9 Non-perturbative e↵ects and holomorphy In this lecture we will start our program regarding the study of the non-perturbative regime of supersymmetric theories. The main point of this first lecture will be to introduce holomorphy,orbetterputholomorphy,whichisanintrinsicpropertyof supersymmetric theories, at work. Before doing that, however, there are a few standard non-perturbative field theory results we need to review. 9.1 Instantons and anomalies in a nutshell Gauge theories might contain a ✓-term, which is ✓ 1 = YM d4x Tr F F˜µ⌫ where F˜µ⌫ = ✏µ⌫⇢F . (9.1) S✓ 32⇡2 µ⌫ 2 ⇢ Z This term is a total derivative. Indeed 1 2 d4x Tr F F˜µ⌫ = d4x✏µ⌫⇢@ Tr A @ A + A A A , (9.2) 2 µ⌫ µ ⌫ ⇢ σ 3 ⌫ ⇢ σ Z Z which implies that the ✓-term does not have any e↵ect on the classical equations of motion. However, when one quantizes a theory one has to average over all fluctua- tions, not just those satisfying the classical equations of motions and therefore the ✓-term can in fact be relevant in some cases. One such configurations are instantons,classicalsolutionsoftheEuclideanaction that approach pure gauge for x (so to make the action finite). In this case | |!1 the relevant integral is in fact an integer number, the so-called instanton number of the configuration (aka winding number) ✓YM 4 ˜µ⌫ ✓ = d x Tr Fµ⌫F = n✓YM where n Z . (9.3) S 32⇡2 2 Z The instanton number is a topological quantity, in the sense that it does not change upon continuous deformations of the gauge field configuration. Moreover, since the i action enters the path integral as φe S✓ ,the✓-angle indeed behaves as an angle, D in the sense that the shift R ✓ ✓ +2⇡, (9.4) YM ! YM is a symmetry of the theory. An instanton field configuration interpolates between di↵erent vacua of the gauge theory. Both vacua are gauge equivalent to the usual vacuum with zero gauge poten- tial but the corresponding gauge transformation cannot be deformed to the identity 1 (these are known as large gauge transformations). If this were the case it would have been possible to let the field strength being vanishing in all space-time, contradict- ing eq. (9.3). The fact that Fµ⌫ cannot vanish identically for configurations with n = 0 implies that there is a field energy associated with the gauge field configura- 6 tion interpolating between the di↵erent vacua. An energy barrier and an associated quantum mechanical tunneling amplitude proportional to e−SE where is the Eu- SE clidean action of a field configuration which interpolates between the di↵erent vacua. Instantons are nothing but just such interpolating field configurations. Notice that configurations related by large gauge transformations are weighted di↵erently in the action, because of eq. (9.3), implying that they should not be identified as physically equivalent configurations. Instantons have an intrinsic non-perturbative nature. Recall that the RG-equation for the gauge coupling g reads @g b µ = 1 g3 + (g5) , (9.5) @µ −16⇡2 O where b1 is a numerical coefficient which depends on the theory. The solution of this equation at one loop is 1 b ⇤ = 1 log . (9.6) g2(µ) −8⇡2 µ where the scale ⇤is defined as the scale where the one-loop coupling diverges. It sets the scale where higher-loop and non-perturbative e↵ects should be taken into account. For any scale µ0 we have that 8⇡2 2 ⇤ µ e− b1g (µ0) . (9.7) ⌘ 0 It is important to stress that ⇤does not depend on the energy scale: it is a RG- invariant quantity. Indeed 8⇡2 2 8⇡2 @⇤ 2 8⇡ 2 b1 3 5 2 = e− b1g (µ0) + µ g (µ )+ (g ) e− b1g (µ0) @µ 0 −b g3(µ ) µ 16⇡2 0 O 0 1 0 0 ✓ ◆ 8⇡2 1 8⇡2 b g2(µ ) b g2(µ ) = e− 1 0 + µ0( )e− 1 0 =0. (9.8) −µ 0 up to higher-order corrections. This can be reiterated order by order in perturbation theory, getting the same result, namely that @⇤/@µ0 =0. Let us now consider an instanton field configuration. There exists a lower bound on the Euclidean action of an instanton. Indeed, 2 0 d4x Tr F F˜ = d4x 2TrF F µ⌫ 2TrF F˜µ⌫ (9.9) µ⌫ ± µ⌫ µ⌫ ± µ⌫ Z ⇣ ⌘ Z h i 2 which implies d4x Tr F F µ⌫ d4x Tr F F˜µ⌫ =32⇡2n, (9.10) µ⌫ ≥ µ⌫ Z Z where the last equality holds for an instanton configuration with instanton number n.Thisimpliesthatthereisalowerboundtoaninstantonaction:instanton contributions to amplitudes are suppressed at least by (multiply above equation by 1/4g2) n nb 8⇡2 1 2 ⇤ e−Sinst = e− g (µ) = (9.11) µ ✓ ◆ ✓ ◆ where in the last step we have used eq. (9.6). This shows that