Seiberg–Witten theory

In theoretical physics, Seiberg–Witten theory is a theory that determines an exact low-energy effective action (for massless degrees of freedom) of a ${\displaystyle {\mathcal {N}}=2}$ supersymmetric gauge theory—namely the metric of the moduli space 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 with ${\displaystyle {\mathcal {N}}=2}$ extended supersymmetry, the moduli space of vacua is a special Kähler manifold and its Kähler potential is constrained by above conditions.

In the original approach,[1][2] by Seiberg and Witten, 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 ${\displaystyle SU(2)}$ gauge group.

More generally, consider the example with gauge group SU(n). The classical potential is

${\displaystyle V(x)={\frac {1}{g^{2}}}\operatorname {Tr} [\phi ,{\bar {\phi }}]^{2}\,}$

(1)

This vanishes on the moduli space, so the vacuum expectation value of ${\displaystyle \phi }$ can be gauge rotated into Cartan subalgebra, making it a traceless diagonal complex matrix ${\displaystyle a}$.

Because the fields ${\displaystyle \phi }$ no longer have vanishing vacuum expectation value, other fields become heavy due to the Higgs effect. They are integrated out in order to find the effective ${\displaystyle {\mathcal {N}}=2}$ Abelian gauge theory. Its two-derivative, four-fermions low-energy action can be expressed in terms of a single holomorphic function ${\displaystyle {\mathcal {F}}}$, as follows:

${\displaystyle {\frac {1}{4\pi }}\operatorname {Im} {\Bigl [}\int d^{4}\theta {\frac {d{\mathcal {F}}}{dA}}{\bar {A}}+\int d^{2}\theta {\frac {1}{2}}{\frac {d^{2}{\mathcal {F}}}{dA^{2}}}W_{\alpha }W^{\alpha }{\Bigr ]}\,}$

(3)

${\displaystyle {\mathcal {F}}={\frac {i}{2\pi }}{\mathcal {A}}^{2}\operatorname {\ln } {\frac {{\mathcal {A}}^{2}}{\Lambda ^{2}}}+\sum _{k=1}^{\infty }{\mathcal {F}}_{k}{\frac {\Lambda ^{4k}}{{\mathcal {A}}^{4k}}}{\mathcal {A}}^{2}\,}$

(4)

The first term is a perturbative loop calculation and the second is the instanton part where k labels fixed instanton numbers. In theories whose gauge groups are products of unitary groups, ${\displaystyle {\mathcal {F}}}$ can be computed exactly, using localization,[3] and the limit shape techniques.[4]

From ${\displaystyle {\mathcal {F}}}$ we can get the mass of the BPS particles.

${\displaystyle M\approx |na+ma_{D}|\,}$

(5)

${\displaystyle a_{D}={\frac {d{\mathcal {F}}}{da}}\,}$

(6)

One way to interpret this is that these variables ${\displaystyle a}$ 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. 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. See Hitchin system.

Seiberg–Witten prepotential via instanton counting

Using supersymmetric localisation techniques, one can explicitly determine the instanton partition function of ${\displaystyle {\mathcal {N}}=2}$ super Yang-Mills theory. The Seiberg-Witten prepotential can then be extracted using the localization approach[5] of Nikita Nekrasov. It arises in the flat space limit ${\displaystyle \varepsilon _{1}}$, ${\displaystyle \varepsilon _{2}\to 0}$, of the partition function of the theory subject to the so-called ${\displaystyle \Omega }$-background. The latter is a specific background of four dimensional ${\displaystyle {\mathcal {N}}=2}$ supergravity. It can be engineered, formally by lifting the 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 ${\displaystyle \varepsilon _{1}}$, ${\displaystyle \varepsilon _{2}}$ of the ${\displaystyle \Omega }$-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 ${\displaystyle \phi \to a}$ at ${\displaystyle x\to \infty }$ 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 ${\displaystyle \varepsilon _{1}}$, ${\displaystyle \varepsilon _{2}}$ approach 0, this sum is dominated by a unique saddle point. On the other hand, when ${\displaystyle \varepsilon _{1}}$, ${\displaystyle \varepsilon _{2}}$ approach 0,

${\displaystyle Z(a;\varepsilon _{1},\varepsilon _{2},\Lambda )=\exp \left(-{\frac {1}{\varepsilon _{1}\varepsilon _{2}}}\left({\mathcal {F}}(a;\Lambda )+{\mathcal {O}}(\varepsilon _{1},\varepsilon _{2})\right)\right)\,}$

(10)

holds.

See also

• Ginzburg–Landau theory

References

1. Seiberg, Nathan; Witten, Edward (1994). "Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory". Nucl. Phys. B. 426: 19–52. arXiv:hep-th/9407087. doi:10.1016/0550-3213(94)90124-4.
2. Seiberg, Nathan; Witten, Edward (1994). "Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD". Nucl. Phys. B. 431: 484–550. arXiv:hep-th/9408099. doi:10.1016/0550-3213(94)90214-3.
3. Nekrasov, Nikita (2002). "Seiberg-Witten Prepotential from Instanton Counting". Advances in Theoretical and Mathematical Physics. 7 (5): 831–864. arXiv:hep-th/0206161. doi:10.4310/ATMP.2003.v7.n5.a4.
4. Nekrasov, Nikita; Okounkov, Andrei (2003). "Seiberg-Witten theory and random partitions". Prog. Math. 244: 525–596. arXiv:hep-th/0306238. doi:10.1007/0-8176-4467-9_15.
5. Nekrasov, Nikita (2002). "Seiberg-Witten Prepotential from Instanton Counting". Advances in Theoretical and Mathematical Physics. 7 (5): 831–864. arXiv:hep-th/0206161. doi:10.4310/ATMP.2003.v7.n5.a4.

• Jost, Jürgen (2002). Riemannian Geometry and Geometric Analysis. Springer-Verlag. ISBN 3-540-42627-2. (See Section 7.2)

External links

• "Monopole Condensation, And Confinement In N=2 Supersymmetric Yang–Mills Theory". arXiv:hep-th/9407087. Missing or empty |url= (help)