Index Theorems and Supersymmetry

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(1)Uppsala University Department of Physics and Astronomy Division of Theoretical Physics. Thesis for the Degree of Master of Science in Physics. Index Theorems and Supersymmetry. Supervisor: Prof. Maxim Zabzine. Author:. Subject Reader:. Andreas Eriksson. Prof. Joseph Minahan. Examiner: Sr Lect. Dr Andreas Korn. Uppsala, Sweden, July 1, 2014. http://uu.diva-portal.org Thesis Series: FYSAST Thesis Number: FYSMAS1019.

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(3) Abstract The Atiyah-Singer index theorem, the Euler number, and the Hirzebruch signature are derived via the supersymmetric path integral. The supersymmetric path integral is a combination of a bosonic and a femionic path integral. The action in the supersymmetric path integral includes bosonic, fermionic, and isospin elds, where the cross terms are eliminated due to scaling of the elds and techniques from spontaneous breaking of supersymmetry. Thus, the supersymmetric path integral is a product of path integrals over the three given elds, respectively, that can be evaluated exactly by means of Gaussian integrals. The closely related Witten index is a measure of the failure of spontaneous breaking of supersymmetry. The basic concepts of supersymmetry breaking are reviewed..

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(5) CONTENTS. CONTENTS. Contents. 1 Introduction. 1. 2 Index Theorems. 3. 2.1. Elliptic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2.2. Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2.3. 2.2.1. The Chern Character . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2.2.2. The Todd Class . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 2.2.3. The Euler Class . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.2.4. ^-genus . The A. 2.2.5. The Hirzebruch L-polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. Index Theorems and Classical Complexes . . . . . . . . . . . . . . . . . .. 8. 2.3.1. A General Formula for Index Theorems . . . . . . . . . . . . . . .. 9. 2.3.2. The de Rham Complex . . . . . . . . . . . . . . . . . . . . . . . .. 10. 2.3.3. The Dolbeault Complex. . . . . . . . . . . . . . . . . . . . . . . .. 11. 2.3.4. The Signature Complex. . . . . . . . . . . . . . . . . . . . . . . .. 12. 2.3.5. The Spin Complex. . . . . . . . . . . . . . . . . . . . . . . . . . .. 14. 3 Path Integrals 3.1. 3.2. 3.3. 3.4. 7. . . . . . . . . . . . . . . . . . . . .. 17. General Formalism of Path Integrals. . . . . . . . . . . . . . . . . . . . .. 17. 3.1.1. The Bosonic Path Integral . . . . . . . . . . . . . . . . . . . . . .. 17. 3.1.2. Gaussian Integrals. 20. 3.1.3. Zeta Function Regularization. 3.1.4. Fourier Series and Path Integrals. 3.1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21 22. Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24. Grassmann Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26. 3.2.1. Grassmann Algebra . . . . . . . . . . . . . . . . . . . . . . . . . .. 26. 3.2.2. Dierentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26. 3.2.3. Integration. 27. 3.2.4. Gaussian Integral of Grassmann Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29. 3.3.1. Fermionic Harmonic Oscillator . . . . . . . . . . . . . . . . . . . .. 30. 3.3.2. Fermionic Coherent States . . . . . . . . . . . . . . . . . . . . . .. 30. 3.3.3. Fermionic Partition Function. . . . . . . . . . . . . . . . . . . . .. 31. . . . . . . . . . . . . . . . . . . . . .. 32. Fermionic Path Integral. The Supersymmetric Path Integral. 4 Spontaneous Breaking of Supersymmetry. 35. 4.1. The Energy Spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35. 4.2. The Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37. 4.3. An Example: The Wess-Zumino Model . . . . . . . . . . . . . . . . . . .. 39. 5 Index Theorems and Supersymmetry 5.1. The Index of the Dirac Operator. 5.2. Trace Formulas. 41. . . . . . . . . . . . . . . . . . . . . . .. 41. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45. 5.2.1. Fermionic Fields with Periodic Boundary Conditions. 5.2.2. Fermionic Field with Anti-periodic Boundary Conditions. . . . . .. 47. 5.2.3. Isospin Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47. 5.2.4. Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48. i. . . . . . . .. 45.

(6) CONTENTS. CONTENTS. 5.3. The Atiyah-Singer Index Theorem . . . . . . . . . . . . . . . . . . . . . .. 49. 5.4. The Euler Number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 52. 5.4.1. Cliord Forms and Dierential Forms . . . . . . . . . . . . . . . .. 52. 5.4.2. The Index as a Topological Invariant. . . . . . . . . . . . . . . . .. 54. 5.4.3. Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55. 5.5. The Hirzebruch Signature. . . . . . . . . . . . . . . . . . . . . . . . . . .. 57. Acknowledgments. 59. Svensk Sammanfattning. 61. A Hamilton's Principle and Supersymmetry. 63. A.1. The Basic Lagrangian. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 63. A.2. The Gauge Field Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . .. 64. B Product Expansion of an Entire Function. 67. C Curvature Tensors. 69. C.1. The Riemann Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . .. 69. C.2. The Field Strength Tensor . . . . . . . . . . . . . . . . . . . . . . . . . .. 70. D Quantum Fluctuations and the Riemann Tensor. 73. References. 75. ii.

(7) 1 Introduction. 1. Introduction. In this thesis we derive index theorems by using techniques from mathematical physics and quantum mechanics. We use mainly the supersymmetrical path integral in the derivations. The path integral describes the time-evolution of a quantum mechanical system given an initial- and a nal position in space-time. There are two kinds of path integrals; the bosonic and the fermionic path integral, where in the former kind we use commutative variables and periodic boundary conditions, while in the latter kind we implement instead anti-commutative variables and anti-periodic boundary conditions. Supersymmetry, on the other hand, treats bosons and fermions on an equal footing, thus the supersymmetrical path integral includes both commutative- and anti-commutative variables and the boundary conditions implemented over both variables are periodic. Index theorems connect analysis to topology by means of the solutions of a dierential equation to a topological invariant, i.e. a topological number. In this thesis we are only concerned with the topological number called the. Euler number, (M ), where M is some Dirac operator. manifold. Given a manifold that admits the spin structure, the index of the leads to the. Atiyah-Singer index theorem. and it is to be considered here as one of the. main derivations using the supersymmetrical path integral. The Atiyah-Singer index theorem originates from the early 1960's and can be considered as a vast generalization of earlier versions of index theorems such as the. signature theorem,. Hirzebruch. also derived here using supersymmetry. In the early 1980's, physicists. realized that the well known results in mathematical index theory could be derived by using relatively simple techniques from supersymmetrical quantum mechanics and thereby, 1. eventually , connect mathematical theory to physics. All the path integrals in the derivations can be solved exactly by using. Gaussian integrals,. thus neither Feynman diagrams,. nor Feynman rules, are needed to yield the solutions. The. Witten index. determines whether it is. not. possible to spontaneously break the. supersymmetry in a supersymmetric model. The index of the Dirac operator is closely related to the Witten index; the Atiyah-Singer index theorem is equal to the Witten index and thus connects index theorems to supersymmetry. A broken supersymmetry implies a mechanism that gives mass to supersymmetric particles (i.e. fermions with integer spin, 2. or bosons with half-integer spin), analogous to the Higgs-mechanism. in the Standard. Model. The aim with this thesis is to present the most necessary preliminaries and to derive index theorems using the supersymmetrical path integral.. Outline of the Thesis The thesis is organized as follows: In. chapter two we introduce the index theorems from. a non-supersymmetric point of view. Mathematical concepts and terminology is briey reviewed. Elliptic dierential operators, such as the Dirac operator in Euclidean metric, and common characteristic classes used in the index theorems are presented.. 1 At. present time of this writing we must consider supersymmerty to be merely a theoretical frameand not a strict physical theory, since yet there are no experimental data that have proofed supersymmetry. 2 The author apologizes for leaving out Brout, Englert, Guralnik, Kibble and possibly other names in the -mechanism. work,. 1.

(8) 1 Introduction. In. chapter three. we review the theory of path integrals.. Various standard tech-. niques used in evaluating path integrals, e.g., Gaussian integrals, are introduced.. The. similarities and dierences in construction of the bosonic- and the fermionic path integral are emphasized. The nal topic of the chapter is the supersymmetric path integral. In. chapter four we review the concept of spontaneous breaking of supersymmetry in. contrast to symmetry breaking in quantum eld theory. The famous Wess-Zumino model serves as an example of whether supersymmetry is broken, and hence describes nature. In the. nal chapter, we use the results from the consecutive chapters to derive the. aforementioned index theorems.. Two extensive examples; the Gauss-Bonnet theorem,. and the winding number, serves as an in depth review on the geometrical and topological meaning of the Euler number and its relation to physics. This chapter can be considered as the main chapter while the previous chapters are preliminaries. Four appendices follow the chapters described above: In. appendix A, we show that. the supersymmetric Lagrangian fullls the principle of least action, by using the supersymmetry transformations and the Bianchi identities for the eld strength- and the Riemann curvature tensors. In. appendix B, we derive an important formula used in the path integrals that are. implemented in the derivation of the index theorems. In. appendix C, we derive the Riemann curvature tensor and the eld strength cur-. vature tensor explicitly. The similarities in construction of the two curvature tensors are emphasized. Finally in. appendix D, a gauge choice, heuristically introduced in the derivation of. the Atiyah-Singer index theorem in chapter ve, is here calculated explicitly.. 2.

