References
139
and thus the Hamiltonian is
H = (L 0 ) cyl + ( ¯
L 0 ) cyl = L 0 + ¯
L 0 −
c + ¯
c
24
.
(6.168)
6.5
Suggested Readings
• The most complete reference on CFTs is [6] but it lacks some recent developments. Two excellent complementary books are [3, 35].
String theory books generally dedicate a fair amount of pages to CFTs:
particularly good summaries can be found in [2, 23, 28, 29].
Finally, a modern and fully algebraic approach can be found in [31,32]. Other
good reviews are [30, 39].
• There are various other books [18, 20, 22, 26] and reviews [4, 10, 12, 34, 37].
• The maps from the sphere and the cylinder to the complex plane are discussed
in [28, sec. 2.6, 6.1].
• Normal ordering is discussed in details in [6, chap. 6] (see also [2, sec. 4.2, 28,
sec. 2.2]).
• Euclidean conjugation is discussed in [6, sec. 6.1.1, [28], p. 202–3]. For a
comparison of Euclidean and BPZ conjugations, see [[40], sec. 2.2, [36], p. 11].
• Normal ordering and difference between the different definitions are described
in [28, chap. 2, 6, sec. 6.5].
References
1. T. Bautista, A. Dabholkar, H. Erbin, Quantum gravity from timelike Liouville theory. J.
High Energy Phys. 2019(10), 284 (2019). https://doi.org/10.1007/JHEP10(2019)284. arXiv:
1905.12689
2. R. Blumenhagen, D. Lüst, S. Theisen, Basic Concepts of String Theory (Springer, Berlin, 2014)
3. R. Blumenhagen, E. Plauschinn, Introduction to Conformal Field Theory: With Applications to
String Theory. Lecture Notes in Physics (Springer, Berlin, 2009). https://www.springer.com/
de/book/9783642004490
4. J. Cardy, Conformal field theory and statistical mechanics (2008). arXiv: 0807.3472
5. G. Delfino, J. Viti, On three-point connectivity in two-dimensional percolation. J. Phys. A
Math. Theor. 44(3), 032001 (2011). https://doi.org/10.1088/1751-8113/44/3/032001. arXiv:
1009.1314
6. P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theory, 2nd edn. (Springer, Berlin,
1999)
7. M. Flohr, On modular invariant partition functions of conformal field theories with logarithmic operators. Int. J. Mod. Phys. A 11(22), 4147–4172 (1996). https://doi.org/10.1142/
S0217751X96001954. arXiv: hep-th/9509166
8. M. Flohr, Bits and pieces in logarithmic conformal field theory. Int. J. Mod. Phys. A 18(25),
4497–4591 (2003). https://doi.org/10.1142/S0217751X03016859. arXiv: hep-th/0111228
9. K. Fujikawa, U. Lindström, N.K. Nielsen, M. Rocek, P. van Nieuwenhuizen, Regularized
BRST-coordinate-invariant measure. Phys. Rev. D 37(2), 391–405 (1988). https://doi.org/10.
1103/PhysRevD.37.391
139
and thus the Hamiltonian is
H = (L 0 ) cyl + ( ¯
L 0 ) cyl = L 0 + ¯
L 0 −
c + ¯
c
24
.
(6.168)
6.5
Suggested Readings
• The most complete reference on CFTs is [6] but it lacks some recent developments. Two excellent complementary books are [3, 35].
String theory books generally dedicate a fair amount of pages to CFTs:
particularly good summaries can be found in [2, 23, 28, 29].
Finally, a modern and fully algebraic approach can be found in [31,32]. Other
good reviews are [30, 39].
• There are various other books [18, 20, 22, 26] and reviews [4, 10, 12, 34, 37].
• The maps from the sphere and the cylinder to the complex plane are discussed
in [28, sec. 2.6, 6.1].
• Normal ordering is discussed in details in [6, chap. 6] (see also [2, sec. 4.2, 28,
sec. 2.2]).
• Euclidean conjugation is discussed in [6, sec. 6.1.1, [28], p. 202–3]. For a
comparison of Euclidean and BPZ conjugations, see [[40], sec. 2.2, [36], p. 11].
• Normal ordering and difference between the different definitions are described
in [28, chap. 2, 6, sec. 6.5].
References
1. T. Bautista, A. Dabholkar, H. Erbin, Quantum gravity from timelike Liouville theory. J.
High Energy Phys. 2019(10), 284 (2019). https://doi.org/10.1007/JHEP10(2019)284. arXiv:
1905.12689
2. R. Blumenhagen, D. Lüst, S. Theisen, Basic Concepts of String Theory (Springer, Berlin, 2014)
3. R. Blumenhagen, E. Plauschinn, Introduction to Conformal Field Theory: With Applications to
String Theory. Lecture Notes in Physics (Springer, Berlin, 2009). https://www.springer.com/
de/book/9783642004490
4. J. Cardy, Conformal field theory and statistical mechanics (2008). arXiv: 0807.3472
5. G. Delfino, J. Viti, On three-point connectivity in two-dimensional percolation. J. Phys. A
Math. Theor. 44(3), 032001 (2011). https://doi.org/10.1088/1751-8113/44/3/032001. arXiv:
1009.1314
6. P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theory, 2nd edn. (Springer, Berlin,
1999)
7. M. Flohr, On modular invariant partition functions of conformal field theories with logarithmic operators. Int. J. Mod. Phys. A 11(22), 4147–4172 (1996). https://doi.org/10.1142/
S0217751X96001954. arXiv: hep-th/9509166
8. M. Flohr, Bits and pieces in logarithmic conformal field theory. Int. J. Mod. Phys. A 18(25),
4497–4591 (2003). https://doi.org/10.1142/S0217751X03016859. arXiv: hep-th/0111228
9. K. Fujikawa, U. Lindström, N.K. Nielsen, M. Rocek, P. van Nieuwenhuizen, Regularized
BRST-coordinate-invariant measure. Phys. Rev. D 37(2), 391–405 (1988). https://doi.org/10.
1103/PhysRevD.37.391