Appendix C: Foundations of Tensors and Differential Geometry¶
Story so far: In Ch. 6, the Einstein equation \(R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^4}T_{\mu\nu}\) appeared, and from Ch. 13 onward, we introduced a metric \(h_{ab}\) on the string worldsheet. The systematic framework of "tensors," "raising and lowering indices," "covariant derivatives," and "curvature" that appears in these equations is covered in detail in General Relativity Chapters 4–General Relativity Ch. 8, Chapters 12–14, and Appendix B.
Goals of this chapter
- Catalog the tools of tensors and differential geometry learned in General Relativity from the "user's perspective," and bridge to contexts specific to string theory—the 2-dimensional worldsheet metric \(h_{ab}\), the Polyakov action, and curvature in 2-dimensional spacetime
- Delegate the details of general tensor analysis (coordinate transformations, derivation of Christoffel symbols, variation of the geodesic equation, symmetries of the Riemann tensor, derivation of the Einstein equation) to General Relativity, avoiding redundancy and focusing on elements newly needed for string theory
🟡 Lina: For those who've already read General Relativity, this appendix is just a quick review of key points. The string-theory-specific parts are only C.3 and C.4, so feel free to focus on those.
🔵 Kai: We did a lot in General Relativity, right? Raising and lowering indices, differentiation in curved spaces… I remember there being quite a lot.
🟡 Lina: That's right. Metric tensor, Christoffel symbols, geodesics, Riemann tensor, Einstein equation—we covered all of it. Here we'll avoid repetition and focus on what's newly needed for string theory—"the geometry of the string worldsheet as a 2-dimensional spacetime." Two dimensions are somewhat special and exhibit different behavior from general relativity.
C.1 Summary of Key Points from General Relativity¶
🟡 Lina: Let me catalog the tools we'll use in this appendix. All detailed derivations, proofs, and worked examples are in General Relativity, so please refer there.
Tensor Basics (General Relativity Chapter 04, Appendix B)¶
- Contravariant vectors \(A^\mu\) (upper indices) and covariant vectors \(A_\mu\) (lower indices) follow opposite transformation rules under coordinate changes:
- \(A'^\mu = \frac{\partial x'^\mu}{\partial x^\nu}A^\nu\) (contravariant)
- \(B'_\mu = \frac{\partial x^\nu}{\partial x'^\mu}B_\nu\) (covariant)
- Einstein summation convention: When the same index appears once up and once down, sum over it
- Tensor products \(\otimes\) construct higher-rank tensors
- Writing physical laws as tensor equations gives them a coordinate-independent form (general covariance)
The Metric Tensor (General Relativity Chapter 06, ch07)¶
- The spacetime line element is \(ds^2 = g_{\mu\nu}\,dx^\mu dx^\nu\)
- Minkowski metric: \(\eta_{\mu\nu} = \mathrm{diag}(-1, +1, +1, +1)\) (the convention used throughout General Relativity and The Quest for Quantum Gravity. Quantum Field Theory uses the opposite \((+,-,-,-)\). This convention was also confirmed in Ch. 5 when introducing light-cone coordinates)
- The inverse metric \(g^{\mu\nu}\) satisfies \(g^{\mu\alpha}g_{\alpha\nu} = \delta^\mu_\nu\)
- Raising and lowering indices: \(A_\mu = g_{\mu\nu}A^\nu\), \(A^\mu = g^{\mu\nu}A_\nu\)
- Representative examples: Schwarzschild metric (spherically symmetric, static), FRW metric (cosmology)
Covariant Derivative and Christoffel Symbols (General Relativity Chapter 12)¶
- The ordinary partial derivative \(\partial_\mu V^\nu\) is not a tensor (extra terms appear under coordinate transformations)
- The covariant derivative \(\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu\rho}V^\rho\) is a tensor
- Christoffel symbols: From metric compatibility \(\nabla_\alpha g_{\mu\nu} = 0\) and vanishing torsion
Mechanically computable from the first derivatives of the metric.
Parallel Transport and Geodesics (General Relativity Chapter 08, ch12)¶
- Parallel transport of a vector \(V^\mu\) along a curve \(x^\mu(\lambda)\): \(\frac{dV^\mu}{d\lambda} + \Gamma^\mu_{\nu\rho}\frac{dx^\nu}{d\lambda}V^\rho = 0\)
- Geodesic equation (the equation of motion for a particle traveling "straight"):
Derived from the variational principle (stationarity of the action \(S = \frac{1}{2}\int d\lambda\, g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}\)).
The Curvature Tensor (General Relativity Chapter 13)¶
- Riemann curvature tensor (from the commutator of covariant derivatives):
- Ricci tensor: \(R_{\mu\nu} = R^\alpha{}_{\mu\alpha\nu}\)
- Scalar curvature: \(R = g^{\mu\nu}R_{\mu\nu}\)
- Geometric meaning: The rotation of a vector after parallel transport around a closed loop
The Einstein Equation (General Relativity Chapter 14, Appendix G)¶
- The left side is the curvature of spacetime (pure geometry), the right side is the distribution of matter and energy (physics)
- From the Bianchi identity \(\nabla^\mu G_{\mu\nu} = 0\), which is consistent with energy-momentum conservation \(\nabla^\mu T_{\mu\nu} = 0\)
🟡 Lina: All the tools above can be computed starting from the metric \(g_{\mu\nu}\)—remember the pipeline: metric → Christoffel → Riemann → Ricci → Einstein equation. And this pipeline applies directly to the worldsheet metric \(h_{ab}\) as well.
⚪ Mei: So we just replace \(g_{\mu\nu}\) with \(h_{ab}\) and the same procedure works.
🟡 Lina: Exactly. We'll see this concretely in C.3 and C.4.
C.2 Exercise Map¶
🟡 Lina: The exercises in this appendix are for reviewing and confirming the content of General Relativity in the context of string theory.
Table C.1: Appendix C Exercise Topics and References
| Exercise | Topic | Reference |
|---|---|---|
| C.1–C.3 | Raising/lowering indices, Einstein summation | General Relativity General Relativity Ch. 4, Appendix B |
| C.4–C.6 | Concrete examples of metric tensors (sphere, polar coordinates, Schwarzschild) | General Relativity General Relativity Ch. 6, ch07 |
| C.7–C.9 | Christoffel symbols, geodesic equation | General Relativity General Relativity Ch. 8, ch12 |
| C.10 | Conservation of the Einstein tensor | General Relativity General Relativity Ch. 14, Appendix G |
📝 All exercises → Appendix C Exercises
C.3 Two-Dimensional Metrics and the Worldsheet for String Theory¶
🟡 Lina: From here on is the string-theory-specific part. The string's worldsheet has a 2-dimensional extent parametrized by time parameter \(\tau\) and spatial parameter \(\sigma\). To measure "distances" on the worldsheet, we use a 2-dimensional metric \(h_{ab}(\tau, \sigma)\). The indices \(a, b\) run over worldsheet coordinates, with \(a, b \in \{0, 1\}\) corresponding to \(\sigma^0 = \tau\), \(\sigma^1 = \sigma\). The 2-dimensional integration measure is written \(d^2\sigma \equiv d\sigma^0\,d\sigma^1 = d\tau\,d\sigma\) (here \(d^2\sigma\) is shorthand for "integration over the two coordinates \(\sigma^a\)," not the square of \(\sigma\)). Note that the letter \(\sigma\) is used both for "the spatial parameter" and "shorthand for all coordinates \(\sigma^a\)"—if it has an index it refers to all coordinates, without an index it means the spatial parameter.
