Introduction to String Theory and D–Branes - School of Natural ...

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9.7 Further Non–Abelian Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 9.8 Further Curvature Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 10 Multiple D–Branes 122 10.1 Dp and Dp ′ From Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 10.2 The BPS Bound for the Dp–Dp ′ System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 10.3 Bound States of Fundamental Strings and D–Strings . . . . . . . . . . . . . . . . . . . . . . . 125 10.4 The Three–String Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 10.5 Aspects of D–Brane Bound States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 10.5.1 0–0 bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 10.5.2 0–2 bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 10.5.3 0–4 bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 10.5.4 0–6 bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 10.5.5 0–8 bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 11 Strings at Strong Coupling 129 11.1 Type IIB/Type IIB Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 11.1.1 D1–Brane Collective Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 11.1.2 S–Duality and SL(2, Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 11.2 SO(32) Type I/Heterotic Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 11.2.1 D1–Brane Collective Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 11.3 Dual Branes from 10D String–String Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 11.3.1 The Heterotic NS–Fivebrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 11.3.2 The Type IIA and Type IIB NS5–brane . . . . . . . . . . . . . . . . . . . . . . . . . . 133 11.4 Type IIA/M–Theory Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 11.4.1 A Closer Look at D0–branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 11.4.2 Eleven Dimensional Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 11.5 E8 × E8 Heterotic String/M–Theory Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 11.6 M2–branes and M5–branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 11.6.1 From D–Branes and NS5–branes to M–Branes and Back . . . . . . . . . . . . . . . . 139 11.7 U–Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 11.7.1 Type II Strings on T 5 and E 6(6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 12 Concluding Remarks 141 References 141 1 Introductory Remarks These lectures are intended to serve as an introduction to some of the basic techniques, language and concepts in string theory and D–branes. by no stretch of the imagination will you learn everything about these topics from these notes. You should be able to use them to get a good sense for how things work, and this will allow you to understand the issues which are discussed in the other lecture courses in this school, which will assume a knowledge of basic string theory, and some previous experience with some of the “post–Second– Revolution” ideas. Once you get to a certain stage, the notes will need to be supplemented with more careful treatments of the technology, and you will be curious to know more about some of the topics and further applications that I did not have time or space to cover here. I recommend the excellent text of Polchinski[1], and also that of Green, Schwarz and Witten[2], which is still a brilliant text for many aspects of the subject. You might want to consult the recent book listed in ref.[4], in order to learn more about D–branes and 4
further aspects of their applications. There are also many other sources, on the web (e.g., www.arXiv.org) and elsewhere, of detailed reviews of various specialized topics, even whole string theory books[3]. These notes grew out of a series of notes, including those of ref.[5, 6, 7, 8], and I have also borrowed from parts of the book “D–Branes”[4]. I’ve included the basic material on stings and T–duality, and tried to include a little more of the conformal field theory language than I have used before in notes of this sort. It allows the language of the study of toroidal compactification to be developed quite efficiently and (I hope) clearly. The payoff is that is becomes easier to see how the heterotic strings work, and how they fit in with the other strings quite nicely. Furthermore, it allows one to explain a little about non–critical strings, a topic of considerable interest. It also sets up the important language used in string compactifications, and prepares one for concepts such as U–duality later on. There’s a touch more geometry covered than I usually do, such as some basic tools for working on the tangent space any curved spacetime manifold. This comes into its own in later sections when I describe the anomalies which arise in the low energy effective actions of the various superstrings. The Green–Schwarz anomaly cancellation mechanism (the foundation of the First Revolution) is studied in gory detail. This is done for fun in section 6, but there is a bonus in section 8 when we look a bit more closely at gauge bundles and characteristic classes, allowing us to revisit the language of anomalies, now on the world–volumes of D–branes. The Green–Schwarz mechanism is then seen to be a basic paradigm for how extended objects with interesting massless fields on their world–volumes actually fit into the various field theories. I suspect that there will be some resonance with the lectures of Jeff Harvey in this area. There is the obligatory section on strings at strong coupling. I’ve put in the standard classic D–brane probes of the duality, and outlined other aspects of how the D–branes get involved in the whole story, including how you can deduce the existence of other important branes, the NS5–branes and the M–branes. This is important basic lore, but I don’t go too far into this treatment in order to save space. There are plenty of excellent reviews of this material in the literature I’ve already mentioned, and also in (for example) ref.[175]. I’ve included a table of contents for you which might help make this a useful reference. Enjoy, —cvj. 2 Relativistic Strings This section is devoted to an introduction to bosonic strings and their quantization. There is no attempt made at performing a rigourous or exhaustive derivation of some of the various formulae we will encounter, since that would take us well away from the main goal. That goal is to understand some of how string theory incorporates some of the familiar spacetime physics that we know from low energy field theory, and then rapidly proceed to the point where many of the remarkable properties which make strings so different from field theory are manifest. That will be a good foundation for appreciating just what D–branes really are. The careful reader who needs to know more of the details behind some of what we will introduce is invited to consult texts devoted to the study of string theory. 