The Bethe/Gauge Correspondence

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4 Bethe/gauge 614.1 The correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.1.1 Main example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.1.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1.3 Dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Further topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Conclusion 69References 74iv
PrefaceThe Bethe/gauge correspondence was formulated by Nekrasov and Shatashvili in two papers[1, 2] that appeared in 2009. These publications were originally intended as announcementsof a longer and more detailed paper. To date, rather than spelling out the details of thesetwo papers, Nekrasov and Shatashvili have generalized the correspondence and investigated itsconsequences [3, 4].In this thesis we review the Bethe/gauge correspondence as presented in [1, 2]. Our aim isto explain the main observation of the correspondence, namely that the vacuum equations ofcertain two-dimensional gauge theories with N = (2, 2) supersymmetry coincide with the Betheequations of some quantum integrable model.OutlineThis thesis is set up as follows. In Chapter 1 we discuss the general picture of the Bethe/gaugecorrespondence. We introduce the ‘Bethe side’ of the story via the prototypical quantum integrablesystem, Heisenberg’s xxx 1/2 model, and sketch the method of the (coordinate) BetheAnsatz. To get acquainted with the ‘gauge side’ we take a first look at supersymmetry. Finally,when we understand the context of the Bethe/gauge correspondence, we discuss the basic ideabehind the correspondence.The theory of quantum integrability is treated in more detail in Chapter 2. In this chapterwe treat most of §3 of [1] and §2.5–§2.7 of [2]. We review the algebraic Bethe Ansatz, whichalso gives more insight into the meaning of quantum integrability. We introduce some relevantgeneralizations of the xxx 1/2 model and look at the ‘Yang-Yang function’.Chapter 3 contains a more in-depth discussion of the gauge side of the story, focussing onsupersymmetry in two dimensions. This chapter essentially covers §2 (except for §2.2.2–§2.2.3)of [1] and §2.1 and §2.3 of [2]. We describe two dimensional superspace and its symmetries andthe precise field content and Lagrangian we’re interested in. Then we go on to the low-energyeffective theory, derive the vacuum equations, and compute the quantum corrections to theeffective twisted superpotential.After all these preparations, we are ready to come back to the Bethe/gauge correspondencein Chapter 4 and discuss it in more detail. We treat §4–§4.1 of [1] and §2.8 of [2], and presentthe idea of §2.2.2–§2.2.3 of [1]. We conclude with a summary and outlook.The division into ‘Bethe’, ‘gauge’ and the correspondence, as well as the logical dependencebetween the sections and chapters, can be represented asBethe ←→ GaugeSections 1.1 and 1.3 Sections 1.2 and 1.3Section 1.4Chapter 2 Chapter 3Chapter 4Conclusionv
Page 1 and 2: Institute for Theoretical PhysicsMa
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 and 26: 1.4. Bethe/gauge: the general idea
Page 27 and 28: Chapter 2BetheIn this chapter we ta
Page 29 and 30: 2.1. Algebraic Bethe Ansatz 21on ou
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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
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Page 46 and 47: 38 Chapter 3. Gaugewhere we can in
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46 Chapter 3. Gaugeunder U(1) G as
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48 Chapter 3. Gauge˜σ k¯f= ˜m k
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50 Chapter 3. GaugeVacuum manifolds
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52 Chapter 3. GaugeThis (almost) is
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54 Chapter 3. GaugeLooking at the t
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56 Chapter 3. Gaugethe effective tw
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58 Chapter 3. GaugeMore matter repr
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Chapter 4Bethe/gaugeNow that we hav
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4.1. The correspondence 63the spin
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4.2. Further topics 654.1.3 Diction
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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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