Worldsheet Metric and the Polyakov Action¶
🟡 Lina: The Polyakov action explicitly contains the worldsheet metric:
Here \(T\) is the string tension (introduced in Ch. 13 as \(T = \frac{1}{2\pi\alpha'}\)), \(h = \det(h_{ab})\) is the determinant of \(h_{ab}\) (for a 2×2 matrix, \(h = h_{\tau\tau}h_{\sigma\sigma} - h_{\tau\sigma}^2\)), and \(h^{ab}\) is the inverse metric of \(h_{ab}\). \(X^\mu(\tau, \sigma)\) is the function specifying where each point on the string sits in spacetime, with \(\mu = 0, 1, \ldots, D-1\) labeling spacetime coordinates. The lower \(\mu\) in \(\partial_b X_\mu\) means the index is lowered with the spacetime metric, i.e., \(\partial_b X_\mu \equiv \eta_{\mu\nu}\partial_b X^\nu\) (in flat spacetime), and by the Einstein summation convention \(\partial_a X^\mu \partial_b X_\mu = \eta_{\mu\nu}\partial_a X^\mu \partial_b X^\nu\).
🔵 Kai: So there are two metrics appearing—the worldsheet metric \(h_{ab}\) and the spacetime metric \(\eta_{\mu\nu}\) (or more generally \(g_{\mu\nu}\)).
🟡 Lina: Exactly. It's crucial not to confuse them. \(h_{ab}\) is the metric on the worldsheet, a function of \(\tau, \sigma\). Meanwhile \(g_{\mu\nu}(X)\) is the metric on spacetime, the background spacetime metric where the string lives. When treating string theory in flat spacetime, \(g_{\mu\nu} = \eta_{\mu\nu}\), but when the string propagates in curved spacetime, \(g_{\mu\nu}(X)\) becomes coordinate-dependent.
Components of the Worldsheet Metric¶
As a 2-dimensional symmetric matrix, it has 3 independent components:
Here \(\det(h_{ab}) = h_{\tau\tau}h_{\sigma\sigma} - h_{\tau\sigma}^2\). When the worldsheet has a time direction \(\tau\) and a spatial direction \(\sigma\), the signature of the metric is \((-,+)\) (called Lorentzian). In this case \(\det(h_{ab}) < 0\), so \(\sqrt{-h}\) appearing in the action is real.
Weyl Transformation and Gauge Freedom¶
🟡 Lina: A very important property of 2-dimensional metrics is the freedom under Weyl transformations:
This is the operation of scaling the entire metric by an arbitrary function \(\omega(\tau, \sigma)\). Remarkably, the Polyakov action is invariant under this transformation.
🔵 Kai: Invariant? But both \(\sqrt{-h}\) and \(h^{ab}\) change—you're saying they cancel each other out? Why?
🟡 Lina: Precisely. Let's verify explicitly. Under \(h_{ab} \to e^{2\omega}h_{ab}\), every component is multiplied by the same factor \(e^{2\omega}\). When you multiply all entries of an \(n \times n\) matrix by the same value \(c\), the determinant gets multiplied by \(c^n\). Let's see this concretely for the \(2 \times 2\) case: \(\det\begin{pmatrix} ca & cb \\ cc & cd \end{pmatrix} = (ca)(cd) - (cb)(cc) = c^2(ad - bc) = c^2 \det\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)—each term is a product of 2 entries, so multiplying all entries by \(c\) multiplies each term by \(c^2\). The same logic applies for general \(n \times n\) (the determinant consists of terms each being a product of \(n\) entries, so multiplying all entries by \(c\) multiplies each term by \(c^n\)). Here \(c = e^{2\omega}\) is a function of coordinates \((\tau, \sigma)\), but the determinant calculation is performed independently at each point \((\tau, \sigma)\). Focusing on a single point, \(e^{2\omega}\) is just a constant, so the property "multiplying all entries by the same constant multiplies the determinant by \(c^n\)" applies directly. For 2×2, \(\det(e^{2\omega}h_{ab}) = (e^{2\omega})^2\det(h_{ab}) = e^{4\omega}\det(h_{ab})\). Therefore \(\sqrt{-h} \to \sqrt{e^{4\omega}}\sqrt{-h} = e^{2\omega}\sqrt{-h}\). And the inverse metric must maintain \(h^{ab}h_{bc} = \delta^a_c\), so \(h^{ab} \to e^{-2\omega}h^{ab}\).
⚪ Mei: \(\sqrt{-h}\) picks up a factor of \(e^{2\omega}\), \(h^{ab}\) picks up \(e^{-2\omega}\)—multiplied together they cancel exactly.
🟡 Lina: Right. Since \(\partial_a X^\mu\) doesn't depend on the metric, it doesn't change. Putting it all together, \(e^{2\omega} \cdot e^{-2\omega} = 1\), and the entire product is invariant.
🔵 Kai: I see, the exponents are \(+2\omega\) and \(-2\omega\), which add to zero—so they cancel. But does this only work because it's 2-dimensional? In 3 dimensions, wouldn't the transformation of \(\sqrt{-h}\) be different?
🟡 Lina: Sharp. In 3 dimensions, \(\sqrt{-h} \to e^{3\omega}\sqrt{-h}\), so \(e^{3\omega} \cdot e^{-2\omega} = e^{\omega} \neq 1\) and they don't cancel. Weyl invariance is a property unique to 2 dimensions. This Weyl symmetry is foundational to string theory and serves as the starting point for Conformal Field Theory (CFT), which we'll cover in detail in Ch. 14.
✅ Comprehension Check: Explain concisely why the Polyakov action is invariant under the Weyl transformation \(h_{ab} \to e^{2\omega}h_{ab}\), using the transformation properties of \(\sqrt{-h}\) and \(h^{ab}\).
Answer
In 2 dimensions, \(\sqrt{-h} \to e^{2\omega}\sqrt{-h}\) (because the determinant picks up a factor of \(e^{4\omega}\)), and the inverse metric transforms as \(h^{ab} \to e^{-2\omega}h^{ab}\). Since \(\partial_a X^\mu\) is independent of the metric, it is invariant. Multiplying these together gives \(e^{2\omega} \cdot e^{-2\omega} = 1\), so the entire action is invariant.
