Projektarbeit. Supersymmetric Landau-Ginzburg model

Transcription

1 Projektarbeit Supersymmetric Landau-Ginzburg model Ausgeführt am Institut für Theoretische Physik der Technischen Universität Wien unter der Anleitung von Univ.Ass. Dipl.-Ing. Dr.rer.nat. Johanna Knapp und Privatdoz. Dipl.-Ing. Dr.techn. Herbert Balasin durch Philipp Stanzer Matrikelnr.: Mollgasse 13/ Wien Wien, am

2 Abstract A supersymmetric field theory can be tested by calculating the so called Witten-Index. This number gives information about the existence of supersymmetric ground states. If there is no supersymmetric ground state, it can be assumed that the supersymmetry is broken. This thesis will focus on a supersymmetric Landau-Ginzburg model in one dimension. The Witten index will be calculated, including technical details about the symmetries of the action and the supercharges. 1

3 Contents 1 Introduction Supersymmetry Witten index Example: 1D - Landau-Ginzburg model Starting point The fields and anti-)commutation relations The Lagrangian The transformations Equations of motion and conjugate momenta of the fields Variation of the Lagrangian Supercharges Noether procedure Anticommutation relations Hamilton operator Witten index of LG-model Summary and conclusion 22 A Variation 23 B Supercharges 24 B.1 Derivation B.2 Anticommutation relations

4 1 Introduction 1.1 Supersymmetry The idea, that every bosonic particle has a fermionic partner was created in the 1970 s and is called supersymmetry. In the following decades many people worked on this concept and figured out several important consequences. On one hand, supersymmetry makes it possible to address many problems e.g. the hierarchy problem) and on the other hand it is necessary for string theory to be consistent. Another point is, that the Coleman-Mandula theorem states, that all interactions vanish, if the symmetry group of a theory mixes spatial and internal symmetries. According to this theorem, supersymmetry is the only possible extension of Poincaré invariance in 3+1 or higher) dimensions. [1] The foundation of supersymmetry is, that the Hilbert space of states can be decomposed in a bosonic and a fermionic space. The generator of the symmetry relates bosonic states with fermionic ones and thus one can interpret it as an anticommuting, spinorial operator relating the elements of the two subspaces. They satisfy a symmetry algebra consisting of anticommutation and commutation relations. The irreducible representations of this algebra are called supermultiplets. Every single particle state forms a supermultiplet with its superpartner, such that all bosonic states are paired with fermionic ones and vice versa. It is also possible to introduce more than one supersymmetry generator. This is called extended supersymmetry, but in four dimensions it is not implemented in nature for several reasons. If one considers higher dimensional models with convenient compactification of some dimensions, extended supersymmetry can possibly describe nature. [2] As long as the supersymmetry is unbroken, both of the paired states have the same energy and same quantum numbers except spin). If this was the case, some of the superpartners of known particles should have been found by experimentalists. This lack of evidence, although supersymmetric field theories are consistent, leads to the assumption, that supersymmetry is broken at tree level). This means, that there are no supersymmetric ground states. This work is organized as follows: in the second part of the introduction we give a short overview about the Witten index. In the main part we discuss a supersymmetric one dimensional Landau-Ginzburg model. Therefore we define the fields 3

5 and the Lagrangian of the theory an show that supersymmetry holds on-shell. Then we calculate the Noether charges i.e. supercharges) and show their connection to the Hamiltonian. Finally we compute the Witten index for this symmetry, using the path integral formalism. 1.2 Witten index Given a supersymmetric theory, one needs a method to determine, whether supersymmetry is broken or not. Already in the early 80 s Witten [3, 4] introduced a method, using an operator ) F. This operator distinguishes bosonic and fermionic states ) F bos = + bos, 1) ) F fer = fer. 2) F represents the fermion number, which is 1 for fermionic states and 0 for bosonic states 1. The generators of supersymmetry, the so called supercharges Q and Q, transform bosonic states into fermionic ones and vice versa, as mentioned in the previous section 1.1. In the zero momentum sector of the Hilbert space, we can express the Hamiltonian in terms of the supercharges 3) H = {Q, Q}. 4) We will see that in section for the one dimensional Landau-Ginzburg model. The Hamiltonian commutes with 1) F, while the supercharges anticommute with it. Taking the trace of this operator over all possible states in the Hilbert space gives some information about the breaking of supersymmetry. As all bosonic states with positive energy have fermionic partners with the same energy, their contributions to the trace cancel due to the extra minus sign from ) F. For states with zero energy i.e. ground states, GS) the situation is a bit different, because H 0 = E 0 = 0, 5) 1 Combining commuting and anticommuting elements into one algebra requires a Z 2 grading, bosons are the even elements and fermions are the odd ones. Such an algebra is called Lie superalgebra [5]. 4

