Worster Defects and Dualities. Phil Saad With Eric Mintun and Joe Polchinski

Transcription

1 Worster 2014 Defects and Dualities Phil Saad With Eric Mintun and Joe Polchinski

2 Table of Contents Overview QFT Renormalization Dualities Application of Dualities

3 Quick Overview-What and Why

4 Quick Overview-What and Why We began by studying a class of systems with Electromagnetism interacting with a 1-dimensional defect

5 Quick Overview-What and Why What are we looking at?- Dualities and Renormalization Group behavior.

6 Quick Overview-What and Why What are we looking at?- Dualities and Renormalization Group behavior. A Duality is about finding two descriptions of the same system. These can make hard problems much easier.

7 Quick Overview-What and Why Renormalization Group behavior tells you about how the system behaves at different scales (energy/length).

8 Quick Overview-What and Why Why do we care about these defect systems?

9 Quick Overview-What and Why Why do we care about these defect systems? Simple, solvable models will hopefully give a deeper understanding of the principles behind dualities.

10 Quick Overview-What and Why Why do we care about these defect systems? Simple, solvable models will hopefully give a deeper understanding of the principles behind dualities. The simple dualities can hopefully be used to investigate more complicated ones

11 Table of Contents Overview QFT Renormalization Dualities Application of Dualities

12 Some Jargon Quantum Field Theory (QFT)

13 Some Jargon Quantum Field Theory (QFT) Particle number not conserved

14 Some Jargon Quantum Field Theory (QFT) Particle number not conserved Coupling Constant

15 Some Jargon Quantum Field Theory (QFT) Particle number not conserved Coupling Constant Strength of interaction (electric charge)

16 Some Jargon Quantum Field Theory (QFT) Particle number not conserved Coupling Constant Strength of interaction (electric charge) Can only calculate at small coupling constant

17 Table of Contents Overview QFT Renormalization Dualities Application of Dualities

18 More Jargon Renormalization Group

19 More Jargon Consider measuring the charge of an electron. Measure electric field near the electron

20 More Jargon Consider measuring the charge of an electron. Measure electric field near the electron Heisenberg Uncertainty Principle- Electron Positron pairs appear in the space around the electron

21 More Jargon Consider measuring the charge of an electron. Measure electric field near the electron Heisenberg Uncertainty Principle- Electron Positron pairs appear in the space around the electron These pairs block some of the electric field

22 More Jargon Electron-positron pairs screen the charge of the electron. Far from the electron, it looks like it has a much smaller charge.

23 More Jargon The measured charge depends on how close you get/how high energy you are

24 More Jargon Coupling Strength Energy

25 Renormalization and Us How does this apply to the problems we consider? We have photons (electromagnetism) interacting with particles on the defect, the interactions have a strength g

26 Renormalization and Us Instead of quantum effects, we have the defect causing scale dependence

27 Renormalization and Us Instead of quantum effects, we have the defect causing scale dependence When the defect lives in 3 dimensions, we found what we expected

28 Renormalization and Us When the defect lives in 2 dimensions, we expected a fixed point A fixed point is a point where the physics is drawn to constant coupling

29 Renormalization and Us Coupling Strength Energy

30 Renormalization and Us The usual methods of renormalization didn t see this

31 Renormalization and Us The usual methods of renormalization didn t see this Due to some simplifying properties of the defect models, we found the fixed point using more complicated methods

32 Renormalization and Us Coupling Strength Energy

33 Table of Contents Overview QFT Renormalization Dualities Application of Dualities

34 Jargon Part III-Dualities When doing quantum mechanics, we usually quantize a classical system

35 Jargon Part III-Dualities When doing quantum mechanics, we usually quantize a classical system Can you quantize different classical systems and get the same quantum system?