Conformal Gauge¶
🟡 Lina: As we just saw, there are 3 independent components. On the other hand, coordinate transformations \(\tau \to \tau'(\tau, \sigma)\), \(\sigma \to \sigma'(\tau, \sigma)\) provide 2 arbitrary functions that can fix 2 components to any desired form. Furthermore, a Weyl transformation can fix the remaining 1 component. In total \(3 - 2 - 1 = 0\), so we can completely fix the metric to a flat form:
This is called the conformal gauge. Here \(\eta_{ab}\) is the 2-dimensional Minkowski metric (\(a, b \in \{\tau, \sigma\}\)), the 2-dimensional version of the 4-dimensional \(\eta_{\mu\nu} = \mathrm{diag}(-1,+1,+1,+1)\) from C.1. In this gauge, the Polyakov action becomes
\(\eta^{ab}\) is the inverse matrix of \(\eta_{ab}\), but the inverse of a diagonal matrix just has the reciprocal of each diagonal entry (\((-1)^{-1} = -1\), \((+1)^{-1} = +1\)), so \(\eta^{ab} = \mathrm{diag}(-1, +1)\), the same form as the original. By the Einstein summation convention, summing over \(a, b\): \(\eta^{ab}\partial_a X^\mu \partial_b X_\mu = \sum_{a}\sum_{b}\eta^{ab}\partial_a X^\mu \partial_b X_\mu\). Since \(\eta^{ab}\) is diagonal, the off-diagonal components \(\eta^{\tau\sigma} = \eta^{\sigma\tau} = 0\) vanish, and only diagonal components survive: \(\eta^{\tau\tau}\partial_\tau X^\mu \partial_\tau X_\mu + \eta^{\sigma\sigma}\partial_\sigma X^\mu \partial_\sigma X_\mu = (-1)\partial_\tau X^\mu \partial_\tau X_\mu + (+1)\partial_\sigma X^\mu \partial_\sigma X_\mu\).
🔵 Kai: Only the diagonal components survive, and then you just plug in the signs.
🟡 Lina: Right. Including the overall \(-T/2\), we get \(S_P = -\frac{T}{2}\int d^2\sigma\,[(-1)\partial_\tau X^\mu \partial_\tau X_\mu + (+1)\partial_\sigma X^\mu \partial_\sigma X_\mu]\). Since \((-T/2) \times (-1) = +T/2\) and \((-T/2) \times (+1) = -T/2\):
⚪ Mei: So in conformal gauge, both \(\sqrt{-h}\) and \(h^{ab}\) disappear, leaving only the difference between \(\tau\)-derivatives and \(\sigma\)-derivatives—a clean form.
⚠️ Don't confuse the two metrics: Here \(\partial_\tau X^\mu \partial_\tau X_\mu = \eta_{\mu\nu}\partial_\tau X^\mu \partial_\tau X^\nu = -(\partial_\tau X^0)^2 + (\partial_\tau X^1)^2 + \cdots + (\partial_\tau X^{D-1})^2\), where the spacetime metric \(\eta_{\mu\nu} = \mathrm{diag}(-1,+1,\ldots,+1)\) makes the time component \(X^0\) contribute with a negative sign. Therefore \(\partial_\tau X^\mu \partial_\tau X_\mu\) is not necessarily positive. The \(\eta_{ab} = \mathrm{diag}(-1, +1)\) that appeared just before is the worldsheet metric (indices \(a, b\)), while \(\eta_{\mu\nu}\) here (indices \(\mu, \nu\)) is the spacetime metric—they live in different spaces, so don't confuse them.
🔵 Kai: This form looks familiar. The square of the \(\tau\)-derivative minus the square of the \(\sigma\)-derivative… it's like the action version of the wave equation, isn't it?
🟡 Lina: Good intuition. Indeed, it has a structure similar to "kinetic energy \(-\) potential energy" being the Lagrangian in mechanics, where you can read the \(\tau\)-derivative as "kinetic-energy-like" and the \(\sigma\)-derivative as "potential-energy-like." Let's derive the equation of motion from this. In mechanics, we obtained the equation of motion \(\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} - \frac{\partial L}{\partial q} = 0\) by making the action \(S = \int dt\,L\) stationary. In field theory, instead of "a particle's position \(q(t)\)" we have "a field \(X^\mu(\tau, \sigma)\)," and the number of independent variables increases from just \(t\) to two: \((\tau, \sigma)\). The action is written as \(S = \int d\tau\,d\sigma\,\mathcal{L}\), where \(\mathcal{L}\) is called the Lagrangian density.
🔵 Kai: How is it different from the Lagrangian \(L\) in mechanics?
🟡 Lina: In mechanics, \(S = \int dt\,L\) was an integral over time only. In field theory, we also integrate over the spatial direction, giving \(S = \int d\tau\,d\sigma\,\mathcal{L}\). \(\mathcal{L}\) is a "density" defined at each point \((\tau, \sigma)\).
⚪ Mei: So while \(L\) was "one for the whole system," \(\mathcal{L}\) "exists at each point"—it becomes the total action only after integration.
🟡 Lina: Exactly. In mechanics, we derived the equation of motion by slightly shifting \(q(t)\) to \(q(t) + \delta q(t)\) and requiring that the change in action vanishes. In field theory we do the same thing—shift \(X^\mu(\tau, \sigma)\) slightly to \(X^\mu + \delta X^\mu\) and demand \(\delta S = 0\). In mechanics we used integration by parts to move \(\frac{d}{dt}\) onto \(\delta q\), but now with two independent variables \(\tau\) and \(\sigma\), integration by parts is needed in each direction. As a result, the Euler-Lagrange equation takes the form where the mechanical \(\frac{d}{dt}\) is replaced by \(\sum_a \partial_a\) (\(a = \tau, \sigma\)):
(Sum over \(a = \tau, \sigma\). In mechanics we integrated by parts once to move \(\frac{d}{dt}(\cdots)\) off \(\delta q\); now with two independent variables \(\tau\) and \(\sigma\), we integrate by parts to move both \(\partial_\tau(\cdots)\) and \(\partial_\sigma(\cdots)\) off \(\delta X^\mu\)—that's why \(\sum_a \partial_a\) appears. For details of the derivation, also see the variational principle chapter in General Relativity.)
⚪ Mei: Comparing with the mechanics version \(\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q}\) that Professor Lina wrote earlier, \(\frac{d}{dt}\) is replaced by \(\sum_a \partial_a\) and \(L\) is replaced by \(\mathcal{L}\).
🔵 Kai: Oh, but let me confirm one thing. In mechanics, differentiating with respect to \(q\) and differentiating with respect to \(\dot{q}\) were separate things, right? In field theory, does that mean differentiating with respect to \(X^\mu\) and differentiating with respect to \(\partial_a X^\mu\) are separate? But \(\partial_a X^\mu\) is the derivative of \(X^\mu\)—is it okay to treat it as an independent variable?
🟡 Lina: Good question. In mechanics too, \(q\) and \(\dot{q}\) were treated as independent—"the position and velocity at a given instant \(t\)." In field theory it's the same—\(X^\mu\) and \(\partial_a X^\mu\) are regarded as independent variables, "the field value and its gradient at a given point \((\tau, \sigma)\)." In the variational framework, when you shift \(X^\mu\) slightly, \(\partial_a X^\mu\) changes along with it, but at the stage of deriving the Euler-Lagrange equation, we formally treat them as independent and take partial derivatives—and the correct equation of motion comes out. It's the same "formally treat as independent" rule as in mechanics.
🔵 Kai: I see, the same rule from mechanics carries over directly. So what happens when we actually apply this formula to our action?
