Original Atiyah–Segal axioms Atiyah suggested a set of axioms for topological quantum field theory, inspired by
Segal's proposed axioms for
conformal field theory (subsequently, Segal's idea was summarized in ), and Witten's geometric meaning of supersymmetry in . Atiyah's axioms are constructed by gluing the boundary with a differentiable (topological or continuous) transformation, while Segal's axioms are for conformal transformations. These axioms have been relatively useful for mathematical treatments of Schwarz-type QFTs, although it isn't clear that they capture the whole structure of Witten-type QFTs. The basic idea is that a TQFT is a
functor from a certain
category of
cobordisms to the category of
vector spaces. There are in fact two different sets of axioms which could reasonably be called the Atiyah axioms. These axioms differ basically in whether or not they apply to a TQFT defined on a single fixed
n-dimensional Riemannian / Lorentzian spacetime
M or a TQFT defined on all
n-dimensional spacetimes at once. Let Λ be a
commutative ring with 1 (for almost all real-world purposes we will have Λ =
Z,
R or
C). Atiyah originally proposed the axioms of a topological quantum field theory (TQFT) in dimension
d defined over a ground ring Λ as following: • A finitely generated Λ-module
Z(Σ) associated to each oriented closed smooth d-dimensional manifold Σ (corresponding to the
homotopy axiom), • An element
Z(
M) ∈
Z(∂
M) associated to each oriented smooth (
d + 1)-dimensional manifold (with boundary)
M (corresponding to an
additive axiom). These data are subject to the following axioms (4 and 5 were added by Atiyah): •
Z is
functorial with respect to orientation preserving
diffeomorphisms of Σ and
M, •
Z is
involutory, i.e.
Z(Σ*) =
Z(Σ)* where Σ* is Σ with opposite orientation and
Z(Σ)* denotes the dual module, •
Z is
multiplicative. •
Z(\emptyset) = Λ for the d-dimensional empty manifold and
Z(\emptyset) = 1 for the (
d + 1)-dimensional empty manifold. •
Z(
M*) = (the
hermitian axiom). If \partial M = \Sigma^*_0 \cup \Sigma_1 so that
Z(
M) can be viewed as a linear transformation between hermitian vector spaces, then this is equivalent to
Z(
M*) being the
adjoint of
Z(
M).
Remark. If for a closed manifold
M we view
Z(
M) as a numerical invariant, then for a manifold with a boundary we should think of
Z(
M) ∈
Z(∂
M) as a "relative" invariant. Let
f : Σ → Σ be an orientation-preserving diffeomorphism, and identify opposite ends of Σ ×
I by
f. This gives a manifold Σ
f and our axioms imply : Z(\Sigma_f) = \operatorname{Trace}\ \Sigma(f) where Σ(
f) is the induced automorphism of
Z(Σ).
Remark. For a manifold
M with boundary Σ we can always form the double M\cup_\Sigma M^* which is a closed manifold. The fifth axiom shows that : Z\left(M\cup_\Sigma M^*\right) = |Z(M)|^2 where on the right we compute the norm in the hermitian (possibly indefinite) metric.
Relation to physics Physically (2) + (4) are related to relativistic invariance while (3) + (5) are indicative of the quantum nature of the theory. Σ is meant to indicate the physical space (usually,
d = 3 for standard physics) and the extra dimension in Σ ×
I is "imaginary" time. The space
Z(Σ) is the
Hilbert space of the quantum theory and a physical theory, with a
Hamiltonian H, will have a time evolution operator
eitH or an "imaginary time" operator
e−tH. The main feature of
topological QFTs is that
H = 0, which implies that there is no real dynamics or propagation along the cylinder Σ ×
I. However, there can be non-trivial "propagation" (or tunneling amplitudes) from Σ0 to Σ1 through an intervening manifold
M with \partial M = \Sigma^*_0 \cup \Sigma_1; this reflects the topology of
M. If ∂
M = Σ, then the distinguished vector
Z(
M) in the Hilbert space
Z(Σ) is thought of as the
vacuum state defined by
M. For a closed manifold
M the number
Z(
M) is the
vacuum expectation value. In analogy with
statistical mechanics it is also called the
partition function. The reason why a theory with a zero Hamiltonian can be sensibly formulated resides in the
Feynman path integral approach to QFT. This incorporates relativistic invariance (which applies to general (
d + 1)-dimensional "spacetimes") and the theory is formally defined by a suitable
Lagrangian—a functional of the classical fields of the theory. A Lagrangian which involves only first derivatives in time formally leads to a zero Hamiltonian, but the Lagrangian itself may have non-trivial features which relate to the topology of
M.
