Perturbative and non-perturbative infrared behavior of ...

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5. N = 2 Chern-Simons matter theories The second generalization we consider is to change the Chern-Simons levels so that they are no longer equal and opposite in sign. In other words, we are interested in U(N)k1 × U(M)k2 Chern-Simons matter theories studied in [79]. Along the lines of the construction of the ABJM model, one is lead to the following action SN=3 = k1 4π SCS(V1) + k2 4π SCS(V2) + d 4 xd 4 θ Tr e −V1 † A ieV2Ai + e −V1Bie V2 † B i + d 4 xd 2 2π θ Tr (BiAi) 2 + 2π Tr (AiBi) 2 (5.2.5) k1 k2 Due to the unequal Chern-Simons levels, the SU(2) × SU(2) global symmetry of the ABJM model is broken to the diagonal SU(2) that rotates the Ai and Bi simultaneously. The action (5.2.5) still has an SO(3)R ≃ SU(2)R R-symmetry. Therefore, it enjoys N = 3 superconformal symmetry. We conclude that unequal CS levels break some supersymmetries. One can further break supersymmetry by allowing a more general superpotential: W = c (5.2.6) d 4 xd 2 θ ǫ ij ǫ kl AiBkAjBl This is the most general SO(4) global symmetry preserving superpotential we can write. Note that the coupling c is now free to assume any complex value; therefore, the theory is no longer superconformal, and supersymmetry is broken to N = 2. If we want to preserve N = 2 supersymmetry without adding any new superfields, (5.2.6) can only be modified by breaking the SO(4) global symmetry, thus broadening the parameter space one can study. We discuss this in details in the next chapter. The addition of fundamental flavor fields [82, 83, 84] to the ABJM action (5.1.4) broadens the class of field theories one can study. In addition to the N = 4 vector multiplets (Vi,Φi) and the bifundamental hypermultiplets (Ai,B † i ) we introduce 2Nf fundamental hypermultiplets (Q1r, ˜ Q † 1r ) and (Q2r, ˜ Q † 2r ) with r = 1,... ,Nf with superpotential Wflavor = α1 ˜ Q1rΦ1Q1r + α2 ˜ Q2rΦ2Q2r (5.2.7) which preserves N = 2 supersymmetry for arbitrary values of the coupling constants α1 and α2. The component fields of the N = 4 vector multiplets are all auxiliary fields and can be integrate out giving rise to (we suppress flavor indices) Wflavor = 4πα1 k ˜Q1(AiB i )Q1 − 4πα2 k ˜Q2(B i Ai)Q2 + 2πα2 1 k (Q1 ˜ Q1) 2 − 2πα2 2 k ( ˜ Q2Q2) 2 (5.2.8) For the particular values α1 = −α2 = 1 supersymmetry is enhanced to N = 3. As in the ABJM model, this is accompanied by an enhancement of the U(1)R R-symmetry to SU(2)R. In addition, the theory also inherits the SU(2)D diagonal subgroup of the SU(2) × SU(2) of (5.1.4). In this case the two SU(2)’s commute with each other. Note that this symmetries survive when we choose the two CS levels to be unequal. 84
5.2. Generalizations The superpotential (5.2.8) clearly exhibits an U(Nf) × U(Nf) flavor symmetry. Given its structure, one can easily extend it to U(Nf) ×U(N ′ f ) with Nf = N ′ f . There is one more operator one can add while preserving all the global symmetries ∆W = Q1 ˜ Q1Q2 ˜ Q2 (5.2.9) In the next chapter we study the quantum effects of the theory obtained by adding (5.2.6), (5.2.8) and (5.2.9) and modifying them so that only the U(Nf)×U(N ′ f ) global symmetry survives. 85
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Università degli Studi di Milano-B
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Riassunto della tesi dimensionale.
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CONTENTS 4.3.2 The superpotential p
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List of Figures 3.1 New vertices pr
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Introduction and outline The advent
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Introduction and outline nonanticom
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Part I Nonanticommutative theories
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1. Basics and motivations ingredien
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1. Basics and motivations to obtain
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1. Basics and motivations 8
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2. Nonanticommutative superspace wh
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2. Nonanticommutative superspace wh
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2. Nonanticommutative superspace op
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3. The Wess-Zumino model We stress
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3. The Wess-Zumino model Φ 2 Φ U
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3. The Wess-Zumino model dim U(1)R
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3. The Wess-Zumino model • ω2 =
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4. Gauge theories ever, a modified
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4. Gauge theories They can be expre
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4. Gauge theories of the U(1) field
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4. Gauge theories and the pure gaug
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4. Gauge theories Since for the chi
Page 48 and 49: 4. Gauge theories Figure 4.1: Gauge
Page 50 and 51: 4. Gauge theories 4.2.2 Three-point
Page 52 and 53: 4. Gauge theories 4.2.4 (Super)gaug
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Page 56 and 57: 4. Gauge theories where the trace o
Page 58 and 59: 4. Gauge theories Figure 4.4: One-l
Page 60 and 61: 4. Gauge theories fore, one-loop re
Page 62 and 63: 4. Gauge theories α ˙α Γ dim R-
Page 64 and 65: 4. Gauge theories 3. Gauge sector.
