World-Sheet Stability of (0,2) Linear Sigma Models

View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server March 7, 2003 hep-th/0303066 EFI-03-09 World-sheet Stability of (0,2) Linear Sigma Models Anirban Basu1 and Savdeep Sethi2 Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA Abstract We argue that two-dimensional (0; 2) gauged linear sigma models are not destabilized by instanton generated world-sheet superpotentials. We construct several examples where we show this to be true. The general proof is based on the Konishi anomaly for (0; 2) theories. 1email address: [email protected] 2email address: [email protected] 1 Introduction One of the basic questions we ask about quantum field theory is whether the classical vacua are stable. Often, the structure of the quantum moduli space is significantly different from the classical moduli space. The theories about which we can usually say the most are supersymmetric. In these cases, we can often make exact statements, either perturbative or non-perturbative, because of non-renormalization theorems. In cases where the vacuum structure is not renormalized at any finite order in perturbation theory, non-perturbative effects, like instantons, can still generate superpotentials which modify or destabilize perturbative vacua. Numerous examples of this kind have been studied in various dimensions; for example, N=1 supersymmetric QCD in four dimensions [1]. The aim of this work is to study the stability of two-dimensional gauge theories, both massive and massless, with (0, 2) world-sheet supersymmetry. On the string world-sheet, the terminology (p, q) supersymmetry refers to theories with p left-moving and q right-moving supersymmetries. Conformal field theories with (0, 2) supersymmetry are a key ingredient in building perturbative heterotic string compactifications (for a review, see [2]). Unlike their (2, 2) cousins, theories with (0, 2) supersymmetry that are conformal to all orders in perturbation theory can still be destabilized by world-sheet instantons. The usual phrasing of this problem is that world-sheet instantons generate a space-time superpotential [3, 4]. However, the general belief is that this destabilization is generic. Under special conditions described in [5,6] for non-linear sigma models, there can be extra fermion zero modes in an instanton background which kill any non-perturbative superpotential. We consider those (0, 2) models which can be constructed as IR limits of gauged linear sigma models [7]. This is a rather nice class of models which can be conformal or non- conformal, and which can flow to theories with IR descriptions like sigma models or Landau- Ginzburg theories. For perturbatively conformal cases, some criteria for the absence of a space-time superpotential have been described in [8–10]. Our interest is in whether a world- sheet superpotential is generated. In perturbatively conformal cases, the two questions should be related in a way that we will describe. In the following section, we consider the stability of (0, 2) theories without tree level superpotentials. We construct several examples of non-conformal (0, 2) models without tree level superpotentials for which we show that no world-sheet superpotential is generated by instantons. This result surprised us initially since we were looking for a model with an 1 instanton generated superpotential! In section three, we give a general argument based on the Konishi anomaly [11, 12] that this is true for all gauged linear sigma models without tree level superpotentials. This argument is inspired in part by recent progress in four- dimensional gauge theories [13,14]. We then extend the argument to cases with a tree level superpotential. In all cases, it appears that a non-perturbative superpotential is forbidden. Lastly, we consider the implication of our results for the space-time superpotential. Based on the absence of a non-perturbative world-sheet superpotential, we argue that there is no corresponding space-time instability. Some related observations will appear in [23]. 2 Some (0,2) Examples In this section we construct examples of (0, 2) gauged linear sigma models [7] without tree level superpotentials. We will show that no superpotential is generated by non-perturbative instanton or anti-instanton effects. It is actually sufficient to consider one instanton contri- butions. Higher instanton numbers generate more fermion zero modes which obstruct the generation of a superpotential. 