nLab small instanton

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.

Related concepts

brane intersections/bound states/wrapped branes/polarized branes

• D-branes and anti D-branes form bound states by tachyon condensation, thought to imply the classification of D-brane charge by K-theory

• intersecting D-branes/fuzzy funnels:

  • Dp-D(p+2)-brane bound state and Yang-Mills monopoles

    • D0-D2 brane bound state

    • D1-D3 brane bound state

    • D2-D4 brane bound state

    • D3-D5 brane bound state

    • D4-D6 brane bound state

    • D6-D8-brane bound state

  • Dp-D(p+4)-brane bound state and Yang-Mills instantons

    • D0-D4 brane bound state

    • D1-D5 brane bound state

    • D2-D6 brane bound state

    • D3-D7 brane bound state

    • D4-D8 brane bound state

    • D5-D9 brane bound state

      small instanton

• Dp-D(p+6) brane bound state

  • D0-D6 brane bound state

  • D1-D7 brane bound state

  • D2-D8 brane bound state

• NS5-D4-D2 bound states

• Hanany-Witten effect

S-duality$\,$bound states:

• (p,q)-string

• (p,q)5-branes

• string network

• 5-brane web

intersecting$\,$M-branes:

• M5-MO9 brane bound state

• M2-M5 brane bound state

  • M-string, E-string

• membrane triple junction

References

General

• Andrew Strominger, Heterotic solitons, Nuclear Physics B, Volume 343, Issue 1, Pages 167-184, 1990 (doi:10.1016/0550-3213(90)90599-9)

• Edward Witten, Small Instantons in String Theory, Nucl. Phys. B460:541-559, 1996 (arXiv:hep-th/9511030)

• Eric Gimon, Joseph Polchinski, Consistency Conditions for Orientifolds and D-Manifolds, Phys. Rev. D54:1667-1676, 1996 (arxiv:hep-th/9601038)

• Ori Ganor, Amihay Hanany, Small $E_8$ Instantons and Tensionless Non-critical Strings, Nucl. Phys. B474 (1996) 122-140 (arXiv:hep-th/9602120)

• Paul Aspinwall, Mark Gross, The $SO(32)$ Heterotic String on a K3 Surface, Phys. Lett. B387 (1996) 735-742 (arXiv:hep-th/9605131)

• Paul Aspinwall, David Morrison, Point-like Instantons on K3 Orbifolds, Nucl. Phys. B503 (1997) 533-564 (arXiv:hep-th/9705104)

• Clifford Johnson, Études on D-Branes, in: Mike Duff et. al. (eds.) Nonperturbative aspects of strings, branes and supersymmetry, Proceedings, Trieste, Italy, March 23-April 3, 1998 (arXiv:hep-th/9812196, spire:481393)

Discussion within AdS-CFT duality:

• Tony Gherghetta, Alex Pomarol, Small Instantons in Weakly-Gauged Holographic Models (arXiv:2110.01762)

$SU(2)$-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):

• Abhijit Gadde, Babak Haghighat, Joonho Kim, Seok Kim, Guglielmo Lockhart, Cumrun Vafa, Section 4.2 of: 6d String Chains, J. High Energ. Phys. 2018, 143 (2018) (arXiv:1504.04614, doi:10.1007/JHEP02(2018)143)

• Kantaro Ohmori, Section 2.3.1 of: Six-Dimensional Superconformal Field Theories and Their Torus Compactifications, Springer Theses 2018 (springer:book/9789811330919)

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:

• Amihay Hanany, Alberto Zaffaroni, Branes and Six Dimensional Supersymmetric Theories, Nucl.Phys. B529 (1998) 180-206 (arXiv:hep-th/9712145)

• Ilka Brunner, Andreas Karch, Branes at Orbifolds versus Hanany Witten in Six Dimensions, JHEP 9803:003, 1998 (arXiv:hep-th/9712143)

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)

See also:

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