Broken (super) conformal Ward identities at finite temperature
Enrico Marchetto, Alessio Miscioscia, Elli Pomoni
Abstract
A bstract When a (super) conformal field theory is placed on a non-trivial manifold, the (super) conformal symmetry is broken. However, it is still possible to derive broken Ward identities for these broken symmetries, which provide additional constraints on the theory. We derive and apply the broken Ward identities associated with the (super) conformal group on the thermal manifold $$ {\mathcal{M}}_{\beta }={S}_{\beta}^1\times {\mathbb{R}}^{d-1} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>β</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>β</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> and $$ \mathcal{M}={T}^2\times {\mathbb{R}}^{d-2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> . The novel constraints not only systematically reproduce known results, including an implicit formulation of the generalized Cardy formula, but also elegantly relate the thermal energy spectrum with the conformal spectrum.