2.1 Classical Bosonic Strings Turning to strings, we parameterize the “world–sheet” which the string sweeps out with coordinates (σ 1 , σ 2 ) = (τ, σ). The latter is a spatial coordinate, and for now, we take the string to be an open one, with 0 ≤ σ ≤ π running from one end to the other. The string’s evolution in spacetime is described by the functions X µ (τ, σ), µ = 0, · · ·,D − 1, giving the shape of the string’s world–sheet in target spacetime (see figure 1). 2.1.1 Two Actions A natural object which arises from embedding the string into spacetime using the functions (or “map”) X µ (τ, σ) is a two dimensional “induced” metric on the world–sheet: hab = ∂aX µ ∂bX ν ηµν . (1) 5
Page 1 and 2: Introduction to String Theory and D
Page 3: 6 D-Brane Tension and Boundary Stat
Page 7 and 8: µ Pσ dσ Figure 2: The infinitesi
Page 9 and 10: 2.3 Further Aspects of the Two-Dime
Page 11 and 12: The two dimensional metric γab is
Page 13 and 14: So we have the equal time Poisson b
Page 15 and 16: where our infinite constant is set
Page 17 and 18: the condition is for the spurious s
Page 19 and 20: One can even ask that both strings
Page 21 and 22: τ ζµν σ α µ ~ ν −1 α−1
Page 23 and 24: 2.11 Unoriented Strings 2.11.1 Unor
Page 25 and 26: (a) (b) Figure 13: (a) Constructing
Page 27 and 28: β B µν = α ′ − 1 β Φ =
Page 29 and 30: x e2(x) e (x) 1 Figure 16: The loca
Page 31 and 32: where the curved space Γ-matrices
Page 33 and 34: Operation Action Generator translat
Page 35 and 36: 3.3 The Operator Product Expansion
Page 37 and 38: φ y z τ φ z σ Figure 17: Comput
Page 39 and 40: We can use here our definition (139
Page 41 and 42: fixed point, such non-marginal oper
Page 43 and 44: 3.5.3 Further Aspects of Conformal
Page 45 and 46: 3.7 The Closed String Partition Fun
Page 47 and 48: states of the form α−n|0>, (α
Page 49 and 50: and so we have: α 25 0 = ˜α 25 0
Page 51 and 52: and this radius is the “self-dual
Page 53 and 54: and [J a n, J b m] = if ab cJ c n+m
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where the pL,R are given in (205).
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The action for a Dirac fermion, Ψ
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parts on their own cannot be modula
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0 2 πR Figure 21: Open strings wit
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that the hyperplanes can fluctuate
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The orientifold construction was di
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which, upon solving it for va and s
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5.2.1 World-Volume Actions from Til
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5.5 Non-Abelian Extensions For N D-
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What does this mean? Well, recall t
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state” formalism, which we will d
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for momentum k. The reader might re
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The factor in braces is One thus fi
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Now we can compare to the open stri
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As before, there is a physical stat
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Figure 28: One-loop diagrams displa
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7.2 Closed Superstrings: Type II Ju
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perturbative open string theory (Ch
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where “f” denotes the fundament
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7.5 The Massless Spectrum In the ca
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takes one from the type I Lagrangia
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is only one way of getting a scalar
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Recall now that we have two twisted
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Here, dΩ2 is the metric on a roun
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8.1.1 T-Duality of Type II Superstr
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8.2 D-Branes as BPS Solitons Let us
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f4(q) 8 , as they ought to since th
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on a Dirac “string” (see also i
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9.1 Tilted D-Branes and Branes with
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ψ θ φ N(+) S(−) Figure 30: Con
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c0 = 1 , c1 = cj = n i1
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and the Hirzebruch ˆ L-polynomial
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The trick is then to simply pick ou
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where we see that the result of act
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a net parity transformation. Since
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(a) (b) (c) Figure 31: (a) A parall
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Is this at all consistent with what
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For n D0-branes and one D4-brane, t
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11.1.2 S-Duality and SL(2, Z) The f
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“F5-branes”, since they are mag
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D2−brane NS5−branes Figure 35:
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11.5 E8 × E8 Heterotic String/M-Th
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and so is the minimum allowed by th
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In fact, the U-duality group for th
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[33] C. Lovelace, Phys. Lett. B34 (
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[79] G. W. Gibbons and S. W. Hawkin
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[130] E. Witten, Nucl. Phys. B500,
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[174] D. R. Morrison and C. Vafa, N
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