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Polyakov action

The Polyakov action is a foundational action in bosonic string theory that describes the quantum dynamics of a string's worldsheet—a two-dimensional surface embedded in a higher-dimensional target spacetime—through a path integral formulation that incorporates both the embedding coordinates and an auxiliary worldsheet metric as dynamical variables. Introduced by Alexander Polyakov in 1981, it provides a reparameterization-invariant framework equivalent to the earlier Nambu–Goto action but better suited for quantization, as it explicitly reveals conformal symmetries and enables integration over all possible worldsheet metrics and topologies. Mathematically, the Polyakov action in Euclidean signature takes the form
S[X, h] = \frac{1}{4\pi\alpha'} \int d^2\sigma \, \sqrt{h} \, h^{ab} \partial_a X^\mu \partial_b X^\nu G_{\mu\nu}(X),
where X^\mu(\sigma) (with \mu = 0, \dots, D-1) are the embedding coordinates mapping the worldsheet parameters \sigma^a ( a=1,2 ) into the D-dimensional target spacetime equipped with metric G_{\mu\nu}, h_{ab} is the induced worldsheet metric (with determinant h = \det h_{ab}), and \alpha' is the string tension parameter (often set to 1 in natural units). This quadratic form in the derivatives contrasts with the nonlinear square-root structure of the Nambu–Goto action, arising from varying the Polyakov action with respect to the metric, which recovers the induced metric h_{ab} \propto \partial_a X^\mu \partial_b X_\mu at the classical level. The action is invariant under diffeomorphisms (reparameterizations of the worldsheet coordinates) and Weyl rescalings h_{ab} \to e^{2\omega} h_{ab}, making it a constrained system where these gauge symmetries must be fixed for quantization. In the path integral approach, the Polyakov action enables the summation over all possible worldsheet configurations, including metrics, to compute string scattering amplitudes as integrals over the moduli space of Riemann surfaces, with the measure determined by the action's symmetries. Weyl invariance plays a crucial role: quantum anomalies vanish only in the critical dimension D=26, where the central charge of the matter fields (from the D free scalars X^\mu) cancels that of the ghosts introduced in the conformal gauge, ensuring a consistent Lorentz-invariant theory. For non-critical dimensions, the action leads to the Liouville theory, describing coupled two-dimensional gravity and matter, which Polyakov originally connected to sums over random surfaces in gauge theories like QCD. This formulation has been pivotal in advancing superstring theories and dualities, providing the geometric foundation for modern developments in quantum gravity and conformal field theory.

Fundamentals

Definition and notation

The Polyakov action serves as the foundational Lagrangian for the bosonic string in string theory, describing the dynamics of a string propagating through a target spacetime as a two-dimensional worldsheet embedded in that space. In this formulation, the string is modeled as a collection of D scalar fields X^\mu(\sigma^a), where \mu = 0, 1, \dots, D-1 labels coordinates in the D-dimensional target spacetime, typically taken as flat Minkowski space with metric \eta_{\mu\nu} = \operatorname{diag}(-1, +1, \dots, +1). The worldsheet is a two-dimensional pseudo-Riemannian manifold with Lorentzian signature parameterized by local coordinates \sigma^a with a = 0, 1 (often denoted as \sigma^0 = \tau for the time-like parameter and \sigma^1 = \sigma for the spatial direction), and it is equipped with an auxiliary metric tensor h_{ab}(\sigma). The explicit form of the Polyakov action S[X, h] is given by S[X, h] = -\frac{1}{4\pi \alpha'} \int d^2 \sigma \, \sqrt{-h} \, h^{ab} \partial_a X^\mu \partial_b X_\mu, where h = \det(h_{ab}), h^{ab} is the inverse worldsheet metric, \partial_a = \frac{\partial}{\partial \sigma^a}, and X_\mu = \eta_{\mu\nu} X^\nu with summation over repeated indices implied (Einstein convention). Here, \alpha' is the Regge slope parameter, a fundamental constant with dimensions of length squared that sets the scale of string tension via T = 1/(2\pi \alpha'). This action integrates over the worldsheet to yield a dimensionless quantity, and the overall sign convention ensures a positive-definite kinetic term for spacelike directions in the target space. In standard notation, Greek indices \mu, \nu run over the target spacetime dimensions, while lowercase Latin indices a, b refer to the worldsheet; the integral measure d^2 \sigma is over the worldsheet coordinates, often with \sigma \in [0, 2\pi) for closed strings and appropriate boundary conditions for open strings. The bosonic string context assumes D = 26 for consistency under quantization, though the action itself is defined for general D. This formulation contrasts with the Nambu–Goto action, which directly measures the worldsheet area induced by the embedding.

