Worldsheet
In string theory, the worldsheet is the two-dimensional surface traced out by a one-dimensional string as it propagates through a higher-dimensional spacetime, serving as the fundamental arena for describing the string's dynamics and quantization.[1] This surface, parameterized by a timelike coordinate τ (evolving the string's position over time) and a spacelike coordinate σ (along the string's length), embeds into a target spacetime of dimension D, typically 26 for bosonic strings or 10 for superstrings, via embedding functions X^μ(τ, σ) that map worldsheet coordinates to spacetime coordinates.[2] The worldsheet's geometry is governed by actions such as the Polyakov action, which incorporates an auxiliary worldsheet metric γ_{αβ} and the spacetime metric g_{μν}, or the Nambu-Goto action, which directly uses the induced metric on the surface to model the string as a relativistic object with tension T.[1][2] The worldsheet formulation unifies point-particle mechanics with field theory principles, where the string's vibrational modes on this surface correspond to the spectrum of particles in spacetime, including gravitons and other fundamental forces, addressing ultraviolet divergences in quantum gravity.[1] Quantization occurs via a two-dimensional conformal field theory on the worldsheet, ensuring anomaly cancellation and consistency only in specific spacetime dimensions, with the central charge c=0 for critical strings.[2] Background fields like the Kalb-Ramond antisymmetric tensor B_{μν} couple to the worldsheet, influencing interactions and dualities, while various topologies, including strips or disks for open strings and cylinders or tori for closed strings depending on the process, determine the theory's perturbative expansion.[1] This framework extends classical general relativity by replacing point-like worldlines with extended worldsheets, providing a candidate for a unified theory of quantum gravity and particle physics.[2]Fundamentals
Definition
In string theory, the worldsheet is defined as a two-dimensional Lorentzian manifold that represents the surface traced out by a one-dimensional string as it propagates through higher-dimensional spacetime.[3] This surface generalizes the concept of a point particle's worldline, which is a one-dimensional curve, to the extended object of a string, where the string's motion sweeps out a (1+1)-dimensional embedding.[4] The worldsheet is parametrized by two coordinates: a timelike parameter \tau, which tracks the evolution of the string along its "proper time," and a spacelike parameter \sigma, which runs along the length of the string.[3] For open strings, \sigma typically ranges from 0 to \pi, while for closed strings, it is periodic with period $2\pi.[3] This parametrization embeds the worldsheet into a target spacetime of dimension D, where D = 26 for bosonic string theory and D = 10 for superstring theory.[3][4] A fundamental property of the worldsheet is its reparametrization invariance, arising from the redundancy in the choice of \sigma and \tau coordinates, which allows for arbitrary diffeomorphic transformations without altering the physical description.[3] This gauge symmetry underscores the worldsheet's role as a dynamical entity in string theory, where the intrinsic geometry is independent of the specific parametrization used.[4]Coordinates and Parametrization
The worldsheet in string theory is parametrized by two coordinates: a timelike parameter \tau, which describes the evolution of the string over time, and a spacelike parameter \sigma, which parametrizes the position along the string's length. These coordinates, often denoted collectively as \xi^\alpha = (\tau, \sigma) with \alpha = 0, 1, endow the worldsheet with a two-dimensional Lorentzian structure, analogous to spacetime coordinates in relativity. The choice of \tau as timelike ensures that it ranges over all real numbers, facilitating the description of the string's dynamical history, while \sigma captures the spatial extension.[5][1] The embedding of the worldsheet into a D-dimensional target spacetime is achieved through functions X^\mu(\sigma, \tau), where \mu = 0, 1, \dots, D-1 labels the spacetime coordinates. These functions map points on the worldsheet to positions in the ambient spacetime, effectively describing how the string traces its path. For closed strings, \sigma ranges from $0 to $2\pi, with periodic boundary conditions X^\mu(\sigma + 2\pi, \tau) = X^\mu(\sigma, \tau), reflecting the string's topology as a loop without endpoints. In contrast, open strings have \sigma ranging from $0 to \pi, subject to boundary conditions at the endpoints \sigma = 0 and \sigma = \pi, such as Neumann conditions \partial_\sigma X^\mu = 0 (free endpoints) or Dirichlet conditions X^\mu = constant (fixed endpoints).[5][1] A key feature of this parametrization is the reparametrization freedom, arising from the diffeomorphism invariance of the worldsheet theory. This allows arbitrary coordinate transformations \sigma' = f(\sigma, \tau), \tau' = g(\sigma, \tau), which preserve the physical content of the embedding without altering the geometry of the worldsheet. Such invariance underscores the coordinate-independent nature of the formulation, enabling gauge choices that simplify computations while maintaining equivalence. These coordinates also play a role in defining the induced metric on the worldsheet from the target spacetime geometry.[5][1]Actions
