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68. When a single spatial dimension is compacti ed on a circle of radius R, there are extra terms in the expression used to determine the mass of the state. What factors are these Is the mass increased or decreased 69. What is the effect of compacti cation on the center of mass momentum 70. Consider the compacti cation of a single spatial dimension to a circle of radius R and consider the limiting behavior as R 0. Why do the winding states go to the continuum limit 71. How does T-duality relate winding and momentum states 72. How does T-duality relate distance scales in different theories 73. What string theories are related by T-duality 74. A certain string theory has orientable strings. What does this mean 75. What is the mass of a tachyon state in bosonic string theory 76. How is Type I string theory different from all the other superstring theories 77. What symmetry groups are associated with heterotic string theory 78. How does the Kasner metric describe the behavior of space-time as the universe evoles 79. What is the effect of the dilaton eld on the Kasner metric 80. What is the self-dual point 81. In the Horava-Witten model, how are the uncompacti ed dimensions laid out 82. What condition on the beta function of the dilaton eld must be satis ed for scale invariance 83. What property of space-time is implied by conformal invariance as related to the dilaton eld 84. How is the vanishing of the beta function related to general relativity 85. Einstein s relativity allows for the description of extreme black holes that carry electric charge. How is this extended in string theory 86. Let K be the Kaluza-Klein excitation and n the winding number. How is a state ( K , n) in a theory with a compacti ed dimension of radius R transformed under T-duality 87. How are the number operators transformed under T-duality 88. A state has mass m. How is the mass changed under T-duality 89. How are the equations of motion related for the compacti ed dimension under T-duality between a theory and its dual
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90. Consider an open bosonic string under a T-duality transformation. What happens to the momentum of the dual string along the compacti ed dimension 91. The string momenta along a certain spatial direction are found to be fractional. How are the elds described in the dual theory 92. What type of boundary condition is X
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93. In type II A superstring theory, what dimensions are allowed for Dp-branes 94. If the value of the elds X are speci ed on the boundary, what type of problem is being posed In Probs. 95 96 let D = 10 and suppose that there is a pure electric background eld F0 i = Ei. 95. What is the form of the Born-Infeld action 96. What is the maximum value of the electric eld 97. How is the dilaton eld related to string coupling gs The nal problems will be very challenging for many readers. For problems 98, 99, and 100, suppose that you have closed strings with periodic boundary conditions for = 0,1,..., 24 but X 25 ( , ) = X 25 ( + s , ). Use the light-cone gauge (based on Polchinski 1.9). 98. How does the modal expansion change for X 25 as opposed to the usual modal expansion for closed bosonic strings 1 1 p+ 99. Using = ( 2 1)2 nd the hamiltonian X and ( n ) = 24 8 n =1 s 1 d 2 i i + X i X i . 2 4 p 0 100. What is the mass spectrum Note that the level-matching condition is still satis ed. from H =
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