- Page 2 and 3: Contents 0. Introduction, Overview,
- Page 4 and 5: 13.8. Generating functional for S-m
- Page 6 and 7: The developments we shall describe
- Page 8 and 9: This exactly solvable spacetime bac
- Page 10 and 11: where P is a polynomial and the exp
- Page 12 and 13: where the constant C is determined
- Page 14 and 15: for arbitrary α 0 . For particular
- Page 16 and 17: 2.2. Path integral approach to 2D E
- Page 18 and 19: where S L is again the Liouville ac
- Page 20 and 21: (consistent with the required overa
- Page 22 and 23: order multicritical point of the on
- Page 24 and 25: 3. Brief Review of the Liouville Th
- Page 26 and 27: 3.2. Classical Uniformization The c
- Page 28 and 29: Here a † n = a −n , b † n = b
- Page 30 and 31: |ψ| V ϕ Fig. 3: Particle wavefunc
- Page 32 and 33: 3.5. Semiclassical States The semic
- Page 34 and 35: c) Show that the solution correspon
- Page 36 and 37: 3.7. Semiclassical Amplitudes In th
- Page 38 and 39: Semiclassical Seiberg Bound [7,8] T
- Page 40 and 41: 3.8. Operator Products in Liouville
- Page 42 and 43: of amplitudes as “functions of s,
- Page 44 and 45: The differential equation (3.66) ha
- Page 46 and 47: The mapping (3.71) carries a circle
- Page 48 and 49: in (3.80) is obtained by comparing
- Page 50 and 51: From our experience with conformal
- Page 52 and 53:
4.2. Canonical Quantization of 2D E

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Using the KPZ formula (2.26) writte

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the p → 0 behavior of amplitudes

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Therefore the chiral cohomology con

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Remark: In general there will be

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. . A . . 3 − 2 − 1 − 2 , 2

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If we isolate one Euclidean coordin

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where κ is the string coupling. Co

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It should come as no surprise to fi

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ii) We must work with macroscopic s

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3) There are indications that under

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The condition (5.30) is of course o

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in Euclidean space. As in 1D, we sh

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vertices and is counted three times

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indices is represented in fig. 13 b

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(The powers of N associated with th

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6.4. A first look at the double sca

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The now-standard method for solving

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(The 10 rrr terms start with r n (r

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(after a suitable rescaling of u an

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In matrix notation, we write this a

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We see that double lines connecting

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7.7. Continuum Solution of the Matr

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Commuting both sides with K and usi

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Exercise. Scaling of Lax operators

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8.2. Precise definition of the cont

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we approach singular values and def

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for which one may draw a similar pi

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Therefore, in order to define the d

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9.3. Double-Scaled Fermi Theory Mor

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contribution of the edge of the eig

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In particular, continuum loop ampli

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More generally, we may calculate th

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Exercise. The general amplitude Usi

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It is extremely interesting to note

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The asymptotic expansion in κ of t

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Thus we are studying the quantum me

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To see how singularities of F 0 can

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Evaluating the singular part of the

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may take the double scaling limit o

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Using Wick’s theorem, we evaluate

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c) Three macroscopic loops: κ The

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Treated as a field, W has a vacuum

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the formula (4.12), we see that the

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By upper triangular transformations

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To interpret (12.4) as a field theo

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Dirichlet boundary conditions imply

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Exercise. Variations on WdW a) Deri

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The basis generators are parametriz

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Exercise. Poisson brackets Use the

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where s = 1, 2, . . . and q ∈ IR

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An incoming or outgoing boson of en

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13.2. On the Violation of Folklore

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since from (13.14) we see that δσ

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is convenient to work in Euclidean

- Page 162 and 163:
ebosonized. The amplitude for this

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Finally, we must relate these nonpe

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since the leading term in amplitude

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particle − # hole number, i.e., n

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where p m ≡ (m + 1 2 )/β. Thus d

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Note that in (13.56) there is a cha

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The relevant density on moduli spac

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sense if we rotate φ → it (as we

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in [156] it is shown that the “le

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kinematic regimes at µ ≠ 0 using

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example has provided some very inte

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Tachyon Modules: Away from the self

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Exercise. Missing lessons Determine

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15.3. Future prospects and Open Pro

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A.2. Asymptotics Define Φ(µ) ≡

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[19] P. Ginsparg and G. Moore, “L

- Page 194 and 195:
[57] I. Kostov and M. Staudacher,

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[97] K. Demeterfi, A. Jevicki, and

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and Eigenstates for Integrable Coll

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[161] I.R. Klebanov, “Ward Identi