Nambu–Goto Action
The Nambu–Goto action was proposed by Yoichiro Nambu in 1970 during lectures at the Copenhagen Summer Symposium[6] and independently formalized by Tetsuo Goto in 1971[7] as a relativistic invariant model for extended string-like objects in the context of strong interactions and hadron structure. It was further proposed independently by Fernando Lund and Tullio Regge in 1976[8] as an action principle minimizing the worldsheet area for soliton-like string and vortex configurations. This formulation generalizes the relativistic action for point particles to one-dimensional extended objects, capturing their dynamics through geometric principles. The action is expressed as S = -T \int d^2\xi \, \sqrt{-\det \gamma_{ab}}, where T denotes the string tension (with dimensions of energy per length), the integration is over the two-dimensional worldsheet parametrized by coordinates \xi^a = (\tau, \sigma), and \gamma_{ab} = g_{\mu\nu} \partial_a X^\mu \partial_b X^\nu is the induced metric on the worldsheet from the embedding functions X^\mu(\xi) mapping into the target spacetime with metric g_{\mu\nu}. This expression arises naturally as the analog of the proper area for a two-dimensional surface in a pseudo-Riemannian manifold. Physically, the Nambu–Goto action is proportional to the proper area spanned by the string's worldsheet in spacetime, which enforces full relativistic invariance under arbitrary reparametrizations of the worldsheet coordinates while preserving Poincaré symmetry of the embedding space. In the limiting case where the spatial extent parametrized by \sigma contracts to zero (effectively integrating over the string length), the action reduces to that of a point particle, S = -m \int ds, with the effective mass m proportional to T times the contracted length, thereby recovering the standard relativistic particle dynamics. A key challenge in working with the Nambu–Goto action lies in its non-polynomial dependence on the embedding coordinates X^\mu due to the square-root determinant, which complicates canonical quantization procedures and the construction of a path-integral formulation, as the measure lacks the Gaussian structure needed for straightforward evaluation. The Nambu–Goto action is classically equivalent to the Polyakov action, which reparametrizes the formulation using an auxiliary worldsheet metric to address these quantization difficulties.Polyakov Action
The Polyakov action provides a reparametrization-invariant formulation for the dynamics of a bosonic string's worldsheet, serving as an alternative to other geometric descriptions. Introduced by Alexander Polyakov in 1981[9], it addressed challenges in quantizing bosonic string theory by enabling a path integral approach over both embedding coordinates and an auxiliary metric. The action is formulated as S = -\frac{T}{2} \int d^2\xi \, \sqrt{-h} \, h^{ab} \partial_a X^\mu \partial_b X_\mu, where T is the string tension, \xi^a (with a,b = 0,1) are worldsheet coordinates, X^\mu(\xi) (with \mu = 0, \dots, D-1) embed the worldsheet into D-dimensional spacetime with Minkowski metric \eta_{\mu\nu}, and h_{ab} is an auxiliary metric on the worldsheet with determinant h = \det(h_{ab}). This auxiliary metric h_{ab} is independent of the embedding and allows for additional symmetries beyond reparametrization invariance. The Polyakov action is classically equivalent to the Nambu–Goto action on-shell, meaning they yield the same equations of motion and solutions. This equivalence arises because varying the Polyakov action with respect to h^{ab} enforces the constraint that h_{ab} is proportional to the induced metric \gamma_{ab} = \partial_a X^\mu \partial_b X_\mu from the embedding.[5] A key advantage of the Polyakov action lies in its quadratic dependence on the embedding coordinates X^\mu, which simplifies functional integration for quantization. Unlike formulations reliant solely on the induced geometry, it permits path integrals over both X^\mu and h_{ab}, facilitating the treatment of worldsheet gravitational degrees of freedom and resolving anomalies in bosonic string quantization. Varying the action with respect to X^\mu produces the equations of motion \partial^a \partial_a X^\mu = 0, corresponding to the wave equation for transverse string oscillations, while variation with respect to the auxiliary metric h_{ab} imposes constraints that ensure consistency with the induced geometry.[5]Bosonic String Theory
