Contents

Contents

Idea

An instanton in plain Yang-Mills theory on a 4-dimensional manifold $K^{(4)}$ necessarily has a positive characteristic radius. Indeed, in the moduli space of instantons the would-be point of vanishing characteristc radius corresponds to a singularity which is unphysical, in that it is not part of the phase space of the theory.

But also the gauge field of heterotic string theory admits instanton-configurations on a transversal 4-space, and these are pullbacks of ordinary Yang-Mills instantons on the Euclidean 4-manifold $K^{(4)}$ to a $(6+4)$-dimensional product manifold-spacetimes

(1)$X^{(9+1)} \;\simeq\; Q^{(5+1)} \times K^{(4)}$

along the projection map $Q^{(5+1)} \times K^{(4)} \overset{pr_2}{\longrightarrow} K^{(4)}$.

Therefore it makes sense to ask if, after embedding Yang-Mills instantons into heterotic string theory this way, the singularity in the moduli space of instantons becomes part of the phase space of the theory (now that quantum gravity-effects are supposedly included) so that instantons of vanishing radius do acquire “stringy” physical meaning after all – then to be called small instantons.

In HET-O. A circumstantial but widely accepted argument (folklore) due to Witten 96 says that this is indeed the case, and that these small instantons in SemiSpin(32)-heterotic string theory are identified with heterotic NS5-branes wrapped on the $Q^{(5+1)}$-factor in (1) and carrying an exotic Sp(1)-gauge field on their worldvolume, identified with the SU(2)-structure group of a single instanton.

Indeed, solutions of heterotic supergravity involving positive-radius Yang-Mills instantons had been interpreted as black NS5-branes of non-vanishing “thickness” in Strominger 90. This makes it plausible that these vanishing-radius small instantons correspond to fundamental hererotic 5-branes of vanishing thickness (Witten 96, top of p. 9)

More precisely, this $Sp(1)$-gauge field is argued to be a non-perturbative effect invisible in perturbative string theory, hence distinct from the SemiSpin(32)- or E8$\times$E8-gauge field famously seen in perturbative heterotic string theory.

There this interpretation of small instantons as wrapped 5-branes really pertains to M5-branes in heterotic M-theory.

In F-theory. Under duality between F-theory and heterotic string theory the argument of Witten 96 was claimed to be confirmed in Aspinwall-Gross 96, Aspinwall-Morrison 97.

In Type I. On the other hand, under duality between type I and heterotic string theory one expects to see dually as a phenomenon visible in perturbative type I string theory, now pertaining to D5-D9-brane bound states. Indeed, in general Dp-D(p+4)-brane bound states are argued to exhibit instantons in the worldvolume gauge theory of the $D(p+4)$-brane, and specifically for D5-D9-brane bound states in type I string theory one finds (Witten 96. Section 3) from orientifold-analysis (Gimon-Polchinski 96, p. 7) that the gauge group on a single D5 is $Sp(2)$.

In HET-E. Similarly under T-duality of the SemiSpin(32)- to the E8$\times$E8-heterotic string the small instantons should also correspond to heterotic 5-branes there. That this is the case was argued, indirectly, in Ganor-Hanany 96, appealing now to orientifold image pairs under the Horava-Witten $\mathbb{Z}_2$-action.

brane intersections/bound states/wrapped branes/polarized branes

S-duality$\,$bound states:

intersecting$\,$M-branes:

References

$SU(2)$-flavor symmetry on heterotic M5-branes

Emergence of SU(2) flavor-symmetry on single M5-branes in heterotic M-theory on ADE-orbifolds (in the D=6 N=(1,0) SCFT on small instantons in heterotic string theory):

Argument for this by translation under duality between M-theory and type IIA string theory to half NS5-brane/D6/D8-brane bound state systems in type I' string theory:

Reviewed in:

• Santiago Cabrera, Amihay Hanany, Marcus Sperling, Section 2.3 of: Magnetic Quivers, Higgs Branches, and 6d $\mathcal{N}=(1,0)$ Theories, JHEP06(2019)071, JHEP07(2019)137 (arXiv:1904.12293)

The emergence of flavor in these half NS5-brane/D6/D8-brane bound state systems, due to the semi-infinite extension of the D6-branes making them act as flavor branes:

Reviewed in:

• Fabio Apruzzi, Marco Fazzi, Section 2.1 of: $AdS_7/CFT_6$ with orientifolds, J. High Energ. Phys. (2018) 2018: 124 (arXiv:1712.03235)

• Amihay Hanany, Noppadol Mekareeya, The Small $E_8$ Instanton and the Kraft Procesi Transition, JHEP07 (2018) 098 (arXiv:1801.01129)