6 implies that each supercharge annihilates the state Q 0 = 0, 6) Q 0 = 0. 7) Thus the ground states do not have to be paired, because they are invariant under the supersymmetry transformations generated by Q and Q. Therefore the Witten index counts the difference between bosonic and fermionic zero energy states T r ) F = #GS bos #GS fer. 8) If this number is not zero, supersymmetry is unbroken [6], because there exists a supersymmetric ground state. We know that this state is a supersymmetric ground state, because it can not be lifted to positive energy, because it has no superpartner. It is worth noting, that the converse is not necessarily true, because it is possible, that there exist as many bosonic as fermionic ground states, although supersymmetry is not broken. The operator ) F is not a trace class operator 2, so actually one needs to write the Witten index like T r [ ) F e βh], 9) with β R. A calculation in [8] shows that the Witten index is independent of this factor. Using the path integral formalism, the partition function on a circle with raduis β can be written as Z = T re βh ) = DφDψD ψ AP e Sφ,ψ, ψ), 10) where φ represents bosonic fields and ψ, ψ fermionic ones. The subscript AP refers to antiperiodic boundary conditions of the fermions on the circle [8]. To get an expression for the Witten index, we need to insert ) F, which changes the boundary conditions to periodic indicated by P ), because of the extra minus sign for fermions T r [ ) F e βh] = DφDψD ψ P e Sφ,ψ, ψ). 11) Summarizing one can say the calculation of the Witten index is, compared to other quantities, a quite simple possibility to check a theory for supersymmetry breaking. 2 In mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis [7]. 5

7 2 Example: 1D - Landau-Ginzburg model The model we focus on is formally similar to a well known model in solid state physics, and therefore called Landau-Ginzburg model. In string theory on the other hand it is possible to end up with this kind of model in higher dimensions. Specific, non-geometric loci in the parameter space of Calabi-Yau compactifications can lead to certain supersymmetric Landau-Ginzburg models. Choosing a one dimensional toy model makes it easier to get familiar with supersymmetric quantum field theory and to calculate certain results. Following [8] we discuss a one dimensional, supersymmetric Landau-Ginzburg theory with complex variables. 2.1 Starting point The fields and anti-)commutation relations To get a complex theory, we write 2m real bosonic variables x I as m complex variables and their complex conjugates 3 z i = 1 2 x i + iy i ), 12) z i = 1 2 x i iy i ), 13) where the x i represent the x I for I = 1...m and the y i represent the x I for I = m + 1)...2m. The same is done with the 2m fermionic variables ψ I and their conjugates ψ I, so we have 3 ψ i = 1 2 ψ i x + iψ i y), ψi = 1 2 ψ i x + i ψ i y), 14) ψī = 1 2 ψ i x iψ i y), ψī = 1 2 ψ i x i ψ i y), 15) where the ψ i x represent the ψ I for I = 1...m and the ψ i y represent the ψ I for I = m + 1)...2m and analogously for ψ. These complex quantities can be conjugated, which leads to 15). The hermitian conjugates are on. ψ i ) = ψī, 16) 3 Compared to the definition in [8] we use a factor 1 2, which avoids some inconsistencies later 6

8 ψ i ) = ψī. 17) Now that we have defined the fields of the Landau-Ginzburg model, we can take a look at the anti-)commutation relations. In a theory with real fields we have [z i, z j ] = 0, 18) {ψ I, ψ J } = 0, 19) {ψ I, ψ J } = δ IJ. 20) From this we can calculate the relations in our complex theory [z i, z j ] = 0, 21) [ z i, z j ] = 0, 22) [z i, z j ] = 0, 23) as all bosonic variables commute. For the fermionic ones, it is a bit more difficult {ψ i, ψ j ) } = 1 2 {ψi x + iψ i y, ψ j x i ψ j y} = = 1 2 {ψi x, ψ j x} {ψi x, i ψ j y} {iψi y, ψ j x} {iψi y, i ψ j y} = = 1 2 δij δij = δ ij, 24) {ψ i, ψ j } = 1 2 {ψi x + iψ i y, ψ j x + i ψ j y} = = 1 2 {ψi x, ψ j x} {ψi x, i ψ j y} {iψi y, ψ j x} {iψi y, i ψ j y} = = 1 2 δij δij = 0, 25) {ψ i, ψ j } = 0, 26) using the relations above and the nilpotency of anticommuting objects. In analogous calculations we find { ψ i, ψ j ) } = δ ij, 27) {ψ i ), ψ j ) } = 0, 28) {ψ i, ψ j ) } = 0, 29) { ψ i, ψ j ) } = 0, 30) { ψ i, ψ j } = 0, 31) {ψ i ), ψ j ) } = 0, 32) 7

