REVIEW
1+1 Dimensional NCOS and its U(N) Gauge Theory Dual
High Energy Physics - Theory · Submitted 12 Jun 2000 · 23 references
Pith's verdict: A tight, honest consistency check on the 1+1 NCOS duality: the tree-level vanishing and the wound-string tension are real results, and the authors themselves flag the higher-loop decoupling as unfinished.
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Referee report
The paper
The paper claims
A string theory built on a D1-brane at a critical electric field passes two sharp tests against its proposed two-dimensional $U(N)$ Yang-Mills dual: massless modes along the field decouple, and compactifying that direction produces one-wayw
The reading
What carries the argument
The load-bearing device is the boundary two-point function of the light-cone combinations $X^± = X^0 ± X^1$ on a D1-brane at critical electric field, $⟨X^+(t)X^-(0)⟩ = 4α'_e(log|t| + iπ sign(t)/2)$, which the authors read as $log t$ with a definite choice of branch cut. Because a massless vertex operator satisfies $p^+p^-=0$, one can take $p^-=0$ and deform its integration contour off the boundary of the disk into the interior, where the holomorphic logarithm has no cut, so the amplitude collapses to zero. The second device is the open-string loop diagram with its Moyal phase $e^{2i p∧q}$; Poisson resummation over the discrete loop momentum $q_1=n/R$ turns the sum into a sum over winding number $m$, and the quadratic term in $p_0$ cancels against the $B_{01}$ term in the closed-string on-shell condition, leaving a non-relativistic dispersion relation in which only $m>0$ states have positive energy.
Extended reading notes
Core claim
The central claim is that 1+1 dimensional NCOS passes two sharp tests against its proposed $U(N)$ Yang-Mills dual. First, the massless open-string states carrying momentum in the direction of the electric field decouple: each ordering of vertex operators on the disk gives a non-zero contribution on its own, but the Moyal phases weight the orderings so that their sum cancels, and the explicit four-scalar amplitude in Appendix A, proportional to $[cos(π s)-cos(sθ/2α'_e)]/sin(π s)$, vanishes identically at $θ=2π α'_e$. This is the string-side image of $U(N)=U(1)×SU(N)/Z_N$ splitting into a free $U(1)$ and a gapped $SU(N)/Z_N$. Second, when the electric-field direction is compactified on a circle, the non-planar one-loop diagram develops poles from closed strings that wind with only one orientation, and their dispersion relation $p_0 = mR/(2α'_e) + α'_e k^2/(2mR) + (N_L+N_R)/(mR)$ matches the Higgs-branch energy $E(1,N-K)$ of the gauge theory, including the factor of two that makes the wound-string tension $1/(4π α'_e)$ rather than the naive $1/(2π α'_e)$.
Load-bearing premise
The vanishing-amplitude argument assumes that the contour deformation and branch-cut choice which works for a single massless insertion on a disk boundary also works when the massless state is inserted on every boundary of a higher-genus worldsheet; the paper states that one seems to need to sum over insertions on all boundaries and that this deserves a more precise analysis, but does not carry it out.
Editorial extensions
If this is right
- If the decoupling holds, the free $U(1)$ of the two-dimensional gauge theory is realized as a free sector of a weakly coupled string theory, with the string coupling fixed by the flux number through $G_0^2 = 1/N$, giving a concrete large-$N$ string/gauge correspondence in two dimensions.
- Compactified NCOS contains wound closed strings of tension $1/(4π α'_e)$ that can wind in only one sense; their low-energy interactions should be those of the orbifold $Sym^K(R^8)$ deformed by a dimension-3 operator whose coefficient is of order $1/g_{YM}$, and the parameter matching $G_0^2/√{α'_e}$ against $1/g_{YM}$ confirms it.
- Raising the temperature produces a massless closed string at $T = 1/(π√{8α'_e})$, matching the Hagedorn temperature read off from the open-string density of states.
- Purely massive amplitudes fall off faster than in the commutative string: away from the poles the $2→2$ cross-section scales as $E^{-4}$, the same energy behaviour as perturbative two-dimensional Yang-Mills.
- With $M$ coincident D1-branes the couplings become $g_{YM}^2 = N/(2M^2α'_e)$ and $G_0^2 = M/N$, so the open-string coupling can take any rational value and the physics should change discontinuously as that value varies.
