Chiral rings in N = 2 superconformal theories

Abstract

We investigate the properties of chiral operators in N = 2 superconformal theories. In particular, we study the spectral flow of such models under a one-parameter family of twists generated by the U(1) current, and use this to deduce various properties of the ring of chiral primary fields. We furthermore investigate under what conditions a given superconformal theory can be represented as the fixed point of an N = 2 Landau-Ginzburg theory and show how to determine the superpotential. We also investigate the coset models of Kazama and Suzuki and find a simple cohomological characterization for the elements of the chiral primary ring. Moreover we show how some of them can be represented as LG models. * This is the analog of the Hodge decomposition for differential forms. See also the discussion in sect. 3. * This can be seen by considering the contour integral of G+(z) about a pair of chiral fields, and noting that the definition of chiral state demands that the contour integral of G+(z) encircling a chiral field vanishes. ** The ring is commutative up to +_ signs, due to the fact that the ring is defined before the GSO projection. **'* This is not to be confused with the commutative ring which appears for rational conformal field theories [9]. In particular the ring we obtain here is nilpotent, which is not the case for RCFT operator algebra. * For the behavior of Tr(1) F under asymmetric spectral flow see ref. [12]. ** These formulas generalize to relations between the full N = 2 characters for all levels [11,13]. * Toroidal compactification [25] and orbifolds based on them [26] could also be viewed as orbifolds of Landau-Ginzburg models. ** Our conventions in this section are opposite to that of sect. 3, where the (c, c) corresponded to Kghler deformations.

Non-compact Gepner models, Landau-Ginzburg orbifolds and mirror symmetry

“…A crucial observation is that for the Calabi-Yau examples we consider in this paper, (c, c) states are in one-to-one correspondence with Ramond-Ramond ground states [37]. More precisely, a Ramond-Ramond ground state becomes a (c, c) state under the action of half-unit left-right symmetric spectral flow.…”

“…For instance, for conformal field theories with central charge c < 3, it is argued in [37] that there always exist a chiral primary field in the conformal field theory with conformal dimension h = c/6. That is a consequence of the existence of the unit operator in the theory (combined with spectral flow).…”

“…An NS-sector field of conformal dimension h and R-charge Q is a chiral primary if and only if h = Q/2 and anti-chiral if and only if h = −Q/2 [37].…”

“…All states in the NS-sector with conformal dimension h and R-charge Q satisfy the inequality h ≥ |Q|/2 [37].…”

“…We can summarize the chiral ring of the Landau-Ginzburg model by specifying a Poincare polynomial which is the trace over the chiral-chiral ring weighted by the U (1) R-charges [37]:…”