Completing the curvilinear-coordinate formalism for instantons and multi-instantons: existence of the compensating gauge rotation, compact multi-instanton field strengths, and singularity effects on gauge-variant quantities

In 2000 A. A. Abrikosov, Jr. generalized 't Hooft's multi-instanton solution to curvilinear coordinates in flat four-dimensional Euclidean space by combining the change of variables with a gauge rotation that replaces the coordinate-dependent 't Hooft eta-symbols by constant vierbein xi-symbols, at the price of a compensating pure-gauge addition to the gauge potential. The construction leaves three linked gaps. (1) The existence of the required gauge rotation is only constrained by necessary duality-matching (parity/even-permutation) conditions; there is no complete classification of the coordinate frames and vierbeins for which the rotation exists. (2) Compact field-strength formulas, in which all connection terms cancel, are derived only for solitary pseudoparticles; general multi-instanton (Jackiw-Nohl-Rebbi-type) configurations still require the general covariant formula. (3) The demonstrated redistribution of the gauge-dependent Chern-Simons number onto the singularities of the compensating field in 4-spherical coordinates has no general characterization across frames, even though geometry-adapted confinement models (QCD strings, bags) were the application that motivated the construction.

Background

Instantons (also called pseudoparticles) are finite-action solutions of the classical Yang-Mills equations in four-dimensional Euclidean space. They satisfy the (anti-)selfduality condition $\hat F_{\mu\nu}=\pm\tfrac12\epsilon_{\mu\nu\lambda\sigma}\hat F_{\lambda\sigma}$, where $\epsilon$ is the totally antisymmetric symbol, which automatically solves the field equations and bounds the action by the topological charge $q=\tfrac{1}{32\pi^2}\int d^4x\,\epsilon^{\mu\nu\lambda\sigma}\operatorname{tr}\hat F_{\mu\nu}\hat F_{\lambda\sigma}$, an integer counting how the gauge field winds over the sphere at infinity. The one-instanton solution was found by Belavin, Polyakov, Schwartz and Tyupkin; the matrix-valued gauge potential $\hat A_\mu$ can be written compactly using the 't Hooft $\eta$-symbols, constant antisymmetric tensors $\eta_{\mu\nu}^{a}$ ($a=1,2,3$ a gauge index, $\mu,\nu=1,\dots,4$ Lorentz indices) that project (anti-)selfdual antisymmetric tensors onto the $su(2)$ algebra. In terms of the $2\times2$ matrices $\tau_\mu=(\tau^a,i)$ built from the Pauli matrices $\tau^a$, the hatted symbols $\hat\eta_{\mu\nu}^{\pm}=\tau^a\eta(\bar\eta)_{\mu\nu}^{a}$ enter both the regular-gauge potential $\hat A_\mu=\tfrac12\hat\eta_{\mu\nu}^{+}\partial_\nu\ln\Pi_{\rm reg}$ and the singular-gauge potential $\hat A_\mu=-\tfrac12\hat\eta_{\mu\nu}^{-}\partial_\nu\ln\Pi_{\rm sing}$, where the scalar profile $\Pi$ carries the instanton data. 't Hooft's ansatz, and its conformal extension by Jackiw, Nohl and Rebbi (JNR), give multi-instanton configurations with $5N$ parameters by superposing poles in $\Pi$; they are the workhorses of instanton-based phenomenology.

All of this is formulated in Cartesian coordinates. Abrikosov's paper addresses what happens in curvilinear coordinates $q^\alpha$ (4-spherical, $(2+2)$- and $(3+1)$-cylindrical frames), which are the natural frames for confinement-inspired backgrounds: a QCD string (a phenomenological model in which a quark-antiquark pair is joined by a string-like color flux tube) is naturally described in $(2+2)$-cylindrical coordinates, and an MIT bag (a model in which quarks are confined to a finite region of space by a boundary condition at its surface) in a $(3+1)$-frame. Under a change of variables the $\eta$-symbols become coordinate-dependent, $\hat\eta_{\mu\nu}\to e_\mu^a e_\nu^b\,\hat\xi_{ab}$, where $e_\alpha^a$ is a vierbein (a field of orthonormal frames converting coordinate indices to frame indices) and $\hat\xi_{ab}^{\pm}=-i\tau_{[a}^{\dagger}\tau_{b]}$ (respectively $-i\tau_{[a}\tau_{b]}^{\dagger}$) are constant frame analogues of the $\eta$-symbols. The paper shows that a coordinate-dependent gauge rotation $\Omega(q)$ can restore constant symbols, at the price of a compensating addition to the gauge potential: the (anti-)selfdual projection of the spin connection (the connection induced on the frames by the Levi-Civita connection) onto $SU(2)$, $\hat A_\alpha^{\pm}=\tfrac14 B_\alpha^{mn}\hat\xi_{mn}^{\pm}$. In flat space written in curvilinear coordinates the Riemann tensor vanishes, so this compensating field is pure gauge; its curvature satisfies $\hat F_{\alpha\beta}^{\pm}=-\tfrac14 R^{\alpha\beta\gamma\delta}\hat\xi_{\gamma\delta}^{\pm}$ and hence vanishes. The resulting pseudoparticle potential is the sum of an instanton-like term and the compensator, the general multi-instanton field strength is given by a covariant formula (eq. (50) of the paper) valid for any profile $\Pi(q)$, and for a solitary instanton all connection terms cancel, giving the compact forms (eqs. (53) and (55) of the paper). The compensator typically has coordinate singularities (polar axes, origins), and the paper demonstrates in the 4-spherical example that these singularities, though physically harmless, redistribute the Chern-Simons number - the gauge-dependent quantity $N_{CS}=\int d^3x\,K^\alpha$ built from the Chern-Simons current $K^\alpha$, whose boundary flux reproduces the topological charge - onto the singular loci: applying Gauss's theorem to the domain $\{0\le r\le\infty,\,0\le\chi\le\pi,\,0\le\theta\le\pi,\,0\le\varphi\le2\pi\}$ decomposes the unit charge into four addends (eq. (70) of the paper), each contributing $\tfrac14$ to $q=1$ - the faces $r=\infty$ and $r=0$, which together constitute $Q(\infty)-Q(0)$, and the singular hyperstrips at $\theta=\pi$ and $\theta=0$; the naive expectation $Q(\infty)=\tfrac12$, $Q(0)=-\tfrac12$ fails because the singular line $\theta=0$ carries Chern-Simons number as well.

