AP08_2.vp 1 Introduction Supersymmetric Quantum Mechanics [1] is under inten- sive development and remarkable new features have been dis- covered in recent years. This attention is due both to the wide range of applicability of one-dimensional supersymmetric theories and especially superconformal quantum mechanics [2] for extremal black holes [3], in the AdS-CFT correspon- dence [4] (when setting AdS2), in investigating partial break- ing of extended supersymmetries [5, 6], as well as for its underlying mathematical structures. It is well known that large N (up to N � 32, starting from the maximal, eleven- -dimensional supergravity) one-dimensional supersymmetric quantum mechanical models are automatically derived [7] from dimensional reduction of higher-dimensional super- symmetric field theories. Large N one-dimensional super- symmetry on the other hand (possibly in the N � � limit) even emerges in condensed matter phenomena. Controlling one-dimensional N-extended supersymmetry for arbitrary values of N (that is, the nature of its representation theory, how to construct manifestly supersymmetric invariants, etc.) is a technical, but challenging program with important conse- quences in many areas of physics, see e.g. the discussion in [8] concerning the nature of on-shell versus off-shell representa- tions, for its implications in the context of the supersymmetric unification of interactions. Over the years, progress has come from two lines of attack. In the pivotal work of [9] irreducible representations were in- vestigated to analyze supersymmetric quantum mechanics. The special role played by Clifford algebra was pointed out [10]. Clifford algebras were also used in [11] to construct representations of the extended one-dimensional supersym- metry algebra for arbitrarily large values of N. Another line of attack involved using superspace, so that manifest invariants could be constructed through superfields. For low values of N this is indeed the most convenient approach. However, with increasing N, the associated superfields become highly reduc- ible and require the introduction of constraints to extract irre- ducible representations. This approach soon becomes im- practical for large N. Indeed, only very recently a manifestly N � 8 superfield formalism for one-dimensional theory has been introduced, see [12] and references therein. A manifest superfield formalism is however lacking for larger values of N. In this work we discuss our results [13], [14], [15] concern- ing the classification of linear irreducible representations real- ized on a finite number of time-dependent, bosonic and fermionic, fields. The connection with Clifford algebras and division algebras is discussed, as well as the construction of off-shell invariant actions and some associations with graph theory. Several important topics that have appeared recently in the literature, like the nature of the non-linear representa- tions will not be discussed here. There are reviews ([16]) that cover this and other aspects. Similarly, the quite important connection with supersymmetric integrable systems in (1+1) dimensions (such as the supersymmetric extension of the KdV equation, will not be discussed since they have been cov- ered elsewhere [17]). The scheme of the paper is as follows. The next Section deals with the relevance of one-dimensional Supersymmetric Quantum Mechanics for understanding higher-dimensional supersymmetric field theory. Some selected examples of di- mensional reductions are pointed out. The relation between irreducible representations of one-dimensional N Extended Supersymmetry Algebra and Clifford algebras is explained in Section 3. Section 4 reviews the classification of Clifford alge- bras and their relation with division algebras, following [18]. In Section 5 the results of [14] concerning the classification of irreducible representations with length-4 field content are reported. Section 6 computes off-shell invariant actions of one-dimensional sigma models within a manifestly super- symmetric formalism which does not require the introduction of superfields. In Section 7 an N � 8 invariant action con- structed in terms of octonionic structure constants is pre- sented. The classification in [15] of nonequivalent N � 5 6, supersymmetry transformations with the same field content is given in Section 8 and 9. A graphical presentation of super- symmetry transformations in terms of N-colored oriented graphs is discussed. Section 10 introduces the fusion algebra produced by tensoring irreducible representations and pres- ents it in graphical form. 58 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 Extended Supersymmetries in One Dimension F. Toppan This work covers part of the material presented at the Advanced Summer School in Prague. It is mostly devoted to the structural properties of Extended Supersymmetries in One Dimension. Several results are presented on the classification of linear, irreducible representations realized on a finite number of time-dependent fields. The connections between supersymmetry transformations, Clifford algebras and division algebras are discussed. A manifestly supersymmetric framework for constructing invariants without using the notion of superfields is presented. A few examples of one-dimensional, N-extended, off-shell invariant sigma models are computed. The relation between supersymmetry transformations and graph theory is outlined. The notion of the fusion algebra of irreps tensor products is presented. The relevance of one-dimensional Supersymmetric Quantum Mechanics as a way