3.3 1+1-dimensional periodic scalar theory
Rather than explaining the general theory of a QFT on a open manifold, let us jump to a concrete example: 1+1-dimensional (relativistic) massless periodic scalar \(\phi\). The theory should be explained in any good textbooks on string theory, e.g. [17]. Here we give a lightening review.
3.3.1 The model
The action of the free massless periodic scalar theory (on the Minkowski flat space) is (in Polchinski’s normalization, in particular \(\hbar =1\) as usual) \[\begin{equation} \tag{3.4} S = -\frac{1}{4\pi\alpha'}\int_{M_2} \mathrm{d}\tau\mathrm{d}\sigma \,\eta^{ab}\partial_a X \partial_b X. \end{equation}\] The scalar field \(X\) is periodic and we set the periodicity to be \(2\pi R\): \(X \sim X + 2\pi R\). The equation of motion is the massless Klein-Gordon one \[\begin{equation} \tag{3.5} \partial^2 X = 0. \end{equation}\] The canonical momentum \(\Pi(\sigma,\tau)\) is \[\begin{equation} \tag{3.6} \Pi = \frac{1}{2\pi \alpha'} \partial_\tau X, \end{equation}\] and the Hamiltonian is \[\begin{equation} \tag{3.7} H = \frac{1}{2\pi\alpha'}\int\, \mathrm{d}\sigma \delta^{ab}\partial_a X \partial_b X. \end{equation}\]
3.3.2 Quantization on \(S^1\) and \(U(1)^2\) symmetry
Let us quantize the theory on \(S^1\), where we set the periodicity of the spacial coordinate \(\sigma \sim \sigma + 2\pi\). We write the general solution to the EOM (3.5) as \[\begin{equation} \tag{3.8} X^w(\sigma,\tau) = x + R w \sigma + 2\pi \alpha' p \, \tau + \mathrm{i}\bigl(\frac{\alpha'}{2}\bigr) \sum_{m = -\infty}^{\infty} \frac1m\Bigl(\frac{\alpha_m}{z^m_+}+\frac{\tilde\alpha_m}{z^m_-}\Bigr), \end{equation}\] where \(z = \exp(- \mathrm{i}(\tau \pm \sigma))\). The second term is possible because both \(\sigma\) and \(X\) is periodic, and the winding number \(w\) is quantized (discretized): \(w\in \mathbb{Z}\). When there is such topologically distinct classical solutions to the EOM exists, we build the Hilbert space \(\mathcal{H}_w\) around each solution labeled by \(w\), and the take the direct sum \(\mathcal{H} = \bigotimes_w \mathcal{H}_w\) as the total Hilbert space. The canonical commutation relation gives \[\begin{equation} \tag{3.9} [x,p] = \mathrm{i}, \quad [\alpha_m,\alpha_n] = [\tilde{\alpha}_m,\tilde{\alpha}_n] = m \delta_{m,-n}, \end{equation}\] and the Hamiltonian is (up to a divergent constant coming from commuting \(\alpha\)’s) \[\begin{equation} \tag{3.10} H = \frac{\alpha'}{4} \bigl(\frac{n^2}{R^2}+\frac{w^2 R^2}{\alpha'^2}\bigr) + \sum_{n=1}^{\infty} \bigl(\alpha_{-n}\alpha_n + \tilde{\alpha}_{-n}\tilde{\alpha}_n\bigr), \end{equation}\] where \(n = R p\) is the quantized momentum whose eigenvalues takes in \(\mathbb{Z}\), as in the case of the particle on \(S^1\). A general eigenstate can be constructed by acting finite number of operators \(\alpha_{-i}\) and \(\tilde{\alpha}_{-j}\) on a state \(\ket{n,w}\): \[\begin{equation} \tag{3.11} \alpha_{-i_1}^{t_1}\alpha_{-i_2}^{t_s} \cdots \tilde\alpha_{-j_1}^{s_1}\tilde\alpha_{-j_2}^{s_2} \cdots \ket{n,w}, \end{equation}\] where \(i_a,j_b,t_c,s_d\) are positive integers, and the state \(\ket{n,w}\) satisfies \[\begin{align} \alpha_i \ket{n,w} = \tilde{\alpha}_j \ket{n,w} = 0, \end{align}\] and has the corresponding quantum number \(n\) and \(w\).
