In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, Seiberg–Witten theory is a theory that determines an exact low-energy effective action (for massless degrees of freedom) of a
supersymmetric gauge theory—namely the metric of the
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
of vacua.
Seiberg–Witten curves
In general, effective Lagrangians of supersymmetric gauge theories are largely determined by their holomorphic properties and their behavior near the singularities. In particular, in
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
with
extended supersymmetry
In theoretical physics, extended supersymmetry is supersymmetry whose infinitesimal generators Q_i^\alpha carry not only a spinor index \alpha, but also an additional index i=1,2 \dots \mathcal where \mathcal is integer (such as 2 or 4).
Extende ...
, the moduli space of vacua is a special
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...
and its Kähler potential is constrained by above conditions.
In the original approach, by
Seiberg and
Witten
Witten () is a city with almost 100,000 inhabitants in the Ennepe-Ruhr-Kreis (district) in North Rhine-Westphalia, Germany.
Geography
Witten is situated in the Ruhr valley, in the southern Ruhr area.
Bordering municipalities
* Bochum
* Dortmu ...
, holomorphy and electric-magnetic duality constraints are strong enough to almost uniquely
constrain the prepotential, and therefore the metric of the moduli space of vacua, for theories with
gauge group.
More generally, consider the example with gauge group SU(n). The classical potential is
This vanishes on the moduli space, so the vacuum expectation value of
can be gauge rotated into Cartan subalgebra, making it a traceless diagonal complex matrix
.
Because the fields
no longer have vanishing
vacuum expectation value
In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. ...
, other fields become heavy due to the Higgs effect. They are integrated out in order to find the effective
Abelian gauge theory. Its two-derivative, four-fermions low-energy action can be expressed in terms of a single holomorphic function
, as follows:
The first term is a perturbative loop calculation and the second is the
instanton
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
part where
labels fixed instanton numbers. In theories whose gauge groups are products of unitary groups,
can be computed exactly using localization and the limit shape techniques.
From
we can get the mass of the
BPS particles.
One way to interpret this is that these variables
and its dual can be expressed as
periods of a meromorphic differential on a Riemann surface called the Seiberg–Witten curve.
Relation to integrable systems
The special Kähler geometry on the moduli space of vacua in Seiberg–Witten theory can be identified with the geometry of the base of complex completely
integrable system
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
. The total phase of this complex completely integrable system can be identified with the moduli space of vacua of the 4d theory compactified on a circle. The relation between Seiberg–Witten theory and integrable systems has been reviewed by
Eric D'Hoker and
D. H. Phong. See
Hitchin system
In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies on the crossroads of algebraic geometry, the th ...
.
Seiberg–Witten prepotential via instanton counting
Using supersymmetric localisation techniques, one can explicitly determine the instanton partition function of
super Yang–Mills theory. The Seiberg–Witten prepotential can then be extracted using the localization approach
of
Nikita Nekrasov
Nikita Alexandrovich Nekrasov (russian: Ники́та Алекса́ндрович Некра́сов; born 10 April 1973) is a mathematical and theoretical physicist at the Simons Center for Geometry and Physics and C.N.Yang Institute for Th ...
. It arises in the flat space limit
,
, of the partition function of the theory subject to the so-called
-background.
The latter is a specific background of four dimensional
supergravity. It can be engineered, formally by lifting the
1)">super Yang–Mills theory to six dimensions, then compactifying on 2-torus, while twisting the four dimensional spacetime around the two non-contractible cycles. In addition, one twists fermions so as to produce covariantly constant spinors generating unbroken supersymmetries. The two parameters
,
of the
-background correspond to the angles of the spacetime rotation.
In Ω-background, we can integrate out all the non-zero modes, so the path integral with the boundary condition
at
can be expressed as a sum over instanton number of the products and ratios of fermionic and bosonic determinants, producing the so-called
Nekrasov partition function.
In the limit where
,
approach 0, this sum is dominated by a unique saddle point.
On the other hand, when
,
approach 0,
holds.
See also
*
Ginzburg–Landau theory
In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenol ...
*
Donaldson theory
In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. It was started by Simon Donaldson (1983) who proved Donaldson's theorem restricti ...
References
* (''See Section 7.2'')
External links
*
{{DEFAULTSORT:Seiberg-Witten theory
Supersymmetric quantum field theory
Gauge theories