(9) 2 Index Theorems. 2. Index Theorems. In this chapter, elementary concepts and terminology of the theory of index theorems are presented. In chapter ve below the same expressions of the index theorems presented here are derived using the supersymmetric path integral. Here, however, we follow closely the seminal articles [1]. The aim of this chapter is to state the major results of index theory in a rather non-technical review. For a more mathematical exposure, we refer to the aforementioned reference. Complementary references to the review given here include [3, 7, 9, 10, 12]. The mathematical preliminaries are, more or less, omitted here and we refer instead to the review article [3] for a more comprehensive exposure. The hallmark of index theorems is that they give information about dierential equations, provided that we understand the topology of the ber bundles upon which the dierential operators are dened.. We outline several examples below, illustrating the. connection between the index of an operator and the related topological numbers.. 2.1. Elliptic Operators. In this section we review the theory of elliptic operators. Elliptic operators on compact manifolds are important in dening index theorems, since the dimension of the kernel of the operator is nite, thus the. analytical index. problem of the generic operator. p; Op ! = n !, where. p (M ). Op. is well dened. Consider the eigenvalue. is the space of. p-forms.. Op ker Op = f!; Op ! = 0g:. are the eigenvalues and the kernel of. is dened as the set of dierential forms. As an example of an elliptic operator we take the and is dened on compact Riemannian manifolds requires a metric geometry. The. g (x). ! 2 p (M ) of order The constants n , for n = 0; 1; : : : ,. acting on some dierential form. Laplacian, p , which act on p-forms. M. of dimension. n.. The Laplacian. for its denition, hence we have a link between analysis and. Hodge-de Rham theorem. yields topological information of the Laplacian. dim ker p = dim H p (M ; R); dR. p. H (M ; R) is the de Rham cohomology group. Next, we dene the Fourier transformation Fff (x)g of a function f (x) by the formula. where. dR. Z. Fff (x)g = (21 )n dnx exp(ix)f (x) =: f^( ): The Laplacian (in Cartesian coordinates) is dened as. = and with. @2 @x21. @ ; @x 2 n 2. acting on f (x) under the inverse Fourier transform yields the equation Z. The leading. 1 f (x) = dn [12 + + n2 ]f^( ) exp( ix): (2)n symbol, denoted by (), of the dierential operator is L. part of its Fourier transform:. 3. the highest order.

(10) 2.1 Elliptic Operators. 2 Index Theorems. () = 12 + + n2 ; L. and for. () equal to a constant we obtain the equation of a sphere. We can generalize ai in the coordinates xi , accordingly, L. the Laplacian by a change of scale. X. L= then the symbol of. i. ai. @2 ; @x2i. L set equal to a constant c is given by a1 12 + + an n2 = c;. which is the equation of an ellipsoid in. Rn ,. hence the name. elliptic operator.. A more. (x; ) is x in Rn , then the associated dierential operator is called elliptic. example of an elliptic operator, consider the Bessel's equation of order . formal denition of ellipticity is formulated as follows; if the leading symbol. L. always non-zero for all As a counter. given by the dierential equation. x2. d2 u(x) du(x) +x + (x2 2 dx dx. 2 )u(x) = 0;. 2 R;. which have the leading symbol. (x; ) = x2 2 ; that vanish at the at the origin x = 0. L. It is common in the literature to use operator, dened in. Rn ,. of order. m. L=. multi-index X. j jm. notation. Let. L be a linear dierential. a (x)D : P. n-tuple = ( 1 ; : : : ; n ), where i 0, is called a multi-index and j j = i is its . j j : : : (@=@x ) n , length. Furthermore, we have p = p1 p2 : : : pn n and D = ( i) (@=@x1 ) n The. 1. 2. 1. thus the linear dierential operator is given by. L=. X. j jm. a( ;::: n ) (x)( i)j j 1. Lu(x) =. j jm. a. (x)D u(x). X. = =. j jm Z. Rn. 1. 1. m (x; ):. Using the Fourier transform, we get the symbol. X. @ @ n : : : : @x 1 @x nn Z. a (x) dn exp( ix)^u( ) Rn. dn [m (x; )] exp( ix)^u( );. hence,. m (x; ) =. X. j jm 4. a (x) :.

(11) 2 Index Theorems. 2.2 Characteristic Classes. The leading symbol is then equal to. (x; ) = L. X. j j=m. We are here mainly interested in the cases. a (x) :. m=1. (Dirac operator) and. m=2. (the. Laplacian). Elliptic operators on compact manifolds are called. Fredholm operators, and we assume. from now on that all dierential operators are Fredholm, unless it is stated as nonFredholm in a certain case.. 2.2. Characteristic Classes. A ber bundle is a manifold that locally looks like a direct product of two topological. S 1 and some non-zero interval I = [a; b], is a cylinder denoted by S 1 I . The manifold M = S 1 is called the base space and F = I the ber. A collection of all the bers is called a ber bundle. Since the cylinder. spaces. As an example, a direct product of a circle. can be expressed as a direct product, locally as well as globally, it is a so called trivial bundle. A Möbius strip, on the other hand, cannot be a direct product as in the case for a cylinder, since it is twisted globally (if wee zoom in and merely look at a small segment of its surface, it is indeed a direct product that looks like. R2 ).. Characteristic classes. measure the non-triviality, or twisting, of a bundle. The measure of the twisting is equal to an integer, a topological constant, expressed as an integral involving the curvature of the ber bundle. In this section we present the most important characteristic classes that appear in the index theorems in the subsequent sections and in chapter ve. Several examples of integrals over characteristic classes are given in the next section, used in the evaluated index theorems.. 2.2.1 The Chern Character E be a complex vector bundle, whose ber is Ck . Given a gauge potential A (x) and 1 a eld strength curvature two-form, F = F dx ^ dx , we dene the total Chern class 2 Let. by. . iF c(F ) = det I + 2. . = 1 + c1 (F ) + c2 (F ) + : : : ;. cj (F ) is the j th Chern class and I is a unit matrix. In an m-dimensional base space M , the Chern class cj (F ) with 2j > m vanish, thus the series terminates at ck (F ) = det(iF =2) and cj (F ) = 0 for j > k. The Chern classes are given, explicitly,. where. by. 5.

(12) 2.2 Characteristic Classes. 2 Index Theorems. c0 (F ) = 1. i Tr F 2 1 i 2 c 2 (F ) = [Tr F ^ Tr F 2 2. c 1 (F ) =. . . .. . i ck (F ) = 2. E. If we now let. Pontrjagin class. k. Tr(F ^ F )]. det F :. be a real vector bundle with rank. dimR E = k,. we dene the total. by. p(F ) = det. . F I+ = 1 + p (F ) + p (F ) + : : : : 1. 2. 2. The relation between the Pontrjagin classes and the Chern classes is given by. pj (E ) = ( i)j c2j (EC ); where. EC. denotes the complexication of the real vector bundle. Finally, the total. Chern character . iF ch(F ) = Tr exp 2. . E , i.e., E R C = EC .. is dened by. 1 = k + c1 (F ) + [c2 (F )2 2. 2c2 (F )] + : : : :. 2.2.2 The Todd Class Let. E. now be a complex vector bundle of rank. Todd class. of. E. xj 's. We dene the total. by. td(E ) =. where the. k, i.e. dimR E = k.. k Y. xj 1 e j =1. 1 1 =1 + c1 (E ) + [c1 (E )2 + c2 (E )] + : : : 2 12 1 1 =1 p1 (E ) + [3p (E )2 p2 (E )] + : : : ; 12 720 1. xj. comes from the. Whitney sum of. splitting principle ;. n complex line bundles,. the bundle. E = L1 L2 Ln :. E. can be written as a. The Whitney sum of the Chern class is, given a direct sum E = E1 E2 , equal to c(E1 E1 ) = c(E1 ) ^ c(E1 ). The Chern class ci (E ) = 0 for k1 + 1 i k1 + k2 , where k1 = dimR E1 and k2 = dimR E2 . For the sum of n complex line bundles L dened above, we get the wedge product. c(E ) = c(L1 ) ^ c(L2 ) ^ ^ c(Ln ): 6.

(13) 2 Index Theorems. The of. 2.2 Characteristic Classes. rth Chern class cr (L) = 0 for r 2 since dimR Li = 1, thus we write the Chern class. Li as. c(Li ) = 1 + c1 (Li ) 1 + xi ; and the total Chern class is now expressed as. c(E ) =. n Y i=1. (1 + xi ):. The Chern character behaves well under Whitney sums; and. ch(E F ) = ch(E ) ^ ch(F ). ch(E F ) = ch(E ) ch(F ), and they are an important property in evaluating the. index theorems as will be demonstrated below.. 2.2.3 The Euler Class M Sbe a 2l-dimensional orientable Riemannian manifold. The real tangent bundle T M = p2M (Tp M ) of M is the collection of all the tangent spaces Tp M of M . We dene the Euler class as the square root of the highest Pontrjagin class: Let the base space. pk=2 (E ) = e2 (E ); where k = 2l is the rank of the real vector bundle E = T M . EC the Euler class is equal to the top Chern class:. For a complex vector bundle. ck (EC ) = e(EC ):. If the rank. where. A. k is even, k = 2l say, the Euler class can be associated to the Pfaan : P f (A) =. p. det(A);. is an even dimensional, skew-symmetric matrix of the form. 0. A=. B B B B B @. 0 x1 : : : x1 0 : : : . . .. . . .. ... .. 1 C C C C: C xk A. 0 xk 0. The Pfaan is dened only for matrices of even order.. For an odd-dimensional skew-. symmetric matrix, the Pfaan vanishes, thus the Euler class for an odd-dimensional manifold. M is equal to zero.. In chapter three we dene the Pfaan in terms of a Gaussian. integral, and in chapter three and ve, Gaussian integrals are used in evaluating path integrals.. 2.2.4 The A^-genus The. A^-genus (called A-roof genus or, common in physics literature, the Dirac genus) is. dened by. A^(F ) =. k Y. xj =2 =1 sinh( x = 2) j j =1. 1 1 p1 + (7p2 24 5760 1 7. 4p2 ) + : : : ;.