to extract information on higher dimensional supersymmetric field theories is discussed. Keywords: Supersymmetric Quantum Mechanics, M-theory. ` 2 N Extended supersymmetries in D � 1 and dimensional reduction of supersymmetric theories in higher dimensions One important motivation for investigating N Extended Supersymmetries in one dimension is the fact that their rich algebraic setting can furnish useful information concern- ing the construction of supersymmetric theories in a higher dimension (such as super-Yang-Mills, supergravity, etc.) Su- persymmetric quantum mechanics with large number N encodes large information of these theories. The simplest way to see this is through dimensional reduction, where all space-dimensions are frozen and the only remaining depend- ence is in terms of a time-like coordinate. The usefulness of this procedure is due to the fact that in such a framework we can make use of powerful mathematical tools (essential- ly based on the available classification of Clifford algebras) which are not available in higher dimensions. It should be remembered that a four-dimensional field theory with N extended supersymmetries corresponds, once it is dimensionally reduced to one-dimension, to a super- symmetric quantum mechanics with four times (4N) the number of the original extended supersymmetries [7]. The most interesting case, in the context of the unification pro- gram, corresponds to eleven-dimensional supergravity (the low-energy limit of M-theory), which is reduced to an N � 8 four-dimensional theory and later to an N � 32 one-dimen- sional supersymmetric quantum mechanical system. In this section we will discuss the dimensional reduction of supersymmetric theories from D � 4 to D �1 in some specific examples. We will prove how certain D � 4 problems can be reformulated in a D �1 language. It is convenient to start with a dimensional analysis of the following theories: i) the free particle in one (time) dimension (D �1) and, for the ordinary Minkowski space-time (D � 4), iia) the scalar boson theory (with quartic potential � �4 4 ! ), iib) the Yang-Mills theory and, finally, iic) the gravity theory (expressed in the vierbein formalism). We further make a dimensional analysis of the above three theories when dimensionally reduced (a la Scherk) to a one (time) dimensional D �1 quantum mechanical system. In the following we will repeat the dimensional analysis for the supersymmetric version of these theories. Case i) – the D �1 free particle It is described by a dimensionless action S given by S m dt� � 1 2 �� . (2.1) The dot denotes, as usual, the time derivative. The dimen- sionality of the time t is the inverse of the mass; we can there- fore set ([ ]t � �1). By assuming � being dimensionless ([ ]� � 0), an overall constant (written as 1 m ) of mass dimension �1 has to be inserted to make S non-dimensional. Summarizing, we have, for the above D �1 model, [ ] , [ ] , [ ] , [ ] , [ ] . t t m S D D D D D � � � � � � � � � � � 1 1 1 1 1 1 1 0 1 0 � � � (2.2) The suffix D �1 has been added for later convenience, since the theory corresponds to a one-dimensional model. Case iia) – the D � 4 scalar boson theory The action can be presented as S d x M� � �� � � �� 4 2 2 4 1 2 1 2 1 4 � � �� �� � � � ! (2.3) A non-dimensional action S is obtained by setting, in mass dimension, [ ] , [ ] , [ ] , [ ] . � � D D D D M � � � � � � � � 4 4 4 4 1 1 1 0 � � (2.4) Case iib) – the D � 4 pure QED or Yang-Mills theories The gauge-invariant action is given by S e d x F F� � 1 2 4 Tr( )�� �� , (2.5) where the antisymmetric stress-energy tensor F�� is given by F D D�� � �� [ , ], (2.6) with D� the covariant derivative, expressed in terms of the gauge connection A� D eA� � ��� � . (2.7) e is the charge (the electric charge for QED). The action is non-dimensional, provided that [ ] , [ ] , [ ] . A F e D D D � �� � � � � � � 4 4 4 1 2 0 (2.8) Case iic) – The pure gravity case The action is constructed, see [19] for details, in terms of the determinant E of the vierbein e a� and the curvature scalar R. It is given by S G d x ER N � � � 6 8 4 . (2.9) The overall constant (essentially the inverse of the gravita- tional constant GN) is now dimensional ([ ]GN D� � �4 2). The non-dimensional action is recovered by setting [ ] , [ ] . e R a D D � � � � � 4 4 0 2 (2.10) Let us now discuss the dimensional reduction from D D� �4 1. Let us suppose that the three space dimensions belong to some compact manifold M (e.g. the three-sphere S3) and let us freeze the dependence of the fields on the space-dimen- © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 59 Acta Polytechnica Vol. 48 No. 2/2008 sions (application of the time derivative �0 leads to non-van- ishing results, while application of the space-derivatives �i, for i �1 2 3, , , gives zero). Our space-time is now given by R � M. We get that the integration over the three space variables con- tributes just to an overall factor, the volume of the three-di- mensional manifold M. Therefore d x Vol dtM 4 � �� . (2.11) Since [ ]VolM D� � �4 3 (2.12) we can express Vol m � 13 , where m is a mass-term. A factor 1 m contributes as an overall factor in one-dimensional theory, while the remaining part 12m can be used to rescale the fields. We have, e.g., for dimensional reduction of the scalar boson theory that � �D Dm� � �1 4 1 . (2.13) The dimensional reduction of the scalar boson theory ii a) is therefore given by S m dt M D� � � � � � ��� 1 1 2 1 2 1 4 2 2 2 1 4 � ! � � � � (2.14) where we have [ ] , [ ] , [ ] . � � D D D M � � � � � � 1 1 1 1 0 1 2 (2.15) The D �1 