This system has two \(U(1)\) symmetries, whose charges are \(n\) and \(w\) respectively. The former is called the momentum \(U(1)\), and the latter called the winding \(U(1)\). The current operators are \[\begin{equation} \tag{3.12} j^p_\mu = R\partial_\mu X, \quad j^w_\mu = \frac{1}{2\pi R}\epsilon_{\mu\nu}\partial^\nu X. \end{equation}\] We write the corresponding unitaries as \(U^p_\lambda\) and \(U^w_\lambda\), where \(\lambda\) is the parameter of the symmetry transformation, so that \[\begin{equation} \tag{3.13} U^p_\lambda \ket{n,w} = e^{\mathrm{i}\lambda n }\ket{n,w},\quad U^w_\lambda \ket{n,w} = e^{\mathrm{i}\lambda w }\ket{n,w}. \end{equation}\] Also, the momentum \(U(1)\) is basically the canonical momentum \(p\) to the average coordinate \(x = \int X \mathrm{d}\sigma\), it causes the translation in the space of \(X\): \[\begin{equation} \tag{3.14} (U^p_{\lambda})^\dagger\, f(X(\sigma,\tau)) \, U^p_{\lambda} = f(X(\sigma,\tau) + \lambda R), \end{equation}\] where \(f\) is a periodic function. With this boundary condition,
3.3.3 Projective phases on the Hilbert spaces on an interval
Now let us see how the two \(U(1)\) symmetries acts on the Hilbert spaces defined on a manifold with boudnaries. In 1d, there is only one (topology type) of a compact manifold with boundaries: the interval \(I = [0,2\pi]\). To define the theory on the interval, we have to pick boundary conditions for each of the boundaries: \(\sigma =0\) (left boundary) and \(\sigma =2\pi\) (right boundary). Here we pick the Dirichlet boundary condition \(\mathcal{D}(\theta_L)\) and \(\mathcal{D}(\theta_R)\) for the left and right boundaries, defined by \[\begin{equation} \tag{3.15} X(\sigma = 0,t) = R \theta_L, \quad X(\sigma = 2\pi, t)= R \theta_R. \end{equation}\] The expansion of the periodic field \(X\) in this case is \[\begin{equation} \tag{3.16} X = R\theta_L + R w \sigma + \mathrm{i}\bigl(\frac{\alpha'}{2}\bigr) e^{-\mathrm{i}\tau} \sum_{m = -\infty}^{\infty} \frac1m \alpha_m \sin(m \sigma/2), \end{equation}\] where the winding \(w\) now has to takes values in \(w \in (\theta_R-\theta_L)/2\pi + \mathbb{Z}\). We write the Hilbert space of states on the interval with the boundary condition \(\mathcal{D}(\theta_L,\theta_R)\) as \(\mathcal{H}_{\theta_L,\theta_R}\), and the state with winding \(w\) in it as \(\ket{w}_{\theta_L,\theta_R}\). We only consider the ground states for the oscillation modes but the story can be applied to the states with these modes without any change.
How does the \(U(1)^2\) symmetries acts on the states? The momentum symmetry translate \(X\), and breaks the boundary condition (3.15). But it still defines a linear map between the Hilbert spaces: \[\begin{align} \tag{3.17} U^p_\lambda : \mathcal{H}_{\theta_L,\theta_R} &\to \mathcal{H}_{\theta_L+\lambda,\theta_R+\lambda}\\ \ket{w}_{\theta_L,\theta_R} &\mapsto \ket{w}_{\theta_L+\lambda,\theta_R+\lambda}. \end{align}\] On the other hand, the winding \(U(1)\) symmetry preserves the Dirichlet boundary condition and the naive extension of the action (3.13) to this case is \[\begin{align} \tag{3.18} \widetilde{U}^w_\lambda : \mathcal{H}_{\theta_L,\theta_R} &\to \mathcal{H}_{\theta_L,\theta_R}\\ \ket{w}_{\theta_L,\theta_R} &\mapsto e^{\mathrm{i}\lambda w}\ket{w}_{\theta_L,\theta_R}. \end{align}\] Here we put the tilde on \(U^w_\lambda\) because there is one problem with this action, i.e. \(\widetilde{U}^w_\lambda = e^{\mathrm{i}(\theta_R-\theta_L)}\widetilde{U}^w_{\lambda+2\pi}\). This projective phase can be easily removed by redefining the unitary as \[\begin{align} \tag{3.19} U^w_\lambda : \mathcal{H}_{\theta_L,\theta_R} &\to \mathcal{H}_{\theta_L,\theta_R}\\ \ket{w}_{\theta_L,\theta_R} &\mapsto e^{\mathrm{i}\lambda (w+(\{\theta_L\}-\{\theta_R\})/2\pi)}\ket{w}_{\theta_L,\theta_R}, \end{align}\] where \(\{\theta\}\) is the number in \([0,2\pi)\) and \(\{\theta\} = \theta \mod 2\pi\). We use this notation to emphasize that the boundary condition \(\mathcal{D}(\theta)\) and \(\mathcal{D}(\theta+2\pi)\) are identified in equations e.g. (3.17). Note that the modification is not \(\{\theta_L-\theta_R\}\), so that the modification is defined locally – it is a sum of independent contributions from the left and right boundaries. However, this redefinition produces a projective phase in the product of \(U^p_{\lambda_p}\) and \(U^w_{\lambda_w}\): \[\begin{equation} \tag{3.20} U^w_{\lambda_w}U^p_{\lambda_p}|_{\theta_L,\theta_R} = e^{\mathrm{i}\lambda_w(-[\lambda_p+\theta_R] + [\lambda_p+\theta_L])/2\pi} U^p_{\lambda_p}U^w_{\lambda_w}|_{\theta_L,\theta_R}, \end{equation}\] where \([\theta] = \theta - \{\theta\} \in 2\pi \mathbb{Z}\). The \(|_{\theta_1,\theta_2}\) denotes the boundary conditions on which the operator just left of the vertical bar acts. For the other operator composed with this one, the domain is chosen so that the composition makes sense. This is the projective phase characterizing the mixed anomaly between the two \(U(1)\) symmetry (or, the anomaly for the (anti-)diagonal of \(U(1)\)’s). This characterization is very different from the conventional one you find in e.g. [17], and the correspondence will be explained in Sec. 3.5.
Advantages of characterizing the anomaly in this way are
- it is clear that this is some kind of generalization of Wigner’s projective phase, and
- applicable to discrete case.
For example, we can take the \(\mathbb{Z}_2\) subgroup of the diagonal of \(U(1)\)’s whose unitary is \(U_{\mathbb{Z}_2}=U^w_\pi U^p_\pi\) and then the projective phase is \[\begin{equation} \tag{3.21} U_{\mathbb{Z}_2}^2|_{\theta_L,\theta_R} = e^{\mathrm{i}(-[\pi+\theta_R]+[\pi+\theta_L])/2}\mathrm{id}|_{\theta_L,\theta_R}. \end{equation}\] The conventional description of the anomaly by anomalous Ward identity does not immediately tell how it restricts to a discrete subgroup.
Next we see that this projective phase can be captured by the third group cohomology.