🟡 Lina: Let's do it. The Lagrangian density is \(\mathcal{L} = \frac{T}{2}(\partial_\tau X^\nu \partial_\tau X_\nu - \partial_\sigma X^\nu \partial_\sigma X_\nu)\). First, \(X^\mu\) itself doesn't appear directly in \(\mathcal{L}\) (only derivatives \(\partial_a X^\mu\) appear), so \(\frac{\partial \mathcal{L}}{\partial X^\mu} = 0\). Next, let's compute the derivative with respect to \(\partial_\tau X^\mu\). We want to differentiate \(\partial_\tau X^\nu \partial_\tau X_\nu = \eta_{\rho\lambda}\partial_\tau X^\rho \partial_\tau X^\lambda\) with respect to \(\partial_\tau X^\mu\). This has the form \(\eta_{\rho\lambda} \cdot (\partial_\tau X^\rho) \cdot (\partial_\tau X^\lambda)\), where \(\partial_\tau X^\mu\) appears in two places—in \(\partial_\tau X^\rho\) when \(\rho = \mu\), and in \(\partial_\tau X^\lambda\) when \(\lambda = \mu\). Using the product rule (same idea as \((fg)' = f'g + fg'\)), from the \(\rho = \mu\) term we get \(\eta_{\mu\lambda}\partial_\tau X^\lambda\), and from the \(\lambda = \mu\) term we get \(\eta_{\rho\mu}\partial_\tau X^\rho\). Since \(\eta\) is symmetric (\(\eta_{\mu\lambda} = \eta_{\lambda\mu}\)) and dummy indices can be freely relabeled, the two contributions are the same, giving \(2\eta_{\mu\lambda}\partial_\tau X^\lambda = 2\partial_\tau X_\mu\). Multiplying by \(T/2\): \(\frac{\partial \mathcal{L}}{\partial(\partial_\tau X^\mu)} = T\,\partial_\tau X_\mu\).
🔵 Kai: The \(\sigma\) part works the same way? Just with a sign change?
🟡 Lina: Exactly. The \(\sigma\) part of \(\mathcal{L}\) is \(-\frac{T}{2}\partial_\sigma X^\nu \partial_\sigma X_\nu\), and differentiating \(\partial_\sigma X^\nu \partial_\sigma X_\nu\) with respect to \(\partial_\sigma X^\mu\) gives \(2\partial_\sigma X_\mu\) by the same procedure. Multiplying by the coefficient \(-T/2\): \(\frac{\partial \mathcal{L}}{\partial(\partial_\sigma X^\mu)} = -\frac{T}{2} \times 2\partial_\sigma X_\mu = -T\,\partial_\sigma X_\mu\). Since \(\frac{\partial \mathcal{L}}{\partial X^\mu} = 0\), the Euler-Lagrange equation simplifies to \(\sum_a \partial_a\frac{\partial \mathcal{L}}{\partial(\partial_a X^\mu)} = 0\). Substituting: \(\partial_\tau(T\,\partial_\tau X_\mu) + \partial_\sigma(-T\,\partial_\sigma X_\mu) = 0\), i.e., \(T(\partial_\tau^2 - \partial_\sigma^2)X_\mu = 0\). In flat spacetime \(\eta_{\mu\nu}\) is constant, so raising the index gives the same equation, yielding the 2-dimensional wave equation \((\partial_\tau^2 - \partial_\sigma^2)X^\mu = 0\) that we saw in Appendix A.3.
✅ Comprehension Check: The 2-dimensional worldsheet metric has 3 independent components, yet it can be completely fixed to \(h_{ab} = \eta_{ab}\) in conformal gauge. Why? What symmetries are being used?
Answer
Coordinate transformations (diffeomorphisms) use 2 arbitrary functions to fix 2 components, and then a Weyl transformation fixes the remaining 1 component. In total \(3 - 2 - 1 = 0\), eliminating all independent components and allowing the metric to be completely fixed to the flat form \(\eta_{ab} = \mathrm{diag}(-1, +1)\).
🔵 Kai: There were many computational steps, but I can see that ultimately it reduces to just the wave equation \((\partial_\tau^2 - \partial_\sigma^2)X^\mu = 0\). But what concerns me is—we fixed the gauge and threw away all the information in \(h_{ab}\), right? Isn't there some condition we're missing?
🟡 Lina: Sharp. In fact, the equation of motion for \(h_{ab}\) before gauge fixing remains as a constraint condition. This is the Virasoro constraint, which we'll cover in detail in Ch. 14.
⚪ Mei: So the wave equation alone isn't sufficient—only solutions satisfying additional conditions are physically allowed.
🔵 Kai: What form do these constraints take specifically? Among solutions to the wave equation, which ones are "not allowed"…?
🟡 Lina: Good question. The specific form will be derived in Chapter 14, but intuitively, think of it as "restrictions on how energy and momentum are distributed." There are constraints on how much energy can be allocated to each vibrational mode of the string. Look forward to it.
🔵 Kai: Constraints per vibrational mode… but honestly, when you say "restrictions on energy distribution," I don't quite grasp what that means yet. Do I have to wait until the specific formulas appear in Chapter 14?
🟡 Lina: Yes, the specific form will be derived in Chapter 14, so for now this is just a preview. But let me give one analogy—it looks like you can freely distribute energy among the string's vibrational modes, but actually "the energy of left-moving waves" and "the energy of right-moving waves" must be equal. This is called the level-matching condition and is part of the Virasoro constraints. And as an even more remarkable consequence—when you investigate the conditions for the Virasoro constraints to hold consistently at the quantum level, the dimensionality of the spacetime in which the string lives gets fixed to a specific value—the critical dimension.
🔵 Kai: Wait, the dimension of spacetime is determined by conditions on string vibrations? Why would vibrations affect the number of dimensions?
🟡 Lina: In quantum mechanics, probabilities must always sum to 1, right? But when you quantize the string, if the spacetime dimension isn't a specific value, the probabilities deviate from summing to 1, or states with negative probability refuse to decouple—pathological behavior that's physically unacceptable emerges. The condition that avoids this determines the dimension—details in Chapter 14.
C.4 Special Properties of Curvature in 2-Dimensional Spacetime¶
🟡 Lina: Two-dimensional spacetime has special properties compared to 4-dimensional spacetime. These become important in string theory.
The Einstein Tensor Identically Vanishes in 2 Dimensions¶
🟡 Lina: In 4 dimensions, the Riemann tensor has 20 independent components, the Ricci tensor has 10, and the Einstein equation \(G_{\mu\nu} = 8\pi G T_{\mu\nu}\) is a nontrivial equation. However, in 2 dimensions the number of independent components is much smaller.
🔵 Kai: Why do the independent components decrease in 2 dimensions?
🟡 Lina: The indices of the Riemann tensor \(R^\rho{}_{\sigma\mu\nu}\) each range over \(n\) values (the dimension), so fewer dimensions mean fewer combinations. Furthermore, the Riemann tensor has symmetries—"antisymmetry of index pairs," "exchange symmetry between pairs," and "the first Bianchi identity" (see General Relativity Chapter 13)—which generate relations between components. For example, antisymmetry \(R_{abcd} = -R_{bacd}\) means "swapping \(ab\) flips the sign," so \(R_{0101}\) and \(R_{1001}\) are not independent; once one is known, the other is determined. Accounting for all such relations, the number of independent components reduces to \(\frac{n^2(n^2-1)}{12}\) (derivation in General Relativity Chapter 13). For \(n = 4\): \(\frac{16 \times 15}{12} = 20\) components; for \(n = 2\): \(\frac{4 \times 3}{12} = 1\) component.
