D-Branes, Open-Closed Duality, and Gauge Sectors

A D-brane is simultaneously a boundary condition for open strings and a charged, tensionful source for closed-string fields. Coincident branes therefore generate non-Abelian gauge sectors while backreacting on spacetime. Open/closed channel duality relates descriptions of the same perturbative amplitude, but it does not by itself prove that a brane gauge theory is exactly equivalent to a closed-string background.

Required background. Worldsheet sigma models and spacetime consistency supplies the conformal boundary-condition framework.

Helpful background. BPS solitons, walls, strings, and defects supplies the spacetime interpretation of saturated objects. BPS shortening bounds explains the stability of D-brane charge and tension data.

Chan–Paton factors and brane sources

Dirichlet boundary conditions fix the endpoints of an open string in directions transverse to a D$p$-brane and leave Neumann conditions along its $(p+1)$-dimensional worldvolume Dai, Leigh, and Polchinski 1989. For $N$ coincident branes, endpoint labels form $N\times N$ Chan–Paton matrices. The massless vector state therefore becomes a $U(N)$ gauge field; transverse scalar matrices describe brane positions, with off-diagonal strings becoming light at coincidence.

The same object carries Ramond–Ramond charge $\mu_p$ and tension

$$T_p=\frac{1}{(2\pi)^p g_s(\alpha')^{(p+1)/2}}.$$

The $1/g_s$ scaling makes a D-brane nonperturbative from the closed-string viewpoint, while disk amplitudes yield its leading open-string interactions. Polchinski’s identification of D-branes as Ramond–Ramond charge carriers fixed their normalization and BPS interpretation Polchinski 1995.

First application: the annulus in two channels

The one-loop vacuum amplitude of an open string stretched between two branes is an annulus integral over proper time $t$. A modular transformation $t\mapsto1/t$ rewrites the same surface as tree-level propagation of closed strings between boundary states. At large separation, the long closed-string tube is dominated by massless NS–NS attraction and Ramond–Ramond repulsion; for parallel supersymmetric branes these cancel Polchinski 1998, Vol. 2, Ch. 13.

This equality is exact order by order for the specified worldsheet background. It shows that an open-string loop already contains closed-string exchange and fixes the brane’s long-range fields. For $N$ branes, planar open-string diagrams scale with Chan–Paton powers of $N$, anticipating a closed-string genus organization in the ‘t Hooft limit. Yet a full gauge/gravity duality additionally requires a decoupling limit and a complete dictionary.

At energies $E\sqrt{\alpha'}\ll1$, the brane worldvolume reduces to supersymmetric Yang–Mills plus higher-dimension operators. Whether closed strings decouple depends on $p$, the scaling of $g_s$, and the energy variable; low energy alone does not guarantee an autonomous gauge theory in every brane setup.

Adversarial control: backreaction without control

Increase $g_sN$ until brane backreaction is order one while keeping the curvature radius comparable to $\sqrt{\alpha'}$. The open-string description is strongly coupled, and the leading supergravity background has order-one $\alpha'$ corrections. Open/closed channel duality still constrains amplitudes, but neither a weak open-string calculation nor classical geometry is reliable. Alternatively, retain finite coupling to asymptotic closed strings: the gauge sector is then an open subsystem rather than a dual complete theory.

The strongest conclusion is a perturbatively exact relation between worldsheet channels plus protected charge data. It establishes why gauge and gravitational descriptions are related, not that they are interchangeable in the absence of decoupling. Near-horizon brane dictionaries adds the required limiting and geometric information.

The chapter overview compares these assumptions and limitations.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Dai, J., Leigh, R. G., and Polchinski, J. (1989), “New Connections Between String Theories,” Modern Physics Letters A 4, 2073–2083. doi:10.1142/S0217732389002331.
• Polchinski, J. (1995), “Dirichlet-Branes and Ramond–Ramond Charges,” Physical Review Letters 75, 4724–4727. arXiv:hep-th/9510017.
• Polchinski, J. (1998), String Theory, Vol. 2: Superstring Theory and Beyond, Cambridge University Press. doi:10.1017/CBO9780511618123.