Physikalisches Institut Exercise 5

(1)

Physikalisches Institut Exercise 5

Universit¨ at Bonn 20 May 2015

Theoretische Physik SS 2015

Exercises on Advanced Topics in String Theory

Priv.-Doz. Dr. Stefan F¨ orste

http://www.th.physik.uni-bonn.de/people/forste/exercises/strings15

–Home Exercises– Due to: 03.06.2015

H 5.1 Two point function for free fermion (15 points) The action for a free Majorana fermion reads

S = 1 4πg

Z

dx ⁰ dx ¹ p

|h|(−i)Ψγ ^α ∂ _α Ψ, (1) where g is a constant, Ψ = Ψ ^† γ ⁰ , h _αβ = diag(1, −1), and the gamma matrices are given by

γ ⁰ = 0 1

1 0

, γ ⁰ =

0 1

−1 0

(2) (a) What is the Majorana condition on the components ψ, ψ of Ψ? (2 point s ) (b) Perform a Wick rotation x ₁ → ix ₁ and define z := x ⁰ + ix ¹ to rewrite the action as

S = 1 4πg

Z

dzdz ψ(z, z)∂ψ(z, z) + ψ(z, z)∂ψ(z, z)

. (3)

(3 point s ) (c) Calculate the equation of motion for ψ and ψ. What do they imply? (1 point ) (d) By imposing invariance of the action (3) under conformal transformations, calculate

the conformal weighrs (h, h) of ψ and ψ. (2 point s )

(e) Next we want to calculate the correlator hΨ _i (z, z), Ψ _j (z ⁰ , z ⁰ )i where i, j = 1, 2 label the comoponents of Ψ. To do so, express the kinetic terms of the components in (3) as a matrix A _ij and write down the differential equation for the Green‘s function.

(2 point s )

(f) We claim that the Green‘s function G _ij (z, z ⁰ ) for the equation in (e) is given by G = 2g

∂ _z−z ¹

0

0 0 ∂ _z−.z ¹

0

(4) Prove this by using the techniques you already learned for the bosonic case. (5 point s)

1

Physikalisches Institut Exercise 5

Physikalisches Institut Exercise 5

Universit¨ at Bonn 20 May 2015

Theoretische Physik SS 2015

Exercises on Advanced Topics in String Theory

Priv.-Doz. Dr. Stefan F¨ orste

http://www.th.physik.uni-bonn.de/people/forste/exercises/strings15

–Home Exercises– Due to: 03.06.2015

H 5.1 Two point function for free fermion (15 points) The action for a free Majorana fermion reads

S = 1 4πg

Z

dx 0 dx 1 p

|h|(−i)Ψγ α ∂ α Ψ, (1) where g is a constant, Ψ = Ψ † γ 0 , h αβ = diag(1, −1), and the gamma matrices are given by

γ 0 = 0 1

1 0

, γ 0 =

0 1

−1 0

(2) (a) What is the Majorana condition on the components ψ, ψ of Ψ? (2 point s ) (b) Perform a Wick rotation x 1 → ix 1 and define z := x 0 + ix 1 to rewrite the action as

S = 1 4πg

Z

dzdz ψ(z, z)∂ψ(z, z) + ψ(z, z)∂ψ(z, z)

. (3)

(3 point s ) (c) Calculate the equation of motion for ψ and ψ. What do they imply? (1 point ) (d) By imposing invariance of the action (3) under conformal transformations, calculate

the conformal weighrs (h, h) of ψ and ψ. (2 point s )

(e) Next we want to calculate the correlator hΨ i (z, z), Ψ j (z 0 , z 0 )i where i, j = 1, 2 label the comoponents of Ψ. To do so, express the kinetic terms of the components in (3) as a matrix A ij and write down the differential equation for the Green‘s function.

(2 point s )

(f) We claim that the Green‘s function G ij (z, z 0 ) for the equation in (e) is given by G = 2g

∂ z−z 1

0 0 ∂ z−.z 1

(4) Prove this by using the techniques you already learned for the bosonic case. (5 point s)

1

dx ⁰ dx ¹ p

|h|(−i)Ψγ ^α ∂ _α Ψ, (1) where g is a constant, Ψ = Ψ ^† γ ⁰ , h _αβ = diag(1, −1), and the gamma matrices are given by

γ ⁰ = 0 1

, γ ⁰ =

(2) (a) What is the Majorana condition on the components ψ, ψ of Ψ? (2 point s ) (b) Perform a Wick rotation x ₁ → ix ₁ and define z := x ⁰ + ix ¹ to rewrite the action as

(e) Next we want to calculate the correlator hΨ _i (z, z), Ψ _j (z ⁰ , z ⁰ )i where i, j = 1, 2 label the comoponents of Ψ. To do so, express the kinetic terms of the components in (3) as a matrix A _ij and write down the differential equation for the Green‘s function.

(f) We claim that the Green‘s function G _ij (z, z ⁰ ) for the equation in (e) is given by G = 2g

∂ _z−z ¹

0 0 ∂ _z−.z ¹