🔵 Kai: Just 1! So there's almost no curvature information?
🟡 Lina: Right. Checking explicitly in 2 dimensions: the indices \(a, b\) can only take values 0 or 1, so by antisymmetry \(R_{abcd} = -R_{bacd} = -R_{abdc}\), the pairs \(ab\) and \(cd\) each have only the single possibility \((01)\), and the only independent component is \(R_{0101}\)—consistent with the formula. With just 1 component, the scalar curvature \(R\) alone completely determines the Riemann tensor.
Specifically, in 2 dimensions the Riemann tensor can be written as
(\(R_{abcd}\) is the fully covariant form, defined by lowering indices with the metric in the same way as "raising and lowering indices" in C.1: \(R_{abcd} = g_{a\rho}R^\rho{}_{bcd}\)—the upper index \(\rho\) is lowered to \(a\) using the metric \(g_{a\rho}\)). Let's compute the Ricci tensor from this. Writing the definition from C.1 \(R_{\mu\nu} = R^\alpha{}_{\mu\alpha\nu}\) in worldsheet indices: \(R_{ab} = R^c{}_{acb}\) (relabeling \(\alpha \to c\), \(\mu \to a\), \(\nu \to b\), where the upper first index and lower third index are both \(c\) and get contracted). The formula \(R_{abcd} = \frac{R}{2}(g_{ac}g_{bd} - g_{ad}g_{bc})\) is for the fully covariant Riemann tensor, so we first want to rewrite \(R^c{}_{acb}\) in fully covariant form. The relationship to the fully covariant Riemann tensor is defined by \(R_{eacb} = g_{e\rho}R^\rho{}_{acb}\) (\(\rho\) is a dummy index). Contracting both sides with \(g^{ce}\) over \(e\): the left side becomes \(g^{ce}R_{eacb} = g^{ce}g_{e\rho}R^\rho{}_{acb}\). Using the inverse metric definition from C.1, \(g^{\mu\alpha}g_{\alpha\nu} = \delta^\mu_\nu\)—this identity holds regardless of what letters we use for the indices. Since \(\mu, \alpha, \nu\) are just labels, we can relabel \(\mu \to c\), \(\alpha \to e\), \(\nu \to \rho\) to get \(g^{ce}g_{e\rho} = \delta^c_\rho\). Therefore \(g^{ce}g_{e\rho}R^\rho{}_{acb} = \delta^c_\rho R^\rho{}_{acb} = R^c{}_{acb}\) (since \(\delta^c_\rho\) is 1 only when \(\rho = c\) and 0 otherwise, only the \(\rho = c\) term survives in the sum). Thus \(R^c{}_{acb} = g^{ce}R_{eacb}\).
⚪ Mei: That's the operation of raising an index back up with the inverse metric. Now we just need to apply the formula to \(R_{eacb}\).
🟡 Lina: We want to apply the formula to \(R_{eacb}\). Let's match the index positions of the formula's left side with those of \(R_{eacb}\). The indices \((a,b,c,d)\) in the formula \(R_{abcd} = \frac{R}{2}(g_{ac}g_{bd} - g_{ad}g_{bc})\) are just dummy labels representing "slot 1, slot 2, slot 3, slot 4." What we want is "the explicit form of \(R_{eacb}\)"—so we assign the indices of \(R_{eacb}\) to the formula's slots in order. Put \(e\) in slot 1, \(a\) in slot 2, \(c\) in slot 3, \(b\) in slot 4:
Correspondence between the formula's index slots \(R_{abcd}\) and our case
| Formula slot | 1st | 2nd | 3rd | 4th |
|---|---|---|---|---|
| Index assigned | \(e\) | \(a\) | \(c\) | \(b\) |
Let's look at the right side of the formula. \(g_{ac}g_{bd}\) has the structure "\(g\) with the 1st and 3rd indices" × "\(g\) with the 2nd and 4th indices," and \(g_{ad}g_{bc}\) has "\(g\) with the 1st and 4th indices" × "\(g\) with the 2nd and 3rd indices." Substituting indices according to the table:
- First term \(g_{ac}g_{bd}\): slot 1→\(e\), slot 3→\(c\) gives \(g_{ec}\); slot 2→\(a\), slot 4→\(b\) gives \(g_{ab}\). Combined: \(g_{ec}g_{ab}\)
- Second term \(g_{ad}g_{bc}\): slot 1→\(e\), slot 4→\(b\) gives \(g_{eb}\); slot 2→\(a\), slot 3→\(c\) gives \(g_{ac}\). Combined: \(g_{eb}g_{ac}\)
Therefore
🔵 Kai: You just assign them mechanically according to the table. Now multiply by \(g^{ce}\) and contract to get the Ricci tensor?
🟡 Lina: Exactly.
First term: \(g^{ce}g_{ec} = \delta^c_c\) (from the inverse metric definition \(g^{c\alpha}g_{\alpha c} = \delta^c_c\)). Here \(\delta^c_c\) means summing over \(c\) by the Einstein convention: \(\delta^c_c = \delta^0_0 + \delta^1_1 = 1 + 1 = 2\) (in 2 dimensions the index only takes values 0 and 1). So the first term overall is \(\frac{R}{2} \cdot 2 \cdot g_{ab}\). Second term: \(g^{ce}g_{eb} = \delta^c_b\), so \(g^{ce}g_{eb}g_{ac} = \delta^c_b g_{ac} = g_{ab}\); the second term overall is \(\frac{R}{2} \cdot g_{ab}\).
Putting it together:
⚠️ Common mistake: If you match the indices intuitively rather than precisely as \((a,b,c,d) \to (e,a,c,b)\), terms like \(g^{ce}g_{ea}g_{cb}\) can appear, leading to the incorrect result "\(R_{ab} = 0\)?" The key is to precisely match the 4 indices of \(R_{eacb}\) to the left side \(R_{abcd}\) of the formula. In particular, the fact that \(g^{ce}g_{ec} = n\) (the dimension) in the first term is the crucial point in 2 dimensions.
🔵 Kai: The index relabeling is quite tricky… But basically, in 2 dimensions \(g^{ce}g_{ec} = 2\) is what matters, and ultimately it reduces to the proportionality relation \(R_{ab} = \frac{R}{2}g_{ab}\)?
🟡 Lina: Exactly. In 4 dimensions, \(g^{ce}g_{ec} = 4\) and the result would be different, so \(R_{ab} = \frac{R}{2}g_{ab}\) is an identity specific to 2 dimensions.