Geometric interpretation

The Polyakov action provides a geometric framework for describing the dynamics of a bosonic string as an extended object propagating in spacetime, where the string traces out a two-dimensional worldsheet. This worldsheet is parameterized by coordinates \sigma^a, and the action is interpreted as a functional that measures the effective area of this surface, weighted by the determinant of an auxiliary metric h_{ab} on the worldsheet. Unlike direct area functionals, which depend solely on the embedding of the worldsheet into spacetime via maps X^\mu(\sigma), the Polyakov formulation incorporates the auxiliary metric to facilitate a more flexible geometric description, allowing the area to be computed in a conformally invariant manner. In this interpretation, the action extremizes the worldsheet area, analogous to how the proper time is extremized for a point particle in relativistic mechanics, but extended to the two-dimensional geometry of the string. For instance, in the static gauge where the worldsheet coordinates align with spacetime time and the string's spatial extent, the Polyakov action reduces to an expression proportional to the string's energy or length, capturing the classical dynamics of transverse oscillations. This geometric role underscores the string as a fundamental object whose motion minimizes the swept-out area, providing a natural generalization of geodesic principles to extended structures. A central feature of this formulation is the introduction of the auxiliary worldsheet metric h_{ab}, which is varied independently of the embedding maps X^\mu. This allows the action to treat the worldsheet geometry on equal footing with the spacetime embedding, enabling the isolation of conformal factors and reparameterizations without altering the physical area. By doing so, the Polyakov action reveals the intrinsic two-dimensional geometry of the worldsheet, where the auxiliary metric effectively rescales the induced metric from the embedding, ensuring the overall functional remains invariant under local rescalings.

Historical background

Origins in string theory

In the late 1960s, theoretical physicists sought models for strong interactions that could reproduce the Veneziano amplitude, a scattering amplitude proposed in 1968 that exhibited crossing symmetry and Regge behavior for hadron processes. This amplitude, derived from the Euler beta function, suggested an underlying structure of resonating states interpretable as vibrations of one-dimensional objects, marking the inception of string theory as a framework for describing particle interactions. Initially confined to hadron physics, string models gained traction in the early 1970s as potential descriptions of quark confinement and meson Regge trajectories. The Nambu–Goto action emerged in 1970 as the foundational relativistic invariant action for these string models, generalizing the area of a worldsheet embedded in spacetime to capture the dynamics of a string-like object. Developed by Yoichiro Nambu in the context of dual resonance models for hadrons, it was independently formulated by Tetsuo Goto, drawing inspiration from the Dirac–Born–Infeld action for extended objects but simplified to the induced metric's determinant for a one-dimensional string. This action portrayed strings as fundamental entities in hadron physics, with tension analogous to surface tension in condensed matter, aiming to model linear Regge trajectories observed in particle scattering. By the mid-1970s, as quantum chromodynamics solidified as the theory of strong interactions following the discovery of the charm quark, string theory's focus shifted from hadrons to a candidate for unifying all forces, including gravity, after the identification of a massless spin-2 excitation as the graviton. However, quantizing the Nambu–Goto action proved challenging due to its nonlinear square-root structure, which enforced reparameterization invariance through constraints that complicated the Hamiltonian formulation and led to issues like negative-norm states in early covariant quantization attempts. Additionally, the action lacked manifest Weyl invariance, obscuring the conformal symmetry essential for consistent quantum anomaly cancellation in higher dimensions. These obstacles highlighted the need for an alternative formulation to facilitate quantization.

