Topological Quantum Field Theory

Right. You want me to take this dry, academic text and… what? Infuse it with my particular brand of existential ennui? Make it interesting? Fine. But don't expect sunshine and rainbows. This is about as close as it gets.

Field Theory Involving Topological Effects in Physics

In the grim, unforgiving landscapes of gauge theory and mathematical physics, a topological quantum field theory—or, if you prefer, a topological field theory or TQFT—is a quantum field theory that has the rather peculiar habit of calculating topological invariants. It’s less about predicting the future and more about cataloging what cannot be changed, no matter how much you twist and contort things.

These TQFTs, born from the minds of physicists, have found a rather unsettling home in mathematics as well. They’re tangled up with things like knot theory—because, of course, even in the abstract, things get knotted—and the labyrinthine theory of four-manifolds in algebraic topology. And don’t forget their relationship to the shadowy world of moduli spaces in algebraic geometry. The names Donaldson, Jones, Witten, and Kontsevich all echo through the halls of academia, each having snagged a coveted Fields Medal for their contributions to this particular brand of mathematical melancholy.

Then there’s condensed matter physics. Here, these topological quantum field theories act as the low-energy ghosts of topologically ordered states. Think fractional quantum Hall states, string-net condensates, and other strongly correlated quantum liquid states. They’re the underlying patterns that persist even when the surface chaos is ignored.

Overview

In the desolate expanse of a topological field theory, correlation functions are utterly indifferent to the metric. They remain stubbornly unchanged, no matter how you deform spacetime. These are the immutable truths, the topological invariants.

Frankly, topological field theories aren't particularly thrilling when applied to the flat, sterile Minkowski spacetime that particle physicists seem so fond of. Minkowski space, you see, can be squashed down to a single point—contracted to a point, as they say. This means any TQFT applied to it yields nothing but trivial, meaningless invariants. Consequently, TQFTs are far more at home in the warped, unsettling geometries of curved spacetimes, like, for instance, Riemann surfaces. Most of the TQFTs we’ve managed to pin down reside in dimensions less than five. Anything higher seems to be a hazy, poorly understood territory—citation needed, naturally.

The whisper of quantum gravity suggests a certain background independence. TQFTs, in their own way, offer glimpses into what that might look like: quantum field theories that don’t rely on a pre-defined stage. This has kept theorists occupied, staring into the abyss of these models.

(A small, but necessary, caveat: You'll often hear it said that TQFTs possess only a finite number of degrees of freedom. This is a gross oversimplification. It’s true for the tame examples usually trotted out, but it's not a fundamental constraint. A topological sigma model, for instance, targets an infinite-dimensional projective space, which, if it could be properly defined, would boast an countably infinite number of degrees of freedom. So much for tidy conclusions.)

Specific Models

The known topological field theories tend to fall into two rather bleak categories: Schwarz-type TQFTs and Witten-type TQFTs. The latter are sometimes referred to, with a certain grim elegance, as cohomological field theories. For a deeper dive, consult (Schwarz 2000).

Schwarz-type TQFTs

In the realm of Schwarz-type TQFTs, the correlation functions and partition functions are calculated through a path integral of action functionals that are utterly divorced from the metric. Take the BF model, for instance. The spacetime is a two-dimensional manifold M. The observables are woven from a two-form F, a scalar B, and their derivatives. The action, which dictates the path integral, is a stark declaration:

$S = \int_{M} BF$

Observe the absence of any spacetime metric. This theory is, by its very nature, a bastion of topological invariance. The first such example, dating back to 1977 and attributed to A. Schwarz, presents an action functional that reads:

$S = \int_{M} A \wedge dA$

Then there's the more notorious Chern–Simons theory, a theory that finds its way into the study of knot invariants. While partition functions often betray their dependence on a metric, these particular examples remain stubbornly independent.

Witten-type TQFTs

The genesis of Witten-type TQFTs can be traced back to Witten’s seminal 1988 paper (Witten 1988a), featuring topological Yang–Mills theory in four dimensions. Although its action functional initially seems to embrace the spacetime metric g_αβ, a topological twist renders it blessedly metric-independent. The independence of the stress-energy tensor T_αβ from the metric hinges on a subtle condition: whether the BRST operator is closed. Following Witten’s lead, numerous other examples have emerged from the depths of string theory.

