CP7 Arbeitsgruppenseminar 2023/24

What: This year’s CP7 Arbeitsgruppenseminar will be an introduction to topological quantum field theories (TQFTs).

A TQFT is a QFT of a special kind: its correlation functions do not depend on the metric of space-time. Therefore, TQFTs compute topological invariants; they have proved very valuable in many areas of physics and mathematics. The development of quantum field theories in the 60s and 70s – and in particular, non-abelian gauge theories a.k.a. Yang–Mills theories (1954) – led to exciting new connections between physics and topology [Ati89]:

• Instantons are finite-action Euclidean gauge-field configurations of a topological nature introduced by Belavin, Polyakov, Schwartz and Tyupkin in 1975. Their importance was emphasized by ’t Hooft in 1976, and a systematic construction was proposed by Atiyah, Drinfeld, Hitchin and Manin in 1978. Donaldson theory [Don83] uses moduli spaces of anti-self-dual instantons on smooth 4-manifolds to define new topological invariants of the latter; these led e.g. to the discovery of an exotic structure on R^4. Donaldson theory was later reformulated and extended in a language developed by Seiberg and Witten [SW94a, SW94b], leading to Seiberg–Witten invariants. Instantons are also central in Floer’s work [Flo88].

• In field theory, a classical symmetry is anomalous if it is actually not a symmetry in the corresponding quantum field theory. Such anomalies arise in particular in chiral theories, i.e. theories that are not invariant under parity. For example, the axial vector current of quantum electrodynamics is anomalous (it suffers a so-called ABJ anomaly (1969)), and this explains the surprisingly fast decay rate of the π_0-meson, compared to the one of its charged cousins π_+ and π_−. Such ABJ anomalies are of a topological nature, and they famously relate to the Atiyah–Singer index theorem [AS69, Ati85, Ati86]. Other chiral anomalies such as gauge anomalies and ’t Hooft anomalies are also explained in terms of topology.

• Scrutiny of the topological aspects of supersymmetry was initiated by Witten [Wit82]. In the case of quantum mechanics, supersymmetry connects with Morse theory; the generalization to quantum field theories led to the development of Floer’s homologies [Flo87].

• Hitchin’s article [Hit87] on self-dual Yang–Mills equations over Riemann surfaces led to defining Higgs bundles, and to the subsequent discovery of higher Teichmüller spaces.

Atiyah’s impulse (see [Wit89]) led to a series of three seminal articles written by Witten. In [Wit88a], he proposes a natural physical interpretation of Donaldson theory as a topologically twisted quantum field theory. In [Wit88b] topologically twisted sigma models are introduced, explaining in particular Gromov’s work on symplectic invariants via pseudo-holomorphic curves [Gro85] in a physical language. Last, [Wit89] shows that Jones polynomials of knot theory are natural observables of Chern–Simons theories, hence giving those an intrinsically three-dimensional definition. These articles together with a bunch of others such as [Ati89] are the cornerstones of the study of TQFTs. Atiyah’s approach in [Ati89] is axiomatic, and provides a minimalist and categorical perspective on TQFTs which has also proved very useful. TQFTs also play an important role in condensed matter physics, where they appear as effective low-energy theories in vacua with topological order – such as superconductors or fractional quantum Hall states in 2d metals [Wen90]. Topological order is characterized by topology-dependent ground state degeneracy, fractional statistics and emergent gauge fields, perfect conducting edge states and long-range entanglement. For example, fractional quantum Hall states are effectively described by Chern–Simons theories. Chern–Simons-like couplings are also ubiquitous in string theory, in which they appear through generalized Green–Schwarz mechanisms. The latter are responsible for the unusual definitions of the field strengths of the string gauge potentials. In general, coupling QFTs to TQFTs can have dramatic consequences, despite the fact that TQFTs have only non-dynamical degrees of freedom [KS14]. This is studied systematically in the modern approach to symmetries that has been initiated in [GKSW15], and which is nowadays a rapidly-growing field in the string theory and condensed matter physics communities (see e.g. [ABGE+23, McG21]). One usually distinguishes two kinds of TQFTs.

• The ones of the first class are defined by an action functional which does not depend on a metric, such as BF theory in d-dimensional space-time, whose fields are differential p-forms A(p) and (d − p − 1)-forms B(d−p−1) with action

SBF = Z M(d) B (d−p−1) ∧ dA (p),

or U(1) d = 3 Chern–Simons theory with fields differential 1-forms A(1) and action

SCS =ZM(3)A(1) ∧ dA(1),

with generalizations for non-abelian groups and odd-dimensional space-times. The fact that there is no need to consider a metric on M(d) to define these theories implies that they are topological (though there are subtleties when going from classical to quantum field theory). These TQFTs are usually said to be of Schwarz type, as Chern–Simons theory was first considered in a series of articles by Albert Schwarz. This is the type of TQFT appearing e.g. in [Wit89].

• Those of the second class are defined by an action functional S which depend on a metric gαβ, but such that the energy-momentum tensor (i.e. δS/δgαβ) is BRST-trivial. The observables – BRST closed by definition – are therefore independent of the metric, meaning that the theory is indeed topological. These TQFTs are said to be of Witten type, and were introduced in [Wit88a, Wit88b].

For a rough overview of possible topics, please click here