9 { ψ i ), ψ j ) } = 0. 33) All commutators between bosonic and fermionic fields also vanish. As we consider a one dimensional theory it should be noted, that all fields depend on time only and not on any spatial coordinates The Lagrangian Now we can take a look at the classical) Landau-Ginzburg Lagrangian L LG = m i=1 1 2 ż 2 + i ψ i t ψī + i ψī t ψ i 14 iw 2 ) m i,j=1 i j W ψ i ψj + ī j W ) ψī ψ j, 34) where the function W is called the superpotential.we will restrict ourselves to holomorphic functions W = W z) and antiholomorphic functions W = W z). In m the following, the Einstein summation convention will be used: x i x i = x i x i. By i we denote the derivative with respect to z i, ī denotes the derivative with respect to z i. We will write i instead of ī. For convenience, we will write the Landau-Ginzburg Lagrangian in the following form L LG = ż i z i + i ψ i t ψ i ) + iψ i ) t ψi 1 ) 4 iw i W 1 i j W ψ i 2 + i j W ψi ) ψ j ) ) = 35) =L LG,bos + L LG,fcc + L LG,fnc + L LG,pot + L LG,mix + L LG,mhc. 36) Each contribution to the Lagrangian 35) is abbreviated term by term as in 36). i= The transformations As we are talking about supersymmetric QFT, we need some supersymmetry transformations. Due to the fact that we have complex fields, we need two constant, complex, fermionic parameters ɛ ±. The transformations read δz i = ɛ + ψi ɛ ψ i, 37) 8

10 δψ i = i ɛ ż i ɛ + i W, 38) δ ψ i = i ɛ + ż i ɛ i W, 39) and for the hermitian conjugates δ z i = ɛ + ψ i ) + ɛ ψ i ), 40) δψ i ) = iɛ z i ɛ + i W, 41) δ ψ i ) = iɛ + z i ɛ i W. 42) 2.2 Equations of motion and conjugate momenta of the fields First we calculate the conjugate momenta for each field, starting with z i and z i p iz) = L ż = z i, i 43) p i z) = L z = i żi. 44) These are the classical momenta, which are replaced by the momentum operators for quantum mechanics Continuing with ψ i and ψ i ) p iz) = z i ˆp iz) = i z = i i, i 45) p i z) = ż i ˆp i z) = i z = i i i, 46) π iψ) = L ψ = i iψi ), 47) π iψ ) = L ψ = 0, i ) 48) where we used that ψ anticommutes with ψ. Finishing with ψ i and ψ i ) π i ψ) = L = 0, ψi 49) π i ψ ) = L ψi ) = i ψ i. 50) 9

11 Now we calculate the equations of motion for this theory by using the Euler- Lagrange equations where φ stands for any of the fields z, ψ, = L φ d L i dt φ 51) i For z i we find For z i we find For ψ i we find z i = 1 4 i j W j W 1 2 i j k W ψ j ψk. 52) z i = 1 4 jw i j W 1 2 i j k W ψj ) ψ k ). 53) t ψi ) = i 2 i j W ψ j. 54) For ψ i ) we find t ψi = i 2 j i W ψj ). 55) For ψ i we find t ψ i ) = i 2 j i W ψ j. 56) For ψ i ) we find t ψ i = i 2 i j W ψ j ). 57) 2.3 Variation of the Lagrangian If the Lagrangian is supersymmetric, the variation of the action with respect to the fields must vanish up to the equations of motion δs = δ L LG = δl LG = 0. 58) 10

12 To check this, we start with the variation of the bosonic part of the Lagrangian δl LG,bos = δż i z i ) = δż i ) z i + ż i δ z i ) = = t δzi ) z i + ż i t δ zi ) = = ɛ+ t ψi ɛ ψ i) z i + ż i ɛ+ t ψ i ) + ɛ ψ i ) ) = = ɛ + t ψ i ) z i ɛ t ψi ) z i ż i ɛ + t ψ i ) + ż i ɛ t ψi ). 59) Next we calculate the variation of the complex-conjugate) fermionic part. As the ɛ ± are fermionic, one must be careful commuting them with any ψ δl LG,fcc = δ i ψ i t ) ψ i ) = iδ ψ i ) t ψ i ) + i ψ i t δ ψ i ) = = i ) i ɛ + ż i ɛ i W t ψ i ) + i ψ i iɛ+ z i ɛ i W ) = t = ɛ + ż i t ψ i ) iɛ i W t ψ i ) + ɛ + ψi z i + i ɛ ψi t iw ), 60) and the non-complex-conjugate) fermionic part δl LG,fnc = ɛ z i t ψi ) i ɛ + i W t ψi ) + ɛ ψ i ) z i + iɛ + ψ i ) t i W ). 61) Because we will need this later on, we take a look at the variation of the superpotential δw z a ) = m W z a )δz m, 62) δ i j W z a )) = m i j W z a )δz m, 63) δ W z a ) = m W z a )δ z m, 64) δ i j W z a )) = m i j W z a )δ z m. 65) Now we need to calculate the variation of the superpotential term δl LG,pot =δ 1 ) 4 iw i W = 1 ) δ i W ) i W + i W δ i W ) = 4 = 1 [ m i W δz m i W + i W 4 m i W ] δ z m = = 1 [ m i W 4 i W ɛ+ ψm ɛ ψ m) + + i W m i W ɛ+ ψ m ) + ɛ ψ m ) )] = 11

