Topological quantum field theory

The topological quantum field theory (TQFT) is a connection of the quantum field theory with topology , which emerged in the late 1980s ( Edward Witten , Michael Atiyah ). Connections between quantum theory and topology had existed before. Important quantities of the TQFT are independent of the metric of the manifolds on which the quantum fields are defined. They are therefore of interest as quantum field theoretical models that provide topological invariants of the underlying manifolds and were therefore also used in pure mathematics, for example in knot theory , in the topology of four-dimensional manifolds and in the theory of modular spaces in algebraic geometry. In physics, for example, they were viewed as a model for quantum gravity, but also as effective field theories in solid-state physics, where topological invariants, for example in the quantum Hall effect, are important.

definition

The functional integral formalism of quantum field theories, which are given by scalar action functionals in the fields , is considered. These are in turn defined as functions on a Riemannian manifold with metric . One considers (for example as observables) functionals of the fields and their vacuum expectation values : ${\displaystyle S(\phi _{i})}$ ${\displaystyle \phi _{i}}$ ${\displaystyle g_{\mu \nu }}$ ${\displaystyle O_{j}(\phi )}$

${\displaystyle \langle O_{1}(\phi _{i})\,O_{2}(\phi _{i})\dotsm O_{n}(\phi _{i})\rangle =\int D[\phi _{i}]\,O_{1}(\phi _{i})\dotsm O_{n}(\phi _{i})e^{-S(\phi _{i})}}$

The quantum field theory is called topological if the vacuum expectation values ​​of the products of operators do not depend on the metric, they are invariant under variation of the metric on M:

${\displaystyle {\frac {\delta }{\delta g^{\mu \nu }}}\langle O_{1}(\phi _{i})\,O_{2}(\phi _{i})\dotsm O_{n}(\phi _{i})\rangle =0}$

The corresponding operators are called observables.

There are basically two ways of implementing a TQFT. In the simplest case, the action functional and the operators are independent of the metrics, then one speaks of TQFT of the Schwarz type (after Albert S. Schwarz ). One example is the Chern-Simons gauge theory. In many cases, however, this turned out to be too restrictive. In the case of Witten-type TQFT (or cohomological TQFT) it is required that a scalar symmetry exists (whose infinitesimal transformations are denoted by, but this should not be confused with the variation), under which the action functional and the observables are invariant: ${\displaystyle \delta }$

${\displaystyle \delta S(\phi _{i})=0}$ and ${\displaystyle \delta O_{k}(\phi _{i})=0}$

In addition, for the energy-momentum tensor, defined by

${\displaystyle T_{\mu \nu }(\phi _{i})={\frac {\delta }{\delta g^{\mu \nu }}}S(\phi _{i})}$,

be valid:

${\displaystyle T_{\mu \nu }=\delta G_{\mu \nu }(\phi _{i})}$,

where is a tensor. This ensures that the vacuum expectation values ​​of the products of observables vanish when the metric varies. ${\displaystyle G_{\mu \nu }}$

An example of Witten-type TQFT is the Donaldson-Witten theory.

history

The first approaches to TQFT come from Albert S. Schwarz , who in 1978 expressed Ray-Singer torsion, a topological invariant, through a distribution function of a quantum field theory. Fundamental for further development - not only at TQFT - was Edward Witten's 1982 work on supersymmetry and morse theory . In 1988 Witten succeeded in his article Topological Quantum Field Theory, two important mathematical developments, the theories of Simon Donaldson on topological invariants of 4-manifolds (which used essentially self-dual Yang-Mills theories on 4-manifolds and their instantons ) and of Andreas Floer ( Floer homology ) for 3-manifolds to give an interpretation with the help of a TQFT. Around the same time Michael Atiyah developed an axiomatic approach to TQFT, interpreting the theories of Witten, Donaldson, Floer and others. A high point of development was soon afterwards the calculation of knot invariants by Witten with the Chern-Simons theory.

Axiomatic definition

Axiomatic approaches to characterizing topological quantum field theories stem largely from Michael Francis Atiyah , who was inspired by axioms for conformal field theories by Graeme Segal and Edward Witten's (aforementioned) geometric interpretation of supersymmetry. They are particularly useful for black-type TQFT and it is unclear whether they include all Witten-type TQFT. The basic idea is to define a topological quantum field theory through a functor of a certain category of cobordisms in the category of vector spaces .