Polyakov's formulation

In 1981, Alexander Polyakov published the seminal paper "Quantum Geometry of Bosonic Strings" in Physics Letters B, introducing a metric-dependent action for the bosonic string that revolutionized its quantization via the path integral formalism. This formulation addressed longstanding challenges in quantizing the earlier Nambu–Goto action, particularly by incorporating the worldsheet metric as an independent dynamical field, which allowed for a more tractable treatment of the string's geometry in quantum mechanics. The key innovation of Polyakov's approach lay in elevating the auxiliary worldsheet metric to a full-fledged quantum variable, enabling the fixing of the conformal gauge and a systematic analysis of conformal anomalies central to consistent string quantization. This built directly on prior work by Stanley Deser and Bruno Zumino, who in 1976 had proposed a similar metric-dependent action in the context of induced gravity on the worldsheet for spinning strings, providing a Lagrangian framework that Polyakov adapted for the non-supersymmetric bosonic case. By resolving difficulties inherent in the light-cone gauge quantization of the Nambu–Goto action—such as the loss of manifest Lorentz covariance—Polyakov's action offered a pathway to fully covariant quantization without restricting to specific gauges. The immediate impact of Polyakov's 1981 formulation was profound, as it facilitated the first consistent covariant path integral quantization of the bosonic string in 26 dimensions, where anomalies cancel, thereby establishing a rigorous quantum framework for the theory. This breakthrough not only solidified the bosonic string model but also laid essential groundwork for extending quantization techniques to superstring theories, influencing subsequent developments in the field throughout the 1980s.

Symmetries

Global symmetries

The Polyakov action preserves the global isometries of the target spacetime metric G_{\mu\nu}(X) at the classical level. In the case of flat Minkowski spacetime, these reduce to Poincaré symmetries, encompassing translations c^\mu and Lorentz rotations \Lambda^\mu{}_\nu, which act linearly on the embedding coordinates as X^\mu(\sigma) \to \Lambda^\mu{}_\nu X^\nu(\sigma) + c^\mu. The invariance arises because the action is constructed using the target metric G_{\mu\nu}(X), ensuring that the contraction \partial_\alpha X^\mu \partial^\alpha X^\nu G_{\mu\nu}(X) remains unchanged under these transformations. Unlike local symmetries, which vary over the worldsheet coordinates, these isometries are rigid, with parameters constant across the entire worldsheet. This global nature distinguishes them as spacetime symmetries that treat the string's embedding uniformly. As Noether symmetries, the global isometries generate conserved currents on the worldsheet. In flat spacetime, integrating these currents over the spatial boundary yields the total momentum P^\mu and angular momentum M^{\mu\nu} of the string, leading to their conservation in time and reflecting the physical isolation of the string.

Reparametrization invariance

The Polyakov action is invariant under reparameterizations of the worldsheet coordinates, which are arbitrary smooth diffeomorphisms \sigma^a \to \sigma'^a(\sigma). Under such transformations, the embedding maps X^\mu(\sigma) transform as scalars, so X'^\mu(\sigma') = X^\mu(\sigma), while the auxiliary worldsheet metric h_{ab}(\sigma) transforms as a covariant tensor, h'_{a'b'}(\sigma') = \frac{\partial\sigma^c}{\partial\sigma^{a'}} \frac{\partial\sigma^d}{\partial\sigma^{b'}} h_{cd}(\sigma). This invariance holds because the partial derivatives \partial_a X^\mu transform as covectors, and the measure in the action combines the determinant factor \sqrt{|h|} with the inverse metric h^{ab} to form a tensor density of weight +1, which compensates exactly for the Jacobian determinant of the coordinate transformation in d^2\sigma, leaving the integral unchanged. The reparameterization symmetry is a local gauge symmetry generated by the infinite-dimensional Lie group of diffeomorphisms \mathrm{Diff}(\Sigma) on the two-dimensional worldsheet \Sigma, parameterized by arbitrary smooth vector fields on \Sigma. As a consequence, this gauge freedom permits the choice of specific coordinate systems, such as the static gauge where \sigma^0 = \tau and \sigma^1 = \sigma for open strings, or the conformal gauge where the metric is proportional to the Minkowski metric, facilitating quantization by eliminating redundant degrees of freedom.

