PHY2406H

This is due in two weeks – or whenever you get to it; but please do hand it in before the date of the presentations. That way everyone’s grades will be filed on-time.

N.B.: querying the prof about homework questions will not reduce your grade.

Question 1: T-duality

Z18.4 and Z18.5

Question 2: S-duality

(a) Consider the gravitational fields of Dp-branes (p<7) of Type IIB superstring theory. Show that under S-duality D3-branes remain untouched while D1-branes switch roles with fundamental strings and D5-branes switch roles with NS5-branes. Note: you will need to keep track of what happens to the spacetime metric, the dilaton, and the fluxes. (Hint: you may ignore D7-branes and D9-branes.)

(b) Starting from the BPS M2-brane solution of D=11 M theory, prove explicitly that dimensional reduction of the D=11 M2-brane along an M-circle parallel to a worldsheet direction gives the D=10 fundamental string solution of Type IIA.

(c) Starting from the BPS M2-brane solution of D=11 M theory, prove explicitly that dimensional reduction of the D=11 M2-brane along an M-circle transverse to its worldsheet gives the D=10 Type IIA D2-brane solution. (Hint: you may want to consult this section of my TASI notes to see how to “smear” brane geometries in order to force an isometry direction prior to Kaluza-Klein reduction on a circle.)

Question 3: gravity/gauge duality for p=0

Consider the classical D=10 supergravity solution for the geometry of N>>1 D0-branes, in the decoupling limit (where the 1 in the harmonic function is lost).

(a) First let us analyze the closed string picture of this D0-brane system in the decoupling limit. Using Maple (or your other fave computer algebra package), show explicitly a calculation of the Ricci scalar curvature of the D0-brane geometry. In what regime of distance (or, using the string IR/UV relation, in what regime of energy) is this geometry becoming strongly curved in string units, invalidating neglect of string tension corrections to the classical geometry?

(b) Where does the dilaton become strong in this D0-brane geometry, invalidating neglect of string loop corrections to the classical geometry? What description of the physics will take over in the regime of strong dilaton? Why?

(c) Using the above info from (a) and (b), prove that for N>>1 there is a wide range of energy/distance in which the D=10 supergravity solution is a good approximation to the physics of N>>1 D0-branes.

(d) Now we switch to consider the open string picture. Using dimensional analysis, write down the dimensionless ‘t Hooft gauge coupling of the d=0+1 supersymmetric nonabelian quantum mechanics living on the D0-branes. Where does this ‘t Hooft coupling become strong? How does the power-law running behaviour you see fit in with what you learned in part (a) and your QFT courses?

(e) Comment on the physical significance of how different pieces of this picture fit together in a consistent way.

Question 4: stringy/cosmological

Show – at whatever level of sophistication you are comfortable with (!) – whether or not four-dimensional de Sitter spacetime (times some other six-manifold) is a solution of superstring theory.

(Hint: pre-KKLT, this story was a no-go theorem…)