The properties of conformal blocks, the AGT hypothesis, and knot polynomials
- PDF / 2,111,761 Bytes
- 63 Pages / 612 x 792 pts (letter) Page_size
- 92 Downloads / 147 Views
e Properties of Conformal Blocks, the AGT Hypothesis, and Knot Polynomials A. A. Morozov Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, 127051 Russia National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow, 115409 Russia e-mail: [email protected] Abstract—Various properties of correlators of the two-dimensional conformal field theory are discussed. Specifically, their relation to the partition function of the four-dimensional supersymmetric theory is analyzed. In addition to being of interest in its own right, this relation is of practical importance. For example, it is much easier to calculate the known expressions for the partition function of supersymmetric theory than to calculate directly the expressions for correlators in conformal theory. The examined representation of conformal theory correlators as a matrix model serves the same purpose. The integral form of these correlators allows one to generalize the obtained results for the Virasoro algebra to more complicated cases of the W algebra or the quantum Virasoro algebra. This provides an opportunity to examine more complex configurations in conformal field theory. The three-dimensional Chern–Simons theory is discussed in the second part of the present review. The current interest in this theory stems largely from its relation to the mathematical knot theory (a rather well-developed area of mathematics known since the 17th century). The primary objective of this theory is to develop an algorithm that allows one to distinguish different knots (closed loops in three-dimensional space). The basic way to do this is by constructing the so-called knot invariants. DOI: 10.1134/S106377961605004X
CONTENTS 1. INTRODUCTION 1.1. Outline of the Review 2. CONFORMAL FIELD THEORY 2.1. Free Field Theory 2.2. Free Theory with c ≠ 1 2.3. Correlators in the Free Theory 2.4. Four-point Conformal Block 2.5. Shapovalov Form 2.6. Triple Vertices 2.6.1. Triple Vertices Γ 2.6.2. Triple Vertices Γ 2.7. Diagram Technique 2.8. Calculated Triple Vertices 2.9. W(3) Algebra 2.9.1. Triple Vertices in the W(3) Algebra 2.9.2. Calculations in the Free Field Theory 2.9.3. Examples of Triple Vertices 3. AGT RELATION 3.1. Nekrasov Function 3.2. AGT Relation for Conformal Blocks on a Sphere 3.2.1. U(1) Factor 3.2.2. Four-point Conformal Block
776 780 781 783 784 784 785 787 787 788 788 788 789 790 791 793 802 804 804 806 806 806
3.2.3. Five-point Conformal Block 3.2.4. Six-point Conformal Block 3.2.5. n-Point Conformal Block 3.2.6. Symmetries 3.2.7. Selection of Diagrams 3.2.8. Explicit Calculations for the AGT Relation 3.3. AGT Relation for Conformal Blocks on a Torus 3.3.1. Large-mass Limit 4. FREE FIELD THEORY AND SELBERG INTEGRALS 4.1. C αα1α+α 2 + bN at Level One 1 2
808 810 811 812 812 813 815 815 816 818
2 + bN 4.2. C αα11α+α at Level Two 820 2 4.3. Generalization to Higher Levels 820 4.4. Transition from the Operator Expansion to a Conformal Block 821 4.5. Selberg Integrals and Their Generalization 821 5. CH
Data Loading...
