In 1996, Seiberg wrote a fundamental paper about the basics of five-dimensional supersymmetric field theories. [1. Here we will consider the minimal amount of supersymmetry, which is 8 real supercharges. ] In five dimensions, the gauge coupling constant is dimensionful : for $\frac{1}{g_{YM}^2} F^2$ to have mass-dimension five, we need $\frac{1}{g_{YM}^2}$ to have the dimension of a mass. So naively, those theories are not very interesting: at low energy, the gauge coupling vanishes, while in the UV it diverges, signaling that the theory is non-renormalizable. Nevertheless, we will see that those theories can be seen as relevant perturbations of a $5d$ UV-complete SCFT, which corresponds to the infinite-coupling limit. This surprising construction is the object of this short note.

First, what is the field content of the theory? Well, the only constraint is that it should fit in representations of the superalgebra. We will focus on massless fields, and this leaves us with two types of representations:

  • The hypermultiplet (this contains two $4d$ $\mathcal{N}=1$ chiral multiplets, or $4$ real scalars and one $5d$ spinor)
  • The vector multiplet $\Phi$ (one vector, one real scalar $\phi$ and one $5d$ spinor), or equivalently the tensor multiplet (dualizing the vector to a $2-$form).

Let us first focus on the hero of our story, the vector multiplet, without whom there is no gauge theory. Arguments related to the $4d$ theory obtained by compactification show that the Lagrangian is determined by a prepotential of the form $$\mathcal{F} = \frac{1}{g_{YM}^2} \mathrm{tr} \Phi^2 + \frac{\kappa}{6}\mathrm{tr} \Phi^3 \, .  $$ From this we obtain a Lagrangian as follows:

  • The term $\mathcal{F} = \frac{1}{g_{YM}^2} \mathrm{tr} \Phi^2$ gives the usual $5d$ YM Lagrangian. In the infinite coupling limit, it will disappear.
  • The cubic term gives:
    1. One Chern-Simons term $\mathcal{L}_{CS} \sim \kappa \, \mathrm{tr} A \wedge F \wedge F + \dots$
    2. One Yang-Mills term $\mathcal{L}_{\kappa} \sim \mathrm{tr} \phi \mathcal{L}_{YM}$.

We see that $\kappa$ is absolutely crucial to have a non-trivial theory. In fact, this term is generated by an anomaly, at one loop. Moreover, although the theory is a priori not UV-finite, $\kappa$ is finite and doesn't depend on the coupling constant nor on the cutoff. Let us consider a single vector multiplet with gauge group $SU(2)$. One computes that the anomaly generates $\kappa = 2(8-N_f)$, where $N_f$ is the number of  quarks and $8$ is the contribution of the vector multiplet. Moreover, in this theory the effective coupling constant is given by $$\frac{1}{g_{eff}^2} =\frac{1}{g^2} + 16 \phi - \sum |\phi - m_i| - \sum |\phi + m_i| \, , $$ where the sum is over the quarks with masses $m_i$. The crucial observation is that for $N_f < 8$, the right-hand side is bounded from below on the Coulomb branch $\mathbb{R}_+$. This means that starting from $g=0$ we can increase $g \rightarrow \infty$ and still have a finite value of $g_{eff}$, provided $\phi$ is sufficiently large.

This theory can be realized on a brane setup, and in the infinite coupling limit, the theory is at a non-trivial fixed point where the global symmetry $SO(2N_f) \times U(1)$ is enhanced to $E_{N_f+1}$. To understand how this is done, we just need to recall below a few things about type I and I' string theories, and then about the appearance of $E_8$ in string theory.

Construction of type I string theory

First let us consider type I theory. As a theory of open strings, it can be seen as living on some number of space-filling $D9$ branes. But then we need something to cancel the charges these branes create, and for this we have to look at the closed strings sector. For this, we take type IIB string theory and project it on the parity-invariant states (this is the $\Omega$ projection) -- or equivalently, we put a space-filling $O9$ orientifold plane into type IIB. This theory in itself is anomalous, and the anomaly could be canceled by a Yang-Mills sector with $\mathcal{N}=1$ supersymmetry and gauge group $SO(32)$ or $E_8 \times E_8$. Those two observations fit nicely together: putting 16 $D9$ branes create the $SO(32)$ gauge group, and the charge created by these branes is exactly compensated by the charge of the orientifold plane.

Let us finally note that the physics of type I theory is locally that of type IIB. In order to investigate $5d$ gauge theories, we need to have $D4$ branes, which exist only in type IIA. Hence the need of $T$-dualizing our type I theory.

T-duality on type I theory

The action of $T$-duality in the open and unoriented type I string theory (with its $O9$ and 16 $D9$) produces a theory on an interval $S^1 / \mathbb{Z}_2$ with one $O8$ orientifold plane at each end of the segment, and 16 $D8$ branes. This is the theory called I'. On the interval we can then add a $D4$-brane, on which lives generically a $U(1)$ gauge theory in $5d$, which is enhanced to $SU(2)$ when the $D4$ coincides with the $O8$. The quarks are generated by strings from the $D4$ to one of the $D8$-branes.

How E8 appears

The gauge coupling in the $5d$ theory is related to the type I' coupling constant $\lambda$ by $$\frac{1}{g^2} = \frac{M_s}{\lambda} \, . $$ This $\lambda$ can be a space-dependent field (it is just the dilaton). It is known that if the coupling diverges at one orientifold where we have $N_f$ $D8$ branes, then the symmetry is enhanced to $E_{N_f +1}$ due to the appearance of new massless states at strong coupling.

References

  • The original paper by Seiberg.
  • Johnson's book on D-branes.