Be a commutative ring with 1 (usually , or ). The axioms of a topological quantum field theory on a manifold of the dimension defined over the ring are based on the following objects: ${\displaystyle \Lambda }$ ${\displaystyle \Lambda =\mathbb {Z} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle d}$ ${\displaystyle \Lambda }$

• A finitely generated - module , smooth each oriented closed d dimensional manifold is assigned ${\displaystyle \Lambda }$ ${\displaystyle Z(\Sigma )}$ ${\displaystyle \Sigma }$
• An element associated with any oriented smooth ( d +1) -dimensional bounded manifold (with boundaries ) ${\displaystyle Z(M)\in \mathrm {Z} (\partial M)}$ ${\displaystyle M}$ ${\displaystyle \partial M}$

The axioms are then:

• 1. is funcional in relation to the orientation-preserving diffeomorphisms of and . ${\displaystyle \mathrm {Z} }$ ${\displaystyle \Sigma }$ ${\displaystyle M}$
• 2. is involutorial, that is = , where the manifold denotes the opposite orientation and the module to be dual . ${\displaystyle \mathrm {Z} }$ ${\displaystyle Z(\Sigma ^{*})}$ ${\displaystyle Z(\Sigma )^{*}}$ ${\displaystyle \Sigma ^{*}}$ ${\displaystyle \Sigma }$ ${\displaystyle Z(\Sigma )^{*}}$ ${\displaystyle Z(\Sigma )}$
• 3. is multiplicative. For disjoint d -dimensional manifolds , we have . ${\displaystyle \mathrm {Z} }$ ${\displaystyle \Sigma _{1}}$ ${\displaystyle \Sigma _{2}}$ ${\displaystyle Z(\Sigma _{1}\cup \Sigma _{2})=Z(\Sigma _{1})\otimes Z(\Sigma _{2})}$
• 4. = for the d -dimensional empty manifold and for the ( d +1) -dimensional empty manifold . ${\displaystyle \mathrm {Z} (\Phi )}$ ${\displaystyle \Lambda }$ ${\displaystyle \Phi }$ ${\displaystyle \mathrm {Z} (\Phi )=1}$ ${\displaystyle \Phi }$
• 5. = . Equivalent to this, is disjoint to . ${\displaystyle \mathrm {Z} (M^{*})}$ ${\displaystyle {\overline {\mathrm {Z} (M)}}}$ ${\displaystyle \mathrm {Z} (M^{*})}$ ${\displaystyle \mathrm {Z} (M)}$

Axioms 4 and 5 were added by Atiyah.

Physical interpretation

The second and fourth points are associated with the underlying manifold (in the case of spacetime with relativistic invariance), while the third and fifth represent the quantum mechanical properties.

${\displaystyle \Sigma }$is the physical space (i.e. three-dimensional in the usual physical theories) and the additional dimension in the product represents an "imaginary time", i.e. the usual time variable with the imaginary unit as a prefactor. The module is the Hilbert space of the quantum theory under consideration. The Hamilton operator results in the time evolution operator acting in the Hilbert space of quantum mechanical states or the time evolution operator for "imaginary time" . The most important properties of topological field theories are that there are no dynamics or propagation along the cylinder . But it can be a tunneling Σ ₀ to Σ ₁ through an intervening manifold with give. ${\displaystyle I}$ ${\displaystyle \Sigma \times I}$ ${\displaystyle i}$ ${\displaystyle Z(\Sigma )}$ ${\displaystyle H}$ ${\displaystyle e^{itH}}$ ${\displaystyle e^{-tH}}$ ${\displaystyle H=0}$ ${\displaystyle \Sigma \times I}$ ${\displaystyle M}$ ${\displaystyle \partial M=\Sigma _{0}^{*}\cup \Sigma _{1}}$

For the vector in Hilbert space represents the vacuum state on . For a closed manifold is the vacuum expectation value. ${\displaystyle \partial M=\Sigma }$ ${\displaystyle Z(M)}$ ${\displaystyle Z(\Sigma )}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle Z(M)}$

The development of a three-dimensional manifold (for example a node) in four-dimensional spacetime (or more generally a d -dimensional manifold in ( d +1) -dimensional spacetime) corresponds to a cobordism . Is a space (for example, a node) at the time and the space at the time , then the global surface of these spaces a Cobordism with A and B as its edges: . The associated Hilbert spaces of A and B are mapped to one another in the TQFT by operators that only depend on the topology of . ${\displaystyle A}$ ${\displaystyle t_{\text{1}}}$ ${\displaystyle B}$ ${\displaystyle t_{\text{2}}}$ ${\displaystyle C}$ ${\displaystyle \partial C=A\cup B}$ ${\displaystyle C}$