Weyl invariance

The Polyakov action is invariant under local Weyl transformations, which consist of rescaling the worldsheet metric h_{ab}(\sigma) \to e^{2\omega(\sigma)} h_{ab}(\sigma), where \omega(\sigma) is an arbitrary smooth function on the worldsheet, while the embedding coordinates X^\mu(\sigma) remain unchanged. This symmetry holds at the classical level and is a key feature distinguishing the Polyakov formulation from the Nambu–Goto action. To see the invariance explicitly, consider the kinetic term in the Polyakov action, \int d^2\sigma \, \sqrt{h} \, h^{ab} \partial_a X^\mu \partial_b X^\nu G_{\mu\nu}(X). Under the Weyl rescaling, the determinant scales as \sqrt{h} \to e^{2\omega} \sqrt{h}, while the inverse metric scales as h^{ab} \to e^{-2\omega} h^{ab}. The product \sqrt{h} \, h^{ab} thus remains unchanged, ensuring that the entire integrand is invariant, as the partial derivatives \partial_a X^\mu and G_{\mu\nu}(X) are unaffected. This cancellation demonstrates the classical Weyl invariance of the action without requiring any adjustment to the embedding fields. The presence of Weyl invariance is crucial for gauge fixing in the theory. Combined with reparameterization invariance, it allows the metric to be fixed in the conformal gauge h_{ab} = e^{2\phi(\sigma)} \eta_{ab}, where \eta_{ab} is the flat Minkowski metric and \phi(\sigma) is the remaining conformal factor. This choice reduces the infinite-dimensional redundancy of the general metric to a single degree of freedom represented by \phi, simplifying the analysis of the equations of motion and facilitating quantization, while preserving the essential dynamics of the string. At the classical level, this invariance ensures the action describes a physically consistent theory of extended objects propagating in spacetime.

Relation to Nambu–Goto action

Classical equivalence

The Nambu–Goto action, which describes the classical dynamics of a relativistic string as the area of its worldsheet, is given by S_{\mathrm{NG}} = -T \int d^2\sigma \, \sqrt{ -\det\left( \partial_a X^\mu \partial_b X_\nu \eta_{\mu\nu} \right) }, where T = 1/(2\pi \alpha') is the string tension, \alpha' is the Regge slope parameter, X^\mu(\sigma^a) are the embedding coordinates into target spacetime with Minkowski metric \eta_{\mu\nu}, and \sigma^a = (\tau, \sigma) parametrize the worldsheet with induced metric components G_{ab} = \partial_a X^\mu \partial_b X_\nu \eta_{\mu\nu}. The Polyakov action, an alternative formulation, is S_{\mathrm{P}} = -\frac{T}{2} \int d^2\sigma \, \sqrt{-h} \, h^{ab} G_{ab}, where h_{ab} is an auxiliary worldsheet metric independent of the embedding X^\mu. To establish classical equivalence, consider the equations of motion obtained by varying S_{\mathrm{P}} with respect to h^{ab}. The variation yields \frac{\delta S_{\mathrm{P}}}{\delta h^{ab}} = -\frac{T}{2} \sqrt{-h} \left( \frac{1}{2} h_{ab} h^{cd} G_{cd} - G_{ab} \right) = 0, implying that the auxiliary metric is proportional to the induced metric on-shell: h_{ab} = \gamma G_{ab} for some conformal factor \gamma(\sigma). Substituting this on-shell condition back into the Polyakov action recovers the Nambu–Goto action up to boundary terms that vanish for closed strings or appropriate boundary conditions. Specifically, the determinant transforms as \det h = \gamma^2 \det G, so \sqrt{-h} = \gamma \sqrt{-\det G}, and the contraction h^{ab} G_{ab} = \gamma^{-1} \times 2 (where 2 is the worldsheet dimension) simplifies the integrand to $2 \sqrt{-\det G}, yielding S_{\mathrm{P}} \big|_{\mathrm{on-shell}} = S_{\mathrm{NG}}. This equivalence holds at the classical level for any target dimension, confirming that both actions describe the same geodesic worldsheet motion.