(14) 2.3 Index Theorems and Classical Complexes. where the. 2 Index Theorems. xj 's are the eigenvalues of the eld strength curvature two form, put in block. A. diagonal form similar to. above. The index of the Dirac operator is the Atiyah-Singer. index theorem and it is equal to an integral of fold. M. must admit a spin structure and the. manifolds. For a real bundle. A^(T M ). over a manifold. Stiefel-Whitney classes. M.. The mani-. singles out all such. E , we dene the total Stiefel-Whitney class by. w(E ) = 1 + w1 (E ) + w2 (E ) + : : : ; where only the rst two classes are important in order to determine whether a manifold allows spin structure.. If the base space is orientable, the rst Stiefel-Whitney class. w1 (T M ) is zero. The manifold is a spin-manifold if the second Stiefel-Whitney class w2 (T M ) is also zero, this means that parallel transport of spinors can be globally dened on E = T M if and only if w1 (T M ) = w2 (T M ) = 0. We give here two examples of spin-manifolds; (i) the complex projective spaces of odd dimension, denoted. CP 1 , CP 3 ; : : : , and (ii) any sphere S n .. 2.2.5 The Hirzebruch L-polynomial k = dimR E. Let. be the rank of a real bundle. Hirzebruch L-polynomial. L(x) =. over an. n-dimensional manifold M .. The. is dened by. k Y. 1 1 xj = 1 + p1 + ( p21 + 7p2 ) + : : : : tanh xj 3 45 j =1. An alternative denition of the. L-polynomial can be found in the literature:. L(x) = 2k In the. E. Hirzebruch signature theorem,. k Y. xj =2 : tanh( x = 2) j j =1. only the highest order term is evaluated and both. terms are equal, as can be realized by expanding the former denition up to order. k.. Hence either denition can be used in the signature theorem. The lower order terms, on the other hand, are sensitive to which denition is used.. 2.3. Index Theorems and Classical Complexes. First we state a general index theorem formula, expressed in terms of the characteristic classes outlined in the previous section. We then apply the index theorem on. complexes,. a nite sequence of elliptic dierential operators acting on ber bundles. The order of the operators in a complex is important so that we get a certain chain of operators in contrast to a partial derivative where the order can be chosen arbitrary. The index theorem of the de Rham complex yields the. Gauss-Bonnet theorem.. The Dolbeault. complex can be considered as the complex variable analogue to the de Rham complex and leads to the. Riemann-Roch theorem.. The. Hirzebruch signature theorem. is derived. in the context of the signature complex and, nally, from the spin complex we get the. Atiyah-Singer index theorem. 8.

(15) 2 Index Theorems. 2.3 Index Theorems and Classical Complexes. 2.3.1 A General Formula for Index Theorems We are already familiar with the concept of ber bundles from the previous section.. M , the ber F and the total space E , E is a collection of all bers, i.e., a ber bundle. A map f : A ! B that maps every element in the domain A to every element in the target B (not necessary one-to-one) is a surjective map, or a surjection. The surjection : E ! M is called the projection and 1 its inverse (p) = Fp is the ber at p 2 M and it is one-to-one and onto to F , hence an isomorphism denoted by Fp = F . A (cross) section s : M ! E satises s = idM , 1 the identity map idM : M ! M . A section of our trivial bundle S I introduced above 1 is just a fraction of the circle M = S , or the entire circle depending on how many bers. Dened more formally, we have the base space where. one chooses to take the cross section of. A generic dierential operator and sections. Let. Dy. , by. D. can now be dened in terms of ber bundles. E! M. (M; E ) denote the set of sections on M , thus we dene D, and is dual. D: Dy :. (M; E 0 ) ! (M; E 1 ); (M; E 1 ) ! (M; E 0 ); 0 1 y where E and E are vector bundles over M . The kernels of D and D. The operator tion. Ds = 0. D. are given by. ker D fs 2 (M; E 0 ); Ds = 0g; ker Dy fs 2 (M; E 1 ); Dy s = 0g: carries analytical information, from the solutions of the dierential equa-. , hence the analytical index is dened by. index(D) = dim ker D dim ker Dy : A nite sequence of operators. Di. is given by. ! (M; E 0) D! (M; E 1) D! D!n (M; E n+1) ! 0 and is called an elliptic complex if the composition Di Di 1 = 0 for any i. 0. 0. A generalization of the denition of. classes, is given by the formula. 1. index(D) above, given in terms of characteristic. index(D) = ( 1)n fch(L (D))td(T MC )g[T M ] where T MC is the complexication of the tangent bundle T M , i.e., T MC = M R C. The expression [T M ] is an abbreviation of taking the integral of the characteristic classes over T M . The right hand side can be generalized even further by rewriting the Chern character of the leading symbol as a fraction of the Chern character of an alternating sum of ber bundles and the Euler class:. index(D) = (. 1)n(n+1)=2. P. ch(. i(. 1)i E i )td(T MC ) [M ]: e(T M ). (2.1). The latter index formula dened above is valid only for even dimensional and orientable manifolds. M.. The Euler class vanishes for odd dimensions and consequently the index is. dened to be equal to zero in the case when the dimension is odd. Next, we apply the generalized index formula (2.1) over four dierent complexes.. 9.

(16) 2.3 Index Theorems and Classical Complexes. 2 Index Theorems. 2.3.2 The de Rham Complex The (complexied) de Rham complex is dened by. d r r+1 (M )C r! : : : dr! r 1(M )C dr! r (M )C d! p p where (M )C = (M; ^ T MC ) is the vector space of p-forms, d is the exterior derivative +1. 1. 2. T MC is the complexied co tangent bundle (which is dual to T MC ). For M an even dimensional manifold, n = 2l and l 0, we write the right hand side of the generalized and. index formula (2.1) as. !. n X. ( 1)l(2l+1) ch. r=0. ( 1)r E r. td(T MC ) [M ]: e(T M ). The Chern character in the index formula can be written as an alternating sum of Chern characters of vector bundles:. n X. ch with. E r = ^r T MC .. r=0. !. 1)r E r. (. For a line bundle. Li. =. n X r=0. ( 1)r ch(E r ). ch(Li ) = exp(xi ), where xi = c1 (Li ),. we have. and using the splitting principle we get the characteristic classes. ch. n X r=0. (. 1)r. ^. r. T M. !. C =. td(T MC ) = e(T M ) =. n Y. (1 e. i=1 n Y. xi 1 e i=1. l Y i=1. xi )(T M. C );. xi (T MC );. xi (T MC ):. Substituting the Chern character, the Todd class, and the Euler class into the index formula we arrive at the. topological index. (given by the integral in the far right hand. side). index(d) =. Z. (. 1)l(2l+1) (. 1)l. M. l Y i=1. !. xi (T MC ) =. Z. e(T M );. M. where in the rst integral we used the following relation between the Euler class and the top Chern class. cn (T MC ) = x1 x2 : : : xn :. cn (T MC ) = ( 1)n=2 e(T M T M ) = ( 1)n=2 e2 (T M ): d : r (M ) ! r+1 (M ) is not Fredholm in the space (M ), r thus we have to dene d in the de Rham cohomology group H (M ) instead. Hence the The exterior derivative. analytical index. dR. is (given by the expressions in the rst and second equality). 10.

(17) 2 Index Theorems. 2.3 Index Theorems and Classical Complexes. index(d) = =. n X. ( 1)r dim H r (M ; C) dR. r=0 n X r=0. ( 1)r dim H r (M ; R) = (M ) dR. where the second equality follows from the de Rham's theorem and the third equality from the Euler-Poincaré theorem, via Hodge's theorem. The topological constant. Euler number. The Gauss-Bonnet theorem. is the. is the index of the de Rham operator. Z. (M ). d:. e(T M ) = (M ):. M. 2.3.3 The Dolbeault Complex 3. Without going into too many details , the Dolbeault complex is analogous to the de Rham complex, using instead complex variables of the form conjugate. n=2.. z = x iy .. The manifold. z = x +iy and its complex. M is now a complex manifold of complex dimension d = @ + @, where the Dolbeault operator @ , and. The exterior derivative is dened as. its dual. @, is given by. @ = dz ^ @=@z ;. z : @ = dz ^ @=@. The complex analogue of the de Rham sequence is. @! p;q (M ) @! p;q+1 (M ) @! : : : ; @! p;q (M ) @! p+1;q (M ) @! : : : : The Dolbeault complex is obtained with. p = 0:. @! 0;q (M ) @! 0;q+1 (M ) @! : : : : Using similar arguments as in the de Rham case above, we have the characteristic classes. cn=2 (T M ) = ( 1)n=2 cn=2 (T M ) = ( 1)n=2 e(T M ); td(T MC ) = td(T M T M ) = td(T M )td(T M ); ch( ) = L. n=2 X q=0. ch(^q T M ) =. cn=2 (T M ) : td(T M ). The index formula reduces to. ( 1)l e(T M ) td(T M )td(T M )[M ] = td(T M )[M ]: index(@) = ( 1)l(2l+1) e(T M )td(T M ) 3 See. for instance. Kähler Geometry. in [7], or. Complex Manifolds. 11. in [3] or in [10]..