13 = ɛ + 4 m i W i W ψm + ɛ 4 m i W i W ψ m + + ɛ + 4 iw m i W ψm ) ɛ 4 iw m i W ψ m ), 66) and the first term of the mixed fermionic part with the superpotential involved δl LG,mix = ɛ + 2 m i j W ψ m ψ i ψj + ɛ 2 m i j W ψ m ψ i ψj i ɛ 2 i j W ż i ψj + ɛ + 2 i j W i W ψj i ɛ + 2 i j W ψ i ż j ɛ 2 i j W ψ i j W. 67) The last term is the hermitian conjugate of the above one δl LG,mhc = ɛ + 2 m i j W ψm ) ψ i ) ψ j ) ɛ 2 m i j W ψ m ) ψ i ) ψ j ) iɛ + 2 i j W z i ψ j ) + ɛ 2 i j W i W ψ j ) iɛ 2 i j W ψi ) z j ɛ + 2 i j W ψi ) j W. 68) Remembering equation 36) we can write the complete variation as δl LG =δl LG,bos + δl LG,fcc + δl LG,fnc + δl LG,pot + δl LG,mix + δl LG,mhc. 69) Since all ɛ are independent, we can split the variation in four parts, which have to vanish separately. Terms with ɛ + : Terms with ɛ : A = t ψ i ) z i 1 4 m i W i W ψm 1 2 m i j W ψ m ψ i ψj i j W i W ψj + ψ i z i + iψ i ) t i W ) i 2 i j W z i ψ j ). 70) B = t ψi ) z i + z i t ψi ) m i W i W ψ m m i j W ψ m ψ i ψj Terms with ɛ + : 1 2 i j W ψ i j W i i W t ψ i ) i 2 i j W ψi ) z j. 71) C = ż i t ψ i ) + ż i t ψ i ) i i W t ψi ) iw m i W ψm ) 12

14 Terms with ɛ : i 2 i j W ψ i ż j m i j W ψm ) ψ i ) ψ j ) 1 2 i j W ψi ) j W. 72) D = ż i t ψi ) + i ψ i t iw ) + ψ i ) z i 1 4 iw m i W ψ m ) i 2 i j W ż i ψj 1 2 m i j W ψ m ) ψ i ) ψ j ) i j W i W ψ j ). 73) To show that the variation is zero, we use the antisymmetry of the fermionic fields ψ m ψj with the symmetry of the derivations m j and rewrite some of the terms A = t ψ i ) z i + z m ψm 1 4 m i W i W ψm i j W ψ j i W + + iψ i ) t i W ) i 2 i j W z i ψ j ). 74) Writing i j W z i as t j W ) yields Using equation 54) leads to A = t ψ i ) z i + z m ψm m i W i W ψm + + iψ i ) t i i W ) 2 t j W )ψ j ). 75) A = t ψ i ) z i + z m ψm + i 2 t ψi ) i i W + 2 ψi ) t i W ). 76) To see, that the action will indeed be zero, we can write this as a total timederivative Using the same methods see Appendix A), we find A = t ψ i z i + i 2 ψi ) i W ). 77) B = t i 2 ψ j ) j W ), 78) C = t i 2 ψi i W ), 79) D = t [ψi ) ż i + i 2 ψ i i W ], 80) So one can see, that the variation of the action vanishes, as there is no boundary δs = dtδl LG = dtɛ + A + ɛ B + ɛ + C + ɛ D) = 0. 81) 13