Advantages in quantization

The Polyakov action provides significant advantages for quantizing the bosonic string compared to the Nambu–Goto action, stemming from its inclusion of an auxiliary worldsheet metric that renders the action quadratic in the derivatives of the embedding coordinates X^\mu. This quadratic form avoids the nonlinear square-root structure of the Nambu–Goto action, which hinders the definition of a suitable path integral measure and perturbative expansion. Instead, the Polyakov formulation permits a straightforward Gaussian integration over fluctuations around a flat background metric, facilitating perturbative calculations in the quantum theory. A key benefit lies in the path integral quantization, formulated as Z = \int \mathcal{D}X \, \mathcal{D}h \, e^{i S_P[X, h]}, where the integration over both embedding fields X^\mu and metrics h_{ab} explicitly incorporates the gauge symmetries of diffeomorphisms and Weyl rescalings.90743-7) Gauge fixing these redundancies via the Faddeev-Popov method introduces anticommuting ghost fields, ensuring the path integral sums over physical configurations without overcounting. This approach manifests the reparametrization and Weyl invariances more transparently than in the Nambu–Goto case, where such symmetries are less evident and harder to implement quantum mechanically, paving the way for interpreting the worldsheet dynamics as a two-dimensional conformal field theory. The Polyakov action's structure also simplifies the analysis of quantum anomalies, particularly the Weyl anomaly arising from the regularization of the path integral measure.90743-7) This anomaly vanishes only in the critical dimension D=26, where the central charge of the D free scalar matter fields (c=D) cancels the central charge of the b-c ghost system (c=-26), preserving conformal invariance. Equivalently, requiring the beta functions of the nonlinear sigma model to vanish at one loop enforces the same dimensionality condition, ensuring consistency of the quantum theory without tachyons or other inconsistencies. These features make the Polyakov formulation indispensable for deriving the spectrum and interactions of the bosonic string in a Lorentz-invariant manner.90743-7)

Equations of motion

Derivation from variation

The equations of motion for the embedding coordinates X^\mu in the Polyakov action are derived by performing a variational principle on the action functional with respect to X^\mu, treating the worldsheet metric h_{ab} as fixed. The Polyakov action takes the form S = -\frac{1}{4\pi\alpha'} \int d^2\sigma \, \sqrt{-h} \, h^{ab} \partial_a X^\mu \partial_b X_\mu, where \alpha' is the Regge slope parameter, \sigma^a = (\tau, \sigma) parameterize the worldsheet, and indices are raised and lowered with the Minkowski metric \eta_{\mu\nu} in the target spacetime. The first-order variation of the action under an infinitesimal change \delta X^\mu yields \delta S = -\frac{1}{2\pi\alpha'} \int d^2\sigma \, \sqrt{-h} \, h^{ab} \partial_a X^\mu \partial_b (\delta X_\mu) + \text{boundary terms}, where the factor of 2 arises from the symmetry of the kinetic term in the action, and the boundary terms vanish assuming appropriate conditions at the worldsheet boundary (e.g., fixed endpoints for open strings or periodicity for closed strings). This expression follows from differentiating the quadratic form in \partial_a X^\mu. To obtain the equations of motion, integrate the variation by parts with respect to the derivative acting on \delta X_\mu, shifting the derivatives onto the original fields. This procedure results in \delta S = \frac{1}{2\pi\alpha'} \int d^2\sigma \, \delta X_\mu \left[ \frac{1}{\sqrt{-h}} \partial_a \left( \sqrt{-h} \, h^{ab} \partial_b X^\mu \right) \right] + \text{surface terms}, where the surface terms are again assumed to vanish under suitable boundary conditions. For the action to be stationary, the coefficient of \delta X_\mu must vanish for arbitrary variations, leading to the equations of motion \frac{1}{\sqrt{-h}} \partial_a \left( \sqrt{-h} \, h^{ab} \partial_b X^\mu \right) = 0. This is the covariant form of the wave equation for each X^\mu on the curved two-dimensional worldsheet geometry defined by h_{ab}. The resulting equation describes the propagation of X^\mu as a massless scalar field on the worldsheet, analogous to the d'Alembertian operator \nabla^a \partial_a X^\mu = 0 in the curved metric h_{ab}, which governs geodesic-like motion of the string embedding in the target spacetime. In practice, explicit solutions are often sought in the conformal gauge, where h_{ab} = e^{2\phi(\sigma)} \eta_{ab} for some conformal factor \phi, simplifying the equation to the flat-space wave equation \partial^a \partial_a X^\mu = 0 while preserving the essential dynamics.