Examples

Chern-Simons theory

The Chern-Simons theory is a TQFT of the Schwarz type with a calibration group and associated main fiber bundle with connection . The effect functional is the integral of the Chern-Simons 3 form over the three-dimensional manifold : ${\displaystyle G}$ ${\displaystyle A}$ ${\displaystyle M}$

${\displaystyle S={\frac {k}{4\pi }}\int _{M}{\text{Sp}}\,(A\wedge dA+{\tfrac {2}{3}}A\wedge A\wedge A)}$

It stands for the formation of tracks in the calibration group. The Chern-Simons theory offers a field theoretical framework for describing three-dimensional knots and links (left). Witten showed that the vacuum expectation values ​​of operators given by Wilson loops result in topological node invariants . Wilson loops to a loop in are defined as ${\displaystyle {\text{Sp}}}$ ${\displaystyle K}$ ${\displaystyle M}$

${\displaystyle W_{R}(K)={\text{Sp}}_{R}\,{\mathcal {P}}\,\exp {i\oint _{K}A}}$,

where denotes an irreducible representation of and the order of the exponential. A link is viewed as a union of loops and the correlation function of the associated Wilson loops is proportional to the HOMFLY polynomial (for the special case of the Jones polynomial ), for the Kauffman polynomials . The Vassiliev invariants of the knot theory (after Viktor Anatoljewitsch Wassiljew ) can also be calculated perturbatively in the Chern-Simons theory (they are coefficients of the perturbative development of the correlation functions). ${\displaystyle R}$ ${\displaystyle G}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle G=SU(N)}$ ${\displaystyle N=2}$ ${\displaystyle G=SO(N)}$

In addition to the three-dimensional theory considered here, there are also higher-dimensional generalizations.

The Chern-Simons theory is the non-Abelian generalization of the simplest Schwarz-type TQFT:

${\displaystyle S=\int _{M}{\text{Sp}}\,(\,A\wedge dA\,)}$

BF theory

Another topological quantum field theory is the BF theory (BF stands for Background Field ). It is the only topological field theory that can be consistently formulated in every dimension. Their effect is defined as follows:

${\displaystyle S=\int _{M}{\text{Sp}}(\mathbf {B} \wedge \mathbf {F} )}$

Again there is a main fiber bundle ( principal bundle ) with connection to the calibration group . There is also a dynamic field , which in the case of a four-dimensional manifold M considered here is a 2-form with values ​​in the adjoint representation of the gauge group (in the general -dimensional case it is a -form). The track formation takes place in the selected representation of the calibration group. ${\displaystyle G}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle d}$ ${\displaystyle (d-2)}$

The curvature is given by ${\displaystyle F}$

${\displaystyle \mathbf {F} \equiv d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} }$.

The BF theory generalizes the case of the Yang-Mills theory , in which is given by , with the Hodge star operator *, which assigns the -form to the 2-form ( is the dimension of the underlying manifold M). The effective functional is proportional in the Yang-Mills theory: ${\displaystyle B}$ ${\displaystyle *F}$ ${\displaystyle F}$ ${\displaystyle (d-2)}$ ${\displaystyle *F}$ ${\displaystyle d}$

${\displaystyle S\sim \int _{M}{\text{Sp}}(*\mathbf {F} \wedge \mathbf {F} )}$

A certain formulation of the Einstein-Hilbert effect of gravity has the form of a BF theory.

Wittensche field theories

The Donaldson-Witten theory is one of the most important representatives of Wittens field theories. It offers an opportunity to study 4-dimensional, smooth manifolds . Seiberg-Witten invariants are important here.

The first version of such a field theory was published by Edward Witten in 1988 in the form of a topological Yang-Mills theory , a twisted version of a supersymmetric Yang-Mills theory in four space-time dimensions. Witten was able to interpret invariants from Simon Donaldson for four-dimensional manifolds and from Andreas Floer ( Floer homology ) in the context of a TQFT.

As shown above, Witten-type TQFT are based on the existence of a scalar symmetry : ${\displaystyle \delta }$

1. The effect of the theory satisfies a symmetry, that is, if a symmetry transformation denotes (for example a Lie derivative ), then it remains . ${\displaystyle S}$ ${\displaystyle \delta }$ ${\displaystyle \delta S=0}$

2. The symmetry transformation is exact, that is . ${\displaystyle \delta ^{2}=0}$

3. There are observables that are sufficient for everyone . ${\displaystyle O_{1},\dotsc ,O_{n}}$ ${\displaystyle \delta O_{i}=0}$ ${\displaystyle i\in \{1,\dotsc ,n\}}$

4. The energy-momentum tensor is given in the form for any tensor . ${\displaystyle T^{\alpha \beta }=\delta G^{\alpha \beta }}$ ${\displaystyle G^{\alpha \beta }}$

Metric independence

The topological quantum field theory is invariant to the metrics and coordinate transformations of spacetime, which means that, for example, the correlation functions of field theory do not change with changes in spacetime geometry. Therefore topological field theories on classical Minkowski spaces from special relativity and elementary particle physics are not particularly useful, since the Minkowski space is a contractible space from a topological point of view and all topological invariants are trivial. Therefore, topological field theories are mostly only considered on curved Riemann surfaces or on curved spacetime and are of interest for the study of models of quantum gravity , the formulation of which should be background-independent with regard to the metric.