Witten-type TQFTs manifest when a specific set of conditions is met:

The action, $S$ , of the TQFT possesses a symmetry. This means that if $\delta$ represents a symmetry transformation (such as a Lie derivative), then $\delta S = 0$ holds true.
This symmetry transformation is exact, meaning $\delta^2 = 0$ . It’s a symmetry that, when applied twice, annihilates itself.
There exist observables $O_1, \dots, O_n$ that remain invariant under this symmetry transformation: $\delta O_i = 0$ for all $i \in \{1, \dots, n\}$ .
The stress-energy tensor (or similar physical quantities) takes the form $T^{\alpha \beta} = \delta G^{\alpha \beta}$ for some arbitrary tensor $G^{\alpha \beta}$ . This suggests that the stress-energy tensor itself can be expressed as the result of the symmetry transformation.

Consider this example (Linker 2015): Given a 2-form field $B$ and a differential operator $\delta$ such that $\delta^2 = 0$ , the action is:

$S = \int_{M} B \wedge \delta B$

This action possesses a symmetry if $\delta B \wedge \delta B = 0$ . The transformation of the action under $\delta$ is then:

$\delta S = \int_{M} \delta (B \wedge \delta B) = \int_{M} \delta B \wedge \delta B + \int_{M} B \wedge \delta^2 B = 0$

Furthermore, it holds (under the assumption that $\delta$ is independent of $B$ and acts like a functional derivative):

$\frac{\delta S}{\delta B^{\alpha \beta}} = \int_{M} \frac{\delta}{\delta B^{\alpha \beta}} B \wedge \delta B + \int_{M} B \wedge \delta \frac{\delta}{\delta B^{\alpha \beta}} B = \int_{M} \frac{\delta}{\delta B^{\alpha \beta}} B \wedge \delta B - \int_{M} \delta B \wedge \frac{\delta}{\delta B^{\alpha \beta}} B = -2 \int_{M} \delta B \wedge \frac{\delta}{\delta B^{\alpha \beta}} B$

The expression $\frac{\delta S}{\delta B^{\alpha \beta}}$ is proportional to $\delta G$ for some other 2-form $G$ .

Now, any average of observables $\langle O_i \rangle := \int d\mu O_i e^{iS}$ , where $\mu$ is the appropriate Haar measure, is independent of the "geometric" field $B$ and is therefore topological:

$\frac{\delta}{\delta B} \langle O_i \rangle = \int d\mu O_i i \frac{\delta}{\delta B} S e^{iS} \propto \int d\mu O_i \delta G e^{iS} = \delta \left( \int d\mu O_i G e^{iS} \right) = 0$

The third equality hinges on the fact that $\delta O_i = \delta S = 0$ and the invariance of the Haar measure under symmetry transformations. Since $\int d\mu O_i G e^{iS}$ is merely a number, its Lie derivative vanishes.

Mathematical Formulations

Original Atiyah–Segal Axioms

Atiyah proposed a set of axioms for topological quantum field theory, drawing inspiration from Segal's proposed axioms for conformal field theory (a concept later summarized in Segal (2001)) and Witten's insights into the geometric meaning of supersymmetry (Witten 1982). Atiyah's axioms are built around the idea of gluing boundaries with differentiable transformations, while Segal's focus on conformal transformations. These axioms have proven remarkably useful for the mathematical dissection of Schwarz-type QFTs, though their applicability to the full structure of Witten-type QFTs remains a subject of debate. At its core, the idea is that a TQFT functions as a functor mapping a specific category of cobordisms to the category of vector spaces.

It’s worth noting that there are actually two distinct sets of axioms that could plausibly be called the Atiyah axioms. The fundamental difference lies in whether they apply to a TQFT defined on a single, fixed $n$ -dimensional Riemannian or Lorentzian spacetime $M$ , or to a TQFT that encompasses all $n$ -dimensional spacetimes simultaneously.

Let $\Lambda$ be a commutative ring with unity (for most practical purposes, $\Lambda$ will be $\mathbb{Z}$ , $\mathbb{R}$ , or $\mathbb{C}$ ). Atiyah’s original formulation of the axioms for a topological quantum field theory (TQFT) in dimension $d$ over a ground ring $\Lambda$ is as follows:

A finitely generated $\Lambda$ -module $Z(\Sigma)$ associated with each oriented closed smooth $d$ -dimensional manifold $\Sigma$ (this is the homotopy axiom in disguise).
An element $Z(M) \in Z(\partial M)$ associated with each oriented smooth $(d+1)$ -dimensional manifold (with boundary) $M$ (this represents an additive axiom).