36 Jargon Part III-Dualities When doing quantum mechanics, we usually quantize a classical system Can you quantize different classical systems and get the same quantum system? Yes, and the classical systems are called Dual

37 Jargon Part III-Dualities These dualities can be very useful Imagine that system is physically interesting, but hard to solve quantum mechanically

38 Jargon Part III-Dualities These dualities can be very useful Imagine that system is physically interesting, but hard to solve quantum mechanically If there is an easy dual system, you can just translate the results

39 Jargon Part III-Dualities S-Duality - systems with a large coupling constant are dual to systems with a small coupling constant

40 Jargon Part III-Dualities S-Duality - systems with a large coupling constant are dual to systems with a small coupling constant Solve weakly coupling system, translate to solve strongly coupling system

41 Jargon Part III-Dualities What sort of dualities did we find?

42 Jargon Part III-Dualities The most interesting one is connected to a well known string theory duality.

43 Jargon Part III-Dualities The most interesting one is connected to a well known string theory duality. In string theory, you can have a string living on a sort of cylinder, with radius R R

44 Jargon Part III-Dualities This string is dual to a string living on a cylinder of radius 1/R R 1/R

45 Jargon Part III-Dualities The equations describing the physics on the defect are very similar to the physics describing a string

46 Jargon Part III-Dualities The particular types of particles we consider make it similar to a string living on a cylinder, and so we can use the string duality Also changes the way it interacts with the electromagnetism

47 Jargon Part III-Dualities Duality has its roots in electromagnetism. Consider Maxwell s equations

48 Jargon Part III-Dualities Duality has its roots in electromagnetism. Consider Maxwell s equations

49 Jargon Part III-Dualities Adding magnetic charges makes it more symmetric

50 Jargon Part III-Dualities Adding magnetic charges makes it more symmetric

51 Jargon Part III-Dualities Switching electric and magnetic charges, as well as electric and magnetic fields, leaves equations unchanged

52 Jargon Part III-Dualities Switching electric and magnetic charges, as well as electric and magnetic fields, leaves equations unchanged

53 Jargon Part III-Dualities Green=Magnetic Orange=Electric

54 Jargon Part III-Dualities Quantum mechanically, electric and magnetic charges are inversely proportional

55 Jargon Part III-Dualities Quantum mechanically, electric and magnetic charges are inversely proportional Electric magnetic duality inverts all of the charges- this is S-duality!

56 Table of Contents Overview QFT Renormalization Dualities Application of Dualities

57 Applications of Dualities The original motivation for this project was to investigate S-duality in string theory

58 Applications of Dualities The original motivation for this project was to investigate S-duality in string theory Before I can discuss our results here, I have to talk a bit about string theory

59 Applications of Dualities The original motivation for this project was to investigate S-duality in string theory Before I can discuss our results here, I have to talk a bit about string theory IIB string theory-one of five string theories

60 Applications of Dualities Strings- Open or Closed. Indivisible, infinitely thin

61 Applications of Dualities The different ways a string can vibrate corresponds to different types of particles

62 Applications of Dualities String theory has one coupling constant, g

63 Applications of Dualities String theory has one coupling constant, g Like QFT, it s hard to calculate at large g. Much of large g string theory is mysterious, may not even involve strings!

64 Applications of Dualities String theory contains Branes (membranes), extended objects

65 Applications of Dualities String theory contains Branes (membranes), extended objects IIB string theory only has odd dimensional branes (1,3,5,etc)

66 Applications of Dualities String theory contains Branes (membranes), extended objects IIB string theory only has odd dimensional branes (1,3,5,etc) Open strings must end on branes

67 Applications of Dualities

68 Applications of Dualities Ends of strings look like electric charges Unless strings are stretched, don t see net charge

69 Applications of Dualities Ends of strings look like electric charges Unless strings are stretched, don t see net charge

70 Applications of Dualities One dimensional branes are a lot like strings

71 Applications of Dualities One dimensional branes are a lot like strings Their endpoints look like magnetic charges

72 Applications of Dualities At small g, strings are wiggly and branes are stiff At large g, the opposite

73 Applications of Dualities Switching strings and one branes looks like electromagnetic and S- duality- switches charges and coupling