Page 66 and 67: 4. Gauge theories We have introduce
Page 68 and 69: 4. Gauge theories the covariant pro
Page 70 and 71: 4. Gauge theories We note that the
Page 72 and 73: 4. Gauge theories satisfy this set
Page 74 and 75: 4. Gauge theories In order to cance
Page 76 and 77: 4. Gauge theories of [22] the most
Page 78 and 79: 4. Gauge theories By direct calcula
Page 80 and 81: 4. Gauge theories pole coefficients
Page 82 and 83: 4. Gauge theories superfields, in m
Page 84 and 85: 4. Gauge theories 70 φ h ∂ Γ (a
Page 86 and 87: 4. Gauge theories is perfectly cons
Page 88 and 89: 4. Gauge theories The system of lin
Page 90 and 91: 4. Gauge theories terms are no long
Page 92 and 93: 4. Gauge theories 78
Page 95 and 96: Chapter 5 N = 2 Chern-Simons matter
Page 97: 5.2. Generalizations representation
Page 101 and 102: Chapter 6 Quantization, fixed point
Page 103 and 104: 6.1. Quantization of N = 2 Chern-Si
Page 105 and 106: 6.1. Quantization of N = 2 Chern-Si
Page 107 and 108: 6.2. Two-loop renormalization and
Page 109 and 110: 6.2. Two-loop renormalization and
Page 111 and 112: In order to cancel the divergences
Page 113 and 114: 6.3. The spectrum of fixed points W
Page 115 and 116: 6.3. The spectrum of fixed points F
Page 117 and 118: addition of flavor matter [83]. The
Page 119 and 120: 6.4. Infrared stability Figure 6.5:
Page 121 and 122: 6.4. Infrared stability Figure 6.6:
Page 123 and 124: 6.5. A relevant perturbation Figure
Page 125 and 126: 6.5. A relevant perturbation This k
Page 127: Part III Supersymmetry breaking 113
Page 130 and 131: 7. Basics and motivations where the
Page 132 and 133: 7. Basics and motivations It follow
Page 134 and 135: 7. Basics and motivations • the h
Page 136 and 137: 7. Basics and motivations 122
Page 138 and 139: 8. Supersymmetric QCD and Seiberg d
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8. Supersymmetric QCD and Seiberg d
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9. Non-supersymmetric vacua If we a
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9. Non-supersymmetric vacua The Pol
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9. Non-supersymmetric vacua in the
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9. Non-supersymmetric vacua by what
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9. Non-supersymmetric vacua We saw
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9. Non-supersymmetric vacua jumps f
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9. Non-supersymmetric vacua The bou
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9. Non-supersymmetric vacua where t
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9. Non-supersymmetric vacua The phy
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9. Non-supersymmetric vacua conform
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9. Non-supersymmetric vacua the lin
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9. Non-supersymmetric vacua The one
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9. Non-supersymmetric vacua and the
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9. Non-supersymmetric vacua The con
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9. Non-supersymmetric vacua 9.4.4 R
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9. Non-supersymmetric vacua We have
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conclusions 172
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174
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A. Mathematical tools A.2 Useful in
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A. Mathematical tools For SU(N) we
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B. Feynman rules as f = ∇ 2 ∗ V
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B. Feynman rules Matter sector We n
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B. Feynman rules Since terms propor
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B. Feynman rules Φ ∇ α Φ ∇
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B. Feynman rules Here we recognize
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B. Feynman rules and contain an inf
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B. Feynman rules 192 ∂ φ Γ Γ
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C. Details on supersymmetry breakin
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BIBLIOGRAPHY [14] S. Penati and A.
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BIBLIOGRAPHY [47] M. Van Raamsdonk,
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BIBLIOGRAPHY [80] D. Gaiotto and A.
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BIBLIOGRAPHY [114] T. Banks and A.
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