Chiral Algebras of (0, 2) Models: Beyond Perturbation Theory

Abstract

We explore the nonperturbative aspects of the chiral algebras of N = (0, 2) sigma models, which perturbatively are intimately related to the theory of chiral differential operators (CDOs). The grading by charge and scaling dimension is anomalous if the first Chern class of the target space is nonzero. This has some nontrivial consequences for the chiral algebra. As an example, we study the case where the target space is CP 1 , and show that worldsheet instantons trivialize the chiral algebra entirely. Consequently, supersymmetry is spontaneously broken in this model. We then turn to a closer look at the supersymmetry breaking from the viewpoint of Morse theory on loop space. We find that instantons interpolate between pairs of perturbative supersymmetric states with different fermionic numbers, hence lifting them out of the supersymmetric spectrum. Our results reveal that a "quantum" deformation of the geometry of the target space leads to a trivialization of the kernels of certain twisted Dirac operators on CP 1 .

Chiral algebras from Ω-deformation

“…Anomalies in U(1) R may originate from nonperturbative effects in the two-dimensional theory, and break U(1) R down to a finite subgroup. Such a situation was studied in [25,[27][28][29] in the context of N = (0, 2) supersymmetric sigma models, for which U(1) R anomalies are linked to conformal anomalies due to the curvature of the target space. It was shown in [29] that the chiral algebras of models with U(1) R anomalies lack an energy-momentum tensor.…”

“…It was shown in [29] that the chiral algebras of models with U(1) R anomalies lack an energy-momentum tensor. In extreme cases, the entire chiral algebra vanish because worldsheet instantons make the identity operator Q-exact [27][28][29].…”

Making supersymmetric connectedN=(0,2)sigma models

“…They were studied from the mathematical perspective [11][12][13][14][15][16] as well as from the standpoint of physical applications (see [17] and extensive references therein). The (0, 2) connected model we will construct has the bosonic target space CP(N 1 −1)×CP(N 2 −1)× ... ×CP(N n −1) .…”

N=(0,2)deformation of theCP(1)Cui1, Shifman2 2012Phys. Rev. D160200View full textAdd to dashboardCite

“…The reason is that involution is lost upon transition to Euclidean space. No zero mode arises from the background ¼ y zÀz 0 (see also [14]). This means that the superinstanton under consideration has no collective coordinates y and .…”

“…Instead, one gets N ¼ ð0; 2Þ heterotic sigma models with the CPðN À 1Þ target space. This finding gave a strong impetus to explorations of these heterotic models which had been previously discussed only in general terms [11][12][13][14][15].…”