The Bethe/Gauge Correspondence

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18 Chapter 1. Overviewspin at the Bethe side, also is restricted to half-integer values at the gauge side. In Chapter 4we will see that this can be arranged in a quite natural way.Let’s update the dictionary for the main example:BetheGaugeN = 2 symwith massive matterin two dimensionslow energy limitxxx s model ←→ vacuum structurebae ←→ vacuum equationsN-particle sector ←→ G = U(N)rapidities λ n ←→ supersymmetric vacua σ nlength L ←→ L f = L¯f = Lspin s ←→ ˜m l f = ˜ml¯f = −isThis dictionary allows us to translate problems on one side to the other, offering a new point ofview on the problem, and perhaps some steps towards a solution.Now we have a general idea of the Bethe/gauge correspondence and its ingredients. In thenext two chapters we will take a detailed look at both sides of the correspondence, allowing usto go deeper into the correspondence in Chapter 4.
Chapter 2BetheIn this chapter we take a more detailed look at quantum integrable models. We will• learn about a useful method generalizing the coordinate Bethe Ansatz from Section 1.1.2(along the way we will briefly discuss the meaning of ‘quantum integrability’, and we willsee how it can be shown that all of the xxx s systems are quantum integrable);• take a look at relevant extensions of the xxx s models, including quasi-periodic boundaryconditions and inhomogeneities; and• find out about a remarkable feature of the Bethe Ansatz equations, namely that there isa function that serves as their potential.Let’s get started right away.2.1 Algebraic Bethe AnsatzIn the 1980s, the ‘Leningrad School’ (Faddeev et al.) developed an extension of the coordinateBethe Ansatz, known as the algebraic Bethe Ansatz (aba) or quantum inverse scattering method.In this section, we give a brief introduction to the aba. The general idea is to obtain theeigenvectors of the Hamiltonian using creation and annihilation operators. This allows us toexplicitly construct the Fock space of states. Moreover, the formalism of the aba can be usedto prove the quantum integrability of the model.Since the aba and its formalism are rather abstract, we only treat a small part of the generaltheory. After discussing what we need, we will proceed along the lines of Faddeev [11] and applythe aba to a representative example, namely our familiar Heisenberg xxx 1/2 magnet. This willculminate in the actual algebraic Bethe Ansatz, from which the Bethe Ansatz equations andthe Fock space of states are derived. We won’t be very rigourous; for a thorough treatment seee.g. Part II, and especially Chapters VI and VII, of [25].As in Section 1.1.2, our goal will be to find the spectrum of the Hamiltonian. Importantingredients that we will encounter along the way are Lax operators that allow us to constructthe monodromy matrix, which in turn leads to the transfer matrix. All of these objects havecounterparts in the classical theory of integrability; see e.g. Babelon et al. [42]. In practice onewants to show that the transfer matrix commutes with the Hamiltonian, so that they can besimultaneously diagonalized. The creation and annihilation operators are then constructed fromthe monodromy matrix. Again we will do this explicitly for the xxx 1/2 model. At the end ofthis section we will sketch how the aba can be applied to solve the entire family of spin-s xxx ssystems.2.1.1 Quantum integrabilityTo motivate our approach we begin with a brief discussion of the meaning of ‘quantum integrability’.Somehow, it should of course be the quantum analogue of classical integrability, so let’s19
Page 1 and 2: Institute for Theoretical PhysicsMa
Page 4 and 5: 4 Bethe/gauge 614.1 The corresponde
Page 6 and 7: viPrefaceA note on notationMany asp
Page 9 and 10: Chapter 1OverviewIn quantum field t
Page 11 and 12: 1.1. Bethe: quantum integrable mode
Page 17 and 18: 1.2. Gauge: supersymmetry 9providin
Page 19 and 20: 1.2. Gauge: supersymmetry 11Because
Page 21 and 22: 1.2. Gauge: supersymmetry 13The def
Page 23 and 24: 1.3. Physics in two dimensions 15be
Page 25: 1.4. Bethe/gauge: the general idea
Page 29 and 30: 2.1. Algebraic Bethe Ansatz 21on ou
Page 31 and 32: 2.1. Algebraic Bethe Ansatz 23To ge
Page 33 and 34: 2.1. Algebraic Bethe Ansatz 25Accor
Page 35 and 36: 2.2. Generalizations of the xxx s m
Page 37 and 38: 2.3. Yang-Yang function 292.2.3 Fur
Page 39: 2.3. Yang-Yang function 31To see th
Page 42 and 43: 34 Chapter 3. GaugeThe Poincaré gr
Page 44 and 45: 36 Chapter 3. Gaugeexpansion we wil
Page 46 and 47: 38 Chapter 3. Gaugewhere we can in
Page 48 and 49: 40 Chapter 3. GaugeHere K is a real
Page 50 and 51: 42 Chapter 3. GaugeVector superfiel
Page 52 and 53: 44 Chapter 3. GaugeThus, L r is pre
Page 54 and 55: 46 Chapter 3. Gaugeunder U(1) G as
Page 56 and 57: 48 Chapter 3. Gauge˜σ k¯f= ˜m k
Page 58 and 59: 50 Chapter 3. GaugeVacuum manifolds
Page 60 and 61: 52 Chapter 3. GaugeThis (almost) is
Page 62 and 63: 54 Chapter 3. GaugeLooking at the t
Page 64 and 65: 56 Chapter 3. Gaugethe effective tw
Page 66 and 67: 58 Chapter 3. GaugeMore matter repr
Page 69 and 70: Chapter 4Bethe/gaugeNow that we hav
Page 71 and 72: 4.1. The correspondence 63the spin
Page 73 and 74: 4.2. Further topics 654.1.3 Diction
Page 75: 4.2. Further topics 67N = 2 symin f
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70 Chapter 4. Bethe/gauge• descri
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[17] N. Dorey, T. Hollowood, and D.
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[56] U. Lindström, “Supersymmetr
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The Bethe/Gauge Correspondence

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