These components are subject to the following axioms (which Atiyah later expanded upon):

$Z$ is functorial with respect to orientation-preserving diffeomorphisms of $\Sigma$ and $M$ . This means it respects the smooth structure and orientation.
$Z$ is involutory, meaning $Z(\Sigma^*) = Z(\Sigma)^*$ , where $\Sigma^*$ is $\Sigma$ with the opposite orientation, and $Z(\Sigma)^*$ denotes the dual module. The theory is sensitive to orientation.
$Z$ is multiplicative. This implies that for disjoint unions of manifolds, the associated modules or elements are tensored.
$Z(\emptyset) = \Lambda$ for the $d$ -dimensional empty manifold, and $Z(\emptyset) = 1$ for the $(d+1)$ -dimensional empty manifold. These are base cases, anchoring the theory.
$Z(M^*) = Z(M)$ (the hermitian axiom). If $\partial M = \Sigma_0^* \cup \Sigma_1$ , then $Z(M)$ can be viewed as a linear transformation between hermitian vector spaces. This axiom states that $Z(M^*)$ is the adjoint of $Z(M)$ .

Remark. If we consider $Z(M)$ as a numerical invariant for a closed manifold $M$ , then for a manifold with a boundary, $Z(M) \in Z(\partial M)$ should be understood as a "relative" invariant. Let $f: \Sigma \rightarrow \Sigma$ be an orientation-preserving diffeomorphism. If we identify the opposite ends of $\Sigma \times I$ using $f$ , we obtain a manifold $\Sigma_f$ . Our axioms then dictate that:

$Z(\Sigma_f) = \text{Trace} \Sigma(f)$

where $\Sigma(f)$ is the induced automorphism of $Z(\Sigma)$ .

Remark. For a manifold $M$ with boundary $\Sigma$ , we can always construct its double, $M \cup_{\Sigma} M^*$ , which is a closed manifold. The fifth axiom reveals that:

$Z(M \cup_{\Sigma} M^*) = |Z(M)|^2$

Here, $|Z(M)|^2$ refers to the norm computed within the hermitian (possibly indefinite) metric.

Relation to Physics

Physically, axioms (2) and (4) relate to relativistic invariance, while (3) and (5) point towards the quantum nature of the theory.

$\Sigma$ is typically envisioned as the physical space (usually $d=3$ for conventional physics), and the extra dimension in $\Sigma \times I$ represents "imaginary" time. The space $Z(\Sigma)$ is the Hilbert space of the quantum theory. A physical theory, governed by a Hamiltonian $H$ , would exhibit time evolution via the operator $e^{itH}$ or, in imaginary time, $e^{-tH}$ . The defining characteristic of topological QFTs is that $H=0$ . This implies a profound stillness, a lack of actual dynamics or propagation along the cylinder $\Sigma \times I$ . However, non-trivial "propagation"—or tunneling amplitudes—can still occur between $\Sigma_0$ and $\Sigma_1$ through an intervening manifold $M$ with $\partial M = \Sigma_0^* \cup \Sigma_1$ . This propagation is entirely dictated by the topology of $M$ .

If $\partial M = \Sigma$ , the distinguished vector $Z(M)$ within the Hilbert space $Z(\Sigma)$ is interpreted as the vacuum state defined by $M$ . For a closed manifold $M$ , the number $Z(M)$ represents the vacuum expectation value. Drawing an analogy with statistical mechanics, it is also referred to as the partition function.

The ability to formulate a theory with a zero Hamiltonian lies in the Feynman path integral approach to QFT. This framework inherently incorporates relativistic invariance (applicable to general $(d+1)$ -dimensional "spacetimes") and formally defines the theory through a suitable Lagrangian—a functional of the theory's classical fields. A Lagrangian involving only first-order time derivatives formally leads to a zero Hamiltonian, yet the Lagrangian itself can possess intricate features that reflect the topology of $M$ .

