Moduli (physics)

Last updated

In quantum field theory, the term moduli (sg.: modulus; more properly moduli fields) is sometimes used to refer to scalar fields whose potential energy function has continuous families of global minima. Such potential functions frequently occur in supersymmetric systems. The term "modulus" is borrowed from mathematics (or more specifically, moduli space is borrowed from algebraic geometry), where it is used synonymously with "parameter". The word moduli (Moduln in German) first appeared in 1857 in Bernhard Riemann's celebrated paper "Theorie der Abel'schen Functionen". [1]

Contents

Moduli spaces in quantum field theories

In quantum field theories, the possible vacua are usually labeled by the vacuum expectation values of scalar fields, as Lorentz invariance forces the vacuum expectation values of any higher spin fields to vanish. These vacuum expectation values can take any value for which the potential function is a minimum. Consequently, when the potential function has continuous families of global minima, the space of vacua for the quantum field theory is a manifold (or orbifold), usually called the vacuum manifold. [2] This manifold is often called the moduli space of vacua, or just the moduli space, for short.

The term moduli are also used in string theory to refer to various continuous parameters that label possible string backgrounds: the expectation value of the dilaton field, the parameters (e.g. the radius and complex structure) which govern the shape of the compactification manifold, et cetera. These parameters are represented, in the quantum field theory that approximates the string theory at low energies, by the vacuum expectation values of massless scalar fields, making contact with the usage described above. In string theory, the term "moduli space" is often used specifically to refer to the space of all possible string backgrounds.

Moduli spaces of supersymmetric gauge theories

In general quantum field theories, even if the classical potential energy is minimized over a large set of possible expectation values, once quantum corrections are included it is generically the case that nearly all of these configurations cease to minimize the energy. The result is that the set of vacua of the quantum theory is generally much smaller than that of the classical theory. A notable exception occurs when the various vacua in question are related by a symmetry which guarantees that their energy levels remain exactly degenerate.

The situation is very different in supersymmetric quantum field theories. In general, these possess large moduli spaces of vacua which are not related by any symmetry, for example, the masses of the various excitations may differ at various points on the moduli space. The moduli spaces of supersymmetric gauge theories are in general easier to calculate than those of nonsupersymmetric theories because supersymmetry restricts the allowed geometries of the moduli space even when quantum corrections are included.

Allowed moduli spaces of 4-dimensional theories

The more supersymmetry there is, the stronger the restriction on the vacuum manifold. Therefore, if a restriction appears below for a given number N of spinors of supercharges, then it also holds for all greater values of N.

N=1 Theories

The first restriction on the geometry of a moduli space was found in 1979 by Bruno Zumino and published in the article "Supersymmetry and Kähler Manifolds". [3] He considered an N=1 theory in 4-dimensions with global supersymmetry. N=1 means that the fermionic components of the supersymmetry algebra can be assembled into a single Majorana supercharge. The only scalars in such a theory are the complex scalars of the chiral superfields. He found that the vacuum manifold of allowed vacuum expectation values for these scalars is not only complex but also a Kähler manifold.

If gravity is included in the theory, so that there is local supersymmetry, then the resulting theory is called a supergravity theory and the restriction on the geometry of the moduli space becomes stronger. The moduli space must not only be Kähler, but also the Kähler form must lift to integral cohomology. Such manifolds are called Hodge manifolds. The first example appeared in the 1979 article "Spontaneous Symmetry Breaking and Higgs Effect in Supergravity Without Cosmological Constant" [4] and the general statement appeared 3 years later in "Quantization of Newton's Constant in Certain Supergravity Theories". [5]

N=2 Theories

In extended 4-dimensional theories with N=2 supersymmetry, corresponding to a single Dirac spinor supercharge, the conditions are stronger. The N=2 supersymmetry algebra contains two representations with scalars, the vector multiplet which contains a complex scalar and the hypermultiplet which contains two complex scalars. The moduli space of the vector multiplets is called the Coulomb branch while that of the hypermultiplets is called the Higgs branch. The total moduli space is locally a product of these two branches, as nonrenormalization theorems imply that the metric of each is independent of the fields of the other multiplet.(See for example Argyres, Non-Perturbative Dynamics Of Four-Dimensional Supersymmetric Field Theories, pp. 6–7, for further discussion of the local product structure.)