2.1 A bundle over CP3 The (0, 2) superspace and superfield notations are reviewed in Appendix A. We begin by considering a U(1) gauge theory. The (0, 2) action is given by a sum of terms S = Sg + Sch + SF + SDθ + SJ (1) where Sg,Sch,SF are canonical kinetic terms for the gauge-fields, bosonic chiral superfields, and fermionic chiral superfields, respectively. The explicit form of these actions appear in Appendix A. The Fayet-Iliopoulos D-term and the theta angle appear in SDθ while SJ contains any tree level superpotential. For these models, we set SJ =0. As our first example, we construct a linear sigma model whose IR limit is a non-linear sigma model on CP3. This is a cousin of the (2, 2) model studied in [15]. Apart from the U(1) gauge superfields Ψ and V ,wehavebosonicsuperfieldsΦi = φi + ... where i =1,...,4, and a single Fermi superfield Γ. Each Φi carries gauge charge 1 while Γ carries gauge charge 2. Since we do not have a tree level superpotential our action is − S = Sg + Sch + SF + SDθ. (2) 2 Solving for the auxiliary fields gives the following bosonic potential for the φi 2 2 D e 2 2 U = = ( φi r) . (3) 2e2 2 | | − Xi We take r to be positive, and set r = η2. After modding by the U(1) gauge symmetry, we see that the target space is CP3. The Fermi superfield determines the gauge-bundle over CP3 which, in this case, is the line bundle ( 2). These particular gauge charge assignments guarantee gauge anomaly O − cancellation, which is a basic consistency requirement. This can be seen either by com- puting the requisite one loop diagrams, or by checking that the condition for anomaly cancellation [5] ch2(TM)=ch2(V )(4) 3 is satisfied. Here, TM is the tangent bundle of CP , V is the ( 2) line bundle, and ch2 O − is the second Chern character. Using the definition 1 2 ch2(X)= c (X) c2(X) 2 1 − we see that both sides of this equation gives 2J 2,whereJ is the curvature 2-form of the hyperplane bundle over CP3. This theory is massive (like the (2, 2) CP3 model) because the sum of the gauge charges of the right moving fermions is non-zero. The theory does, at the classical level, have a i chiral U(1) symmetry under which (ψ+,λ ,χ ) carry charges (1, 1,q), where q is any − − − integer. This symmetry is anomalous at one-loop for any q = 2.Thechargeofthe 6 − gaugino, λ , does not matter in the anomaly computation because it is not charged in a − U(1) theory. Since this chiral symmetry is anomalous, we can shift the theta angle to any value, and we choose to set it to zero. Let us now consider how instantons modify the perturbative theory. First we construct the one instanton BPS solution. In order to construct an instanton solution, we wick rotate to Euclidean space sending 0 2 y iy ,v01 iv12. →− →− The Euclideanised bosonic action is 2 2 1 2 2 e 2 2 2 S = d y [ v + Dαφi + ( φi η ) ]. (5) Z 2e2 12 | | 2 | | − Xi Xi 3 We construct the well known vortex instanton solution [16,17] of the Abelian Higgs model in two dimensions: take φi=0 for i=2,3,4 and take non-zero φ1 and gauge fields. From now on we refer to Φ1 as Φ for brevity. We also set e = 1. In polar coordinates, the one-instanton configuration is given by iθ vr =0,vθ = v(r),φ= f(r)e (6) where for large r, 1 e ηr v(r) + constant − , (7) ∼ r × √r √2ηr f(r) η + constant e− , (8) ∼ × and v(0) = f(0) = 0. The Bogomolnyi equations are (D1 + iD2)φ =0 (9) and D + v12 =0. (10) On evaluating (5) in this background, we easily obtain the usual instanton action S =2πη2. Next, we are interested in constructing the fermion zero modes in this instanton background. They are explicitly given by ¯0 ¯ ¯ ¯ 0 ψ+ √2(D1 + iD2)φ µ = 0 = − (11) λ D v12 − − χ¯0 = φ2, (12) − and ¯0 ¯ ψ+i = φ (13) for i =2, 3, 4. Note that these zero modes are normalizable because of the exponential fall off of the fields at large distances. The µ0 fermion zero mode is actually the zero mode generated by the broken supersymmetry generator. In order to see this, we must examine the supersymmetry transformations in the instanton background. The supersymmetry transformations become involved because we must also make gauge transformations to preserve Wess-Zumino gauge. The relevent supersymmetry transformations are given by ¯ ¯ δψ+ = i√2(D¯ 0 + D¯ 1)φ , − − 4 δλ = iD + v01 . (14) − − − The supersymmetry parameter, , corresponds to Q+. Wick rotating to Euclidean space − gives the zero mode found in (11). Hence, Q+ is the broken supersymmetry while the ¯ Q+ supersymmetry is still preserved by the instanton background. Using the Bogomolnyi equations, it is not hard to check that the µ0 zero mode does satisfy iD/ µ0 =0whereiD/ is the Dirac-Higgs operator ¯ i(D¯1 iD¯ 2) √2iφ iD/ = − − . (15) √2iφ i∂1 ∂2 − − The fieldχ ¯ is expanded in modes of the Dirac operator (D¯ 1 + iD¯ 2) (note that Γ has − ¯ gauge charge 2).

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