(18) 2.3 Index Theorems and Classical Complexes. There is a relation between the classical. Hodge numbers hp;q :. (M ) =. X q. 2 Index Theorems. Betti numbers bq. ( 1)q bq =. X p;q. = dim H q (M ; R) dR. and the. ( 1)p+q hp;q :. The Hodge numbers can be regarded as a renement of the Betti numbers. If we denote the Dolbeault complex by. " we get the topological index index(@) =. Finally, the. Riemann-Roch theorem Z. X q. ( 1)q h0;q = ("). is given by. td(T M ) = (");. M. where. (") is called the arithmetic genus. of the complex manifold. M.. 2.3.4 The Signature Complex M be an oriented manifold of even B : H l (M ; R) H l (M ; R) ! R by. Let. dimension,. B ( ;

(19) ) where. ;

(20). 2. H l (M ; R),. Z. n = 2l .. We dene a bilinear form. ^

(21) ;. M which is the middle cohomology group. The form. B ( ;

(22) ) is a. symmetric matrix if l is even, where = is the Betti number. If l = 2k (so n is divisible by four) the symmetric form B ( ;

(23) ) has real eigenvalues where + the number of positive (negative) eigenvalues is denoted by b (b ). The Hirzebruch signature of M is dened by bl. bl. bl. dim H l (M ; R). signature(M ) := b+ For. b :. l odd, signature(M ) is dened to vanish. The Hodge star operator. is a duality transformation; : r ! n. r,. and it satises. 2 = 1 when acting on a 2k-form in a 4k-dimensional manifold, hence has eigenvalues 1. We dene an operator D D = d + dy;. = ddy + dy d = D2 (since d2 = (dy )2 = 0). Let Harm2k (M ) = f! 2 2k (M ); D! = 0g be the set of harmonic 2k-forms on M , which is 2k isomorphic to the cohomology groups of order 2k , i.e., Harm (M ) = H 2k (M ; R). Due 2k to the 1 eigenvalues of the operator , the set of harmonic forms Harm (M ) can be which is the square root of the Laplacian. decomposed, accordingly,. Harm2k (M ) = Harm2+k (M ) Harm2k (M ): 2k The Betti numbers are b = dim Harm (M ) and the Hirzebruch signature is given by signature(M ) = dim Harm2+k (M ) dim Harm2k (M ): 12.

(24) 2 Index Theorems. 2.3 Index Theorems and Classical Complexes. When dealing with elliptical complexes we can split the space of forms a similar way as for. Harm2k (M ).. We dene an operator. ,. that acts on. (M ) in r-forms !,. accordingly,. := ir(r 1)+l : r (M ) ! n r (M ); 2 which satises = 1 and D + D = 0. The exterior algebra (M ) is decomposed as (M ) =. D =. The anti-commutativity dual. D. , of the operator. M r. r (M ) = + :. D implies that we can dene a restriction D+, and its. D given by. D+ :+(M ) ! (M ); D : (M ) ! +(M ): On the exterior algebra. 2k. for dimension. n = 4k. we have that. = , and the index of. the signature complex reduces to the Hirzebruch signature:. index(D+ ) = dim ker D+. dim ker D = signature(M ):. The topological index is given by the formula. ( 1)l fch(^+ T M R C) ch(^ T M R C)g. td(T M R C) [M ]: e(T M ). From the splitting principle of the characteristic classes, we get. ch(^. +. T M R C). ch(^. n=2. Y T M R C) = (e xi i=1. exi );. xi xi ; x i 1 e 1 e xi e(T M ) = x1 x2 : : : xn=2 :. td(T M R C) =. Hence, substituting the characteristic classes into the index formula yields:. index(D+ ) = ( 1)n=2 = = =. 8 n=2 xi <Y e :. n=2 Y xi (exi exi i=1 n=2 Y n= 2 2 n=2 Y. xi. i=1. exi. + 1) [M ] 1. xi =2 [M ] tanh(xi =2) i=1. xi [M ]: tanh x i i=1 13. xi xi x i 1 e 1 e xi. 9 = ;. [M ].

(25) 2.3 Index Theorems and Classical Complexes. 2 Index Theorems. As discussed above, the last equality can be realized by expansion of order. n=2.. The. Q. xi = tanh xi up to. n=2-order term coincides with the expression in the penultimate equality. since it is only the highest term that is evaluated in the index. The. Hirzebruch signature theorem. dimension. states that, for a compact oriented manifold of. n, where n is divisible by 4, the signature of M. is given by. L(x) = signature(M ): The integer. L(x) =. Z Y n=2. xi tanh xi i=1. M is called the. L-genus. of. M.. The Hirzebruch signature can be used in order to determine whether a manifold admits a complex structure. In. dimR M = 4 we have the following relations. M. index(@) = ((M ) + (M ))=4: Example :. If. M = S4. is the four-sphere then. (S 4 ) = 2. and. (S 4 ) = 0,. hence the. ) = 1=2 which is not an integer and it means that S 4 arithmetic genus is given by index(@. is not complex. We can draw the same conclusion for the complex projective space, with. index(@) = (3 1)=4 = 1=2. +CP 2 , it is complex; index(@) = (3 + 1)=4 = 1.. the orientation. CP 2 ,. since. For the opposite orientation,. 2.3.5 The Spin Complex Let. TM. ! M. dim M = n = 2l even and M orientable. A M = S 2 as discussed above. We dene the double. be a tangent bundle, where. spin structure can be dened on, e.g., covering by the map. : Spin(n) ! SO(n): The. Spin(2). group is the double covering of. S 2.. Geometrically it is visualized as the. splitting of the sphere into two half-spheres that are covering the upper- and lower hemi-. SO(2), that we can regard as a : S 2 ! S 2 . The two-sphere 1 2 can also be dened as the complex projective space CP = S , with transition functions tij = exp ( i2), where is an angle describing the rotation, i.e., the double covering : 7! 2. Topologically Spin(2) is a latitudinal circle describing spin states on the 2 double cover of S . The set of transition functions denes a spin bundle SM , and the set of sections of SM is denoted by (M ) = (M; SM ). The spin-group is generated by n numbers of Dirac matrices, f g, which satisfy the following conditions spheres, respectively.. The super orthogonal Lie-group. dierentiable manifold, describe rotations in. R3 ,. hence. y = ; f ; g = + = 2g : We dene the gamma matrix of dimension. n + 1 as 14.

(26) 2 Index Theorems. 2.3 Index Theorems and Classical Complexes. n+1. . (i)n=2 1 2 : : : n. ( n+1 )2 = I; where. I. is a. 2n=2 2n=2. . . 0 ; 1. 1 = 0. n = 2 we yield the Pauli matrices 1;2;3 , and 1=2 particle on S 2 in the x-, y- and z -direction,. unit matrix. For. they are related to the rotations of a spinrespectively,. 0 = 2 ; Since the eigenvalues of of the spin bundle. 1 = 1 ;. 2 = i 0 1 = 3 :. n+1 , called the chirality, are equal to 1, the set of sections. (M ) is decomposed into two eigenspaces, accordingly, (M ) = + (M ) (M ):. The spin complex is dened in terms of the. Dirac operator. D. , and its dual. Dy. , by. D : +(M ) ! (M ); Dy : (M ) ! +(M ): The analytical index of the spin complex is. index(D) = dim ker D dim ker Dy = n+ where. n+ (n. ) is the number of zero-energy modes of chirality 4. is elliptic only in Euclidean metric. . ,i.e., g. =. n ; +(. ). The Dirac operator. , which is the ordinary Kronecker delta;. =diag(+1,+1,+1,+1). Thus, on the Riemann sphere M = S 2 we assume that the metric is locally at ; g (x0 ) = and @ g (x0 ) = 0, x0 2 M . This choise of coordinates is called the Riemann normal coordinates (see a diagonal matrix of the form. appendix D for further details). The index theorem for the spin complex is given by the index formula. ( 1)n=2 fch(+ (M ) (M ))g. td(T MC ) [M ]: e(T M ). From the splitting principle we have. (. 1)n=2. fch( (M )) ch( (M ))g = +. n=2 Y i=1. (exi =2. e. xi =2 );. Thus the topological index is equal to. relativistic quantum mechanics the Dirac operator D is dened in the Lorentzian metric given by =diag(-1,1,1,1). The index of D is related to spontaneous breaking of supersymmetry (chapter four), where we are only interested of the physics in the ground state, i.e., the zero energy state. The total energy is E jP j, thus in the ground state the momentum is P = 0. 4 In. 15.

(27) 2.3 Index Theorems and Classical Complexes. index(D) = = The. n=2 xi =2 Y e i=1 n=2 Y i=1. exi =2. e. xi =2. xi. xi e. Atiyah-Singer index theorem. 2 Index Theorems. xi 1 e. xi =2 [M ]. . xi x i 1 e x i [M ]. n=2 Y. xi =2 [M ] = A^(T M )[M ]: sinh( x = 2) i i=1. =. is given by. Z. index(D) = A^(T M ); M. A^-genus contains only 4i-forms, hence the index, as presented above, vanishes unless the dimension of M is a multiple of four. Furthermore, The Dirac operator D can be twisted if the spin bundle SM is replaced by the tensor product SM V , where V is a vector bundle. Using the multiplicativity where the. property of the Chern character, the index theorem applied to the twisted spin complex. DV : +(M ) V ! . (M ) V. is then equal to. index(DV ) = For. A^(T M ) ^ ch(V ):. M. dim M = 2, we have. n+. n =. Z M. where. Z. i ch1 (V ) = 2. Z. Tr(V ). M. Tr(V ) is associated to the trace of the eld strength curvature two-form F , i.e., a. background eld that causes the twisting of the operator. D. .. The Atiyah-Singer index theorem of the twisted Dirac operator is derived in the context of supersymmetry in chapter ve.. 16.

(28) 3 Path Integrals. 3. Path Integrals. In this chapter we review the theory of path integrals and anti-commuting algebra, also called Grassmann algebra. We arrive in the end of this chapter at the path integral for fermions and, nally, the supersymmetric path integral.. The fermionic and supersym-. metric path integral play a crucial role in the proofs of the index theorems, presented in chapter ve below.. 3.1. General Formalism of Path Integrals. 3.1.1 The Bosonic Path Integral The dynamics of a quantum mechanical system can be described by a path integral, 5. which is a sum of all eld congurations. between a given initial point and a nal point. in space-time. We rst consider the case of a system with one degree of freedom, and later generalize to a system with several degrees of freedom. In this section we deal with the bosonic case, hence the variables are commutative, in contrast to anti-commutative in the fermionic case. A picture of the quantum process in space-time is given in gure (1) below.. time. . t00. . t0. x0. space. x00. Figure 1: A path integral is a sum over all eld congurations in space-time, where the paths in the gure describes a dynamical quantum process evolving from an initial point to a nal point.. The initial position is denoted by. 00 evolution to the nal position x. x0. at the initial time. 00 is taking place at time t .. The derivation of the path integral starts with the classical Lagrangian. L = L(x; x_ ) = where. K = (m=2)x_ 2. m 2 x_ 2. t0 ,. and the. L of the form. V (x);. is the kinetic energy of a particle of mass. m under the inuence of. F (x) = dV (x)=dx, and V (x) is the potential energy for the classical trajectory x = x(t). The Hamiltonian H is the sum of the kinetic and potential the time independent force energy. 5 The. terminology sum of all paths, or sum of all histories, can also be found in the literature. Since paths are not well dened in quantum mechanics, due to the Heisenberg uncertainty principle given by xp ~=2, sum over all histories attempts to avoid such terminology. See the discussion on the validity of the path integral, further below in this section. 17.