15 2.4 Supercharges Noether procedure To find the conserved charges of the supersymmetry transformations, the supercharges Q ± and Q ±, we follow the Noether procedure. This means repeating the variation of the Lagrangian with time dependent ɛ ± and ɛ ±, denoted by ˆδ. We should end up with an expression like ˆδS = dtˆδl LG = dt ɛ ) + Q ɛ Q + ɛ + Q + ɛ Q+ + δl LG = 0, 82) where all terms except time derivatives of the variation parameter will vanish, defining the supercharges [8]. Using the same abbreviation 36) as before, the first part yields ˆδL LG,bos =ˆδż i z i ) = ˆδż i ) z i + ż iˆδ z i ) = The second part yields The third part yields = t ˆδz i ) z i + ż i t ˆδ z i ) = = ɛ+ t ψi ɛ ψ i) z i + ż i ɛ+ t ψ i ) + ɛ ψ i ) ) = = ɛ + ψi z i + ɛ + t ψ i ) z i ɛ ψ i z i ɛ t ψi ) z i ż i ɛ + ψ i ) ż i ɛ + t ψ i ) + ż i ɛ ψ i ) + ż i ɛ t ψi ) = = ɛ + ψi z i ɛ ψ i z i ż i ɛ + ψ i ) + ż i ɛ ψ i ) + δl LG,bos. 83) ˆδL LG,fcc = ɛ + ψi z i + i ɛ ψi i W + δl LG,fcc. 84) ˆδL LG,fnc = ɛ ψ i ) ż i + i ɛ + ψ i ) i W + δllg,fnc. 85) The remaining three contributions equations 67), 68) and 66)) do not change, because there is no time derivative involved. To find the supercharges, we are interested in the coefficients of the ɛ. Thus we first focus on all terms involving ɛ +. Using equation 77) and the new terms from above, we find 0 = dt ɛ + 2 ψ i z i + iψ i ) i W ) + ɛ+ t [ ψ i z i + i ) 2 ψi ) i W ]. 86) 14

16 Rewriting the terms using partial integration leads to 0 = dt { ɛ + 2 ψ i z i + iψ i ) i W ) + + t [ɛ + ψ i z i + i 2 ψi ) i W )] ɛ+ [ ψ i z i + i } 2 ψi ) i W ] = = dt ɛ + ψ i z i + i 2 ψi ) i W ) + t [ɛ + ψ i z i + i ) 2 ψi ) i W )]. 87) Comparing this with equation 82) results in our first supercharge Q = ψ i p iz) + i 2 ψi ) i W. 88) Repeating this calculation for the other supercharges see Appendix B.1), leads to This is in agreement with [8]. Q + =ψ i p iz) i 2 ψ i ) i W, 89) Q = ψ i ) p i i 2 ψi i W, 90) Q + =ψ i ) p i + i 2 ψ i i W. 91) Anticommutation relations Once the supercharges are determined, one can calculate anticommutation relations using the relations for the fields and p iz) = i i {Q +, Q + } ={ψ i p iz) i 2 ψ i ) i W, ψ j p jz) i 2 ψ j ) j W } = =p iz) p jz) {ψ i, ψ j } 1 4 i W j W { ψi ), ψ j ) } i 2 p iz) j W {ψ i, ψ j ) } i 2 i W pjz) { ψ i ), ψ j } = = = 0. 92) Here we could pull p iz) and j W and pi z) and j W respectively) out of the anti-commutator, because W and W are anti-)holomorphic and thus, they commute.the same arguments hold for {Q, Q } =0, 93) 15

17 { Q, Q } =0, 94) { Q +, Q + } =0. 95) In the following relation, the p iz) affects j W and so we get {Q +, Q + } ={ψ i p iz) i 2 ψ i ) i W, ψ j ) p j z) + i 2 ψ j j W } = =p iz) p j z) {ψ i, ψ j ) } iw j W { ψi ), ψ j }+ + i 2 {ψi p iz), ψ j j W } i 2 { ψ i ) i W, ψ j ) p j z) } = =p iz) p i z) iw 2 + i 2 ψi p iz) ψ j j W ) + i 2 ψ j j W ψ i p iz) i 2 ψ i ) i W ψ j ) p j z) i 2 ψj ) p j z) [ ψ i ) i W ] = =p iz) p i z) iw 2 + i 2 ψi ψj p iz) j W + + i 2 ψi ψj j W p iz) + i 2 ψ j ψ i j W p iz) i 2 ψ i ) ψ j ) i W pj z) i 2 ψj ) ψ i ) p j z) i W i 2 ψj ) ψ i ) i W pj z) = =p iz) p i z) iw 2 + i 2 ψi ψj i i j W ) + i 2 jw p iz) {ψ i, ψ j } i 2 ψj ) ψ i ) i j i W ) i 2 i W pj z) { ψ i ), ψ j ) } = =p iz) p i z) iw i j W ψ i ψj i j W ψi ) ψ j ) = H, 96) {Q, Q } =H. 97) In the next section we will see, that this is indeed the Hamilton operator. The remaining relations are {Q +, Q } ={ψ i p iz) i 2 ψ i ) i W, ψj ) p j z) i 2 ψj j W } = =p iz) p j z) {ψ i, ψ j ) } 1 4 iw 2 { ψ i ), ψ j } i 2 {ψi p iz), ψ j j W } i 2 { ψ i ) i W, ψj ) p j z) } = =0 0 i 2 ψi p iz) ψ j j W ) i 2 ψj j W ψ i p iz) 16