74 Applications of Dualities This tells us that large g IIB string theory is not mysterious- it s the same as small g!

75 Applications of Dualities Can we see this more explicitly in the equations of string theory?

76 Applications of Dualities For the most part, the equations of string theory aren t known IIB string theory is an exception, but there s a problem

77 Applications of Dualities QFT approximation to string theory At low energies/long distances, strings look like particles-can use QFT

78 Applications of Dualities QFT approximation to string theory At low energies/long distances, strings look like particles-can use QFT Much of what we know about string theory is from this approximation

79 Applications of Dualities QFT for closed strings, no open strings or branes, S-duality obvious

80 Applications of Dualities QFT for closed strings, no open strings or branes, S-duality obvious Hard to see in QFTs for systems of branes

81 Applications of Dualities Consider parallel three-dimensional branes

82 Applications of Dualities Strings and one branes stretch between branes and end on branes

83 Applications of Dualities QFT that describes this is well known

84 Applications of Dualities QFT that describes this is well known Predicted that it has S-duality in 70s, even before string theory

85 Applications of Dualities QFT that describes this is well known Predicted that it has S-duality in 70s, even before string theory Can the exact equations of IIB string theory shed some light on this?

86 Applications of Dualities QFT that describes this is well known Predicted that it has S-duality in 70s, even before string theory Can the exact equations of IIB string theory shed some light on this? These equations are a limit of this same exact QFT!

87 Applications of Dualities Consider three branes intersecting with a one dimensional intersection

88 Applications of Dualities Strings with one end on each brane, or both ends on a single brane

89 Applications of Dualities Eric Mintun and Joe Polchinski found the QFT that describes this The defect models are simpler versions of this, with the defect corresponding to the intersection

90 Applications of Dualities Eric Mintun and Joe Polchinski found the QFT that describes this The defect models are simpler versions of this, with the defect corresponding to the intersection Can we use what we learned from the defect models to show S-duality for this system?

91 Applications of Dualities We have gotten the main part of the process down, and seen the g to 1/g flip, along with the electric and magnetic charge flip

92 Applications of Dualities We have gotten the main part of the process down, and seen the g to 1/g flip, along with the electric and magnetic charge flip There are a couple subtleties left to deal with

93 Applications of Dualities We have gotten the main part of the process down, and seen the g to 1/g flip, along with the electric and magnetic charge flip There are a couple subtleties left to deal with I ll explain the process a bit because it is really neat

94 Applications of Dualities The physics on the intersection looks like the physics of a string living on a strange space Strange Space

95 Applications of Dualities This space has centers spaced equally on a line Strange Space

96 Applications of Dualities This space has a circular dimension that the string can wrap around Strange Space Strange Space

97 Applications of Dualities We can use the duality for strings on a circle, but it is more complicated Strange Space Strange Space

98 Applications of Dualities The new space has the centers periodically arranged around the circles Dual Strange Space

99 Applications of Dualities The spacing along the circle, X, is different from the spacing on the line, Y. The ratio X/Y is g^2 X Y Dual Strange Space

100 Applications of Dualities Another duality switches the spacing ratio, so X/Y=1/g^2. It also switches the position of the electric and magnetic charges X Y Dual Strange Space

101 Applications of Dualities We use the circle duality to bring us back to the original space, but with g switched to 1/g, and charges flipped Strange Space

102 What Next? The next step for us is to get all the details down with this process

103 What Next? The next step for us is to get all the details down with this process There were a couple of very interesting questions raised, and the solutions to these questions should be pretty neat

104 What Next? The next step for us is to get all the details down with this process There were a couple of very interesting questions raised, and the solutions to these questions should be pretty neat Understanding this case might help in showing S-Duality for the parallel brane case, which is very important

105 Thanks! I d like to thank- The Worster family for funding this research Eric Mintun and Joe Polchinski, my mentor and advisor My audience for being patient though these (probably over) twenty minutes of ramblings about strings and defects