1. String Theory Essentials
Replace 0D points with 1D strings. Physics becomes geometry of worldsheets.
What is a string?
A relativistic string sweeps a 2D worldsheet $X^\mu(\sigma^0,\sigma^1)$ in $D$-dimensional target space. Tension $T = 1/(2\pi\alpha')$, with $\alpha' = l_s^2$ the Regge slope.
Nambu–Goto
$$ S_{NG} = -T\int dA = -\frac{1}{2\pi\alpha'}\int d^2\sigma \sqrt{-\det h_{ab}} $$$h_{ab}=\partial_a X^\mu \partial_b X_\mu$ induced metric
Polyakov
$$ S_P = -\frac{1}{4\pi\alpha'}\int d^2\sigma \sqrt{-\gamma}\,\gamma^{ab}\partial_a X^\mu \partial_b X_\mu $$Classically equivalent, better for quantization
Quantization → Particles as notes
Mode expansion gives oscillators $\alpha_n^\mu, \tilde\alpha_n^\mu$. For closed bosonic string:
$$ m^2 = \frac{1}{\alpha'}\big(N + \tilde N - 2\big), \quad N=\sum_{n>0} n\,a_n^\dagger a_n,\; \tilde N = \sum_{n>0} n\,\tilde a_n^\dagger \tilde a_n $$with level-matching $N=\tilde N$. $N=1$ gives massless graviton, dilaton, B-field. Tachyon at $N=0$ is removed by supersymmetry.
Why 10 dimensions? Hilbert space
Conformal anomaly cancels at $D=10$ for superstrings. The physical Hilbert space is a Fock space:
$$ \mathcal{H}_{\text{string}} = \bigoplus_{N,\tilde N} \mathcal{H}_L^{(N)} \otimes \mathcal{H}_R^{(\tilde N)} $$This is exactly the tensor product structure $\mathcal{H}_{\text{in}} \otimes \mathcal{H}_{\text{out}}$ we discussed for black holes — left and right movers are independent quantum subsystems.
Vibrating String Modes
Mode $n$ corresponds to oscillator level $N$. Mass grows with $n$.
Key scales
- • String length: $l_s = \sqrt{\alpha'} \sim 10^{-33}\,\text{cm}$
- • String coupling: $g_s = e^{\langle\Phi\rangle}$
- • Planck length: $l_P^{8} = g_s^2 l_s^8$ (in 10D)
2. T-Duality: Big ↔ Small
A circle of radius $R$ is physically identical to radius $\tilde R = \alpha'/R$.
For a closed string on $S^1_R$: momentum is quantized $p=n/R$, winding $w\in\mathbb{Z}$ counts wraps. Mass:
$$ m^2 = \left(\frac{n}{R}\right)^2 + \left(\frac{wR}{\alpha'}\right)^2 + \frac{2}{\alpha'}(N+\tilde N-2) $$Under $R \to \alpha'/R$, exchange $n \leftrightarrow w$. The spectrum is invariant. This is a concrete duality — two geometries give the same Hilbert space.
Connection to Euler: compactification uses $e^{i p X} = e^{i n X/R}$ which is periodic under $X \to X+2\pi R$. T-duality is a Fourier transform on the circle — swapping momentum and winding is swapping $e^{inx/R} \leftrightarrow e^{iwR\tilde x/\alpha'}$.
Euler Circle $e^{i\theta}$
e^{i 1.00} = 0.540 + 0.841 i
Worldsheet torus partition function: $Z(\tau)=\mathrm{Tr}\,e^{2\pi i\tau(L_0-c/24)}e^{-2\pi i\bar\tau(\bar L_0-c/24)}$ uses exactly this phase.
Physical meaning
Strings cannot probe distances below $l_s$. Trying to shrink $R \to 0$ just makes winding modes light — geometry emerges from the spectrum, not the other way around.
3. S-Duality: Strong ↔ Weak
Coupling $g_s \leftrightarrow 1/g_s$. Perturbation theory on one side is non-perturbative on the other.
Type IIB string has complex coupling:
$$ \tau = C_0 + \frac{i}{g_s}, \quad \text{with } SL(2,\mathbb{Z}):\ \tau \to \frac{a\tau+b}{c\tau+d} $$Key generator: $\tau \to -1/\tau$ sends $g_s \to 1/g_s$ (when $C_0=0$). This exchanges fundamental strings with D1-branes, and NS5 with D5.
Why it matters
- • Gives non-perturbative definition of string theory
- • $g_s \to \infty$ limit is not sick — it’s dual to another weakly coupled theory
- • Preserves the S-matrix: information is not lost, just mapped
- • Direct analog of electric-magnetic duality $\; e \leftrightarrow 1/e$
4. M-Theory: The 11D Mother
Witten (1995): five 10D strings are limits of one theory in 11 dimensions.