18 i 2 ψ i ) i W ψj ) p j z) i 2 ψ j ) p j z) [ ψ i ) i W ] = = i 2 ψi ψ j p iz) j W ) i 2 ψi ψ j j W p iz) i 2 ψj j W ψ i p iz) i 2 ψ i ) i W ψj ) p j z) i 2 ψ j ) ψ i ) p j z) i W ) i 2 ψ j ) ψ i ) i W pj z) = = 1 2 ψi ψ j i j W 1 2 ψ j ) ψ i ) j i W = 0, 98) {Q, Q + } =0, 99) due to the antisymmetry of the fermionic fields and the symmetry of the derivatives. See Appendix B.2 for all relations, which are not written explicitly here Hamilton operator The Hamilton operator can be obtained from the Lagrangian in the following way H = q i p i L, 100) all fields where q i are the fields, p i are their conjugate momenta and L is the Lagrangian. Therefore we get H =ż i p iz) + z i p i z) + ψ i π iψ) ψi ) π i ψ ) L = =2 ż 2 i ψ i ψ i ) i ψi ) ψi ż i z i i ψ i t ψ i ) iψ i ) t ψi iw i W i j W ψ i ψj i j W ψi ) ψ j ) = = ż iw i j W ψ i ψj i j W ψi ) ψ j ) i ψ i ψ i ) i ψi ) ψi i ψ i t ψ i ) iψ i ) t ψi = = ż iw i j W ψ i ψj i j W ψi ) ψ j ) i{ ψ i, ψ i ) } i{ ψi ), ψ i }. 101) To compare this Hamiltonian with the result from the supercharges 96), we need to go on-shell i.e. insert the equations of motion). This is needed, because the 17

19 symmetry of the Lagrangian holds only on-shell and the equations of motion were used in the derivation of the supercharges. We insert equations 54) and 57) into the anticommutator and see, that it is zero. The on-shell Hamiltonian is H on shell = ż iw i j W ψ i ψj i j W ψi ) ψ j ). 102) 2.5 Witten index of LG-model As we saw in the introduction, the Witten index can be calculated via the path integral formalism, using equation 11). Therefore we need to calculate the Euclidean action with a suitable Lagrangian L E. Starting from a standard path integral DφDψD ψ e isφ,ψ, ψ) = = DφDψD ψ e i Lφ,ψ, ψ)dt, 103) we perform a Wick rotation t = iτ DφDψD ψ e is W ickφ,ψ, ψ) = = DφDψD ψ e i L W ick φ,ψ, ψ)d iτ) = = DφDψD ψ e L W ick φ,ψ, ψ)dτ, 104) and compare it to the Witten index 11) Tr 1) F = DφDψD ψ P e S Eφ,ψ, ψ) = = DφDψD ψ P e [ L W ick φ,ψ, ψ)]dτ. 105) The minus sign in front of the integral is needed later on to solve the path integral. Now we can apply the Wick rotation to our Lagrangian 35). As a first step, we decompose the fields into their components L LG = 1 2 t xi + iy i ) t xi iy i ) ψ x i + i ψ y)i i t ψi x iψy)+ i 18

20 + 1 2 ψ i x i ψ i y)i t ψi x + iψ i y) 1 4 iw i W 1 4 i j W ψ i x + iψ i y) ψ j x + i ψ j y) 1 4 i j W ψ i x iψ i y) ψ j x i ψ j y) = =... = 1 2 xi 2 t 2 xi 1 2 yi 2 t 2 yi 1 4 iw i W + + ψ x iδ i ij t + 14 ) i j W + i j W ) ψx+ j + ψ y iδ i ij t 14 ) i j W + i j W ) ψy j 1 4 i j W i j W )iψ i y ψj x + ψ i x ψ j y). 106) Now we can perform the Wick rotation L LG,W ick = 1 2 xi 2 τ 2 xi 1 2 yi 2 τ 2 yi iw i W + + ψ x i δ ij τ 1 ) 4 i j W + i j W ) ψx+ j + ψ y i δ ij τ + 1 ) 4 i j W + i j W ) ψy+ j i j W i j W )iψ i y ψj x + ψ i x ψ j y). 107) The next step is to make some assumptions on the superpotential. As mentioned in section W and W are anti-)holomorphic functions and therefore the critical points W = 0) are the same and we assume these points to be non-degenerate i j W 0). The second derivatives i j W and i j W give the same result. We make an expansion of the superpotential around the critical points z n W z) = W z n ) i j W z n )z i z n )z j z n ) + Oz 3 ). 108) The first derivatives, neglecting the higher order terms, are k W z) = i j W z n )z k z n ). 109) For convenience it is possible to perform a coordinate transformation such that z k = z k z n ) [8].The same works for the complex conjugate W z) = W z n ) i j W zn ) z i z n ) z j z n ) + O z 3 ), 110) k W z) = i j W zn ) z k z n ). 111) 19