coupling constant �1 is related to the D � 4 non-dimensional coupling constant � by the relation � �1 2� m . (2.16) We proceed in a similar way in the case of Yang-Mills the- ory. We can rescale the D � 4 Yang-Mills fields A� to the D �1 fields B A m� � � 1 . The D �1 charge e is rescaled to e e m1 � . We have, symbolically, for the dimensionally reduced action, a sum of terms of the type � �S m dt B e BB e B� � �� 1 2 1 2 1 2 4� � , (2.17) where [ ] , [ ] . B e D D � � � � 1 1 1 0 1 (2.18) The situation is different as far as gravity theory is con- cerned. In that case the overall factor Vol GM N produces the dimensionally correct 1 m overall factor of the one-dimensional theory. This implies that we do not need to rescale the dimensionality of the vierbein e a� and of the curvature. Sum- marizing, we have the following results scalar boson gauge c � � �: [ ] [ ]D D� �� �4 11 0 onnection :[ [ vierbein A A AD D� � �] ]� �� �4 11 0 [ [ electric charge e e eD D� � � : ] ]� �� �4 11 0 :[ [e e eD D] ]� �� �4 10 1 (2.19) Let us now discuss the N �1 supersymmetric version of the three D � 4 theories above. First, we have the chiral multiplet, described in [19], in terms of the chiral superfields �, �. Next the vector multiplet V, the vector-multiplet in the Wess-Zumino gauge, the supergravity multiplet in terms of vierbein and gravitinos and, finally, the gauged supergravity multiplet presenting an extra set of auxiliary fields. The total content of fields is given by the following table, which presents also the D � 4 and respectively the D �1dimensionality of the fields (in the latter case, after dimensional reduction). We have Some comments are in order: the vector multiplet corre- sponds, in D �1language, to the N � 4 “enveloping represen- tation” [14] (1, 4, 6, 4, 1). The latter is a reducible, but non-de- composable representation of the N � 4 supersymmetry. Its irreducible multiplets are split into (1, 4, 3, 0, 0) and (0, 0, 3, 4, 1). The Wess-Zumino gauge, in D �1 language, corresponds to selecting the latter N � 4 irreducible multiplet, whose fields present only non-negative dimensions. The N � 2 four-dimensional super-QED involves cou- pling a set of chiral superfields together with the vector multiplet. Due to the dimensional analysis, the correspond- ing one-dimensional multiplet is the (5, 8, 3) irrep of N � 8 given by (2, 4, 2)�(3, 4, 1). 60 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 chiral multiplet : � �, fields content : (2, 4, 2) D � 4 dimensionality : [ , , ]1 232 4D� D �1 dimensionality : [ , , ]0 112 1D� vector multiplet : V V� † fields content : (1, 4, 6, 4, 1) D � 4 dimensionality : [ , , , , ]0 1 212 3 2 4D� D �1 dimensionality : [ , , , , ]� � �1 0 1 1 2 1 2 1D vector multiplet : V in the WZ gauge fields content : (3, 4, 1) D � 4 dimensionality : [ , , ]1 232 4D� D �1 dimensionality : [ , , ]0 112 1D� supergravity multiplet : e a� � �, fields content : (16, 16) D � 4 dimensionality : [ , ]0 12 4D� D �1 dimensionality : [ , ]0 12 1D� gauged sugra multiplet : e ba i� � �, , fields content : (6, 12, 6) D � 4 dimensionality : [ , , ]0 112 4D� D �1 dimensionality : [ , , ]0 112 1D� (2.20) As far as supergravity theories are concerned, the original supergravity multiplet corresponds to four irreducible N � 4 one-dimensional multiplets, while the gauged supergravity multiplet is obtained, in the D �1viewpoint, in terms of three irreducible N � 4 multiplets whose total number of fields is (6, 12, 6). The multiplet of the physical degrees of freedom of elev- en-dimensional supergravity (44 components for the gravi- ton, i.e. the components of a SO(9) traceless symmetric tensor, the 128 fermionic components of the gravitinos and the 84 components of the three form) can be accommodated into the (44, 128, 84) multiplet of an N-extended one-dimen- sional supersymmetry. As will be shown later, 128 bosons and 128 fermions can accommodate at most 16 off-shell super- symmetries that are linearly realized. It is under question whether an off-shell formulation of eleven-dimensional su- pergravity indeed exists. In any case it would require at least 32768 bosonic (and an equal number of fermionic) degrees of freedom to produce an N � 32 supersymmetry representa- tion in D �1. 3 Supersymmetric quantum mechanics and Clifford algebras In this section we discuss several results, based on ref. [13], concerning the classification of irreducible representations (from now on “irreps”) of the N-extended one-dimensional supersymmetry algebra and their connection with Clifford algebras. The N extended D �1supersymmetry algebra is given by � �Q Q Hi j ij, � � , (3.21) where the Qi’s are the supersymmetry generators (for i j N, , ,�1 � ) and H i t � � � � is a Hamiltonian operator (t is the time coordinate). If the diagonal matrix �ij is pseudo- -Euclidean (with signature (p, q), N p q� � ) we can speak of generalized supersymmetries. For convenience we limit the discussion here (despite the fact that our results can be straightforwardly generalized to pseudo-Euclidean super- symmetries, having applicability, e.g., to supersymmetric spinning particles moving in pseudo-Euclidean manifolds) to ordinary N-extended supersymmetries. Therefore for our purposes � ij ij� . The (D-modules) representations of the (3.21) supersym- metry algebra realized in terms of linear transformations acting on finite multiplets of fields satisfy the following prop- erties. The total number of bosonic fields equal the total number of fermionic fields. For irreps of the N-extended supersymmetry the number of bosonic (fermionic) fields