On the Virasoro crossing kernels at rational central charge

We report novel analytic results for the Virasoro modular and fusion kernels relevant to 2d conformal field theories (CFTs), 3d topological field theories (TQFTs), and the representation theory of certain quantum groups. For all rational values of the parameter b^2∈\mathbb{Q}^{×} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi>b</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">ℚ</mml:mi> <mml:mo>×</mml:mo> </mml:msup> </mml:mrow> </mml:math> – corresponding in 2d CFT to rational central charge values in the domain (-∞,1]\cup[25,∞) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>−</mml:mi> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">]</mml:mo> <mml:mo>∪</mml:mo> <mml:mo stretchy="false" form="prefix">[</mml:mo> <mml:mn>25</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> – we establish two main results. First, in the domain c∈\mathbb{Q}_{[25,∞)} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi mathvariant="double-struck">ℚ</mml:mi> <mml:mrow> <mml:mo stretchy="false" form="prefix">[</mml:mo> <mml:mn>25</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> we show that the modular and fusion kernels derived by Teschner and Teschner-Vartanov respectively can be expressed as a linear combination of two functions, which (i) are themselves admissible crossing kernels, (ii) have square-root branch point singularities in the Liouville momenta, (iii) are not reflection-symmetric in the Liouville momenta. These features illustrate that the space of solutions to the basic shift relations determining these kernels is broader than previously assumed. Second, in the domain c∈\mathbb{Q}_{(-∞,1]} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi mathvariant="double-struck">ℚ</mml:mi> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>−</mml:mi> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">]</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> we derive for the first time the physical modular and fusion kernels for generic values of the Liouville momenta. These can again be written as a linear combination of two other admissible kernels but overall, and unlike the Teschner and Teschner-Vartanov solutions for c≥ 25 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>25</mml:mn> </mml:mrow> </mml:math> , they possess square-root branch point singularities. As a corollary, we demonstrate that timelike Liouville theory at c∈\mathbb{Q}_{(-∞,1]} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi mathvariant="double-struck">ℚ</mml:mi> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>−</mml:mi> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">]</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> is crossing symmetric and modular covariant. Surprisingly, the crossing kernels at any b^2∈\mathbb{Q}^{×} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi>b</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">ℚ</mml:mi> <mml:mo>×</mml:mo> </mml:msup> </mml:mrow> </mml:math> behave as if they were semiclassical and one-loop exact, and we discuss the interpretation of this fact in the context of the 2d conformal bootstrap and the 3d TQFT that captures pure 3d gravity with negative cosmological constant.