21 Inserting this into the Lagrangian results in L LG,W ick xi 2 τ 2 xi 2ỹi τ 2 ỹi i j W z i i j W z i + + ψ x i δ ij τ 1 ) 4 i j W + i j W ) ψx+ j + ψ y i δ ij τ + 1 ) 4 i j W + i j W ) ψy+ j i j W i j W )iψ i y ψj x + ψxi i ψ y) j = = 1 2 xi 2 τ + ) i j W ) 2 x i + 12ỹi τ + i j W ) ψ x i δ ij τ 1 ) 2 i j W ψx j + ψ y i δ ij τ i j W which can be inserted into the path integral 105) T r 1) F = D x i Dỹ i Dψ i Dψ i ) D ψ i D ψ i ) P e S E = = D[ x i ỹ i ψ i ψ i ) ψi ψ i ) ] P e dτ[ 1 2 xi e dτ[ 1 2 ỹi ) 2 τ 2 + i j W ) 2 ỹ 4 i ] e dτ[ ψ i xδ ij τ 1 2 i j W)ψ j x] ) ỹ i + ) 2 τ 2 + i j W ) 2 x 4 i ] ) ψ j y, 112) e dτ[ ψ i yδ ij τ i j W)ψ j y]. 113) As indicated by P we have periodic boundary conditions, which we need in order to be able to evaluate the path integral on a finite volume [9] for instance a circle with radius β = 1 [8]). Expanding the exponential functions in the last two lines, using properties of Grassmann variables and the rules of Grassmann integration, one can integrate the fermionic part T r 1) F = D[ x i ỹ i ] e dτ[ 1 2 xi e dτ[ 1 2 ỹi ) 2 τ 2 + i j W ) 2 x 4 i ] ) 2 τ 2 + i j W ) 2 ỹ 4 i ] detδ ij τ 1 2 i j W ) detδ ij τ i j W ). 114) We use the Gaussian form of the remaining, bosonic path integral to solve it also 20

22 using deta) detb) = detab)) T r 1) F = detδij τ 1 2 i j W ) detδ ij τ i j W ) det τ 2 + i j W ) 2 ) det 4 τ 2 + i j W ) 2 ) 4 = detδij 2 τ i j W )2 4 ) det 2 τ + i j W ) 2 4 ). 115) To evaluate the determinant we need to think of the operator τ acting on the fields. As we are on a circle, we can write the fields with Fourier modes φ n e inτ and insert in the eigenvalue) into the determinant T r 1) F = in) 2 i j W ) 2 ) m 4 n Z in) 2 + i j W ) 2 ) = 4 m = ) n 2 i j W ) 2 m ) 4 = 1) m, 116) n Z n 2 + i j W ) 2 ) 4 where m is the dimension of the complex fields. We also used that the determinant of a diagonal matrix, with identical entries d, is d n where n is its dimension). Therefore the Witten index for every critical point is the same and to find it for the whole theory, we need to sum over all critical points i.e. all possible ground states) = T r 1) F = N z c 1) m. 117) 21

23 3 Summary and conclusion Given a Lagrangian and some supersymmetry transformations, we explicitly check supersymmetry invariance. Using the Noether procedure we calculated the supercharges i.e. the generators of the supersymmetry). We further showed that a defining property of such theories, the relation between the supercharges and the Hamiltonian, is also fulfilled. Furthermore we calculated the Witten index via the path integral formalism, which allowed us to determine whether the supersymmetry is broken for a specific superpotential or not. It turned out that the Witten index depends only on the dimension of the model, if the superpotential meets some assumptions like holomorphy. Another interesting fact is, that the Witten index is the same for every critical point of the superpotential as long as they are non degenerate and disjoint), which corresponds to a possible ground state of the theory. On the other hand, the theory we used for the calculations is just a toy model and for an application in string theory for instance, it will be necessary to use higher dimensional models. Nonetheless, our calculations showed, that the supersymmetry is unbroken, if the superpotential has at least one critical point. 22

24 A Variation In section 2.3 we saw that the variation of the Lagrangian can be written as δl LG = ɛ + A + ɛ B + ɛ + C + ɛ D. 118) Using the same methods as for the first part A equations 74) - 77)), we find for the other ones B = 1 4 m i W i W ψ m 1 2 i j W ψ i j W i i W t ψ i ) i 2 i j W z j ψ i ) = = i 2 t ψ j ) j W i t ψ j ) i j W 2 t i W ) ψi ) = = t i 2 ψ j ) j W ), 119) C = i i W t ψi ) iw m i W ψm ) i 2 i j W ż j ψ i 1 2 i j W ψi ) j W = = i i W t ψi ) + i 2 iw t ψi ) i 2 t iw )ψ i ) = = t i 2 ψi i W ), 120) D = ż i t ψi ) + i ψ i t iw ) + ψ i ) z i 1 4 iw m i W ψ m ) i 2 i j W ż i ψj 1 2 i j W ψ j ) i W = = ż i t ψi ) + ψ i ) z i + i 2 ψ i t iw ) + i 2 t ψ i i W ) = = t [ψi ) ż i + i 2 ψ i i W ]. 121) 23