Dynamics in Bosonic Strings
In bosonic string theory, the worldsheet describes the embedding of a string into a flat target spacetime of 26 dimensions, with no fermionic fields present. The dynamics arise from the Polyakov action, which is equivalent to the Nambu–Goto action upon integrating out the worldsheet metric.[3] This formulation facilitates quantization and reveals the underlying conformal symmetry essential for consistency.[3] Varying the Polyakov action with respect to the embedding coordinates X^\mu yields the equations of motion. In the conformal gauge, where the worldsheet metric is fixed to the Minkowski form, these reduce to the wave equation \partial_\tau^2 X^\mu - \partial_\sigma^2 X^\mu = 0 for each transverse mode \mu = 1, \dots, 24.[3] The longitudinal modes are constrained by gauge choice, ensuring reparametrization invariance.[3] Reparametrization invariance imposes additional constraints known as the Virasoro constraints, derived from the vanishing of the worldsheet stress-energy tensor T_{ab} = 0. In the conformal gauge, these manifest as two conditions: \partial_\tau X \cdot \partial_\tau X + \partial_\sigma X \cdot \partial_\sigma X = 0 and \partial_\tau X \cdot \partial_\sigma X = 0, where the dot denotes the target space Minkowski metric.[3] These constraints eliminate unphysical degrees of freedom and generate the Virasoro algebra upon quantization.[3] Quantum consistency requires anomaly cancellation in the conformal field theory description. The central charge of the bosonic matter sector is c = D, while the ghost sector contributes c = -26; vanishing total central charge demands D = 26.[3] This critical dimension ensures Lorentz invariance and unitarity at the quantum level.[3] The general solution to the equations of motion is given by a mode expansion. For open strings, X^\mu(\sigma, \tau) = x^\mu + p^\mu \tau + i \sum_{n \neq 0} \frac{\alpha_n^\mu}{n} e^{-i n \tau} \cos(n \sigma), where \alpha_n^\mu are the oscillator modes.[3] For closed strings, the expansion includes both left- and right-moving modes: X^\mu(\sigma, \tau) = x^\mu + p^\mu \tau + i \sum_{n \neq 0} \frac{\alpha_n^\mu}{n} e^{-i n (\tau + \sigma)} + i \sum_{n \neq 0} \frac{\tilde{\alpha}_n^\mu}{n} e^{-i n (\tau - \sigma)} + w^\mu \sigma, incorporating winding contributions w^\mu.[3] The Virasoro constraints then impose commutation relations among these modes, forming the foundation for the string spectrum.[3]Worldsheet Metric
In bosonic string theory, the worldsheet is embedded into a D-dimensional Minkowski target spacetime with metric η_{μν} of Lorentzian signature (-, +, ..., +), via embedding functions X^μ(τ, σ) that map worldsheet coordinates to spacetime points. The induced metric on the worldsheet, denoted γ_{ab} where a, b are worldsheet indices, is the pullback of the target metric and is defined as γ_{ab} = ∂_a X^μ ∂b X^ν η{μν}.[3][1] This metric encodes the intrinsic geometry of the worldsheet as determined by its embedding. In the standard parametrization using worldsheet coordinates (τ, σ), where τ is the timelike parameter and σ the spacelike parameter along the string, the components of the induced metric take the explicit form: γ_{ττ} = ∂_τ X · ∂τ X = \dot{X}^2,γ{σσ} = ∂σ X · ∂σ X = (X')^2,
γ{τσ} = γ{στ} = ∂_τ X · ∂_σ X = \dot{X} · X', with the dot denoting contraction using η_{μν}.[3][1][4] These components reflect the squared lengths and angle between the tangent vectors ∂_τ X and ∂_σ X in the target spacetime. The determinant of the induced metric is det γ = γ_{ττ} γ_{σσ} - (γ_{τσ})^2 = \dot{X}^2 (X')^2 - (\dot{X} · X')^2, which is central to the Nambu–Goto action for the bosonic string, S = -T ∫ dτ dσ √{-det γ}, where T is the string tension; this form corresponds to the proper area of the worldsheet.[3][1] The worldsheet inherits a Lorentzian signature (-, +) from the target spacetime, ensuring a causal structure with timelike and spacelike directions.[1][4] In the bosonic string, the induced metric satisfies the Virasoro constraints, which arise from the reparametrization invariance of the theory and set the off-diagonal components to zero—specifically γ_{τσ} = \dot{X} · X' = 0—in orthogonal gauges such as the conformal gauge, while also enforcing relations like \dot{X}^2 + (X')^2 = 0 (up to normalization).[3][4] These constraints ensure the physical consistency of the string dynamics by eliminating unphysical degrees of freedom.