(29) 3.1 General Formalism of Path Integrals. 3 Path Integrals. p2 + V (x); 2m. H = H (p; x) := px_ L =. p = mx_ is the (generalized) momentum. Replacing the variables (x; p) by the time independent operators x ^ and p^ = id=dx in the Hamiltonian above we get the quantum ^ Hamiltonian H where. p^2 ^ H := H (^x; p^) = + V (^x): 2m The time dependent state vector j (t)i describes the physical state of a quantum mechanical system at a given time t, and the time-evolution of the states is governed by the. Schrödinger equation. d j (t)i = H^ j (t)i: dt 0 If we know the state at some initial time t , we then want to compute j (t)i for a nal 00 0 time t > t . Solving the Schrödinger equation i~. i^ H j (t)i = 0; t0 < t < t00 ;. d j (t)i dt. ~. we nd, from the general solution of the dierential equation, the time-evolution operator. U^ (t00 ; t0 ) = exp i.e., the nal state vector is of the form. U^. . . i ^ 00 H (t. t0 ) ;. ~ j (t00)i = U^ (t00; t0)j (t0)i.. The time-evolution. t0 < t1 < t2 < t00 we ^ ; U^ (t00 ; t0 ) = U^ (t00 ; t2 )U^ (t2 ; t1 )U^ (t1 ; t0 ). Since H^ depends on have the composition law of U x^ and p^ we work in the x-representation and p-representation, respectively. Instead of j (t)i we use the state vectors jxi and jpi, having the following properties operator. fullls the Schrödinger equation as well and for, e.g.,. x^jxi = xjxi; hx0 jxi = (x0 p^jpi = pjpi; These properties are the. pleteness relation. the. ZR. R. dxjxihxj = 1; dpjpihpj = 1:. orthogonality of states ;. for x- and p-representation, respectively.. The path integral describes the evolution of the initial state. 00 00 evolving to the nal state jx(t )i = jx i, Feynman Kernel K (x00 ; x0 ; t00 ; t0 ). t0 ,. x);. hp0jpi = (p0 p);. eigenvalue equation ;. Z. .

(30). at time.

(31). . t00 .. and the. com-. jx(t0)i = jx0i at time. Hence, we shall calculate the. .

(32) ^

(33)

(34)

(35) x0 hx00jU^ (t00; t0)jx0i = x00

(36)

(37) exp ~i HT := K (x00 ; x0 ; t00 ; t0 ); T = t00. t0 ; t0 < t < t00 :. The transformation function, in the coordinate to momentum representation, is given by the plane wave. 18.

(38) 3 Path Integrals. 3.1 General Formalism of Path Integrals. hxjpi = p 1 eipx=~: 2 ~. With the transformation function dened above, we compute the matrix element expressed in the classical Hamiltonian. H (p; x):. hxjH^ jpi = p 1 e 2 ~. For small. T = t00. hxjH^ jpi,. ipx=~ H (p; x):. t0 we expand the time-evolution operator up to rst order in T exp. and the matrix element. . . i ^ 00 H (t. i ^ 00 H (t. t0 ) =1. ~. t0 );. ~. hpjU^ (t00; t0)jxi is equal to . . p 1 e ipx=~ 1 ~i H (p; x)(t00 t0) 2 ~ 1 i i 00 0 = p exp px H (p; x)(t t ) : ~ ~ 2 ~. hpjU^ (t00; t0)jxi =. Inserting the completeness relation,. hx00j1U^ (t00; t0)jx0i =. Z. R. R. dpjpihpj = 1, inside the Feynman kernel gives. dphx00 jpihpjU^ (t00 ; t0 )jx0 i . Z. 1 i = dp exp p(x00 2 ~ ~. i. x0 ). R. ~. . H (p; x0 )(t00. t0 ) :. (3.1). The time-evolution operator fullls the composition law as mentioned above, hence in the. U^ (t00 ; t0 ) = U^ (t00 ; tN 1 ) : : : U^ (t1 ; t0 ); a 00 t0 into N steps: We divide the time interval t. right hand side of the kernel we use the composition factorization into. N. factors.. t =. t00. N. t0. 1;. hence we can carry out the integration of the term dependent on the Hamiltonian in (3.1). The time-evolution operator. . U^ (t00 ; t0 ) = 1. U^ (t00 ; t0 ) is now a product, written as. i ^ (t00 H. ~. N. t0 ). N. R. . = exp. dxjxihxj = 1, N 00 0 ^ factor, except the ultimate one, of U (t ; t ) gives. Inserting the completeness relation,. 19. . N. i^ H t. ~. 1. :. times to the right of every.

(39) 3.1 General Formalism of Path Integrals. hx00jU^ (t00; t0)jx0i = = =. Z. R. 3 Path Integrals. dphx00 jpihpjU^ (t00 ; tN 1 )1 : : : U^ (t2 ; t1 )1U^ (t1 ; t0 )jx0 i. ZY N dpi NY1. dxj hxN jpN ihpN jU^ (tn ; tN 1 )jxN. 2 i=1. ZY N dpi. j =1 NY1. R. j =1. R. 2 i=1. i. ~. dxj exp. . i. (p (x ~ N N. 1. i : : : hp1jU^ (t1; t0)jx0i. xN 1 ) + + p1 (x1. x0 )). . (H (pN ; xN 1 ) + + H (p1 ; x0 ))t ;. xN = x00 and x0 = x0 . In the limits N ! 1 and t ! dt, we integrate over pN ! p(t) and (xN xN 1 )=t ! x_ (t) for t0 < t < t00 . The boundary terms of the coordinates 0 0 00 00 are x(t ) = x and x(t ) = x , hence the argument of the exponential transforms into the where. classical action. S=. Z t00. Z t00. t0. t0. dt[p(t)x_ (t) H (p(t); x(t))] =. dtL(x; x_ ):. The measure is a product of Liouville measures; they are all classical quantities,. dp00 NY1 dpi (t)dxi (t) := D p(t)D x(t): 2 i=1 2 In summary, the path integral is given by. K (x00 ; x0 ; t00 ; t0 ) = Since both the measure. Z. D p(t)D x(t)eiS=~:. (3.2). D p(t)D x(t) and the Lagrangian L(x; x_ ) are classical quan-. tities, it might seem to be a contradiction that quantum mechanics can be expressed in terms of classical mechanics. (3.2) is written out. symbolically ;. The path integral expressed in the right hand side of which means that it is to be considered as a limiting. process, valid in the framework of perturbation theory in quantum mechanics.. For a. comprehensive review on path integrals, we refer to [4, 8].. 3.1.2 Gaussian Integrals We often use the Gaussian integral when evaluating path integrals. The Gaussian integral is dened as. Z. F (z; w) = dxe. zx2 +wx. R. =. The one-dimensional Gaussian integral. Z. Fd(M) := dx Rd. 1. r. . . w2 exp ; z; w 2 R; z 6= 0: z 4z. F (z; 0) can be generalized to d-dimensions. : : : dxd exp. d X i;j =1 20. !. xi Mij xj. Z. dxe Rd. xt Mx ;.

(40) 3 Path Integrals. where. M Nt = N. 3.1 General Formalism of Path Integrals. is a real symmetric. dd. We can diagonalize the matrix matrix;. 1. x. accordingly. is a column vector and. M = N MD N MD t. xt. N y = Nx. , where. its transpose.. is an orthogonal. det N = 1. The matrix is diagonal with real, assuming all 1 ; : : : ; d . Hence, for a change of variable , the Gaussian. and. non-zero, eigenvalues. M. matrix,. integral is written as. Fd(M) = det N. Z. Rd. d Z. Y t dye y MD y = dyk e. = d=2 (det MD ). 1=2. =. k=1 R d=2 (det. M). k (yk )2 1=2. = d=2 (1 2 : : : d ). 1=2. :. A more general Gaussian integral is given by. Z. F (M; u) = dxe Rd. xt Mx+ut x+xt u = d=2 (det M). 1=2. euM u : 1. 3.1.3 Zeta Function Regularization When evaluating path integrals via the Gaussian integral we need to solve functional determinants, e.g.. det(d 2 =dt2 ), via an eigenvalue problem.. Imposing Dirichlet (or periodic). boundary conditions on the path integral, we solve eigenvalue equations of the form. d2 x (t) = n xn (t); 0 t T ; xn (0) = xn (T ) = 0: dt2 n The eigenfunctions xn are, due to the boundary values, proportional to sin(nt=T ) and 2 the eigenvalues are n = (n=T ) , n 1. Hence, the functional determinant is equal to det. . 1 1 2 Y Y d2 n < 1: = n = 2 dt T n=1 n=1 . O^ be a generic operator whose eigenvalues are positive denite, 1 2 : : : n > 0, and from the formula det O^ = exp[Tr log O^ ] we have Let. log det O^ = Tr log O^ = We dene the. MP zeta function. 6. , associated to. O^ (s) := Tr O^. s. =. 1 X. 1 X n=1. det O^ =. log n :. O^ , as. 1 ; s n=1 n. where the sum converges for suciently large. i.e.. s 2 C;. <(s). Notice. d ( s ) = log n exp( s log n ) dt n and. 6 The. zeta function of Minakshisundaram and Pleijel. There are several zeta functions; the Riemann zeta function is also referred to in this thesis. 21.