Type IIA ↔ M
$R_{11} = g_s l_s$. At strong coupling, an extra circle grows. D0-branes are KK momentum:
$$ m_{D0} = \frac{1}{g_s l_s} = \frac{1}{R_{11}} = \frac{|n|}{R_{11}} $$Low energy
11D supergravity with M2 and M5 branes. Their tensions:
$$ T_{M2} = \frac{1}{(2\pi)^2 l_p^3},\quad T_{M5} = \frac{1}{(2\pi)^5 l_p^6} $$Unification facts
- • All 5 strings + 11D SUGRA are corners of moduli space
- • Dualities are exact symmetries of $\mathcal{H}$
- • $g_s$ is not a parameter — it's vev of dilaton
- • M-theory has no strings, only membranes
M-theory circle
Take IIA at coupling $g_s$. The 11th dimension radius:
$$ R_{11} = g_s^{2/3} l_p = g_s l_s $$Weak IIA ($g_s\ll1$): circle tiny, looks 10D. Strong IIA: decompactifies to 11D.
5. Connection to Our Previous Conversation
Hilbert space, black holes, Hawking, Euler, and time reversal — now with strings.
Hilbert Space $\mathcal{H}$
String Fock space $\mathcal{H}_L\otimes\mathcal{H}_R$ is the concrete example of $\mathcal{H}_{\text{in}}\otimes\mathcal{H}_{\text{out}}$ for black holes. Left/right movers are entangled across the horizon in the eternal black hole TFD state.
Black Holes: Strominger–Vafa
D1-D5-P system on $K3\times S^1$. Microstate count from CFT gives:
$$ S_{\text{micro}} = 2\pi\sqrt{Q_1Q_5n} = \frac{A}{4G_5} $$Euler characteristic $\chi(K3)=24$ fixes central charge $c=6Q_1Q_5$.
Hawking Radiation
String theory gives a unitary S-matrix. Dualities map the evaporation process to a weakly coupled D-brane decay where information is manifestly preserved — no paradox, just a change of basis in $\mathcal{H}$.
Euler's Formula
Modular invariance of torus: $Z(\tau+1)=Z(\tau)$ because
$$ Z(\tau) = \mathrm{Tr}\, e^{2\pi i\tau(L_0-c/24)} $$uses $e^{2\pi i}=1$. The $SL(2,\mathbb{Z})$ generated by $\tau\to\tau+1$ and $\tau\to-1/\tau$ is exactly S- and T-duality.
Time Reversal
Worldsheet parity $\Omega: \sigma\to -\sigma$ swaps left/right: $\alpha_n \leftrightarrow \tilde\alpha_n$. T-duality does $(X_L,X_R)\to(X_L,-X_R)$, exchanging $n\leftrightarrow w$ — like reversing winding direction. CPT is exact in the worldsheet CFT.
Zeno & Minimum Length
T-duality implies no $R\to0$. The string worldsheet smooths UV divergences — propagator $\sim e^{-\alpha' p^2}$ suppresses $p>1/l_s$. Achilles catches the tortoise because spacetime stops making sense below $l_s$.
6. GNU Octave Playground
Three runnable scripts to see duality in action.
a) T-Duality mass spectrum
% T-duality: R <-> alpha'/R
alpha_prime = 1;
R = linspace(0.2, 3, 400);
n = 1; w = 1;
m2_mom = (n./R).^2;
m2_wind = (w*R/alpha_prime).^2;
m2_total = m2_mom + m2_wind;
figure(1); clf; hold on;
plot(R, m2_total, 'linewidth', 2);
plot(R, m2_mom, '--', 'linewidth', 1.5);
plot(R, m2_wind, '--', 'linewidth', 1.5);
Rdual = alpha_prime ./ R;
plot(Rdual, m2_total, ':', 'linewidth', 1);
xlabel('Radius R / sqrt(alpha'')'); ylabel('m^2 \times alpha''');
title('T-Duality: momentum <-> winding');
legend('total', 'n^2/R^2', 'w^2 R^2', 'dual', 'location', 'north');
grid on; axis tight;b) String excitation levels
% Closed bosonic string spectrum (alpha' = 1)
alpha_prime = 1;
Nmax = 6;
fprintf('N\tNt\tm^2\n');
levels = [];
for N = 0:Nmax
for Nt = 0:Nmax
if N == Nt % level matching
m2 = (N + Nt - 2)/alpha_prime;
fprintf('%d\t%d\t%g\n', N, Nt, m2);
levels(end+1,:) = [N, m2];
endif
endfor
endfor
figure(2); clf;
stem(levels(:,1), levels(:,2), 'filled', 'linewidth', 2);
xlabel('Oscillator level N = Nt'); ylabel('m^2');
title('Closed bosonic string (tachyon at N=0)');
grid on;c) Euler circle & modular transformation
% Euler's formula and SL(2,Z)
theta = linspace(0, 2*pi, 500);
z = exp(1i*theta);
figure(3); clf; subplot(1,2,1);
plot(real(z), imag(z), 'linewidth', 2); axis equal; grid on;
hold on; plot([0 cos(1)], [0 sin(1)], 'r', 'linewidth', 2);
title('e^{i\theta} on unit circle'); xlabel('Re'); ylabel('Im');
% Modular orbit tau -> -1/tau, tau -> tau+1
tau = 0.5 + 0.8i;
orbit = tau;
for k = 1:6
orbit(end+1) = -1/orbit(end);
orbit(end+1) = orbit(end) + 1;
endfor
subplot(1,2,2);
plot(real(orbit), imag(orbit), 'o-', 'linewidth', 1.5, 'markersize', 8);
xlabel('Re \tau'); ylabel('Im \tau'); grid on;
title('SL(2,Z) orbit (S and T generators)');
axis([ -1 1.5 0 1.5 ]);