nLab dual heterotic string theory

Idea

Ordinary heterotic string theory is the study of the string perturbation series of correlators of a certain 2-dimensional SCFT over super Riemann surfaces. The resulting S-matrix is interpreted as encoding the scattering amplitudes of strings propagating in some target space.

Among the gauge fields that these strings are charged under is the Kalb-Ramond field, which is a circle 2-bundle with connection.

By general electric-magnetic duality (see Freed 00 for a formal treatment in differential cohomology) one expects there to be a dual theory theory where the $(p=1)$-dimensional strings are replaced by their magnetic duals, which are $10-(p+2)-2 = 5$-branes.

The study of the corresponding perturbation series over correlators of the 6-dimensional fivebrane worldvolume SCFT over 6-volumes is far from tractable, but a handful of consistency checks exist, that the corresponding dual heterotic string theory makes sense.

Where the quantum anomaly-cancellation for the heterotic string involves the demand for (twisted) string structures, that for the dual theory involves fivebrane structures (which gives these their name).

The dual degree-8 anomaly polynomial in Green-Schwarz mechanism for dual heterotic string theory is (e.g. Freed 00 (3.48))

$$d H_7 \;=\; ch_4(A) - \tfrac{1}{48} p_1(\omega) \wedge ch_2(A) + \tfrac{1}{64} p_1(\omega) \wedge p_1(\omega) - \tfrac{1}{48} p_2(\omega) \,,$$

where $A$ denotes the gauge field connection, $\omega$ the gravity spin connection, $ch_n$ the degree-$n$ component of the Chern character and $p;_i$ the Pontryagin classes.

Related concepts

• string theory

• heterotic string theory

  • Green-Schwarz mechanism

• dual heterotic string theory

References

The 10d sugra + SYM Lagrangian in terms of a six-form potential was constructed in:

• Ali Chamseddine. Interacting supergravity in ten dimensions: The role of the six-index gauge field. Phys. Rev. D 24, 3065 (1981). (doi)

The heterotic NS5-brane (see also at half M5-brane) was found as a black brane in heterotic string theory in

• Andrew Strominger,

  Heterotic solitons, Nucl. Phys. B343 (1990) 167-184 (doi:10.1016/0550-3213(90)90599-9)

  Erratum: Nucl. Phys. B353 (1991) 565-565 (doi:10.1016/0550-3213(91)90349-3, spire:27900)

Discussion of the heterotic NS5-brane as a Green-Schwarz sigma-model and its fermion worldvolume quantum anomaly:

• Kurt Lechner, Mario Tonin, Worldvolume and target space anomalies in the D=10 super–fivebrane sigma–model (arXiv:hep-th/9603094)

• Kurt Lechner, Mario Tonin, World-manifold and target space anomalies in heterotic Green-Schwarz strings and five-branes, in: Gauge Theories, Applied Supersymmetry and Quantum Gravity II, pp. 311-318 (1997) (arXiv:hep-th/9610110, doi:10.1142/9781848160927_0022)

• J. Mourad, Anomalies of the $SO(32)$ five-brane and their cancellation, Nucl.Phys. B512 (1998) 199-208 (arxiv:hep-th/9709012)

• Kurt Lechner, Quantum properties of the heterotic five-brane, Phys. Lett. B693:323-329, 2010 (arxiv:1005.5719)

• J. A. Dixon, Mike Duff, Jan Plefka, Putting the string/fivebrane duality to the test (arXiv:hep-th/9208055)

Review of the history of the dual heterotic string conjecture is in

• Mike Duff, chapter 6 of The World in Eleven Dimensions: Supgergravity, Supermembranes and M-theory IoP 1999

On p. 3 and 4 the latter reviews the worldsheet description of the relevant terms for the string and then look at the corresponding situation for the 5-brane on p. 6.

So this is the EM-dual analog of the Killingback-Witten-computation of the fermionic anomaly that leads to string structures in the heterotic string (see there). Where the former involves cancelling the first fractional Pontryagin class, this involves the second.

For a description of the general mechanism in differential cohomology at work here, see

• Daniel Freed, Dirac charge quantization and generalized differential cohomology, 2000

See also

• Hao-Yuan Chang, Ergin Sezgin, Yoshiaki Tanii. Dualization of higher derivative heterotic supergravities in 6D and 10D. JHEP 10 (2022): 1-31. (doi062))