This gives us the correct result. Under the conventions adopted in this General Relativity—\(R^\rho{}_{\sigma\mu\nu}\) (first index up) and \(R_{\mu\nu} = R^\alpha{}_{\mu\alpha\nu}\)—the 2-dimensional identities are:
- Ricci tensor: \(R_{ab} = \frac{1}{2}g_{ab}R\) (identity)
- Einstein tensor: \(G_{ab} = R_{ab} - \frac{1}{2}g_{ab}R = \frac{1}{2}g_{ab}R - \frac{1}{2}g_{ab}R \equiv 0\) (always zero)
⚪ Mei: Since \(R_{ab}\) and \(\frac{1}{2}g_{ab}R\) are exactly the same, subtracting them leaves nothing—they cancel perfectly.
(For the proof of the starting formula \(R_{abcd} = \frac{R}{2}(g_{ac}g_{bd} - g_{ad}g_{bc})\), see the exercises in General Relativity Chapter 13.)
🔵 Kai: Wait, the left side of the Einstein equation is always zero in 2 dimensions? What happens if you try to build a theory of gravity in 2 dimensions?
🟡 Lina: In 2 dimensions, the Einstein-Hilbert action \(S = \int d^2\sigma\sqrt{-h}\,R\) becomes a topological quantity—meaning its value doesn't change under continuous deformations of the surface. This is a result called the Gauss-Bonnet theorem. For surfaces without boundary, the Gauss-Bonnet theorem is originally proven for Euclidean metrics (signature \((+,+)\)). To apply it to a Lorentzian worldsheet metric \((-,+)\), we use an operation called Wick rotation. The goal is "to change the minus sign in the time direction to plus, making it \((+,+)\)"—for this we formally substitute the time coordinate as \(\tau = -i\tau_E\) (\(i\) is the imaginary unit, \(i^2 = -1\)). Looking at how \(\tau\) changes when \(\tau_E\) changes slightly: since \(-i\) is a constant, \(d\tau = -i\,d\tau_E\) (same idea as \(y = cx\) giving \(dy = c\,dx\)). Computing the time part of the line element: \(-d\tau^2 = -(-i\,d\tau_E)^2\). Here \((-i)^2 = (-1)^2 \cdot i^2 = 1 \cdot (-1) = -1\), so \(-(-i\,d\tau_E)^2 = -((-1)\,d\tau_E^2) = +d\tau_E^2\), and the signature flips from \((-,+)\) to \((+,+)\)—exactly as intended (details in Ch. 14).
🔵 Kai: Wait a moment. Making time imaginary… is that really allowed? What's physically happening?
🟡 Lina: Good question. "Imaginary time" doesn't have physical reality. Let me state the motivation first: the Gauss-Bonnet theorem—the powerful result that "the integral of curvature is determined by topology alone"—is proven for metrics with signature \((+,+)\) (Euclidean). Our worldsheet metric is \((-,+)\) (Lorentzian), so we can't directly apply the theorem. So we use the mathematical trick of Wick rotation to temporarily change the signature to \((+,+)\), apply the theorem, and bring back only the conclusion. Since the result (the Euler number being an integer) is a geometric fact independent of the signature, this "go and come back" operation is justified.
⚪ Mei: So it's the technique of "temporarily moving to a world where the theorem applies, obtaining the result there."
🟡 Lina: Exactly. Specifically, let's write the Euclidean metric obtained by Wick rotation as \(g_{ab}^{(E)}\). Since the determinant is now positive, we write \(\sqrt{g^{(E)}}\) instead of \(\sqrt{-h}\). Writing the scalar curvature computed from this Euclidean metric \(g_{ab}^{(E)}\) as \(R^{(E)}\), the Gauss-Bonnet theorem states
This is the remarkable theorem that "summing up the curvature over the entire surface gives an integer determined solely by the shape (topology) of the surface." The Euler number \(\chi\) is an integer determined only by the topology of the surface—specifically, by how many "handles" are attached. To visualize handles, think of a coffee mug's handle—attaching one handle to a sphere (balloon surface) gives the shape of a doughnut (torus). Writing the number of handles as \(\mathfrak{g}\) (genus; written in Fraktur to distinguish from the metric determinant \(g\)), we have \(\chi = 2 - 2\mathfrak{g}\). For a sphere (no handles, \(\mathfrak{g} = 0\)): \(\chi = 2\); for a torus (one handle, \(\mathfrak{g} = 1\)): \(\chi = 0\); for a double torus (two handles, \(\mathfrak{g} = 2\)): \(\chi = -2\).
🔵 Kai: Huh, so adding up all the curvature gives an integer. The fact that the value depends only on the shape means it doesn't depend on how you bend or stretch it, right?
🟡 Lina: Exactly. The \(1/(4\pi)\) in front of the formula is a normalization constant chosen so that the left side equals exactly \(2\) for a sphere (\(\chi = 2\)). Intuitively, a sphere curves outward everywhere so the total curvature is positive, while saddle-shaped surfaces have negative curvature—summing all of this up gives a total that doesn't change no matter how you deform the surface (as long as you don't open or close holes). The key point is that \(\chi\) doesn't change no matter how you change the metric. Recall that equations of motion are obtained by "differentiating the action with respect to a variable and setting it to zero" (in mechanics, \(\delta S/\delta q = 0\)). In gravity theory, the metric \(h_{ab}\) is the variable, so the equation of motion takes the form \(\delta S/\delta h_{ab} = 0\). But if the value of the action is determined solely by topology (\(\chi\)) and doesn't depend at all on the specific form of the metric, then the action doesn't change no matter how you vary the metric—meaning \(\delta S/\delta h_{ab}\) is zero for any \(h_{ab}\). This is the same as the equation "\(0 = 0\)," imposing no constraint on \(h_{ab}\). Therefore, a naive 2-dimensional gravity theory has no dynamical degrees of freedom.
✅ Comprehension Check: The fact that the Einstein tensor \(G_{ab} = R_{ab} - \frac{1}{2}g_{ab}R\) identically vanishes in 2 dimensions—what consequence does this have for the physical role of the worldsheet metric \(h_{ab}\) in string theory?
Answer
Since the Einstein tensor identically vanishes, the 2-dimensional Einstein-Hilbert action becomes a topological quantity (the Euler number), and no dynamical equation of motion arises from varying \(h_{ab}\). Therefore \(h_{ab}\) is not a physical degree of freedom but merely a gauge freedom (redundancy in the description), and can be completely removed by Weyl transformations and coordinate transformations.
🔵 Kai: The Euler number is determined by the number of holes in a doughnut? More holes means lower Euler number?
🟡 Lina: "Handles" is more precise than "holes." As I just explained, as the number of handles \(\mathfrak{g}\) increases, the Euler number decreases via \(\chi = 2 - 2\mathfrak{g}\). In string theory, string interactions are viewed as "changes in worldsheet topology," so this Euler number becomes the parameter of the perturbative expansion—we'll see this from Chapter 14 onward.
🔵 Kai: "Topology changes"—does that mean when strings split or merge, handles get added to the worldsheet?
🟡 Lina: Yes. When viewed on the worldsheet, a process where one string splits into two takes the shape of a pair of pants—called a "pants diagram." When loop corrections enter, handles are added and \(\mathfrak{g}\) increases. Since higher \(\mathfrak{g}\) means smaller contributions, the Euler number controls the order of the perturbative expansion. More in Chapter 14.