is given by d, with N and d linked through N l n d G nl � � � 8 24 , ( ), (3.22) where l � 0 1 2, , ,� and n �1 2 3 4 5 6 7 8, , , , , , , . G(n) appearing in (3.22) is the Radon-Hurwitz function [13] The modulo 8 property of the irreps of the N-extended supersymmetry is a consequence of the famous modulo 8 pro- perty of Clifford algebras. The connection between super- symmetry irreps and Clifford algebras is specified later. The D �1 dimensional reduction of the maximal N � 8 supergravity produces a supersymmetric quantum mechani- cal system with N � 32 extended number of supersym- metries. It is therefore convenient to explicitly report the number of bosonic/fermionic component fields in any given irrep of (3.21) for any N up to N � 32. We get the table The bosonic (fermionic) fields entering an irreducible multiplet can be grouped together according to their dimen- sionality. Sometimes instead of “dimension”, the word “spin” is used to refer to the dimensionality of the component fields. This choice of word finds some justification when discussing the D �1 dimensional reduction of higher-dimensional su- persymmetric theories. The number (equal to l) of different dimensions (i.e. the number of different spin states) of a given irrep, will be referred to as the length l of the irrep. Since there are at least two different spin states (one for bosons, the other for fermions), obtained when all bosons (fermions) are grouped together within the same spin, the minimal length of an irrep is l � 2. A general property of (linear) supersymmetry in any di- mension is the fact that the states of highest spin in a given multiplet are auxiliary fields, whose supersymmetry transfor- mations are given by total derivatives. Just for D �1 total derivatives coincide with the (unique) time derivative. Using this specific property of the one-dimensional supersymmetry it was proven in [13] that all finite linear irreps of the (3.21) supersymmetry algebra fall into classes of equivalence, each class of equivalence being singled out by an associated mini- © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 61 Acta Polytechnica Vol. 48 No. 2/2008 n 1 2 3 4 5 6 7 8 G(n) 1 2 4 4 8 8 8 8 (3.23) N �1 1 N � 9 16 N �17 256 N � 25 4096 N � 2 2 N �10 32 N �18 512 N � 26 8192 N � 3 4 N �11 64 N �19 1024 N � 27 16384 N � 4 4 N �12 64 N � 20 1024 N � 28 16384 N � 5 8 N �13 128 N � 21 2048 N � 29 32768 N � 6 8 N �14 128 N � 22 2048 N � 30 32768 N � 7 8 N �15 128 N � 23 2048 N � 31 32768 N � 8 8 N �16 128 N � 24 2048 N � 32 32768 (3.24) mal length (l � 2) irreducible multiplet. It was further proven that the minimal length irreducible multiplets are in 1-to-1 correspondence with a subclass of Clifford algebras (those which satisfy a Weyl property). The connection goes as fol- lows. The supersymmetry generators acting on a length-2 irreducible multiplet can be expressed as Qi H i i � � � � � � 1 2 0 0 � �~ , (3.25) where � i and ~�i are matrices entering a Weyl type (i.e. block antidiagonal) irreducible representation of the Clifford alge- bra relation �i i i � � � � � 0 0 � �~ , { , }� �i j ij� 2� . (3.26) The Q i’s in (3.25) are supermatrices with vanishing bosonic and non-vanishing fermionic blocks, acting on an ir- reducible multiplet m (thought of as a column vector) which can be either bosonic or fermionic. We conventionally con- sider a length-2 irreducible multiplet as bosonic if its upper half part of component fields is bosonic and its lower half is fermionic. It is fermionic in the converse case. The connec- tion between Clifford algebra irreps of the Weyl type and minimal length irreps of the N-extended one-dimensional supersymmetry is such that D, the dimensionality of the (Eu- clidean, in the present case) space-time of the Clifford algebra (3.26) coincides with the number N of the extended super- symmetries, according to The matrix size of the associated Clifford algebra (equal to 2d, with d given in (3.22)) corresponds to the number of (bosonic plus fermionic) fields entering the one-dimensional N-extended supersymmetry irrep. The classification of Weyl-type Clifford irreps, furnished in [13], can be easily recovered from the well-known classifica- tion of Clifford irreps, given in [20] (see also [21] and [22]). The (3.25) Q i’s matrices realizing the N-extended su- persymmetry algebra (3.21) on length-2 irreps have entries which are either c-numbers or are proportional to the Hamiltonian H. Irreducible representations of higher length (l � 3) are systematically produced [13] through repeated ap- plications of the dressing transformations Q Q S Q Si i k k i k � � ( ) ( ) ( )� �1 (3.28) realized by diagonal matrices S k( )’s (k d�1 2, ,� ) with entries s ij k( ) given by s Hij k ij jk jk ( ) ( )� � � 1 . (3.29) Some remarks are in order [13]: i) The dressed supersymmetry operators �Q i (for a given set of dressing transformations) have entries which are inte- gral powers of H. A subclass of the �Q i s dressed operators is given by the local dressed operators, whose entries are non-negative integral powers of H (their entries have no 1 H poles). A local representation (irreps fall into this class) of an extended supersymmetry is realized by lo- cal dressed operators. The number of the extension, given by � � �N N N( ), corresponds to the number of local dressed operators. ii) The local dressed representation is not necessarily an irrep. Since the total number of fields (d bosons and d fermions) is unchanged under dressing, the local dressed representation is an irrep iff d and �N satisfy the (3.22) requirement (with �N in