The paper leaves the following loose ends. First, Section 3.2 poses "the question ... whether there are solutions" to the conjugation condition $\Omega\hat\xi_{ab}^{\pm}=e_\mu^a e_\nu^b\hat\eta_{\mu\nu}^{\pm}\Omega$ (eq. (36) of the paper): normalization holds automatically, and a necessary condition is that both sides have the same duality, which "is the case if the two sets $\tau_a$ and $\tau_\mu e_\mu^a$ differ by a rotation and/or an even permutation", equivalently that the change of variables respect parity - but no complete classification or sufficiency proof is given. Second, after the compact single-instanton results the paper notes that "for multipseudoparticle solutions one still has to apply the clumsy formula", i.e. the general covariant field strength. Third, the singularity analysis is carried out for one frame only, with the general statement that "in general singular gauge transformations may generate singularities of gauge variant quantities" left undeveloped. The construction was motivated by the unsettled question of the role of instantons in quark confinement, for which the paper's own assessment is that it "will not be solved unless new methods and ideas appear".

Problem Statement

Complete the curvilinear-coordinate (compensating-gauge, vierbein) formalism for instantons and multi-instantons in flat four-dimensional Euclidean space, by closing the three gaps explicitly left open in A. A. Abrikosov, Jr., Nucl. Phys. B 586 (2000) 589 (arXiv:hep-th/9906008):

(1) Existence classification of the compensating rotation. For a curvilinear coordinate system $q^\alpha(x)$ in flat $\mathbb R^4$ and a chosen vierbein $e_\alpha^a(q)$, determine completely for which frames and vierbein choices there exists a gauge rotation $\Omega(q)$ solving the conjugation condition (eq. (36) of the source) $\Omega\hat\xi_{ab}^{\pm}=e_\mu^a e_\nu^b\hat\eta_{\mu\nu}^{\pm}\Omega$, which replaces the coordinate-dependent $\hat\eta$-combinations by the constant frame $\hat\xi$-symbols. The source verifies normalization automatically and states necessary duality-matching conditions (the quadruples $\tau_a$ and $\tau_\mu e_\mu^a$ must differ by a rotation and/or an even permutation; equivalently the change of variables must respect parity). Determine the necessary and sufficient conditions - a characterization of the admissible coordinate changes and vierbein fields - proving them and specializing them to the frames treated in the source (4-spherical, $(2+2)$- and $(3+1)$-cylindrical), including the parity-violating changes of variables (such as inversion) that require the $\hat\eta\to\hat{\bar\eta}$ substitution.

(2) Compact multi-instanton expressions. Find compact closed-form expressions for the gauge potential and field strength of general multi-instanton configurations - 't Hooft and Jackiw-Nohl-Rebbi-type profiles $\Pi(q)$ with instanton number $N\geq1$ - in the geometry-adapted frame with constant $\hat\xi$-symbols, in which the connection terms cancel as they do for the solitary pseudoparticle in eqs. (53) and (55) of the source; or else establish precisely which structural features of the general case (for example, non-degeneracy of the matrices $\mathcal N_\pm$ built from the inverse vierbein and the radial function) obstruct such a compactification, and to what extent.

(3) Gauge-variant consequences of compensating singularities. Characterize, beyond the computed 4-spherical example, how the singularities of the compensating pure-gauge field (coordinate axes, origins, and other singular loci of the frame) redistribute gauge-dependent quantities - in particular the Chern-Simons current and number - across the coordinate domain, for general curvilinear frames admitting the construction.

These three components jointly constitute one objective: a complete, classified, analytically compact, and physically reliable curvilinear-coordinate treatment of (multi-)instantons. Component (1) is the foundation (which frames admit the construction at all), component (2) is its analytic payoff (currently available only for single pseudoparticles), and component (3) is its reliability condition for gauge-dependent observables. The broader question of the role of instantons in quark confinement is the motivating context and intended application of the formalism, not itself part of the completion boundary of this problem.

References

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