⚪ Mei: To summarize: the 2-dimensional Einstein-Hilbert action only gives the topological quantity \(\chi = 2 - 2\mathfrak{g}\) (the Euler number) and doesn't generate equations of motion constraining the metric \(h_{ab}\)—so \(h_{ab}\) has no dynamical degrees of freedom.
🔵 Kai: But wait. If \(h_{ab}\) isn't a dynamical degree of freedom, why is it in the Polyakov action? It's in the action but has no physical meaning—isn't that contradictory?
🟡 Lina: Right. In the Polyakov action, \(h_{ab}\) appears to be a dynamical variable, but it's actually just a gauge freedom—a redundancy in the description. Since Weyl transformations and diffeomorphisms can remove all its information, the only physical degrees of freedom are \(X^\mu\) (the string's position in spacetime).
Infinite-Dimensional Symmetry of Conformal Transformations¶
🟡 Lina: Even after fixing conformal gauge, the gauge freedom isn't completely gone. The remaining symmetry is the conformal transformation.
🔵 Kai: Earlier you said "\(3 - 2 - 1 = 0\) so it's completely fixed," right? How can there still be a remaining symmetry?
🟡 Lina: Good question. The earlier argument was that we can "locally" make the metric \(\eta_{ab}\). But among coordinate transformations, there exist ones that change coordinates while preserving the form \(h_{ab} = \eta_{ab}\)—i.e., coordinate transformations that don't break the condition \(h_{ab} = \eta_{ab}\). These are called "residual gauge symmetries." Let's see this concretely. To solve the wave equation \((\partial_\tau^2 - \partial_\sigma^2)X = 0\), we introduce the variables \(\sigma^+ = \tau + \sigma\), \(\sigma^- = \tau - \sigma\) (already introduced in Appendix A.3). Let's briefly review the factorization here since we'll use it later. Setting \(\sigma^+ = \tau + \sigma\), \(\sigma^- = \tau - \sigma\), from the chain rule: \(\partial_\tau f = \frac{\partial f}{\partial \sigma^+}\frac{\partial \sigma^+}{\partial \tau} + \frac{\partial f}{\partial \sigma^-}\frac{\partial \sigma^-}{\partial \tau} = \partial_+ f \cdot 1 + \partial_- f \cdot 1\), so \(\partial_\tau = \partial_+ + \partial_-\). Similarly, \(\frac{\partial \sigma^+}{\partial \sigma} = 1\), \(\frac{\partial \sigma^-}{\partial \sigma} = -1\), so \(\partial_\sigma = \partial_+ - \partial_-\). Therefore \(\partial_\tau^2 - \partial_\sigma^2 = (\partial_+ + \partial_-)^2 - (\partial_+ - \partial_-)^2\). Expanding: \((\partial_+ + \partial_-)^2 = \partial_+^2 + 2\partial_+\partial_- + \partial_-^2\), \((\partial_+ - \partial_-)^2 = \partial_+^2 - 2\partial_+\partial_- + \partial_-^2\), so taking the difference, \(\partial_+^2\) and \(\partial_-^2\) cancel, leaving \(4\partial_+\partial_-\). The wave equation becomes \(4\partial_+\partial_- X = 0\), and dividing by \(4 \neq 0\) gives \(\partial_+\partial_- X = 0\).
⚪ Mei: The second-order PDE factorizes into \(\partial_+\partial_- X = 0\), and the solution is \(X = f(\sigma^+) + g(\sigma^-)\)—a superposition of left-moving and right-moving waves independently.
🟡 Lina: Exactly. Such a "set of variables in which the wave equation factorizes into its simplest form" is called characteristic coordinates (a term from PDE theory, where "characteristic" refers to the specific directions in which waves propagate—corresponding to the solution propagating independently in the \(\sigma^+\) and \(\sigma^-\) directions). In string theory we call these worldsheet light-cone coordinates (in summary, \(\sigma^\pm = \tau \pm \sigma\)). It's the same idea as the spacetime light-cone coordinates \(x^\pm = (t \pm x)/\sqrt{2}\) introduced in Ch. 5, applied to the worldsheet. Note however that spacetime light-cone coordinates are coordinates within \(D\)-dimensional spacetime with index \(\mu\), while worldsheet light-cone coordinates are coordinates within the 2-dimensional worldsheet with index \(a\)—they live in different spaces.
In these coordinates, the conformal gauge line element takes the form \(ds^2 = -d\sigma^+ d\sigma^-\). Let's verify. In conformal gauge \(h_{ab} = \eta_{ab} = \mathrm{diag}(-1, +1)\), so the line element is \(ds^2 = h_{ab}\,d\sigma^a d\sigma^b = -d\tau^2 + d\sigma^2\). On the other hand, \(d\sigma^+ = d\tau + d\sigma\), \(d\sigma^- = d\tau - d\sigma\), so \(d\sigma^+ d\sigma^- = (d\tau + d\sigma)(d\tau - d\sigma) = d\tau^2 - d\sigma^2\). Therefore \(-d\sigma^+ d\sigma^- = -d\tau^2 + d\sigma^2 = ds^2\). (As metric tensor components, \(h_{+-} = h_{-+} = -\frac{1}{2}\). This is because in the double sum \(ds^2 = h_{ab}\,d\sigma^a d\sigma^b\) with \(a, b\) each taking values \(+, -\), both the \((a,b) = (+,-)\) and \((a,b) = (-,+)\) terms contribute. Since the product of infinitesimals is order-independent, \(d\sigma^+ d\sigma^- = d\sigma^- d\sigma^+\), the two terms combine to give \(ds^2 = (h_{+-} + h_{-+})\,d\sigma^+ d\sigma^- = 2h_{+-}\,d\sigma^+ d\sigma^-\) (using metric symmetry \(h_{+-} = h_{-+}\) in the last equality). Matching with \(ds^2 = -d\sigma^+ d\sigma^-\) requires \(2h_{+-} = -1\), i.e., \(h_{+-} = -\frac{1}{2}\).) The key point is that under independent reparametrizations \(\sigma^+ \to f(\sigma^+)\), \(\sigma^- \to g(\sigma^-)\), the infinitesimal changes in the new coordinates are \(d\tilde\sigma^+ = f'(\sigma^+)\,d\sigma^+\), \(d\tilde\sigma^- = g'(\sigma^-)\,d\sigma^-\) (chain rule). The metric in the new coordinates \(\tilde\sigma^\pm = (f(\sigma^+), g(\sigma^-))\) retains the same form \(ds^2 = -d\tilde\sigma^+ d\tilde\sigma^-\)—from the new coordinates' perspective, nothing has changed.
🔵 Kai: Same form in the new coordinates… but what happens from the old coordinates' perspective?
🟡 Lina: Think of it this way. First let's confirm an important distinction. A coordinate transformation merely re-describes the same geometric object in different coordinates, so the value of the line element \(ds^2\) itself doesn't change—component values change but the physical "distance" is the same. A Weyl transformation, on the other hand, is an operation that scales the metric itself, changing the line element value. These two are fundamentally different things.
🔵 Kai: A coordinate transformation is "just changing how you draw the map" while the terrain stays the same, and a Weyl transformation is "inflating or shrinking the terrain itself"—is that the image?