place of N). iii) The dressing changes the dimension (spin) of the fields of the original multiplet m. Under the S k( ) dressing transformation (3.28), m S mk� ( ) , all fields entering m are unchanged apart from the k-th one (denoted, e.g., as �k and mapped to ��k). Its dimension is changed from [ ] [ ]k k� � 1. This is why the dressing changes the length of a multiplet. As an example, if the original length-2 multiplet m is a bosonic multiplet with d spin-0 bosonic fields and d spin-1 2 fermionic fields (in the following such a multiplet will be denoted as ( ; ) ( , )x d di j s� � �0, for i j d, , ,�1 � ), then S mk( ) , for k d� , corresponds to a length-3 multiplet with d �1bosonic spin-0 fields, d spin- 1 2 fermionic fields and a single spin-1 bosonic field (in the following we employ the notation ( , , )d d s� �1 1 0 for such a multiplet). Let us now fix the overall conventions. The most general multiplet is of the form (d d dl1 2, , ,� ), where di for i l�1 2, , ,� specify the number of fields of a given spin s i� �1 2 . The spin s, i.e. the spin of the lowest component fields in the multiplet, will also be referred to as the “spin of the multiplet”. When looking purely at the representation properties of a given multiplet the assignment of an overall spin s is arbitrary, since the supersymmetry transformations of the fields are not affected by s. Introducing a spin is useful for tensoring multi- plets and becomes essential for physical applications, e.g. in the construction of supersymmetric invariant terms entering an action. In the above multiplet l denotes its length, dl the number of auxiliary fields of highest spins transforming as time-deriv- atives. The total number of odd-indexed equal the total num- ber of even-indexed fields, i.e. d d d d d1 3 2 4� � � � � �� � . The multiplet is bosonic if the odd-indexed fields are bosonic and the even-indexed fields are fermionic (the multiplet is 62 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 � of space-time dim. (Weyl-Clifford) � � of extended su.sies (in 1-dim.) D � N (3.27) fermionic in the converse case). For a bosonic multiplet the auxiliary fields are bosonic (fermionic) if the length l is an odd (even) number. Just like the overall spin assignment, the assignment of a bosonic (fermionic) character to a multiplet is arbitrary since the mutual transformation properties of the fields inside a multiplet are not affected by its statistics. Therefore, mul- tiplets always appear in dually related pairs so that to any bosonic multiplet there exists its fermionic counterpart with the same transformation properties (see also [23]). Throughout this paper we assign integer valued spins to bosonic multiplets and half-integer valued spins to fermionic multiplets. As pointed out before, the most general (d d dl1 2, , ,� ) multiplet is recovered as a dressing of its corresponding N-ex- tended length-2 (d, d) multiplet. In [13] it was shown that all dressed supersymmetry operators producing any length-3 multiplet (of the form ( , , )d p d p� for p d� �1 1, ,� ) are of lo- cal type. Therefore, for length-3 multiplets, we have � �N N. This implies, in particular, that the ( , , )d p d p� multiplets are non-equivalent irreps of the N-extended one-dimensional supersymmetry. As concerns length l � 4 multiplets, the gen- eral problem of finding irreps was not addressed in [13]. It was shown, as a specific example, that the dressing of the length-2 (4, 4) irrep of N � 4, realized through the series of mappings ( , ) ( , , ) ( , , , )4 4 1 4 3 1 3 3 1� � , produces at the end a length-4 multiplet ( , , , )1 3 3 1 carrying only three local super- symmetries ( � �N 3). Since the relation (3.22) is satisfied when setting the number of extended supersymmetries acting on a multiplet equal to 3 and the total number of bosonic (fer- mionic) fields entering a multiplet equal to 4, as a conse- quence, the ( , , , )1 3 3 1 multiplet corresponds to an irreducible representation of the N � 3 extended supersymmetry. Based on an algorithmic construction of representatives of Clifford irreps, we present an iterative method for classifying all irreducible representations of higher length for arbitrary N values of the extended supersymmetry. 4 Clifford algebras and division algebras Due to the relation between Supersymmetric Quantum Mechanics and Clifford algebras, we present here a classifica- tion of the irreducible representations of Clifford algebras in terms of an algorithm which allows us to single out, in arbi- trary signature space-times, a representative in each irreduc- ible class of representations of Clifford’s gamma matrices. The class of irreducible representations is unique apart from special signatures, where two non-equivalent irreducible rep- resentations are linked by sign flipping (� �� �� � ). The construction goes as follows. First, we prove that starting from a given spacetime-dimensional representation of Clifford’s Gamma matrices, we can recursively construct D � 2 space- time dimensional Clifford Gamma matrices with the help of two recursive algorithms. Indeed, it is a simple exercise to ver- ify that if �i’s denotes the d-dimensional Gamma matrices of a D p q� � spacetime with (p, q) signature (namely, providing a representation for the C(p, q) Clifford algebra) then 2d-di- mensional D � 2 Gamma matrices (denoted as �j) of a D � 2 spacetime are produced according to either © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 63 Acta Polytechnica Vol. 48 No. 2/2008 1 * 2 * 4 * 8 * 16 * 32 * 64 * 128 * 256 * (1, 0) � (2, 1) � (3, 2) � (4, 3) � (5, 