Atiyah's examples In 1988, M. Atiyah published a paper in which he described many new examples of topological quantum field theory that were considered at that time . It contains some new
topological invariants along with some new ideas:
Casson invariant,
Donaldson invariant,
Gromov's theory,
Floer homology and
Jones–Witten theory. ====
d = 0 ==== In this case Σ consists of finitely many points. To a single point we associate a vector space
V =
Z(point) and to
n-points the
n-fold tensor product:
V⊗
n =
V ⊗ … ⊗
V. The
symmetric group S
n acts on
V⊗
n. A standard way to get the quantum Hilbert space is to start with a classical
symplectic manifold (or
phase space) and then quantize it. Let us extend
Sn to a compact Lie group
G and consider "integrable" orbits for which the symplectic structure comes from a
line bundle, then quantization leads to the irreducible representations
V of
G. This is the physical interpretation of the
Borel–Weil theorem or the
Borel–Weil–Bott theorem. The Lagrangian of these theories is the classical action (
holonomy of the line bundle). Thus topological QFT's with
d = 0 relate naturally to the classical
representation theory of
Lie groups and the
symmetric group. ====
d = 1 ==== We should consider periodic boundary conditions given by closed loops in a compact symplectic manifold
X. Along with holonomy such loops as used in the case of
d = 0 as a Lagrangian are then used to modify the Hamiltonian. For a closed surface
M the invariant
Z(
M) of the theory is the number of
pseudo holomorphic maps in the sense of Gromov (they are ordinary
holomorphic maps if
X is a
Kähler manifold). If this number becomes infinite i.e. if there are "moduli", then we must fix further data on
M. This can be done by picking some points
Pi and then looking at holomorphic maps with
f(
Pi) constrained to lie on a fixed hyperplane. has written down the relevant Lagrangian for this theory. Floer has given a rigorous treatment, i.e.
Floer homology, based on Witten's
Morse theory ideas; for the case when the boundary conditions are over the interval instead of being periodic, the path initial and end-points lie on two fixed
Lagrangian submanifolds. This theory has been developed as
Gromov–Witten invariant theory. Another example is
Holomorphic Conformal Field Theory. This might not have been considered strictly topological quantum field theory at the time because Hilbert spaces are infinite dimensional. The conformal field theories are also related to the compact Lie group
G in which the classical phase consists of a central extension of the
loop group (LG). Quantizing these produces the Hilbert spaces of the theory of irreducible (projective) representations of
LG. The group Diff+(
S1) now substitutes for the symmetric group and plays an important role. As a result, the partition function in such theories depends on
complex structure, thus it is not purely topological. ====
d = 2 ==== Jones–Witten theory is the most important theory in this case. Here the classical phase space, associated with a closed surface Σ is the moduli space of a flat
G-bundle over Σ. The Lagrangian is an integer multiple of the
Chern–Simons function of a
G-connection on a 3-manifold (which has to be "framed"). The integer multiple
k, called the level, is a parameter of the theory and
k → ∞ gives the classical limit. This theory can be naturally coupled with the
d = 0 theory to produce a "relative" theory. The details have been described by Witten who shows that the partition function for a (framed) link in the 3-sphere is just the value of the
Jones polynomial for a suitable root of unity. The theory can be defined over the relevant
cyclotomic field, see . By considering a
Riemann surface with boundary, we can couple it to the
d = 1 conformal theory instead of coupling
d = 2 theory to
d = 0. This has developed into Jones–Witten theory and has led to the discovery of deep connections between
knot theory and quantum field theory. ====
d = 3 ==== Donaldson has defined the integer invariant of smooth 4-manifolds by using moduli spaces of SU(2)-instantons. These invariants are polynomials on the second homology. Thus 4-manifolds should have extra data consisting of the symmetric algebra of
H2. has produced a super-symmetric Lagrangian which formally reproduces the Donaldson theory. Witten's formula might be understood as an infinite-dimensional analogue of the
Gauss–Bonnet theorem. At a later date, this theory was further developed and became the
Seiberg–Witten gauge theory which reduces SU(2) to U(1) in
N = 2,
d = 4 gauge theory. The Hamiltonian version of the theory has been developed by
Andreas Floer in terms of the space of connections on a 3-manifold. Floer uses the
Chern–Simons function, which is the Lagrangian of Jones–Witten theory to modify the Hamiltonian. For details, see . has also shown how one can couple the
d = 3 and
d = 1 theories together: this is quite analogous to the coupling between
d = 2 and
d = 0 in Jones–Witten theory. Now, topological field theory is viewed as a
functor, not on a fixed dimension but on all dimensions at the same time.
