for gluing and coherence.

4. Applications: Deformations, Langlands Duality, and Representation Theory

Two major classes of applications illustrate the power of Extended TFT:

A. Monoidal Deformations of Derived Categories:

Quantum corrections deform the monoidal category structure of $\mathrm{D}(\mathrm{Coh}(Y))$ for a Lagrangian $Y \subset (X, \Omega)$. The deformation is parametrized by a $(0,1)$-form $\beta$ valued in

$\bigoplus_{p=2}^\infty S^p(TY)$

with $\beta$ subject to a Maurer–Cartan-type equation

$\beta + \frac{1}{2}[\beta, \beta] = 0$

and gauge transformations

$$\beta \mapsto \beta + a + [\beta, a], \quad a \in \bigoplus_{p=1}^\infty S^p(TY)$$

The space of solutions (modulo gauge) parametrizes monoidal deformations, constituting a categorification of Kontsevich’s Formality Theorem (Kapustin, 2010).

B. Geometric Langlands Duality:

Extended TFT constructions underpin the geometric Langlands program. A 4d supersymmetric gauge theory with a topological twist, when reduced on a Riemann surface $C$, yields:

• At $t = i$: 2d B-model with branes in $$\mathrm{D}^b\mathrm{Coh}(\mathcal{M}_{\mathrm{flat}}(G_\mathbb{C},C))$$
• At $t = 1$: 2d A-model with branes in the Fukaya--Floer category of the moduli space for the Langlands dual group $LG$

The Montonen–Olive duality gives category equivalences

$$\mathrm{D}^b\mathrm{Coh}(\mathcal{M}_{\mathrm{flat}}(G_\mathbb{C},C)) \cong \mathrm{Fuk}^{\mathrm{symp}}(\mathcal{M}_{\mathrm{flat}}(LG_\mathbb{C},C))$$

and at the level of 2-categories associated to circles,

$(G, i, S^1) \cong (LG, 1, S^1)$

These equivalences, glued along bordisms, provide a categorical foundation for the Langlands program (Kapustin, 2010).

5. Boundary Conditions, Defects, and the Cobordism Hypothesis

Extended TFT formalism captures boundary conditions as higher-categorical objects. For example, in an n-dimensional theory, a (n–1)-category of "boundary conditions" is assigned to codimension-one manifolds; fusion and composition rules for defects are described via multi-morphism composition in the relevant higher category.

The Cobordism Hypothesis formalizes this: an extended TFT is classified (up to equivalence) by its value on the point, provided the assigned object in the target $(\infty,n)$-category is fully dualizable. This sharply constrains and enables the classification of extended and fully extended TFTs, with dualizability conditions corresponding to the algebraic data (e.g., fully dualizable algebras yield 2D TQFTs; fully dualizable fusion categories yield 3D TQFTs) (Schommer-Pries, 2015, Lee, 2014).

6. Advanced Structures: AKSZ and Homotopical Extensions

Recent developments extend the scope of Extended TFTs to AKSZ-type constructions in derived algebraic geometry, where the objects of interest are symplectic derived stacks and their iterated Lagrangian correspondences. Here, integration over oriented manifolds and their bordisms is made functorial using Poincaré–Lefschetz duality for manifolds with boundary (modeled via Verdier duality). The construction associates to each bordism a cospan of spaces (homotopy types) preoriented by fundamental class data and defines derived pushforward/integration on mapping stacks, resulting in symmetric monoidal functors from the oriented bordism $(\infty,n)$-category to higher categories of symplectic derived stacks (Calaque et al., 2021). This enables the AKSZ construction to produce a robust family of extended TFTs with symplectic and Lagrangian data encoded in higher-categorical structures.

7. Implications and Synthesis

Extended TFT provides a unifying and flexible mathematical framework:

• It models physical boundary conditions, domain walls, and defects in quantum field theories and condensed matter systems.
• It enables classification of field theories in terms of higher-categorical algebraic data (e.g., fully dualizable objects, monoidal categories with deformations, modular tensor categories).
• Through concrete models such as Rozansky–Witten theory and implications for the geometric Langlands duality, it has driven connections between mathematical physics, geometric representation theory, and homological algebra.
• Advanced methods, including those based on derived algebraic geometry, allow for the covariant, functorial construction of field theories sensitive to orientation and higher-structure, generalizing classical duality-based constructions to the fully extended, n-categorical context.

These results have catalyzed major advances in both the classification of TFTs and their utility as tools across mathematics and physics, with the formalism of higher categories and dualizability at their core (Kapustin, 2010, Feshbach et al., 2011, Schommer-Pries, 2015, Calaque et al., 2021).