(41) 3.1 General Formalism of Path Integrals. 3 Path Integrals.

(42). d (s)

(43) O0^ (0) = O^

(44)

(45) ds. s=0. 1 X. =. n=1. log n ;. hence we arrive at. . O0^ (s) :. det O^ = exp Thus with the operator. . O^ = d2 =dt2 mentioned as an example above, we nd. d2 =dt2 (s). =. P. 1 2s X n. =. T. n=1. 2s. T . (2s);. 2s (2s) = 1 is the Riemann zeta function, with well-dened (0) = 1=2, n=1 n 0 and (0) = log(2)=2. Finally, we get the derivative of the zeta function at s = 0 equal where. to. 0. d2 =dt2 (0). = 2 log. . T (0) + 2 (2s) = log(2T ): . The nal result of the functional determinant is. . det. . d2 = 2T: dt2. We give an example below on how to evaluate a path integral using the zeta function regularization and Fourier series.. 3.1.4 Fourier Series and Path Integrals Previously, we divided the time period. T. into. N. steps, i.e.. t = (t00. t0 )=N = T=N .. Instead of discretizing the time interval we can evaluate the path integral using a Fourier series. x(t) = where. an. 1 X n=1. fn x by the nite series. are the Fourier coecients and. discretize the trajectory. xN (t). =. t0 < t < t00 ;. an fn (t);. N X n=1. are trigonometric functions.. t0 < t < t00 :. an fn (t);. N QN x (t) are functions of n=1 dan . We can choose. The approximate paths. the Fourier coecients. measure is. here. conditions. D a (N ). x(0) = x(T ) = 0.. Hence, we. t0 = 0, t00 = T. fang,. thus the. and the boundary. Due to the boundary conditions we must use the sine-Fourier. series:. xN The path integral, here denoted. . N X. . nt = an sin : T n=1 F (T ), is then equal to 22.

(46) 3 Path Integrals. F (T ) =. 3.1 General Formalism of Path Integrals. x(ZT )=0. D x(t) exp. x(0)=0. := lim. ". N !1. p 2. . N. . i. ~. (. Z. S [x(t)] =. ". 1 X. . nt D a exp ~ S an sin T n=1. P BCs. . m (N +1)=2 N! 2i~T. #. N Z Y n=1 R. i. dan exp. . i. ~. #). N S [x (t)]. (3.3). The prefactor inside the square brackets in (3.3) is chosen so that in the end we get the result of the free eld case (for which. 1=2. 1=2. (!T ) (sin !T ). harmonic oscillator. V (x). . As an example we compute the path integral of the. osc. !. one-dimensional. whose Lagrangian is given by. L (x; x_ ) = where. 0) multiplied by the trigonometric term. m 2 x_ 2. is the oscillation frequency. Here we. integral by K (x00 ; x0 ; T ):. m 2 2 !x; 2 0 00 use t = 0, t = T. and denote the path. osc. K (x00 ; x0 ; T ) =. x(TZ)=x00. osc. x(0)=x0. D x(t)eiSosc[x(t)]=~;. where the action is equal to. ZT. m S [x(t)] = dt(x_ 2 2 osc. 0. Expanding the variable. ZT. m ! x )= dtx(t) 2 2 2. 0. cl. x. cl. d2 dt2. !. 2. x(t):. x(t) as. x(t) = x (t) + q(t); x (0) = x0 ; x (T ) = x00 ; where. . cl. cl. is the classical trajectory and. q(t). and. q(0) = q(T ) = 0;. is the closed quantum uctuation.. The. exponent of the action is factorized into a classic factor, where the equations of motion is given by the Euler-Lagrange equation, and a quantum factor. F. osc. which is the path. integral of the quantum uctuations:. K (x00 ; x0 ; T ) = exp(iS [x ]=~)F (T ): osc. osc. For closed quantum uctuations integral. F (T ). q(t). cl. osc. we use a sine-Fourier series and obtain the path. osc. F (T ) := K (0; 0; T ) = osc. q(ZT )=0. D q(t)eiSosc[q(t)]=~:. osc. q(0)=0 The nite series approximation. qN (t) of the action S [qN (t)] is equal to osc. 23.

(47) 3.1 General Formalism of Path Integrals. ZT. m dt(q_2 S [qN (t)] = 2. !2q2). osc. 0. ZT. N mX = a2 dt 2 n=1 n N X. mT = 4. n=1. 3 Path Integrals. 0. an 2. . . n 2 2 nt cos T T. n 2. !. T. . ! sin 2. 2. . nt T. . 2. Now we can motivate, more explicitly, the choice of the prefactor in (3.3) by evaluating. F N (T ): osc. F N (T ) = osc. (. N Z Y n=1. = N=2 . mT 4i. dan exp. = p 2. N. n=1. m 2 i ~T. !2 a2n. T. n=1 ( N " N Y mT N=2 Y T. 4i~. n. N=2. 1 N!. ). . N X n 2. 1. n=1 ( " N Y n=1. 1. 2 #) 1=2. . !T n. . !T n. 2 #) 1=2. ;. where in the second equality the Gaussian integral was used in evaluating the path integral, and the two products come from factorizing. FN. osc. F N (T ) = osc. (. m 2i~T. N Y. ". n=1. 1. . !2 ]). ! 1, yields. (T ) in (3.3,) and taking the limit N r. Q. ( [(n=T )2. !T n. 2 #) 1=2. =. r. m 2i~T. r. 1=2. !T sin !T. . Substituting. for. T > 0;. where the rst square root is the free eld result of the path integral. As an example of evaluating a path integral using the result of the zeta function regularization above, we consider the Lagrangian. F0 (T ) = =. L = (1=2)mq_2 and check the free eld case F0 (T ): 2. qZ (t)=0. D q(t) exp 4. q(0)=0 r . m det i~. . d2 dt2. ZT 1 m. dtq(t). 2 i~. 1=2. 0. =. r. . . 3. d q(t)5 dt2 2. m : 2i~T. 3.1.5 Coherent States Previously we introduced the classical Lagrangian. L = (1=2)mx_ 2. (1=2)m!2 x2. for the. simple one dimensional harmonic oscillator. The (quantum) Hamiltonian is then equal to. 24.

(48) 3 Path Integrals. 3.1 General Formalism of Path Integrals. p^2 m!2 x^2 + : H^ = 2m 2 We dene the annihilation and creation operator, respectively, as. r. . r. . . . ip^ ip^ m! m! a= x^ + ; ay = x^ : (3.4) 2~ m! 2~ m! With the commutation relation [^ x; p^] = x^p^ p^x^ = i~, we get the Hamiltonian in terms y of the number operator N = a a: m! 2 p^2 i H^ y aa= x^ + 2 + [^x; p^] = 2~ m! 2~ ~!. 1 ; 2. or. . . 1 H^ = ~! N + : 2 The eigenvalue equation of. (3.5). N , acting on the energy eigenkets jni, is equal to N jni = njni:. The eigenvalues on. n are positive integers,. and the annihilation (creation) operator acting. jni decreases (increases) the energy state by one unit, accordingly p p ajni = njn 1i; ay jni = n + 1jn + 1i:. One can show [13], by using the Heisenberg equations of motion, that the time evolution of. a and ay are. Expressing. x^. and. a(t) = a(0) exp( i!t); ay (t) = ay (0) exp(i!t) p^ in terms of a and ay , by rewriting (3.4), we get x^(t). (3.6) and. p^(t). from. (3.6):. . . p^(0) x^(t) = x^(0) cos !t + sin !t; m! p^(t) = m!x^(0) sin !t + p^(0) cos !t: x^(t) and p^(t) seem to oscillate, analogous to the case in classical mechanics. Notice, however, that x ^(0) a+ay and p^ a+ay ; computing the expectation values hnjx^(t)jni and hnjp^(t)jni gives zero in both cases due to the orthogonality hnjn 1i = 0. In order to observe oscillations of x ^(t) and p^(t) we must use instead a superposition of energy eigenstates, e.g. using j0i and j1i, The variables of. j i = c0j0i + c1j1i; c0; c1 2 C: A. coherent state. is dened by the following eigenvalue equation:. aji = ji; 2 C; 25.

(49) 3.2 Grassmann Algebra. where the eigenket. 3 Path Integrals. ji is a superposition of jni: ji =. The distribution of mean of. n. jfn(n)j. 2. is of. 1 X n=0. fn (n)jni:. Poisson type ;. measurements. For large values of. bell-shaped. Gauss distribution. jfn(n)j2 = (nn=n!) exp( n ), where n is a. n,. the Poisson distribution approaches a. [2].. In summary; a coherent state is an oscillator ground state (a Gauss distribution) that can bounce back and forth by some nite distance in space. The shape of a wave-package translated in space remains in an oscillator ground state, for all time intervals. t, without. spreading in shape. We use coherent states in the derivation of the fermionic path integral below.. 3.2. Grassmann Algebra. The Pauli exclusion principle states that no two electrons with identical quantum numbers can occupy the same quantum state. Consider, e.g., an electron with, say, spin up and is in a state. jni, if another electron is in the same state, then the latter electron must have. a spin down. In the next section the path integral for fermions will be derived. Instead of commuting numbers, as used in the construction of the bosonic path integral, anti-commuting Grassmann numbers are thus imposed in the Lagrangian and the measures.. 3.2.1 Grassmann Algebra Let. f1; : : : ; ng be a set of Grassmann variables, satisfying the anti-commutation relation. fi; j g = ij + j i = 0 8i; j: A set of linear combinations of fi g, with coecients that are complex numbers, is called a. Grassmann number,. e.g. for. n = 2,. f () = f0 + f1 1 + f2 2 + f12 1 2 ; f0 ; f1 ; f2 ; f12 2 C: 2 From the anti-commutative relation above, we have that (i ) = 0. We dene a function of Grassmann numbers as a Taylor expansion. E.g., for n = 1 and a Grassmann variable, a Grassmann function exp( ) is equal to e = 1 + : The exponential of one, or several, Grassmann numbers is a Grassmann function we encounter frequently when evaluating integrals in the following sections of this chapter, and in chapter ve below.. 3.2.2 Dierentiation The dierential operator. @=@i. act acts on a function from the left, in a similar way as. the ordinary dierential operator:. @ = : @i j ij 26.