🟡 Lina: Nice analogy. Exactly. Now, the question here is "what transformations preserve the conformal gauge form \(h_{ab} = \eta_{ab}\)?" In the original coordinates \(\sigma^\pm\), the metric component was \(h_{+-} = -1/2\). Moving to new coordinates \(\tilde\sigma^\pm\), in the new coordinates we also have \(\tilde h_{+-} = -1/2\) (from the new coordinates' perspective, we're still in conformal gauge). However, from the perspective of someone continuing to use the old coordinates \(\sigma^\pm\), expressing the new-coordinate line element in terms of old-coordinate differentials: \(d\tilde\sigma^+ = f'(\sigma^+)\,d\sigma^+\), \(d\tilde\sigma^- = g'(\sigma^-)\,d\sigma^-\), so \(ds^2 = -d\tilde\sigma^+ d\tilde\sigma^- = -f'(\sigma^+)\,g'(\sigma^-)\,d\sigma^+ d\sigma^-\). Reading this as metric components in the old coordinates \(\sigma^\pm\): \(h_{+-}^{\text{new}} = -f'g'/2\), which differs from the original \(h_{+-} = -1/2\) (here \(h_{+-}^{\text{new}}\) is the pulled-back metric component from the coordinate transformation). From the old-coordinates perspective, "the metric component has changed." But to restore something multiplied by \(f'g'\), we just need to multiply by \(1/(f'g')\)—choosing the Weyl transformation \(h_{ab} \to e^{2\omega}h_{ab}\) with \(e^{2\omega} = 1/(f'g')\) gives \(h_{+-} \to e^{2\omega} \cdot (-f'g'/2) = (1/(f'g')) \cdot (-f'g'/2) = -1/2\), restoring \(h_{+-} = -1/2\) in the old-coordinate component representation. So the combination "coordinate transformation + appropriate Weyl transformation" is the operation that preserves the conformal gauge form \(ds^2 = -d\sigma^+ d\sigma^-\) while staying in the old coordinates—this is the symmetry remaining after conformal gauge fixing.
⚪ Mei: So the new-coordinates person thinks "I'm still in conformal gauge," but from the old-coordinates perspective the metric components are multiplied by \(f'g'\)—that's where the Weyl transformation comes in.
🟡 Lina: Exactly. Writing just the coordinate transformation part:
\(f, g\) are arbitrary functions, and for each one, the Weyl transformation with \(e^{2\omega} = 1/(f'g')\) automatically accompanies it. In 2 dimensions, this "coordinate transformation + Weyl transformation" pair constitutes a symmetry with infinitely many parameters. That is, the two arbitrary functions \(f\) and \(g\) play the role of parameters, and since a single function contains infinitely many degrees of freedom (its value at each point), the symmetry has infinitely many parameters. For comparison, in 4-dimensional spacetime, collecting all "angle-preserving transformations" (conformal transformations) gives only a finite-dimensional group with a total of 15 parameters (4 translations + 6 Lorentz transformations + 1 scaling + 4 special conformal transformations—don't worry about the details now). The point is "finitely many." In 2 dimensions this expands to infinite dimensions—this is the source of the power of conformal field theory.
⚪ Mei: A symmetry with infinitely many parameters… compared to the 15 dimensions in 4D, that's on a completely different scale. Does that mean much stronger constraints are imposed on the theory?
🟡 Lina: Yes. In fact, when infinite-dimensional symmetry is present, quantum field theories can sometimes be solved exactly. This is one of the keys that makes string theory quantization possible. When we cover conformal field theory and the Virasoro algebra in Chapters 14–Ch. 15, this infinite-dimensional symmetry plays a decisive role.
✅ Comprehension Check: What is the crucial difference between 2-dimensional conformal symmetry and 4-dimensional conformal symmetry? And why is this important for string theory?
Answer
The 4-dimensional conformal group is 15-dimensional (finite-dimensional), but in 2 dimensions, the light-cone coordinates \(\sigma^+, \sigma^-\) can each be reparametrized by arbitrary functions \(f(\sigma^+), g(\sigma^-)\), expanding the symmetry to infinite dimensions. This infinite-dimensional symmetry imposes extremely strong constraints on the theory, and in some cases allows quantum field theories to be solved exactly. This is one of the keys enabling the quantization of string theory.
Summary: How to Use These Tools in String Theory¶
🟡 Lina: Let me summarize the key points to retain from this appendix.
Table C.2: General relativity tools and their roles in string theory
| Topic | Reference | Role in string theory |
|---|---|---|
| Tensor basics | General Relativity ch04, Appendix B | Handling spacetime vectors \(X^\mu\) and worldsheet tensors \(h_{ab}\) |
| Metric tensor | General Relativity ch06, ch07 | Two-layer structure of worldsheet metric \(h_{ab}\) and background spacetime metric \(g_{\mu\nu}\) |
| Covariant derivative / Christoffel | General Relativity ch08, ch12 | Needed when strings propagate in curved background spacetime |
| Riemann / Ricci / Einstein | General Relativity ch13, ch14 | Einstein tensor identically vanishes in 2 dimensions |
| 2-dimensional specifics | This appendix C.3, C.4 | Weyl symmetry, conformal gauge, infinite-dimensional conformal symmetry |
⚪ Mei: To organize: applying general relativity's tools to "the string worldsheet" is the essence, but 2 dimensions are special—there's an additional Weyl symmetry, and the Einstein tensor identically vanishes. These two points are what make the difference.
🟡 Lina: Right. And these two points are what greatly simplify the treatment of the string worldsheet.
🔵 Kai: Because it's 2-dimensional, we can completely eliminate the metric, and only the string's position \(X^\mu\) remains as physics—the "low dimensionality" is actually an advantage. But conversely, what if instead of a 1-dimensional string, we had a 2-dimensionally extended sheet-like object? Its trajectory would have more dimensions, right? Would this special property no longer apply?
🟡 Lina: Exactly. Such a 2-dimensionally extended object is called a membrane. The trajectory of a membrane—called the worldvolume—is 3-dimensional or higher. There, Weyl transformations can no longer completely eliminate the metric, so the simplification we had for strings doesn't work. This is precisely one of the reasons membrane theory is not as tractable as string theory.
Preview of Next Chapter¶
In Appendix D, we introduce the basics of "group theory," which systematically treats the symmetries lurking behind physical laws. From continuous groups like rotation groups and Lorentz groups to gauge groups that play central roles in string theory—let's build the mathematical framework for why symmetries govern conservation laws and particle classification.
References¶
- General Relativity Chapters 4, 6–8, 12–14, Appendix B — Self-contained exposition of tensors, metrics, covariant derivatives, curvature tensors, and the Einstein equation (hub shared across all 4 volumes)
- David Tong, Lectures on String Theory, Ch.2 — Polyakov action, worldsheet metric, conformal gauge
- Barton Zwiebach, A First Course in String Theory, Ch.6–13 — Relativistic strings, Nambu-Goto action, Polyakov action, conformal invariance
- Joseph Polchinski, String Theory Vol. 1, Ch.1–2 — Classical string theory, worldsheet geometry (advanced)
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