A toy model of Polyakov duality

Polyakov has conjectured that Yang–Mills theory should be equivalent to a noncritical string theory. I pointed out, based on the work of Marchesini, Ishibashi, Kawai and collaborators, and Jevicki and Rodrigues, that the loop operator of the Yang–Mills theory is the temporal gauge string field theory Hamiltonian of a noncritical string theory. In the present note I explicitly show how this works for the one–plaquette model, providing a consistent direct string interpretation of the unitary matrix model for the first time. The naturality of a string interpretation of Yang–Mills theory was first appreciated by Mandelstam [1], and has been elaborated upon with significant insights by many others [2]. The Wilson loop observables of gauge theories satisfy dynamical Schwinger–Dyson equations that have precise geometric interpretations [3]. These loop equations simplify considerably in the limit N ↑ ∞ where N is the rank of the gauge group. This is also natural for the string interpretation following ‘t Hooft’s study of the large N limit of gauge theories, since this is the limit when string loop contributions to amplitudes are suppressed. Thus Wilson loop expectation values in the large N gauge theory satisfy classical equations. The loop equation appears also as the saddle–point equation in the collective field theory setup of Jevicki and Sakita [4]. Strings do not interact in this limit—yet the loop equation has terms corresponding to the splitting of strings, and terms that annihilate strings. This, of course, is completely consistent, but not entirely obvious to the string theorist steeped in the lore of critical string theories in conformal gauge. Indeed, the simplest physical interpretation of the loop equation and its meaning in a string theory equivalent to the Yang–Mills theory is obtained by combining [5] two beautiful results. The first of these is the observation of Marchesini [6] that the Fokker–Planck Hamiltonian that arises in Parisi–Wu stochastic quantization [7] is precisely the loop operator, i.e. the operator H such that 〈H ∏ i W (Ci)〉 = 0 (1) are the Schwinger–Dyson loop equations. The second result is that of Jevicki and Rodrigues [8]: the temporal gauge noncritical string field theory Hamiltonian found in the work of Ishibashi, Kawai and collaborators [9] is the Fokker–Planck Hamiltonian of the matrix model representation of the noncritical string theory. (For earlier relevant work see [11,12].) Combining these results, I found that the loop operator of the Yang–Mills theory 1 can be interpreted as the exact string field theory Hamiltonian of a noncritical string theory, provided a crucial consistency condition is satisfied: the loops must be defined in a manner that preserves the zig–zag symmetry emphasized by Polyakov [13]. The string theory defined by the loop operator is valid at arbitrary N. Now loop equations are notorious for requiring careful regularization [14], so one must make such formal statements precise with well–defined operators. Fortunately, since one is automatically in the framework of stochastic quantization [7,15] there is in fact a continuum regularization available [16]. I derive in detail the noncritical string theories one obtains in this manner in [25], but the aim of the present paper is to present a simple toy model in which all the features mentioned in the previous paragraph are as transparent as can be. In the process, one finds a resolution for the (decidedly minor) puzzle of a string interpretation of the unitary matrix model to which I now turn. The depth of the large N limit is nowhere more apparent than in the model of one Hermitian matrix first solved by Brézin, Itzykson, Parisi and Zuber [17]. The analogue of their model for gauge theories is the one–plaquette model solved by Gross and Witten [18]. This model is defined by a Wilson lattice gauge theory action but involving only one plaquette: Z = ∫