instantons are inherently non-perturbative e↵ects, since they vanish for ⇤ 0, and are very weak, ! if not negligible, in the perturbative regime. Anomalies are classical symmetries of the action which are broken by quantum e↵ects. In other words, we have µ quantum µ @µj =0 @µj =0, (9.12) A −−corrections − − − − − ! A 6 where jA is the current associated to the anomalous symmetry. In what follows we will focus on chiral anomalies, that is anomalies associated to chiral currents. These arise in field theories in which fermions with chiral symmetries are coupled to gauge fields. Recall that local currents cannot be anomalous, since they would imply violation of unitarity of the theory (we only know how to couple spin-1 fields in a way respecting unitarity to conserved currents). Hence a quantum field theory, in order to make sense, should not have any gauge anomaly. On the contrary, global chiral currents can be anomalous. This is what is usually meant as chiral anomaly. Let us review a few basic facts about anomalies. Anomalies get contribution at one loop, only, by evaluation of triangle diagrams like those reported in Figure 9.1. Let us first consider a set of global currents jA. The one-loop three-point function corresponding to diagram a of Figure 9.1 will be jµ (x )j⌫ (x )j⇢ (x ) =Tr(t t t ) f (x ) , (9.13) h A 1 B 2 C 3 i A B C µ⌫⇢ i where the trace comes from contraction of the group generators around the loop. This correlator has important properties, as we will see momentarily, but it does not provide by itself any anomaly: the corresponding classical conservation law is not violated quantum mechanically. 3 j B VB VB + jA ; jjA A VC jC VC a bc Figure 9.1: One-loop diagrams contributing to correlators of one global current with two global or local currents. Diagram a does not provide any anomaly. Diagrams b and c,instead,contributetotheanomalyoftheglobalcurrentjA. Suppose now to gauge some (or all) global currents, by coupling the original Lagrangian with gauge fields as = + V B jµ , (9.14) L Lfree µ B B X and let us compute the correlator j V V .Theone-loopdiagramscontributing h A B C i to such correlator are diagrams b and c of Figure 9.1. By di↵erentiating the result one gets @ jµ Tr (t t ,t ) F µ⌫F˜ , (9.15) µ A ⇠ A { B C } B C,µ⌫ and we do have an anomaly now. We clearly see here that jA should be a global current, since if this were not the case we would have had a violation of unitarity in the quantum theory. From the above formula we see that the anomaly coefficient is proportional to Tr (t t ,t ). This vanishes for real and pseudoreal representations. Indeed, for A { B C } real or pseudoreal representations we have that t = (t )T and it then easily A − A follows that Tr (t t ,t )= Tr (t t ,t ) . (9.16) r A { B C } − r A { B C } Therefore, only massless chiral fermions can contribute to the anomaly coefficient. This result, once applied to local currents, which cannot be anomalous, provides severe restrictions on the massless fermion content of a quantum field theory. Suppose to have a theory with gauge group G with generators tA,aglobalsym- metry group G˜ with generators t˜A,andasetofWeylfermions i,transformingin the representations (ri, r˜i)ofthegaugeandglobalsymmetrygroups,respectively. In this case, the ABJ anomaly computation gives @ jµ Tr t˜ Tr (t t ) F µ⌫F˜ . (9.17) µ A ⇠ r˜i A ri B C B Cµ⌫ i X 4 Since Tr t˜ =0foranysimplealgebra,onlyabelianfactorsU(1) G˜ can be r˜i A ⇢ anomalous. On the other hand, Trri (tBtC )=C(ri)δBC,whereC(ri)isthequadratic invariant (Casimir) of the representation ri.Workingeverythingout,payingatten- tion to numerical coefficients, one finally gets for an abelian group A @ jµ = F µ⌫F˜B , (9.18) µ −16⇡2 B µ⌫ where A = i qiC(ri)istheanomalycoefficient,qi being the U(1) global charges of fermion fields . P i This result shows the connection between anomalies, instantons and the ✓-angle. First, we see that the anomaly is proportional to the instanton number we have previously defined. Indeed, integrating eq. (9.18) in space-time, and comparing with eq. (9.3), we get ∆Q =2An , (9.19) where n is the instanton number and ∆Q the amount of charge violation due to the anomaly. So we see that anomalous symmetries are violated by a specific amount, given by eq. (9.19), in an instanton background. This also shows that anomalies are IR e↵ects, since the violation is very mild at weak coupling.