In theoretical physics, the six-dimensional (2,0)-superconformal field theory is a
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
whose existence is predicted by arguments in
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
. It is still poorly understood because there is no known description of the theory in terms of an
action functional. Despite the inherent difficulty in studying this theory, it is considered to be an interesting object for a variety of reasons, both physical and mathematical.
Applications
The (2,0)-theory has proven to be important for studying the general properties of quantum field theories. Indeed, this theory subsumes a large number of mathematically interesting
effective quantum field theories and points to new dualities relating these theories. For example, Luis Alday,
Davide Gaiotto
Davide Silvano Achille Gaiotto (born 11 March 1977) is an Italian mathematical physicist who deals with quantum field theories and string theory. He received the Gribov Medal in 2011 and the New Horizons in Physics Prize in 2013.
Biography
Gai ...
, and Yuji Tachikawa showed that by compactifying this theory on a
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
, one obtains a four-dimensional quantum field theory, and there is a
duality known as the
AGT correspondence which relates the physics of this theory to certain physical concepts associated with the surface itself. More recently, theorists have extended these ideas to study the theories obtained by compactifying down to three dimensions.
In addition to its applications in quantum field theory, the (2,0)-theory has spawned a number of important results in
pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications ...
. For example, the existence of the (2,0)-theory was used by
Witten
Witten () is a city with almost 100,000 inhabitants in the Ennepe-Ruhr-Kreis (district) in North Rhine-Westphalia, in western Germany.
Geography
Witten is situated in the Ruhr valley, in the southern Ruhr area.
Bordering municipalities
* Bochum ...
to give a "physical" explanation for a conjectural relationship in mathematics called the
geometric Langlands correspondence
In mathematics, the geometric Langlands correspondence relates algebraic geometry and representation theory. It is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic ...
. In subsequent work, Witten showed that the (2,0)-theory could be used to understand a concept in mathematics called
Khovanov homology
In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial.
It was developed in the late 1990s by Mikhail Khovanov.
Overv ...
. Developed by
Mikhail Khovanov
Mikhail Khovanov (; born 13 January 1972) is a Russian professor of mathematics at Johns Hopkins University who works on representation theory, knot theory, and algebraic topology. He is known for introducing Khovanov homology for links, which ...
around 2000, Khovanov homology provides a tool in
knot theory
In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...
, the branch of mathematics that studies and classifies the different shapes of knots. Another application of the (2,0)-theory in mathematics is the work of Davide Gaiotto,
Greg Moore, and
Andrew Neitzke, which used physical ideas to derive new results in
hyperkähler geometry.
[Gaiotto, Moore, Neitzke 2013]
See also
*
ABJM superconformal field theory
*
N = 4 supersymmetric Yang–Mills theory
''N'' = 4 supersymmetric Yang–Mills (SYM) theory is a relativistic conformally invariant Lagrangian gauge theory describing the interactions of fermions via gauge field exchanges. In ''D''=4 spacetime dimensions, ''N''=4 is the m ...
Notes
References
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{{Quantum field theories
Conformal field theory
Supersymmetric quantum field theory
String theory