Case of a fixed spacetime Let
BordM be the category whose morphisms are
n-dimensional
submanifolds of
M and whose objects are
connected components of the boundaries of such submanifolds. Regard two morphisms as equivalent if they are
homotopic via submanifolds of
M, and so form the quotient category
hBordM: The objects in
hBordM are the objects of
BordM, and the morphisms of
hBordM are homotopy equivalence classes of morphisms in
BordM. A TQFT on
M is a
symmetric monoidal functor from
hBordM to the category of vector spaces. Note that cobordisms can, if their boundaries match, be sewn together to form a new bordism. This is the composition law for morphisms in the cobordism category. Since functors are required to preserve composition, this says that the linear map corresponding to a sewn together morphism is just the composition of the linear map for each piece. There is 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 is a (1+1)-dimensional bordism, which corresponds to a product or coproduct in a 2-dimensional TQFT. To consider all spacetimes at once, it is necessary to replace
hBordM by a larger category. So let
Bordn be the category of bordisms, i.e. the category whose morphisms are
n-dimensional manifolds with boundary, and whose objects are the connected components of the boundaries of n-dimensional manifolds. (Note that any (
n−1)-dimensional manifold may appear as an object in
Bordn.) As above, regard two morphisms in
Bordn as equivalent if they are homotopic, and form the quotient category
hBordn.
Bordn is a
monoidal category under the operation which maps two bordisms to the bordism made from their disjoint union. A TQFT on
n-dimensional manifolds is then a functor from
hBordn to the category of vector spaces, which maps disjoint unions of bordisms to their tensor product. For example, for (1 + 1)-dimensional bordisms (2-dimensional bordisms between 1-dimensional manifolds), the map associated with a
pair of pants gives a product or coproduct, depending on how the boundary components are grouped – which is commutative or cocommutative, while the map associated with a disk gives a counit (trace) or unit (scalars), depending on the grouping of boundary components, and thus (1+1)-dimension TQFTs correspond to
Frobenius algebras. Furthermore, we can consider simultaneously 4-dimensional, 3-dimensional and 2-dimensional manifolds related by the above bordisms, and from them we can obtain ample and important examples.
Development at a later time Looking at the development of topological quantum field theory, we should consider its many applications to
Seiberg–Witten gauge theory,
topological string theory, the relationship between
knot theory and quantum field theory, and
quantum knot invariants. Furthermore, it has generated topics of great interest in both mathematics and physics. Also of important recent interest are non-local operators in TQFT (). If string theory is viewed as the fundamental, then non-local TQFTs can be viewed as non-physical models that provide a computationally efficient approximation to local string theory.
Witten-type TQFTs and dynamical systems Stochastic (partial) differential equations (SDEs) are the foundation for models of everything in nature above the scale of quantum degeneracy and coherence and are essentially Witten-type TQFTs. All SDEs possess topological or
BRST supersymmetry, \delta, and in the operator representation of stochastic dynamics is the
exterior derivative, which is commutative with the stochastic evolution operator. This supersymmetry preserves the continuity of phase space by continuous flows, and the phenomenon of supersymmetric spontaneous breakdown by a global non-supersymmetric ground state encompasses such well-established physical concepts as
chaos,
turbulence,
1/f and
crackling noises,
self-organized criticality etc. The topological sector of the theory for any SDE can be recognized as a Witten-type TQFT. == See also ==