In the case of global N=2 supersymmetry, in other words in the absence of gravity, the Coulomb branch of the moduli space is a special Kähler manifold. The first example of this restriction appeared in the 1984 article Potentials and Symmetries of General Gauged N=2 Supergravity: Yang-Mills Models by Bernard de Wit and Antoine Van Proeyen, while a general geometric description of the underlying geometry, called special geometry, was presented by Andrew Strominger in his 1990 paper Special Geometry.

The Higgs branch is a hyperkähler manifold as was shown by Luis Alvarez-Gaume and Daniel Freedman in their 1981 paper Geometrical Structure and Ultraviolet Finiteness in the Supersymmetric Sigma Model. Including gravity the supersymmetry becomes local. Then one needs to add the same Hodge condition to the special Kahler Coulomb branch as in the N=1 case. Jonathan Bagger and Edward Witten demonstrated in their 1982 paper Matter Couplings in N=2 Supergravity that in this case, the Higgs branch must be a quaternionic Kähler manifold.

N>2 Supersymmetry

In extended supergravities with N>2 the moduli space must always be a symmetric space.

Related Research Articles

Supersymmetry is a theoretical framework in physics that suggests the existence of a symmetry between particles with integer spin (bosons) and particles with half-integer spin (fermions). It proposes that for every known particle, there exists a partner particle with different spin properties. There have been multiple experiments on supersymmetry that have failed to provide evidence that it exists in nature. If evidence is found, supersymmetry could help explain certain phenomena, such as the nature of dark matter and the hierarchy problem in particle physics.

<span class="mw-page-title-main">Vacuum expectation value</span> Type of operator expectation value

In quantum field theory the vacuum expectation value of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by One of the most widely used examples of an observable physical effect that results from the vacuum expectation value of an operator is the Casimir effect.

<span class="mw-page-title-main">Supergravity</span> Modern theory of gravitation that combines supersymmetry and general relativity

In theoretical physics, supergravity is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as the Minimal Supersymmetric Standard Model. Supergravity is the gauge theory of local supersymmetry. Since the supersymmetry (SUSY) generators form together with the Poincaré algebra a superalgebra, called the super-Poincaré algebra, supersymmetry as a gauge theory makes gravity arise in a natural way.

<span class="mw-page-title-main">Minimal Supersymmetric Standard Model</span> Simplest supersymmetric extension to the Standard Model

The Minimal Supersymmetric Standard Model (MSSM) is an extension to the Standard Model that realizes supersymmetry. MSSM is the minimal supersymmetrical model as it considers only "the [minimum] number of new particle states and new interactions consistent with "Reality". Supersymmetry pairs bosons with fermions, so every Standard Model particle has a superpartner. If discovered, such superparticles could be candidates for dark matter, and could provide evidence for grand unification or the viability of string theory. The failure to find evidence for MSSM using the Large Hadron Collider has strengthened an inclination to abandon it.

In quantum field theory, Seiberg duality, conjectured by Nathan Seiberg in 1994, is an S-duality relating two different supersymmetric QCDs. The two theories are not identical, but they agree at low energies. More precisely under a renormalization group flow they flow to the same IR fixed point, and so are in the same universality class. It is an extension to nonabelian gauge theories with N=1 supersymmetry of Montonen–Olive duality in N=4 theories and electromagnetic duality in abelian theories.

In theoretical physics, the Wess–Zumino model has become the first known example of an interacting four-dimensional quantum field theory with linearly realised supersymmetry. In 1974, Julius Wess and Bruno Zumino studied, using modern terminology, dynamics of a single chiral superfield whose cubic superpotential leads to a renormalizable theory.

In string theory, the string theory landscape is the collection of possible false vacua, together comprising a collective "landscape" of choices of parameters governing compactifications.

Erick J. Weinberg is a theoretical physicist and professor of physics at Columbia University.

