A lecture on the Liouville vertex operators

By J. Teschner. 2003

146 citations recorded by the source corpus.

We reconsider the construction of exponential fields in the quantized Liouville theory. It is based on a free-field construction of a continuous family or chiral vertex operators. We derive the fusion and braid relations of the chiral vertex operators. This allows us to simplify the verification of locality and crossing symmetry of the exponential fields considerably. The calculation of the matrix elements of the exponential fields leads to a constructive derivation of the formula proposed by Dorn/Otto and the brothers Zamolodchikov.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/90cee44488889997d8977540050aa7f236505575

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In Algebra and Number Theory

1. Yuji Tachikawa
2. L. Alday
3. Y. Nakayama
4. D. Gaiotto
5. S. Ribault
6. N. Drukker
7. Joaquim Gomis
8. T. Okuda

In Geometry and Topology

1. D. Gaiotto
2. Sergei Gukov
3. Yuji Tachikawa
4. Joaquim Gomis
5. L. Alday
6. N. Drukker
7. Tudor Dimofte
8. J. Teschner

In Statistical and Nonlinear Physics

1. Yuji Tachikawa
2. D. Gaiotto
3. Sergei Gukov
4. Joaquim Gomis
5. Y. Nakayama
6. L. Alday
7. N. Drukker
8. S. Ribault

Citations in the corpus

• A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories (2009, unknown): “(2.5) The correlation function of three primary fields in the Liouville theory was calculated in [6] (see also [7]) and takes the form C(α1, α2, α3) = [ πµγ(b2)b2−2b 2 ] Q−α1−α2−α3 b (2.6) × Υ(b)Υ(2α1)Υ(2α2)Υ(2α3) Υ(α1 + α2 + α3 −Q)Υ(−α1 + α2 + α3)Υ(α1 − α2 + α3)Υ(α1 + α2 − α3) .”
• Liouville Correlation Functions from Four-Dimensional Gauge Theories (2009, methodology): “The three-point function is given by the DOZZ formula [5, 6, 7, 29, 14] 〈Vα1(z1)Vα2(z2)Vα3(z3)〉 = |z12||z23||z31|C(α1, α2, α3) (A.”
• Classical and quantum D-branes in 2D string theory (2003, background): “In order to define the latter we shall take the limit c ↓ 25 of the c > 25 Liouville theory constructed in [10][11].”
• Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I (2010, methodology): “Knowing the modular transformation properties of the Liouville conformal blocks [57, 74, 75, 78] now allows us to investigate and test the S-duality conjectures in these gauge theories, as illustrated in [2,11].”
• Liouville Field Theory — A decade after the revolution (2004, methodology): “With these developments, the consistency of the DOZZ formula as a CFT is finally given (with a mild assumption) by proving the general conformal bootstrap in [152], which is reviewed in [11], [153] and [12].”
• Gauge theory loop operators and Liouville theory (2009, methodology): “Correlators in Liouville theory are invariant with respect to the action of MCG(Cg,n), which relate different sewings of the surface [14, 15, 16].”
• Three-Dimensional Gravity Revisited (2007, unknown): “(Indeed, there has been remarkable progress in learning how to do this and to relate the results to Liouville theory [8-11].”
• Loop and surface operators in $ \mathcal{N} = 2 $ gauge theory and Liouville modular geometry (2009, background): “Fusion and braiding matrices are known to satisfy several identities [22] [13].”
• Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories (2012, methodology): “ies arising from the operator product expansions of the fields in (4.3) are all integrable. Similar ob jects can be defined for more general values of the parameters α, b and s by analytic continuation [T04]. For explicit calculations it may also be useful to replace (Q(z))s in (4.3) by expressions of”
• H3+-WZNW correlators from Liouville theory (2005, methodology): “At the moment , it is Liouville theory which is by far the best understood example of an interacting confo rmal field theory with a continuous set of primary fields, see [4, 5, 6] and references therein.”
• Isomonodromic Tau-Functions from Liouville Conformal Blocks (2014, unknown): “z∆β1+∆α−∆β2Vα β2β1 (z)uniquely in the sense of formal power series in z, Vα β2β1 (z) = z∆β2−∆β1−∆α X∞ n=−∞ zn W α β2β1 (n), Wβ 2β1 (n) : V(k) β1 → V(k+n) β2 . (3.19) It has furthermore been argued in [T03] that the composition Vα2 β3β2 (z)Vα1 β2β1 (w) of such vertex operators exists for |w/z| <1,”