4) � (6, 5) � (7, 6) � (8, 7) � (9, 8) � � (1, 4) � (2, 5) � (3, 6) � (4, 7) � (5, 8) � (6, 9) � (0,3) � (5,0) � (6, 1) � (7, 2) � (8, 3) � (9, 4) � (10, 5) � � (1, 8) � (2, 9) � (3, 10) � (4, 11) � (5, 12) � (0,7) � (9, 0) � (10, 1) � (11, 2) � (12, 3) � (13, 1) � � (1, 12) � (2, 13) � (0, 11) � (13, 0) � (14, 1) � � (1, 16) � (0, 15) � (17, 0) � Table 1: Table with the maximal Clifford algebras (up to d � 256) (4.33) � j i i d d d d p q � � � � � � � � � � � � � � � 0 0 0 1 1 0 1 0 0 1 � � , , ( , ) (� p q� �1 1, ). (4.30) or � j i i d d d d p q � � � � � � � � � � � � � � � 0 0 0 1 1 0 1 0 0 1 � � , , ( , ) (� q p� 2, ). (4.31) It is immediately clear, e.g., that the two-dimensional real-valued Pauli matrices �A, �1, �2 which realize the Clifford algebra C(2, 1) are obtained by applying either (4.30) or (4.31) to the number 1, i.e. the one-dimensional realization of C(1, 0). We have indeed � � �A � � � � � � � � � � � � � � � � � 0 1 1 0 0 1 1 0 1 0 0 11 2 , , . (4.32) All Clifford algebras are obtained by recursively applying algorithms (4.30) and (4.31) to the Clifford algebra C(1, 0) ( )�1 and the Clifford algebras of the series C m( , )0 3 4� (with m non-negative integer), which must be previously known. This is in accordance with the scheme illustrated in the Table1. Concerning the previous table, some remarks are in order. The columns are labeled by the matrix size d of the maximal Clifford algebras. Their signature is denoted by the (p, q) pairs. Furthermore, the underlined Clifford algebras in the table can be named as “primitive maximal Clifford algebras”. The remaining maximal Clifford algebras appearing in the table are “maximal descendant Clifford algebras”. They are obtained from the primitive maximal Clifford algebras by iteratively applying the two recursive algorithms (4.30) and (4.31). Moreover, any non-maximal Clifford algebra is ob- tained from a given maximal Clifford algebra by deleting a certain number of Gamma matrices (as an example, Clifford algebras in even-dimensional spacetimes are always non-maximal). It is immediately clear from the above construction that the maximal Clifford algebras are encountered if and only if the condition p q� �15 8, mod (4.34) is matched. The notion of a Clifford algebra of the generalized Weyl type, namely satisfying the (3.26) condition, has already been introduced. All maximal Clifford algebras, both primitive and descendant, are not of the generalized Weyl type. As already pointed out, the notion of generalized Weyl spinors is based on real-valued representations of Clifford algebras which, for classification purposes, are more convenient to use w.r.t. the complex Clifford algebras that one in general deals with. For this reason generalized Weyl spinors exist also in odd-dimen- sional space-time, see formula (3.26), while standard Weyl spinors only exist in even-dimensional spacetimes. This can be understood by analyzing a single example. The real irrep C(0, 7), with all negative signs, is 8-dimensional, see table (4.33), while the real irrep C(7, 0) is 16-dimensional, but of generalized Weyl type (3.26). Accordingly, Euclidean 8-di- mensional fundamental spinors can be understood either as the 8-dimensional “Non-Weyl” spinors of C(0, 7), or as 8-di- mensional “Weyl-projected” C(7, 0) spinors. In the complex case, the sign flipping C C( , ) ( , )0 7 7 0� can be realized by multiplying all Gamma matrices by the imaginary unit “i”. No doubling of the matrix size of the �’s is found and the notion of Weyl spinors cannot be applied. One faces a similar situa- tion in one-dimensional spacetime. In the complex case we can realize C(1, 0) with 1 and C(0, 1) with i (both one-dimen- sional). On the other hand, in the real case, C(0, 1) can only be realized through the 2-dimensional irrep 0 1 1 0� � � � �, which is block-antidiagonal. Throughout the text Weyl (Non- -Weyl) spinors are always referred to the (3.26) property with respect to real-valued Clifford algebras. Non-maximal Clifford algebras are of the Weyl type if and only if they are produced from a maximal Clifford algebra by deleting at least one spatial Gamma matrix which, without loss of gener- ality, can always be chosen as the one with diagonal entries. Let us now illustrate how non-maximal Clifford algebras are produced from the corresponding maximal Clifford alge- bras. The construction goes as follows. We illustrate at first the example of the non-maximal Clifford algebras obtained from the 2-dimensional maximal Clifford irrep C(2, 1) furnished by the three matrices �1, �2, �A given in (4.32). If we restrict the Clifford algebra to �1, �A, i.e. if we delete �2 from the previous set, we get the 2-dimensional irrep C(1, 1). If we further de- lete �1 we are left with �A only, which provides the 2-dimen- sional irrep C(0, 1) discussed above. On the other hand, delet- ing �A from C(2, 1) leaves us with �1, �2, the 2-dimensional irrep C(2, 0). To summarize, from the 2-dimensional irrep of the ”maxi- mal Clifford algebra” C(2, 1) we obtain the 2-dimensional irreps of the non-maximal Clifford algebras C(1, 1), C(0, 1) and C(0, 2) through a “�-matrices deleting procedure”. Please note that, through deleting, we cannot obtain from C(1, 2) the irrep , since the latter is one-dimensional. In full generality, non-maximal Clifford algebras are pro- duced from the corresponding maximal Clifford algebras ac- cording to the following table, which specifies the number of time-like or space-like Gamma matrices that should be de- 64 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 leted, as well as the