Virasoro constraints

The Virasoro constraints in the Polyakov action emerge from varying the action with respect to the auxiliary worldsheet metric h_{ab}, enforcing the equations of motion for the metric and ensuring consistency of the theory. The Polyakov action is given by S = -\frac{1}{4\pi\alpha'} \int d^2\sigma \, \sqrt{-h} \, h^{ab} \partial_a X^\mu \partial_b X^\nu \eta_{\mu\nu}, where \alpha' is the Regge slope parameter, X^\mu(\sigma^a) are the embedding coordinates in target spacetime with Minkowski metric \eta_{\mu\nu}, and h_{ab} is the worldsheet metric with determinant h = \det(h_{ab}). Varying the action with respect to h^{ab} yields the condition \frac{\delta S}{\delta h^{ab}} = 0, which is equivalent to the vanishing of the worldsheet stress-energy tensor T^{ab}, defined as T^{ab} = -\frac{2}{\sqrt{-h}} \frac{\delta S}{\delta h_{ab}}. Explicit computation gives T^{ab} = \frac{1}{2\pi\alpha'} \left( \partial^a X \cdot \partial^b X - \frac{1}{2} h^{ab} (\partial^c X \cdot \partial_c X) \right), where \partial^a = h^{ab} \partial_b and the dot denotes contraction with \eta_{\mu\nu}. In lowered indices, this becomes T_{ab} = \frac{1}{2\pi\alpha'} \left( \partial_a X \cdot \partial_b X - \frac{1}{2} h_{ab} (\partial^c X \cdot \partial_c X) \right). The trace T = h^{ab} T_{ab} vanishes due to Weyl invariance of the action. Due to the tracelessness, the condition T_{ab} = 0 is equivalent to \partial_a X \cdot \partial_b X = \frac{1}{2} h_{ab} (\partial^c X \cdot \partial_c X). This imposes constraints on the embedding functions X^\mu, ensuring the induced metric on the worldsheet aligns with the dynamical metric h_{ab}. In the conformal gauge, where the metric is fixed to the flat form h_{ab} = \eta_{ab} = \operatorname{diag}(-1, 1) with coordinates (\tau, \sigma), the stress tensor components simplify, yielding the explicit Virasoro constraints: (\partial_\tau X)^2 + (\partial_\sigma X)^2 = 0, \quad \partial_\tau X \cdot \partial_\sigma X = 0. These are often written in light-cone coordinates \sigma^\pm = \tau \pm \sigma as T_{++} = (\partial_+ X)^2 = 0, \quad T_{--} = (\partial_- X)^2 = 0, with the off-diagonal component T_{+-} = \partial_+ X \cdot \partial_- X = 0 following from tracelessness and conservation. The constraints are conserved (\partial_a T^{ab} = 0) and traceless (h^{ab} T_{ab} = 0), reflecting the underlying diffeomorphism and Weyl invariances of the action. These constraints preserve reparametrization invariance after gauge fixing, as the conformal gauge eliminates most of the diffeomorphism freedom but leaves a residual conformal symmetry generated by holomorphic transformations \xi^z = \epsilon(z) and anti-holomorphic ones \xi^{\bar{z}} = \bar{\epsilon}(\bar{z}). The Virasoro constraints generate these residual transformations, ensuring the physical degrees of freedom remain invariant under worldsheet reparametrizations. In the quantum theory, they correspond to the generators of the Virasoro algebra in the mode expansion of X^\mu, named after the central extension of the Witt algebra discovered by Miguel Virasoro.