Exact analytic solution of two-dimensional noncommutative Yang-Mills theory on the noncommutative plane

Gross and Nekrasov (JHEP 0103 (2001) 044) classified the classical solutions of two-dimensional noncommutative Yang-Mills (NCYM) theory and proposed to solve the quantum theory exactly: the naive partition function diverges because the unstable fluxon solutions carry flat moduli, and they regularized it with a gauge-invariant term $\epsilon\,\mathrm{Tr}(DD^\dagger)$, producing a matrix model $Z_\epsilon$ of Kazakov-Kostov-Nekrasov type that is exactly solvable, leaving open whether this yields the full exact solution of 2d NCYM. Later work exactly solved the theory on the noncommutative torus (Paniak-Szabo) and resummed the decompactified fluxon expansion into a closed-form partition function on the noncommutative plane (Griguolo-Seminara-Szabo). What remains open is the completion of the original program on the plane: deriving the solution directly from the $\epsilon$-regularized matrix model (the $\epsilon \to 0$ limit was never computed), and obtaining exact nonperturbative expectation values of the gauge-invariant observables - momentum-space open Wilson lines and the energy operator - at finite gauge-group rank $N$, together with a principled normalization of these operators that produces the physical form factors.

Background

On a Moyal noncommutative plane the spatial coordinates obey $[x^1, x^2] = i\theta$, where $\theta$ is the noncommutativity parameter, and functions are multiplied with the associative star (Moyal) product $(f \star g)(x) = \exp\left(\frac{i\theta}{2}\epsilon_{ij}\,\partial_i\partial'_j\right) f(x)\,g(x')\big|_{x'=x}$, which arises as the low-energy limit of open strings ending on a D-brane (a hypersurface in spacetime on which open strings can end) carrying a constant background B-field - the Seiberg-Witten limit. Noncommutative Yang-Mills (NCYM) theory in two dimensions is therefore naturally written in operator language: the gauge field is encoded in a covariant coordinate operator $D$ (the noncommutative analogue of the covariant derivative, combining the gauge potential with the Fock-space ladder operators) acting on the one-particle harmonic-oscillator (Fock) space, and the theory is equivalent to a matrix model in $D$. Ordinary two-dimensional Yang-Mills (YM) theory is a classic exactly solvable model: on a closed surface its partition function is the Migdal sum over gauge-group representations, and its strong-coupling expansion reorganizes into the Gross-Taylor series, a sum over branched coverings of the surface with the interpretation of a string worldsheet expansion. Gross and Nekrasov showed that the classical solutions of 2d NCYM are labelled by two integers - the rank $N$ of the gauge group (which emerges as a superselection parameter of the operator theory, since the rank is not fixed by the action) and the magnetic charge $q$ - and that the charge-$q$ solutions are fluxons, solitons of finite energy proportional to $q$ that describe D1-strings (D-branes with one spatial direction) piercing a D3-brane, with positions $x^\mu = \theta^{\mu\nu}\lambda_\nu$ given by the eigenvalues of $D$ (the moduli). A central consequence is that the naive Euclidean partition function of 2d NCYM on the plane, $Z = \int \mathcal{D}D\,\mathcal{D}\bar{D}\,\exp\left(-\frac{\pi\theta}{2g^2}\,\mathrm{Tr}\left([D,\bar{D}] - 1\right)^2\right)$ with $g$ the gauge coupling, is divergent: integrating along the fluxon moduli (the eigenvalues $\lambda$) renders it divergent, the noncommutative analogue of the fact that a drop of magnetic flux in two dimensions spreads over all space. Gross and Nekrasov therefore proposed to regularize the partition function by adding the gauge-invariant term $\epsilon\,\mathrm{Tr}(DD^\dagger)$ to the action, obtaining a regulated partition function $Z_\epsilon$ that sums over $N$ with a Penner-type weight $e^{\mu N}$ and is of the type solved exactly by Kazakov, Kostov and Nekrasov ('D-particles, matrix integrals and KP hierarchy'). They stated explicitly that whether this solvable regulated model can lead to a full exact solution of 2d NCYM 'is a question that deserves further study', and in their conclusions: 'we set up the machinery to derive an exact analytic solution of 2 dimensional noncommutative gauge theory. It would be of great interest to complete this construction.' Gauge Morita equivalence - the exact correspondence between noncommutative gauge theories on tori and commutative gauge theories with twisted boundary conditions - together with a double-scaling limit that remains valid at finite $N$, underlies the exact solution of the theory on the noncommutative torus (Paniak-Szabo) and its decompactification to the plane (Griguolo-Seminara-Szabo); the noncommutative loop equation - the Schwinger-Dyson equation for Wilson loop averages - has likewise been formulated on the plane (Dorn-Torrielli), where it remains unsolved at finite $N$. A complete solution must also fix the observables. Gauge-invariant operators of NC gauge theory are necessarily nonlocal: besides closed Wilson loops, there are momentum-space open Wilson lines $W(\vec l)$ - straight Wilson lines of length $|\vec l| \sim \theta|p|$, whose length is tied to the momentum $p$ they carry - and the gauge-invariant energy operator $\mathcal{E}(p) = \mathrm{Tr}\left(e^{D\theta^{\mu\nu}p_\nu} F^2\right)$ constructed by Gross, Hashimoto and Itzhaki, which reduces to a momentum-space Wilson line inserted in the energy density. In the fluxon background these operators exhibit pointlike structure (a sum of position phases localized at the vortex positions), while the fluctuation spectrum shows vortices spread over a size $\sqrt{\theta}$; Gross and Nekrasov noted that the normalization of $W(\vec l)$ and $\mathcal{E}(p)$ is 'somewhat arbitrary' - multiplying them by, for example, $\exp(-p^2\theta)$ would not change the total energy or the closed Wilson loop but would produce the expected smooth form factors (the momentum-space profiles of the extended vortex matter distribution) - and that Gross, Hashimoto and Itzhaki had handled this ambiguity by considering ratios of three- and two-point functions. Extracting physical form factors from these nonlocal operators is thus an inseparable part of what an exact solution must deliver.