generalized Weyl (W) character or not (NW) of the given non-maximal Clifford algebra. We get W NW (4.35) ( mod ) ( mod ) ( , ) ( , ) 0 8 1 8 1 � � �p q p q ( mod ) ( mod ) ( , ) ( , ) 2 8 1 8 1 � � �p q p q ( mod ) ( mod ) ( , ) ( , ) 4 8 5 8 1 � � �p q p q ( mod ) ( mod ) ( , ) ( , ) 3 8 1 8 2 � � �p q p q ( mod ) ( mod ) ( , ) ( , ) 6 8 1 8 3 � � �p q p q ( mod ) ( mod ) ( , ) ( , ) 7 8 1 8 2 � � �p q p q In the above entries x mod 8 specifies the mod 8 residue of t s� for any given ( , )t s spacetime. Non-maximal Clifford al- gebras are denoted by p t� , q s� , while maximal Clifford algebras are denoted by ( , )� �p q , with � �p p, � �q q. The differ- ences � �p p, � �q q denote how many Clifford gamma matrices (of time-like or respectively space-like type) have to be deleted from a given maximal Clifford algebra to produce the irrep of the corresponding non-maximal Clifford algebra. To be spe- cific, e.g., the 6 8mod non-maximal Clifford algebra C(6, 0) is obtained from the maximal Clifford algebra C(9, 0), whose matrix size is 16 according to (4.33), by deleting three gamma matrices. To complete our discussion what it remains to specify the construction of the primitive maximal Clifford algebras for both C n( , )0 3 8� series (which can be named as “quaternionic series”, due to its connection with this division algebra, as we will see in the next section), and also the “octonionic” series C n( , )0 7 8� . The answer can be provided with the help of the three Pauli matrices (4.32). We first construct the 4×4 matri- ces realizing the Clifford algebra C(0, 3) and the 8×8 matrices realizing the Clifford algebra C(0, 7). They are given, respec- tively, by C A A A ( , ) , , . 0 3 1 2 2 � � � � � � � � �1 (4.36) and C A A A A( , ) , , , ,0 7 1 2 2 2 2 1 2 2 1 2 � � � � � � � � � � � � � � � � � � � � � 1 1 1 1 1 A A A A A , , . � � � � � 2 2� � � � 1 (4.37) The three matrices of C(0, 3) will be denoted as �i �1 2 3, , . The seven matrices of C(0, 7) will be denoted as ~ , , ,�i �1 2 7� . In order to construct the remaining Clifford algebras of the two series we first need to apply the (4.30) algorithm to C(0, 7) and construct the 16×16 matrices realizing C(1, 8) (the matrix with a positive signature is denoted as �9, � 9 2 �1, while the eight matrices with negative signatures are denoted as �j, j �1 2 8, , ,� , with � j 2 � �1). We are now in the position to explicitly construct the whole series of primitive maximal Clif- ford algebras C n( , )0 3 8� , C n( , )0 7 8� through the formulas C n i j j( , )0 3 8 9 4 16 4 9 16 4 9 9 � � � � � � � � � � � � � � � � � � � � � � � 1 1 1 1 1 � � � � � � � � � � � � � � � � j j 1 1 1 1 116 4 9 9 16 16 16 9 � � � � � � � � � � , , , , , , (4.38) and similarly C n i j j( , ) ~ 0 7 8 9 8 16 8 9 16 8 9 � � � � � � � � � � � � � � � � � � � � � � � 1 1 1 1 1 9 16 8 9 9 16 16 16 9 � � � � � � � � � � � � � � � � j 1 1 1 1 1� � � � � � � � � � , , , , , j . (4.39) Please note that the tensor product of the 16-dimensional representation is taken n times. The total size of the (4.38) ma- trix representations is then 4×16n, while the total size of (4.39) is 8×16n. With the help of the formulas presented in this section we are able to systematically construct a set of representatives of the real irreducible representations of Clifford algebras in arbitrary space-times and signatures. It is also convenient to explicitly present of Clifford algebras with the division alge- bras of the quaternions (and of the octonions). This relation can be understood as follows. First we note that the three matrices appearing in C(0, 3) can also be ex- pressed in terms of the imaginary quaternions �i satisfying � � i j ij� � � � �ijk �k . (4.40) As a consequence, the whole set of maximal primitive Clif- ford algebras C n( , )0 3 8� , as well as their maximal descen- dants, can be represented with quaternionic-valued matrices. In its turn the spinors now have to be interpreted as quater- nionic-valued column vectors. Similarly, there exists an alternative realization of the basic relations of the generators of the Euclidean Clifford algebra , obtained by identifying its seven generators with the seven imaginary octonions (for a review on octonions see e.g. [24]) satisfying the algebraic relation � � �i j ij ijk kC� � � � (4.41) for i j k, , , ,�1 7� and Cijk the totally antisymmetric octonionic structure constants given by C C C C C C C123 147 165 246 257 354 367 1� � � � � � � (4.42) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 65 Acta Polytechnica Vol. 48 No. 2/2008 and vanishing otherwise. The octonions are non-associative and cannot be represented in matrix form with the usual ma- trix multiplication. On the other hand, a construction due to Dixon allows us to produce the seven 8×8 matrix generators of the C(0, 7) Clifford algebras in terms of the octonionic structure constants. Given a real octonion x x xi ii � � �0 � , with real coefficients x0, xi, for i �1 7, ,� , the left action of the imaginary octonions �i ( � � �x xi� ) is reproduced in terms of the 8×8 Clifford gamma matrix �i, linearly acting on x0, xi’s. 5 The field content of irreducible representations It is now possible to plug the information contained in Clifford algebras and apply the construction outlined in Section 3 to compute the admissible field content for the length-4 multiplets for arbitrary values of N. This construc- tion was done in [14]. We present here the list of length-4 field content up to N �11. Up to N � 8 we have N �1 NO (5.43) N � 2 NO N � 3 (1, 3, 3, 1) N � 4 NO N � 5 (1, 5, 7, 3), (3, 7, 5, 1), (1, 