Reading between the lines
- The contour argument should extend to $p+1$ dimensional NCOS for modes carrying no momentum in the non-commutative spatial directions, which would make the free $U(1)$ a general prediction; it also suggests the argument is really a statement about modes whose support is effectively one-dimensional, so it may say nothing about the genuinely non-commutative sector.
- The persistent factor of two between the flux-line mass and the naive Born-Infeld tension hints that the Born-Infeld description of brane dynamics fails in NCOS not through a calculational slip but because the massless sector is free; a direct string-field-theory derivation of the brane mass would be the cleanest test of that reading.
- If the coupling $G_0^2=M/N$ is quantized, the marginal direction separating $M$ coincident branes at leading order should be lifted at order $1/N^3$; a one-loop computation reproducing that lift would be a third independent check of the duality.
- The analogy drawn with the $SL(2,R)$ WZW model — short strings versus long strings that wind one way only — suggests the closed-string sector of compactified NCOS may be organized by a similar spectral-flow structure, which would be a route to writing down its interacting theory.
Original abstract
We study some aspects of open string theories on D-branes with critical electric fields. We show that the massless open string modes that move in the direction of the electric field decouple. In the 1+1 dimensional case the dual theory is U(N) SYM with electric flux, and the decoupling of massless open strings is dual to the decoupling of the U(1) degrees of freedom. We also show that, if the direction along the electric field is compact, then there are finite energy winding closed string modes. They are dual to Higgs branch excitations of the SYM theory, and their energetics works accordingly. These properties provide new non-trivial evidence for the duality.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We study some aspects of open string theories on D-branes with critical electric fields. We show that the massless open string modes that move in the direction of the electric field decouple. In the 1+1 dimensional case the dual theory is U(N) SYM with electric flux, and the decoupling of massless open strings is dual to the decoupling of the U(1) degrees of freedom. We also show that, if the direction along the electric field is compact, then there are finite energy winding closed string modes. They are dual to Higgs branch excitations of the SYM theory, and their energetics works accordingly. These properties provide new non-trivial evidence for the duality.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study some aspects of open string theories on D-branes with critical electric fields. | contribution 0.83 | 0.81 | 0.04 | asserted 0.66 |
| s1 | We show that the massless open string modes that move in the direction of the electric field decouple. | result 0.99 | 0.93 | 0.18 | asserted 1.00 |
| s2 | In the 1+1 dimensional case the dual theory is U(N) SYM with electric flux, and the decoupling of massless open strings is dual to the decoupling of the U(1) degrees of freedom. | result 0.85 | 0.88 | 0.49 | asserted 1.00 |
| s3 | We also show that, if the direction along the electric field is compact, then there are finite energy winding closed string modes. | result 0.97 | 0.91 | 0.07 | asserted 1.00 |
| s4 | They are dual to Higgs branch excitations of the SYM theory, and their energetics works accordingly. | interpretation 0.64 | 0.83 | 0.13 | asserted 0.99 |
| s5 | These properties provide new non-trivial evidence for the duality. | interpretation 0.99 | 0.65 | 0.41 | asserted 0.81 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:hep-th/0006085 by the authors.
References
How the literature uses it
37 arXiv papers cite this in 69 sentences
Papers citing it · 45 sentences without a confident label
Compares with it 1 paper
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Quantization of string charge then determines $g̃$ as follows: [equation] This agrees with [ref] for the case $n_5=0$.
Holography with Null Boundaries · Imposing string charge quantization · 0.98
Cites it as background 20 papers
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Another limit is the non-relativistic (NR) limit [ref] [ The NR string theory in flat space was studied in [ref] [ref] .], which corresponds to taking a close-up of an 2 string worldsheet.
Super Galilean conformal algebra in AdS/CFT · Introduction · 0.90
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For this reason, closed string states generically survive the limit only if they carry positive longitudinal winding [ref] .
On the Underlying Nonrelativistic Nature of Relativistic Holography · Flat-space NRF1 · 0.86
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One such corner of string theory that enjoys nonrelativistic symmetries was put forward more than twenty years ago, under the name nonrelativistic string theory [ref] .
Torsional Deformation of Nonrelativistic String Theory · Introduction and Conclusions · 0.99
Discussion
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