(50) 3 Path Integrals. 3.2 Grassmann Algebra. f ( ). E.g., taking the derivative of. dened above with respect to the operators. yields. @ @ @1 @2. @ @ f () = f12 : @1 @2 Notice the order of the dierential operators and that 1 2 = 2 1 in the fourth term of f ().. 3.2.3 Integration integration with respect to a Grassmann variable introduce the. Berezin integrals Z. For a general function. d = 1;. Z and. . is equivalent to dierentiation. We. d1 = 0:. f () we have Z. df () =. @f () ; @. i.e.,. Z. d1 d2 : : : dn f (1 ; 2 ; : : : ; n ) =. Since the order of the dierentials. @=@i. @ @ @ : : : f (1 ; 2 ; : : : ; n ): @1 @2 @n. is the same as for. d1 : : : dn ,. one must use. the anti-commutation rule to, if necessary, arrange the Grassmann variables of. f () in a. descending order with respect to the dierentials. Integration under a change of variable. Z. i.e.,. d 0 = (1=a)d.. 0 = a, a 2 C, transform as Z. @f () @f ( 0 =a) 0 0 = = a d f ( =a); df () = 0 @ @ ( =a) Extending to the case of. n. variables;. i. ! i. = aij j ,. gives the. transformation. Z. where. ~ = (1 ; : : : ; n ). Z. 0 0 0 d1 : : : dn f (~ ) = det a d1 : : : dn f (a 1 ~ ); is a column vector and. a = [aij ]. a matrix.. We use the. n-case. change of variables when computing path integrals and Gaussian integrals in a following chapter. The Gaussian integral using Grassmann variables is dened below.. 3.2.4 Gaussian Integral of Grassmann Variables The Gaussian integral is given by. Z. I = d1 d1 : : : dn dn e 27. Pi;j iMij j. :.

(51) 3.2 Grassmann Algebra. The matrix the sets. 3 Path Integrals. M = [M ij ]. 0i i P = i i and Under a change of variables i = j Mij j the Mij = Mji ,. is skew-symmetric, i.e.. fig and fi g are independent.. integral is evaluated as. Z. Pi ii0. I = det M d1 d1 : : : dn dn e 0. 0. Z. = det M d1 d1 e = det M. 0. Z. 1 10. since. 0. : : : dn dn e. n n0. n. 0 0 d d (1 + ). = det M:. Notice the lack of square-root when integrating over two independents sets of variables. The determinant is in the nominator, rather than in the denominator as in the bosonic case, when implementing Grassmann variables into the Gaussian Integral. If the Grassmann variable. . is complex, then. . is the complex conjugate of. a later chapter we introduce an anti-commutative eld dual to the spin eld. ,. .. (In. called the isospin eld which. is then denoted by .). , and the complex conjugate of. We can show that the Gaussian integral vanishes if we have an odd number of factors in the measure. order. 2n as. We dene[10] the. (A) =. Pf. Pfaan. of the anti-symmetric matrix. X 1 sgn(P )ai i ai i : : : ai n 2n n! Permutations of 1 2. 3 4. i. 1 2n. 2. A = [Aij ]. of. :. fi1 ;:::;i2n g. (P ) is the signature of the permutation P . minant Dn of order n: where sgn. Dn = where. X i;j;k;:::. Recall the denition[2] of a deter-. "ijk::: ai bj ck : : :. "ijk::: is the n-dimensional analogue to the Levi-Civita symbol. n = 3:. As an example, we. consider the familiar case.

(52)

(53)

(54)

(55)

(56)

(57).

(58). a1 a2 a3

(59)

(60) D3 = b1 b2 b3

(61)

(62) = +a1 b2 c3 c1 c2 c3

(63). a1 b3 c2. a2 b1 c3 + a2 b3 c1 + a3 b1 c2. (P );. Now we can clearly see the meaning of sgn sgn. (P ) = +1 (sgn(P ) = 1).. a3 b2 c1 :. an even (odd) permutation. P. yields. In the example above, we see that a simple transposition. (123), gives a minus, hence an odd permutation. For instance; (123) ! (213) ! +(231) ! (321), hence sgn(P ) = 1 in the last term of D3 . Notice that there are 3! = 6 terms in the sum n of D3 . From a combinatorical point of view, there are 2 ways of swapping the indices, of a subscript of the matrix elements, with respect to the linear sequence. e.g.,. a i i a i i : : : ai n 1 2. There are. 3 4. 2. i. 1 2n. ! a i i a i i : : : ai n 2 1. 3 4. n! permutations of the pairs of indices, e.g., 28. 2. i. 1 2n. :.

(64) 3 Path Integrals. 3.3 Fermionic Path Integral. a i i a i i : : : ai n 1 2. 3 4. i. 1 2n. 2. ! a i i a i i : : : ai n 3 4. 1 2. 2. i. 1 2n. :. In order to avoid double counting of the terms, there is a fraction of the sum in the denition of the Pfaan.. A. The matrix. accordingly,. 0. NtAN = AD = and the determinant of. A. B B B B B @. . . .. ... in front. can be block diagonalized,. 1. 0 1 : : : 1 0 : : : . . .. 1=(2n n!). .. 0 2n. C C C C; C 2n A. 0. is equal to. det(A) = det(AD ) =. 2n Y. i=1. 2i :. The Pfaan of a block diagonalized matrix is given by. P f (AD ) = ai i ai i : : : ai n 1 2. 3 4. 2. in=. 1 2. 2n Y. i=1. i ;. which yields the relation between the Pfaan and the determinant:. det(A) = [Pf(A)]2 : The Gaussian integral can be expressed in terms of the Pfaan:. ". Z. #. Z. 1 1X i Aij j = n d2n : : : d1 I = d2n : : : d1 exp 2 i;j 2 n!. X i;j. i Aij j. !n. = Pf(A):. Notice here the factor 1/2 in the argument of the exponential in the absence of pairs of. d's.. In the second equality, the exponential is expanded and the only term that saturates. the measure. d2n : : : d1 is of the order n since there are two Grassmann variables i and j. in the sum. The Pfaan vanishes for odd order matrices. As we shall see in chapter ve, the order of the matrix is associated to the dimension of a manifold and the non-vanishing of the analytical index.. 3.3. Fermionic Path Integral. The fermionic path integral is constructed analogous to the bosonic case. We use instead Grassmann variables and arrive at a path integral identical, except for the boundary conditions, to the bosonic path integral. The boundary conditions are now anti-periodic rather than periodic.. 29.

(65) 3.3 Fermionic Path Integral. 3 Path Integrals. 3.3.1 Fermionic Harmonic Oscillator Here we consider a quantum system with a single spin-1/2 particle described by the. matrices. , x. y. and. . z. With. = (. x. Pauli. i )=2 we dene the fermionic annihilation and y. creation operators, respectively,. . . . . c = = 01 00 ; cy = + = 00 10 : The operators. c and cy satisfy the anti-commutation relations. fc; cyg = ccy + cyc = 1; fc; cg = fcy; cyg = 0: Hence. c2 = (cy )2 = 0.. 7. The fermionic harmonic oscillator is described by the Hamiltonian. . 1 1 H^ = (cy c cy c)! = [cy c (1 cy c)]! = ! N 2 2. N = cy c eigenvalue of N. where. is the number operator (cf. is either zero or one;. Let the energy state. j1i = 10 ;. j0i =. cy j0i = j1i, cj0i = cy j1i = 0, and cj1i = j0i.. . H = !(N + 1=2)). The or N (N 1) = 0.. c y c )c = N ,. jni, n = 0 or 1, be an eigenvector of H^ : . then. the bosonic case. N = cy ccy c = cy (1 2. 1 2. . 0 1 ;. Hence the eigenvalues of the Hamilto-. nian are given by the eigenvalue equations. ! ! j 0i; H^ j1i = j1i: H^ j0i = 2 2. 3.3.2 Fermionic Coherent States N has eigenvectors j0i and j1i, hence an arbitrary vector jf i can be jf i = P fnjni = f0j0i + f1j1i. In the fermionic coherent state representation. The number operator written as. we have two basis functions fermionic coherent state. f0 = 1. and. f1 = , . a Grassmann variable, hence the. ji, and its dual hj, are equal to. ji = j0i + j1i; hj = h0j + hj: The coherent states are eigenstates of. c and cy , respectively:. cji = j0i = j0i + 0 = j0i + j1i2 = ji; hjcy = hj: this section we set ~ = m = 1, since it is more convenient to introduce this notation here, in agreement with the notation in chapter ve below. 7 In. 30.

(66) 3 Path Integrals. 3.3 Fermionic Path Integral. 3.3.3 Fermionic Partition Function. Z (

(67) ), derived from the path integral by the replacement t = i , i.e. the imaginary time (or Euclidean time). p The partition function is 1 = i in the exponent; identical to the path integral, except for the absence of a factor ^ )jx i is identical to Z (

(68) ) = hx j exp(

(69) H^ )jx i, where

(70) = iT . e.g. hx j exp( iHT We introduce the partition function. f. i. f. i. The reason we introduce the partition function here is due to the notation used in the index theorems, presented in chapter ve below. First, we dene and compute the partition function of a fermionic harmonic oscillator:. Z (

(71) ) = Tr e.

(72) H^. =. 1 X. n=0. hnje.

(73) H^. jni = e

(74) !=2 + e.

(75) !=2. = 2 cosh(

(76) !=2). This partition function is of great importance in proving the. (3.7). Hirzebruch signature,. as. veried in chapter ve. Using the completeness relation for fermionic coherent states:. Z. . d djihje. = 1;. one can show [10] that the partition function is related to the integral over Grassmann variables, accordingly. Tr e. Z. = d dh je.