In theoretical physics, topological string theory is a version of string theory. Topological string theory appeared in papers by theoretical physicists, such as Edward Witten and Cumrun Vafa, by analogy with Witten's earlier idea of topological quantum field theory.

In theoretical physics, Seiberg–Witten theory is an supersymmetric gauge theory with an exact low-energy effective action, of which the kinetic part coincides with the Kähler potential of the moduli space of vacua. Before taking the low-energy effective action, the theory is known as supersymmetric Yang–Mills theory, as the field content is a single vector supermultiplet, analogous to the field content of Yang–Mills theory being a single vector gauge field or connection.

In theoretical physics, the μ problem is a problem of supersymmetric theories, concerned with understanding the parameters of the theory.

In theoretical physics a nonrenormalization theorem is a limitation on how a certain quantity in the classical description of a quantum field theory may be modified by renormalization in the full quantum theory. Renormalization theorems are common in theories with a sufficient amount of supersymmetry, usually at least 4 supercharges.

Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings.

In theoretical physics, super QCD is a supersymmetric gauge theory which resembles quantum chromodynamics (QCD) but contains additional particles and interactions which render it supersymmetric.

Higher-dimensional supergravity is the supersymmetric generalization of general relativity in higher dimensions. Supergravity can be formulated in any number of dimensions up to eleven. This article focuses upon supergravity (SUGRA) in greater than four dimensions.

This page is a glossary of terms in string theory, including related areas such as supergravity, supersymmetry, and high energy physics.

Ali H. Chamseddine is a Lebanese physicist known for his contributions to particle physics, general relativity and mathematical physics. As of 2013, Chamseddine is a physics Professor at the American University of Beirut and the Institut des hautes études scientifiques.

In theoretical physics, 3D mirror symmetry is a version of mirror symmetry in 3-dimensional gauge theories with N=4 supersymmetry, or 8 supercharges. It was first proposed by Kenneth Intriligator and Nathan Seiberg, in their 1996 paper "Mirror symmetry in three-dimensional gauge theories", as a relation between pairs of 3-dimensional gauge theories, such that the Coulomb branch of the moduli space of one is the Higgs branch of the moduli space of the other. It was demonstrated using D-brane cartoons by Amihay Hanany and Edward Witten 4 months later, where they found that it is a consequence of S-duality in type IIB string theory.

Peter Christopher West, born on 4 December 1951, is a British theoretical physicist at King's College, London and a fellow of the Royal Society.

Olaf Lechtenfeld is a German mathematical physicist, academic and researcher. He is a full professor at the Institute of Theoretical Physics at Leibniz University, where he founded the Riemann Center for Geometry and Physics.

References

  1. ↑ Riemann, Bernhard (1857). "Theorie der Abel'schen Functionen". Journal für die reine und angewandte Mathematik. 54: 101–155.
  2. ↑ Teerthal, Patel (2022-01-16). "Kibble mechanism for electroweak magnetic monopoles and magnetic fields". Journal of High Energy Physics . 2022 (1). Arizona State University: 10. arXiv: 2108.05357 . Bibcode:2022JHEP...01..059P. doi:10.1007/JHEP01(2022)059. S2CID   256034831.
  3. ↑ Zumino, B. (Nov 1979). "Supersymmetry and Kähler manifolds". Physics Letters B. 87 (3): 203–206. doi:10.1016/0370-2693(79)90964-X.
  4. ↑ Cremmer, E.; Julia, B.; Scherk, J.; Ferrara, S.; Girardello, L.; van Nieuwenhuizen, P. (Jan 1979). "Spontaneous symmetry breaking and Higgs effect in supergravity without cosmological constant". Nuclear Physics B. 147 (1–2): 105–131. doi:10.1016/0550-3213(79)90417-6. Archived from the original on 10 Dec 2012.
  5. ↑ Witten, Edward; Bagger, Jonathan (Sep 1982). "Quantization of Newton's constant in certain supergravity theories". Physics Letters B. 115 (3): 202–206. doi:10.1016/0370-2693(82)90644-X.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.