String Theory I Mock Exam

Transcription

1 String Theory I Mock Exam Ludwig Maximilians Universität München Prof. Dr. Dieter Lüst 15 th December :00 18:00 Name: Student ID no.: address: Please write down your name and student ID number on every page. The examination time is 120 min. Additional sheets can be obtained from the supervisor. This is a closed book exam (i.e. no books, notes, electronic devices, etc. are allowed). Please have your student ID and your personal ID available. The exercise scores are indicated. You can maximally reach 65 points. Question: Total Points: Score:

2 Name: Student ID: Page 2/15 1. Open string trajectory Consider the following classical trajectory of an open string, X 0 = B τ, X 1 = B cos(τ) cos(σ), X 2 = B sin(τ) cos(σ), X i = 0, i > 2, (1a) where 0 σ π. Assume the conformal gauge g αβ = η αβ. (a) Show that this configuration describes a solution to the equations of motion for the field X µ (τ, σ) corresponding to an open string with Neumann boundary conditions.

3 Name: Student ID: Page 3/15 (b) Consider a point on this string at fixed σ. Calculate the speed of this point via (dx ) 1 2 ( ) dx dx 0 dx 0 What is the value of the speed of the endpoints of this string?

4 Name: Student ID: Page 4/15 (c) Consider the conserved charges P µ = T J µν = T π 0 π 0 dσ τ X µ, dσ (X µ τ X ν X ν τ X µ ), where T = (2πα ) 1. Compute the energy E = P 0 and the angular momentum J = J 12 of the string. Show that E 2 J = 1 α.

5 Name: Student ID: Page 5/15 (d) The worldsheet energy momentum tensor T αβ is in general given by T αβ = 1 2 αx µ β X µ 1 4 g αβ g γδ γ X µ δ X µ. Show that the above solution (1) satisfies the constraint equation T αβ = 0 Hint: Use the conformal gauge.

6 Name: Student ID: Page 6/15 2. Closed string states and Virasoro operators Consider the closed string in light-cone gauge. (a) Compute the mass squared of the following on-shell closed string states: φ 1 = α 1 i α j 1 0; p, φ 2 = α 1 i α j 1 αk 2 0; p. What can you say about the following closed string state? φ 3 = α 1 i α 2 k 0; p.

7 Name: Student ID: Page 7/15 (b) The Virasoro operators are given by L 0 = 1 2 α 0 α 0 + L m = 1 2 n= α n α n, n=1 α m n α n, m 0, with α µ 0 pµ and the dot product means, as usual, contraction with η µν. They satisfy the Virasoro algebra [L m, L n ] = (m n) L m+n + D 12 (m3 m) δ m+n,0. Use the Virasoro algebra to prove by mathematical induction that if a state is annihilated by L 1 and L 2 it is annihilated by all L n with n 1.

8 Name: Student ID: Page 8/15 (c) Consider the Virasoro operators L 0, L 1 and L 1. Write out the three relevant commutators. Do these operators form a subalgebra of the Virasoro algebra? Calculate the result of acting with each of these three operators on the zero-momentum vacuum state 0; 0.

9 Name: Student ID: Page 9/15 (d) Consider the open string state where χ satisfies ψ = L 1 χ, (L 0 a + 1) χ = 0, L m χ = 0, m > 0. Determine the value of a by demanding that ψ is physical. Compute the norm of ψ.

10 Name: Student ID: Page 10/15 3. Circle compactification Consider the closed string on a circle with radius R, i.e. on R 1,24 S 1. Denote the vacuum state by p; n, m, where p = (p +, p, p i ) is the momentum in the 25 dimensional Minkowski space and n, m Z the Kaluza Klein and winding numbers, respectively. The mass formula is then given by while level matching gives the constraint M 2 = p 2 + w α ( N + N 2 ), For the momentum and winding modes, we have N N = n m. (2) p = n R and w = m R α. (a) Explain why the momentum states p are quantized. [2 Punkte]

11 Name: Student ID: Page 11/15 (b) Given the Virasoro operators L 0 = α 4 pi p i α 0 25 α N, L 0 = α 4 pi p i + 1 2ᾱ0 25 ᾱ N, with p = 1 ) (ᾱ025 + α 025 2α and w = 1 2α (ᾱ025 α 025 ), show that the level matching condition gives us equation (2). Note that for the compactified closed string α 025 ᾱ 025. [3 Punkte]

12 Name: Student ID: Page 12/15 (c) Explain which of the following states survive the level matching: (i) α 2 i p; 0, 0 (ii) ᾱ 1 i α25 1 p; 1, 1 (iii) ᾱ 1 i p; 1, 1 (iv) α 1 i p; 1, 1 (v) p; 2, 0

13 Name: Student ID: Page 13/15 (d) Interpret the surviving state(s) of exercise 3 c) as representation(s) of the Lorentz group in 25 dimensions characterized by spin. Determine the mass(es) that this/these state(s) have?

14 Name: Student ID: Page 14/15 (e) Which are the massless states 3 d) in 25D Minkowski space? Are there states that are only massless at a special radius of the circle?

15 Name: Student ID: Page 15/15 (f) Explain which of the states in 3 d) have a positive-semidefinite mass m 2 (R) and which states can become tachyonic for certain values of R.