25 B Supercharges B.1 Derivation Repeating the calculation in 86) to 88) for the other supercharges leads to 0 = dt ɛ ψ i z i + ɛ t i ) 2 ψ j ) j W ) = = dt ɛ ψ i z i + t [ɛ i2 ψ j ) j W )] ɛ [ i2 ) ψ j ) j W ] = = dt ɛ ψ i z i + i 2 ψ i ) i W ) + t [ɛ ψ i z i + i ) 2 ψ i ) i W )], 122) Q + =ψ i p iz) i 2 ψ i ) i W, 123) 0 = = = dt dt dt t i ) 2 ψi i W ) ɛ + ψ i ) ż i + ɛ + = ɛ + ψ i ) ż i + t [ ɛ + i 2 ψi i W )] ɛ + i ) 2 ψi i W ) = ɛ + ψ i ) ż i + i 2 ψi i W ) + t [ ɛ + i ) 2 ψi i W )], 124) Q = ψ i ) p i z) i 2 ψi i W, 125) 0 = = dt ɛ [2ψ i ) ż i + i ψ i i W ] + ɛ t [ψi ) ż i + i ) 2 ψ i i W ] = dt { ɛ [2ψ i ) ż i + i ψ i i W ] + + t ɛ [ψ i ) ż i + i 2 ψ i i W ]) ɛ [ψ i ) ż i + i } 2 ψ i i W ] = = dt ɛ [ψ i ) ż i + i 2 ψ i i W ] + t ɛ [ψ i ) ż i + i ) 2 ψ i i W ]), 126) Q + =ψ i ) p i z) + i 2 ψ i i W. 127) 24

26 B.2 Anticommutation relations The same arguments as in 92) hold for {Q, Q } ={ ψ i p iz) + i 2 ψi ) i W, ψj p jz) + i 2 ψj ) j W } = =p iz) p jz) { ψ i, ψ j } 1 4 i W j W {ψ i ), ψ j ) }+ + i 2 p iz) j W { ψi, ψ j ) } + i 2 i W pjz) {ψ i ), ψ j } = = = 0, 128) { Q, Q } ={ ψ i ) p i z) i 2 ψi i W, ψ j ) p j z) i 2 ψj j W } = =p i z) p j z) { ψ i ), ψ j ) } 1 4 iw j W {ψ i, ψ j } i 2 p i z) j W { ψ i ), ψ j } i 2 iw p j z) {ψ i, ψ j ) } = = = 0, 129) { Q +, Q + } ={ψ i ) p i z) + i 2 ψ i i W, ψ j ) p j z) + i 2 ψ j j W } = =p i z) p j z) {ψ i ), ψ j ) } 1 4 iw j W { ψ i, ψ j }+ + i 2 p i z) j W {ψ i ), ψ j } + i 2 iw p j z) { ψ i, ψ j ) } = = = ) 25

27 References [1] Philip Argyres. Introduction to supersymmetry. University Lecture, URL argyres/661/susy1996.pdf. University of Cincinnati. [2] Stephen P. Martin. A supersymmetry primer. Adv.Ser.Direct.High Energy Phys., 21:1 153, doi: / [3] Edward Witten. Dynamical breaking of supersymmetry. Nuclear Physics B, 1883): , doi: DOI: / ) URL B6TVC P-53/2/8e3fb1f9cd83bd0b fdbc9. [4] Edward Witten. Supersymmetry and morse theory. Journal of Differential Geometry, 174): , URL [5] Wikipedia. Supersymmetry algebra wikipedia, the free encyclopedia, URL Supersymmetry algebra&oldid= [Online; accessed 27-August- 2014]. [6] Andreas Wipf. Introduction to supersymmetry. University Lecture, URL homepage/wipf/lecturenotes.html. Friedrich-Schiller-University Jena. [7] Wikipedia. Trace class wikipedia, the free encyclopedia, URL title=trace class&oldid= [Online; accessed 27-August-2014]. [8] Kentaro Hori et al. Mirror Symmetry. Clay mathematics monographs. American Mathematical Society, ISBN URL [9] Maximilian Kreuzer. Supersymmetry. University Lecture, Vienna University of Technology. [10] Edward Witten. On the Landau-Ginzburg description of N=2 minimal models. Int.J.Mod.Phys., A9: , doi: /S X X. 26