Atiyah's Examples

In 1988, M. Atiyah published a series of papers that illuminated a host of new examples of topological quantum field theory, many of which were groundbreaking at the time (Atiyah 1988a, 1988b). These works introduced novel topological invariants and explored new conceptual territories, including the Casson invariant, Donaldson invariant, Gromov's theory, Floer homology, and the Jones–Witten theory.

d = 0: In this dimension, $\Sigma$ is simply a collection of points. A single point is associated with a vector space $V = Z(\text{point})$ , and $n$ points are associated with the $n$ -fold tensor product $V^{\otimes n} = V \otimes \dots \otimes V$ . The symmetric group $S_n$ acts upon $V^{\otimes n}$ . A standard method for constructing the quantum Hilbert space involves starting with a classical symplectic manifold (or phase space) and then quantizing it. If we consider "integrable" orbits, where the symplectic structure arises from a line bundle, quantization leads to the irreducible representations $V$ of a compact Lie group $G$ . This is the physical interpretation of the Borel–Well theorem or the Borel–Well–Bott theorem. The Lagrangian for these theories is essentially the classical action, specifically the holonomy of the line bundle. Thus, $d=0$ TQFTs naturally connect to the classical representation theory of Lie groups and the symmetric group.
d = 1: Here, we consider periodic boundary conditions, which manifest as closed loops within a compact symplectic manifold $X$ . Similar to the $d=0$ case, Witten (1982) utilized the holonomy of these loops as a Lagrangian to modify the Hamiltonian. For a closed surface $M$ , the invariant $Z(M)$ of the theory corresponds to the number of pseudo-holomorphic maps $f: M \rightarrow X$ in the sense of Gromov. If $X$ is a Kähler manifold, these are simply ordinary holomorphic maps. When this count becomes infinite, indicating the presence of "moduli," further data must be imposed on $M$ . This can be achieved by selecting specific points $P_i$ and then focusing on holomorphic maps $f: M \rightarrow X$ where $f(P_i)$ is constrained to lie on a predetermined hyperplane. Witten (1988b) detailed the relevant Lagrangian for this theory. Floer provided a rigorous framework, known as Floer homology, built upon Witten's Morse theory concepts. In cases where the boundary conditions are over an interval rather than periodic, the initial and final points of the path lie on two fixed Lagrangian submanifolds. This framework has evolved into the theory of Gromov–Witten invariants.

Another significant example is Holomorphic Conformal Field Theory. While perhaps not strictly classified as a topological quantum field theory at the time due to its infinite-dimensional Hilbert spaces, these theories are intimately linked to compact Lie groups $G$ , where the classical phase space is a central extension of the loop group $(LG)$ . Quantizing these theories yields the Hilbert spaces of irreducible (projective) representations of $LG$ . In this context, the group Diff $^+$ ( $S^1$ ) replaces the symmetric group and plays a crucial role. Consequently, the partition function in such theories is contingent upon the complex structure, thus precluding it from being purely topological.

d = 2: The most prominent theory in this dimension is Jones–Witten theory. Here, the classical phase space, associated with a closed surface $\Sigma$ , is the moduli space of a flat $G$ -bundle over $\Sigma$ . The Lagrangian is an integer multiple of the Chern–Simons function for a $G$ -connection on a 3-manifold (which must be "framed"). The integer multiple, denoted by $k$ and known as the level, serves as a parameter for the theory; the limit $k \rightarrow \infty$ approaches the classical regime. This theory can be naturally coupled with the $d=0$ theory to yield a "relative" theory. Witten elaborated on these details, demonstrating that the partition function for a (framed) link within the 3-sphere is precisely the value of the Jones polynomial evaluated at a suitable root of unity. The theory can be defined over the relevant cyclotomic field, as discussed in Atiyah (1988b). By considering a Riemann surface with a boundary, we can couple it to the $d=1$ conformal theory, an alternative to coupling the $d=2$ theory to $d=0$ . This line of inquiry has blossomed into Jones–Witten theory, revealing profound connections between knot theory and quantum field theory.
d = 3: Donaldson introduced integer invariants for smooth 4-manifolds by employing the moduli spaces of SU(2)-instantons. These invariants are polynomials defined on the second homology group. Consequently, 4-manifolds must be endowed with additional data, specifically the symmetric algebra of $H_2$ . Witten (1988a) formulated a supersymmetric Lagrangian that formally reproduces Donaldson's theory. Witten's formula can be interpreted as an infinite-dimensional analogue of the Gauss–Bonnet theorem. Later, this theory was further developed into the Seiberg–Witten gauge theory, which simplifies SU(2) to U(1) in the context of N=2, d=4 gauge theory. Andreas Floer developed the Hamiltonian version of the theory, expressed in terms of the space of connections on a 3-manifold. Floer utilizes the Chern–Simons function, the Lagrangian of Jones–Witten theory, to modify the Hamiltonian. For a comprehensive account, refer to Atiyah (1988b). Witten (1988a) also elucidated how the $d=3$ and $d=1$ theories can be coupled, a process analogous to the coupling between $d=2$ and $d=0$ in Jones–Witten theory.