Problem Statement

Complete the exact analytic solution of two-dimensional pure (bosonic, without matter or supersymmetry) Yang-Mills theory on the Moyal noncommutative plane $\mathbb{R}^2_\theta$ (the plane with coordinate commutator $[x^1, x^2] = i\theta$), as initiated by Gross and Nekrasov. Specifically: (1) Determine the quantum (Euclidean) partition function of the theory from the regulated operator/matrix model $Z_\epsilon$ obtained by adding the gauge-invariant regulator $\epsilon\,\mathrm{Tr}(DD^\dagger)$ to the action - that is, compute the $\epsilon \to 0$ limit of the exactly solvable matrix model of Kazakov-Kostov-Nekrasov type and establish whether, and in what sense, it reproduces the full partition function of 2d NCYM (two-dimensional noncommutative Yang-Mills theory) on the plane, including the sum over the superselection sectors of gauge-group rank $N$ and over fluxon sectors with their correct measure and quantum fluctuations. (2) Determine exact, nonperturbative expectation values of the gauge-invariant observables on the plane - the momentum-space open Wilson lines $W(\vec l)$ (straight Wilson lines whose length is fixed by the momentum they carry through $\theta$) and the energy operator $\mathcal{E}(p) = \mathrm{Tr}\left(e^{D\theta^{\mu\nu}p_\nu} F^2\right)$ - at finite $N$, and fix the normalization of these operators (for instance, whether a factor such as $\exp(-p^2\theta)$ is required) so that the physical form factors, such as the localized energy density of a fluxon of size $\sqrt{\theta}$, can be extracted unambiguously. The two parts share one objective: a closed-form exact (nonperturbative in the coupling $g^2$, the rank $N$, and the noncommutativity $\theta$) solution of 2d NCYM on the plane with its gauge-invariant observables, of the same status as the Migdal representation-sum solution of ordinary 2d Yang-Mills theory. The result must be exact rather than an expansion truncated at fixed order in $g^2$, $1/\theta$ or $1/N$, and its derivation from the $\epsilon$-regularized model, or its relation to it, must be made precise.

References

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