See also

• Cobordism conjecture

literature

Original works:

• Michael Atiyah : New invariants of three and four dimensional manifolds . In: Proc. Symp. Pure Math. Band 48 . American Math. Soc., 1988, pp. 285-299 .
• Michael Atiyah: Topological quantum field theories . In: Publications Mathématiques de l'IHÉS . tape 68 , 1988, pp. 175–186 , doi : 10.1007 / BF02698547 ( online [PDF]).
• E. Witten: Topological Quantum Field Theory. Comm. Math. Phys., Vol. 117, 1988, pp. 353-386, Project Euclid.
• E. Witten: Quantum Field Theory and the Jones Polynomial. Comm. Math. Phys. Vol. 121, 1989, p. 351, Project Euclid.
• E. Witten: Supersymmetry and Morse Theory. J. Diff. Geom., Vol. 17, 1982, pp. 661-692, Project Euclid.
• E. Witten: Topological Sigma Models. Communications in Mathematical Physics, Volume 118, 1988, pp. 411-449, Project Euclid.

Overviews:

• D. Birmingham, M. Blau, M. Rakowski, G. Thompson: Topological Field Theory. Physics Reports, Volume 209, 1991, pp. 129-340.
• S. Cordes, Gregory W. Moore , S. Ramgoolam: Lectures on 2D Yang-Mills-Theory, equivariant cohomology and topological field theories. Les Houches Lectures, Session 62, Elsevier 1994, Arxiv.
• JMF Labastida, M. Marino: Topological Quantum Field Theory and Four Manifolds. Elsevier 2005.
• JMF Labastida, C. Lozano: Lectures on Topological Quantum Field Theory. In: H. Falomir, R. Gamboa, F. Schaposnik: Trends in Theoretical Physics. AIP, New York 1998, Arxiv.
• JMF Labastida, C. Lozano: Topological Quantum Field Theory. In: Jean-Pierre Françoise, Gregory L. Naber, Tsou Sheung Tsun: Encyclopedia of Mathematical Physics. Elsevier, 2006.
• Ruth Lawrence : An introduction to topological field theory , in: LH Kauffman (Ed.), The interface of knots and physics, Proc. Symp. Applied Math. 51, Amer. Math. Soc., 1996, pp. 89-128
• Cumrun Vafa : Unifying Themes in Topological Field Theories, Conference on geometry and topology in honor of M. Atiyah, R. Bott, F. Hirzebruch and I. Singer, Harvard University 2000, Arxiv
• Albert Schwarz: Topological Quantum Field Theories. UC Davis, 2000, Arxiv.

Web links

• Vladimir G. Ivancevic, Tijana T. Ivancevic: Undergraduate Lecture Notes in Topological Quantum Field Theory. 2008, Arxiv.
• Lexicon of Physics. Topological QFT. Spectrum.

Individual evidence

1. ↑ See Vladimir G. Ivancevic, Tijana T. Ivancevic: Undergraduate Lecture Notes in Topological Quantum Field Theory. 2008, p. 36, Arxiv.
2. ^ Schwarz: The partition function of a degenerate quadratic functional and the Ray-Singer-Invariants. Lett. Math. Phys., Vol. 2, 1978, p. 247.
3. ^ Atiyah: Topological quantum field theories. Pub. Math. IHES 1988, p. 178.
4. ^ Higher dimensional Chern Simons theory. Ncat-Lab.
5. ↑ For example Alberto Cattaneo, Paolo Cotta-Ramusino, Jürg Fröhlich , Maurizio Martellini: Topological BF-Theories in 3 and 4 dimensions. J. Math. Phys., Vol. 36, 1995, pp. 6137-6160, Arxiv.
6. ↑ Gravity as BF theory. Ncat-Lab.
7. ↑ The symmetry operators are related to the Becchi-Rouet-Stora-Tyutin formalism (BRST) of the quantization of gauge field theories (or general quantum theories with constraints) and the nilpotent operators used there. ${\displaystyle \delta }$