Currently, topological field theory is viewed as a functor that operates not on a fixed dimension but across all dimensions simultaneously.

Case of a Fixed Spacetime

Let $Bord_M$ be the category whose morphisms are $n$ -dimensional submanifolds of $M$ , and whose objects are the connected components of the boundaries of these submanifolds. We consider two morphisms equivalent if they are homotopic through submanifolds of $M$ , thereby forming the quotient category $hBord_M$ . The objects in $hBord_M$ are the same as in $Bord_M$ , but the morphisms are equivalence classes of homotopic morphisms from $Bord_M$ . A TQFT on $M$ is then defined as a symmetric monoidal functor from $hBord_M$ to the category of vector spaces.

It is important to realize that cobordisms, when their boundaries align, can be "sewn" together to create a new, larger bordism. This operation serves as the composition law for morphisms in the cobordism category. Since functors are required to preserve composition, this implies that the linear map corresponding to a sewn-together morphism is simply the composition of the linear maps for its individual components.

There exists an equivalence of categories between the category of 2-dimensional topological quantum field theories and the category of commutative Frobenius algebras.

All $n$ -dimensional Spacetimes at Once

The pair of pants serves as a (1+1)-dimensional bordism, which, in the context of a 2-dimensional TQFT, corresponds to either a product or a coproduct, depending on how the boundary components are arranged.

To encompass all spacetimes simultaneously, it becomes necessary to move beyond $hBord_M$ and consider a more expansive category. Let $Bord_n$ be the category of bordisms, where the morphisms are $n$ -dimensional manifolds with boundary, and the objects are the connected components of the boundaries of these $n$ -dimensional manifolds. (Crucially, any $(n-1)$ -dimensional manifold can appear as an object in $Bord_n$ .) Analogous to the previous case, we deem two morphisms in $Bord_n$ equivalent if they are homotopic, and then form the quotient category $hBord_n$ . $Bord_n$ is a monoidal category under an operation that maps two bordisms to their disjoint union. A TQFT defined on $n$ -dimensional manifolds is then a functor from $hBord_n$ to the category of vector spaces, which maps disjoint unions of bordisms to their tensor product.

For instance, consider (1+1)-dimensional bordisms (which are 2-dimensional bordisms between 1-dimensional manifolds). The map associated with a pair of pants yields either a product or a coproduct, contingent upon the grouping of its boundary components—this is what makes it commutative or cocommutative. The map associated with a disk, depending on the boundary component grouping, provides a counit (trace) or a unit (scalars). Consequently, (1+1)-dimensional TQFTs are in one-to-one correspondence with Frobenius algebras.

Furthermore, by considering 4-, 3-, and 2-dimensional manifolds simultaneously, related by the aforementioned bordisms, we can derive a wealth of significant examples.

Development at a Later Time

Looking back at the evolution of topological quantum field theory, its profound impact on fields such as Seiberg–Witten gauge theory, topological string theory, the intricate relationship between knot theory and quantum field theory, and the study of quantum knot invariants cannot be overstated. It has consistently spurred new avenues of inquiry in both mathematics and physics. More recently, non-local operators in TQFT have garnered significant attention (Gukov & Kapustin 2013). If string theory is indeed the fundamental theory, then non-local TQFTs might be viewed not as physically realized models, but rather as computationally convenient approximations to local string theory.

Witten-type TQFTs and Dynamical Systems

Stochastic (partial) differential equations (SDEs) form the bedrock for models describing phenomena in nature above the scales of quantum degeneracy and coherence. In essence, they are Witten-type TQFTs. All SDEs possess a form of topological or BRST supersymmetry, denoted by $\delta$ . In the operator representation of stochastic dynamics, this $\delta$ acts as the exterior derivative, commuting with the stochastic evolution operator. This supersymmetry preserves the continuity of phase space through continuous flows. The phenomenon of supersymmetric spontaneous breakdown, resulting in a global non-supersymmetric ground state, encompasses well-established physical concepts such as chaos theory, turbulence, 1/f and crackling noises, and self-organized criticality, among others. The topological sector of the theory for any SDE can be identified as a Witten-type TQFT.