(77) H^.

(78) H^. jie. :. We emphasize here the anti-periodic boundary conditions (APBCs) over. Tr(). ji, evolving to the nal state j i; the Grassmann. above. The initial state is. variable is. . at. = 0,. . and. at. =

(79) .. The construction of this path integral is. analogous to the bosonic case. With the time step. e. [0;

(80) ] in the trace. =

(81) =N , hence the limit. ^ )N ; = lim (1

(82) H=N.

(83) H^. N !1. and inserting the coherent completeness relation. N 1 times, gives the following partition. function:. Z. d de Z (

(84) ) = Nlim !1 Z. = lim d de N !1. . ^ )N ji h j(1

(85) H=N NY1Z PnN n n k=1. dk dk e. 1 =1. h j(1 H^ )jN 1i : : :. : : : h2j(1 tH^ )j1ih1j(1 H^ )ji ZY N d e PNn n n h j(1 H ^ )jN 1 i : : : h1 j(1 H^ )j N i; = lim d N k k N !1 =1. k=1. = N = 0 , = N = 0 . From the denition of fermionic coherent states we have hk jk 1 i = 1 + k k 1 = exp(k k 1 ) and hk jH^ jk 1i = hk j(k k 1 1=2)!jk i. We now evaluate each one of the matrix elements for k = 0; : : : ; N : where we dene the initial and nal states as. 31.

(86) 3.4 The Supersymmetric Path Integral. 3 Path Integrals. hk j(1 H^ )jk 1i = hk jk 1i. ". h jH^ jk 1i k hk jk 1i. 1. = exp( . k k. 1. . ) exp. #. . ! k k. 1 2. 1. . :. Hence, the partition function is. Z=. e

(87) !=2. lim. N !1. = e

(88) !=2 lim. N !1. =. Z. N Z Y k=1 N Z Y k=1 2. dk dk e. PNn. [n (n n. dk dk e. PNk. [(1 !)n (n n. =1. =1. Z

(89). D D exp 4

(90) !=2. !n n. 1 )+. . d (1 !). 0 where in the rst equality we add and subtract a factor. 1]. =+!n n ]. 1). . 3. d + ! 5 ; d. !n n. (3.8). in the sum of the ex-. ponential, in order to rewrite the argument of the exponential as given in the second equality.. . Finally, in the third equality the time step. is kept in the action due to its. contribution of a factor of two, when evaluating the partition function via zeta function regularization[10] that gives. 3.4. Z (

(91) ) = 2 cosh(

(92) !=2) as in (3.7).. The Supersymmetric Path Integral. We derived one kind of path integral for bosons and another kind for fermions; except from commutativity and anti-commutativity of their variables, respectively, they dier by the boundary conditions imposed on their solutions. To put the bosonic and fermionic path integrals on an equal footing, we impose therefore periodic boundary conditions on the fermionic part partition function and it is given by. Tr(. 1)F e

(93) H^. =. 1 X. n=0 Z. hnj(.

(94) H^. jni. = d dh j( 1)F e.

(95) H^. = d dhje. ;. Z. where. 1)F e.

(96) H^. jie. F = cy c is the fermion number operator, and ( 1)F (. Let the operator. ( 1)F. 1)F. =. . 32. . is dened as. . 1 0 0 1 :. act on a coherent state. condition of that state is changed accordingly. jie. ji. = j0i + j1i,. thus the boundary.

(97) 3 Path Integrals. (. 3.4 The Supersymmetric Path Integral. 1)F. ji =. . 1 0 0 1. . = 1. . . = j0i 1. j 1 i = j i;. i.e., an anti-periodic boundary condition is changed into a periodic one as. h j(. 1)F = hj. in the third equality of the trace above. Thus, in the supersymmetric path integral we combine the bosonic and the fermionic cases into one, unied, path integral in Euclidean time:. Z (

(98) ) = where. D x and D. Z. D xD e. R

(99) dtL 0. ;. (3.9). PBCs. are the measures of the bosonic and the fermionic elds, respectively. (a eld is a variable with innitely many degrees of freedom).. 33.

(100) 3.4 The Supersymmetric Path Integral. 3 Path Integrals. 34.

(101) 4 Spontaneous Breaking of Supersymmetry. 4. Spontaneous Breaking of Supersymmetry. In this chapter we study the trace formula. Tr( 1)F exp(

(102) H ),. rst introduced in the. fermionic partition function in the previous chapter, and relate it to the. analytical index. of an operator. In this review of symmetry breaking we will be rather heuristic, hence no derivations will be found here, with the goal of merely presenting basic facts about. Tr( 1)F. and its physical meaning in the context of supersymmetry. We follow closely. [15, 16] where a more comprehensive study can be found. We do not observe in nature, e.g., neither spin-1 electrons, nor photons with half integer spin, hence supersymmetry must be spontaneously broken. A broken symmetry implies a mechanism that gives mass to particles.. 1)F. Tr(. As will be shown below, the index. takes integer values and determines whether supersymmetry is. other words,. Tr( 1)F. un broken.. In. is a mathematical tool used for identifying, and discarding, super-. symmetrical models that cannot describe nature. First we introduce some terminology that will be used frequently throughout this chapter. By. internal symmetry breaking. we mean the symmetry breaking mechanism in. electroweak theory that gives mass to non-supersymmetric particles. Electroweak theory is a topic usually reviewed in introductory textbooks on quantum eld theory, see for instance [11]. contrast to. The concepts of internal symmetry breaking should be stated in stark. spontaneous supersymmetry breaking, since there are certain conditions where. the latter will occur. Thus, the major topics in this chapter is the formal denition of. Tr( 1)F , and the conditions that forces us to discard a supersymmetrical model. 4.1. The Energy Spectrum. In order to associate. Tr( 1)F. to an index and to determine whether we have unbroken. supersymmetry , we need to dene and study the energy spectrum of the theory. We dene a supersymmetric theory in a volume. V. a few low lying states above. index,. j0i.. The denition of. V ! 1 in the j0i, or zero energy state, and. (and take the limit. end) where we are mainly interested in the ground state. Tr( 1)F ,. which is called the. Witten. is. Tr( 1)F = nEB=0. nEF =0 ;. (4.1). nEB=0 (nEF =0 ) is the number of zero energy bosonic (fermionic) states. In supersymmetric theories the energy E jP j, jP j the magnitude of the momentum, hence P = 0 F for the ground state. Notice that we can regulate Tr( 1) with the kernel exp(

(103) H ), hence Tr( 1) exp(

(104) H ), and let

(105) ! 0 which is the high temperature limit, and thus where. removing high energy states.. We dene the Hamiltonian H in terms of the hermitian supersymmetry Q1 ; Q2 ; : : : ; QK (K = 4 for supersymmetry in 3 + 1 dimensions):. charges. Q21 = Q22 = = Q2K = H; Qi Qj + Qj Qi = 0; for i 6= j:. jbi (jf i) that satises the operator exp(2iJz )jbi = jbi (exp(2iJz )jf i = jf i). The operator exp(2iJz ) rotates a state 1 4 counter clockwise by 2 in the x-y plane. To be more precise, exp(2 iJz ) is (exp( iJz )) ; 2. In four dimensions we dene a bosonic (femionic) state. 35.

(106) 4.1 The Energy Spectrum. 4 Spontaneous Breaking of Supersymmetry. four successive rotations by ninety degrees in the x-y plane. In a nite volume the rotation generator. Jz. 8. is not well dened , but. the matrix. exp( 12 iJz ) is well dened. . Furthermore, we dene. . 1 0 ( = exp(2iJz ) = 0 1 : F In 0 + 1 dimensions we have to dene ( 1) more abstractly (since there are no angular momentum Jz in dimensions less than two) as the commutator and the anti-commutator, 1)F. respectively,. ( 1)F = ( 1)F ; ( 1)F =. ( 1)F ;. and Fermi eld . jbi, and for E 6= 0, the fermionic p (1= E )Qjbi, where Q is now any of the Qi , i = 1; : : : ; K . for some Bose eld. For any bosonic state. interested in the zero energy state is due to the pair. p. state is dened as. p. (4.2). where the second equation follows from the denition of the Hamiltonian. H jbi = E jbi.. =. The reason we are only. Qjbi = E jf i; Qjf i = E jbi; and. jf i. Q2 = H. 0,. The interpretation of (4.2) is that, for every non-zero energy state,. there must exist Bose-Fermi pairs, hence we have the dierence. 0 nE> B. 0 nE> = 0. F. An. energy spectrum is shown in gure (2) where the lowest horizontal line is the ground state, hence the equation (4.1) is equal to one in this particular example.. E. E=0. Figure 2:. The bosons are indicated by circles, and the fermions by lled rectangles,. in the diagram.. 3. 2 = 1,. while. Tr( 1)F = Tr( 1)F 6= 0,. The lowest horizontal line is the zero energy state where. Tr( 1)F = 0. for all states above the ground state. For. supersymmetry is unbroken.. mi. The parameters of the supersymmetric theory is understood as the volume of the particles, and the coupling constants. gi .. V , the mass. Varying the parameters implies that. the energy states are shifted, either up or down, in the energy spectrum. For instance, assume that we are varying some parameter so that the rst state above the ground state in gure (2) is slowly moving down and, eventually, coincide with the ground state. The dierence is now. nEB=0 nEF =0 = 4 3 = 1, hence, the same as in the original conguration.. This invariance is, of course, due to the Bose-Fermi pairs in the non-zero energy states. The important property here is the following conditions:. 8 Rotate. a cube lying on the x-y plane. We put a label on one of its vertical faces and apply a (discrete) rotation. If the label seems to be on the same face, we can't tell whether there has been applied a 2 rotation, or no rotation at all. A ninety degree rotation, on the other hand, will surely distinguish the initial position from the nal position. 36.

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