6, 7, 2), (2, 7, 6, 1), (2, 6, 6, 2), (1, 7, 7, 1) N � 6 (1, 6, 7, 2), (2, 7, 6, 1), (2, 6, 6, 2), (1, 7, 7, 1) N � 7 (1, 7, 7, 1) N � 8 NO Since there are no length-l irreps with l � 5 for N � 9, the above list, together with the already known length-2 and length-3 irreps, provides the complete classification of the admissible field content of the irreducible representations for N � 8. Please note that the length-4 irrep of N � 3, (1, 3, 3, 1), is self-dual under the [14] high � low spin duality, while two of the non-equivalent length-4 N � 5 irreps are self-dual, (2, 6, 6, 2) and (1, 7, 7, 1). The remaining ones are pair- -wise dually related (( , , , ) ( , , , )1 5 7 3 3 7 5 1� and ( , , , )1 6 7 2 � ( , , , )2 7 6 1 ). The N � 9 length-4 irreducible multiplet (d d d d1 2 3 4, , , ) is for simplicity expressed in terms of the two positive integers h d� 1, k d� 4, since d h3 16� � , d k4 16� � . The complete list of N � 9 length-4 fields content is expressed by h, k satisfying the constraint h k� � 8. (5.44) N �10 is the lowest supersymmery admitting length-5 irreps. The field content of its length-4 irreps is given by ( , , , )d d d d1 2 3 4 , expressed in terms of the two positive integers h d� 1, k d� 4, since d h3 32� � , d k4 32� � . If we set r h k� min( , ) (5.45) the non-equivalent length-4 field content is given by the or- dered pair of positive integers h, k satisfying the constraint h k r� � � 24 . (5.46) For N �11 the length-4 fields content ( , , , )d d d d1 2 3 4 is ex- pressed in terms of the two positive integers h d� 1, k d� 4, since d h3 64� � , d k2 64� � . Setting as before r h k� min( , ) and introducing the s(r) function defined through s r r r ( ) , , � � �� ! " # $ 8 0 1 7for otherwise � (5.47) we can express the constraints on h, k as h k r s r� � � �( ) 48. (5.48) 6 The off-shell invariant actions of the N � 4 sigma models In the late 1980’s and early 1990’s, the whole set of off-shell invariant actions of the N � 4 supersymmetries were produced ([5] and references therein), by making use of the superfield formalism. This result was reached after slowly recognizing the multiplets carrying a representation of the one-dimensional N � 4 supersymmetry. The results discussed here allow us to reconstruct, in a unified framework, all off-shell invariant actions of the correct mass-dimension (the mass-dimension d � 2 of the kinetic energy) for the whole set of N � 4 irreducible multiplets. They are given by the (4, 4), (3, 4, 1), (2, 4, 2) and (1, 3, 4) multiplets. We are able to construct the invariants without using a superfield formalism. We use instead a construction which can be extended, how we will prove later, even for large values of N, in the cases where the superfield formalism is not avail- able. We will use the fact that the supersymmetry generators Q i’s act as graded Leibniz derivatives. Manifestly invariant ac- tions of the N-extended supersymmetry can be obtained by expressing them as I t Q Q f x x xN k� � �� d ( ( , , , ))1 1 2� � (6.49) with the supersymmetry transformations applied to an ar- bitrary function of the 0-dimensional fields xi’s, i k�1, ,� entering an irreducible multiplet of the N-extended super- symmetry. Since the supersymmetry generators admit mass- -dimension � 12 (being the “square roots” of the Hamiltonian), we have that (6.49) is a manifestly supersymmetric invariant whose lagrangian density Q Q f x x xN k1 1 2� �� �( , , , ) has a di- mension d N� 2 . For N � 4 the lagrangian density has the cor- rect dimension of a kinetic term. The k variables xi’s can be regarded as coordinates of a k-dimensional manifold. The corresponding actions can therefore be seen as N � 4 supersymmetric one-dimensional sigma models evolving in a k-dimensional target manifold. For each N � 4 irrep we get the following results. In all cases 66 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 below the arbitrary ( )xi function is given by � � f x( ). We get the following list. i) The N � 4 (4, 4) case. We have: Q x x x x xi j j i ij ijk k i ij ijk k( , ; , ) ( , ; � , � � )� � � � � � �� � � � � , ( , ; , ) ( , ; �, � ).Q x x x xj j j j4 � � � �� (6.50) The most general invariant Lagrangian L of dimension d � 2 is given by L x x x x j j j x j j ijk i j � � � � � � �� � � � �� � � � ( )[ � � � � ] [ � � � 2 2 1 2 x x x x x k l l l j j ljk j k ljk j k ] [ � � � � ]� � � � � � � � � � � � � � � � 1 2 1 � 6 � �� � �ljk l k k. (6.51) ii) The N � 4 (3, 4, 1) case. We have: Q x g x g xi j j ij ijk k i ij ijk k i( ; , ; ) ( ; � ; � ; ),� � � � � � �� � � � � Q x g g xj j j j j4( ; , ; ) ( ; , � ; ).� � � �� (6.52) The most general invariant Lagrangian L of dimension d � 2 is given by L x x g x g j j j i ijk j k j k � � � � � � �� � � � � ��� � � ( )[ � � � ] [ � � 2 2 1 2 ) � ] . � � � g xi i j j ijk i j k �� � � � �� � � � 6 (6.53) iii) The N � 4 (2, 4, 2) case. We have: Q x y g h x g h y1 0 1 2 3 0 3 1( , ; , , , ; , ) ( , ; �, , , � ; � , �� � � � � � � �� � � � 2 2 0 1 2 3 3 0 2 ), ( , ; , , , ; , ) ( , ; �, , , �; �Q x y g h y h g x� � � � � � �� � � � , � ), ( , ; , , , ; , ) ( , ; , � �, ; � � � � � � � � � � � � 1 3 0 1 2 3 2 1Q x y g h h y x g � � � , � ), ( , ; , , , ; , ) ( , ; , �, �, � � � � � � � � 3 0 4 0 1 2 3 1 2Q x y g h g x y h; � , � ).� �0 3 (6.54 ) The most general invariant Lagrangian L of dimension d � 2 is given by L x y x y g h y j j x � � � � � � � � �� � � � �� � � � ( , )[ � � � � ] [ � 2 2 2 2 1 2 0 3 ) ) )] [ � ) � � � � � � � � g h x gy �� � � � �� � � � � �� � � � � 2 3 0 1 1 3 0 2 1 2 0 3 � � � � �� � � � � � � � 1 3 0 2 2 3 0 1 0 1 2 3 � � � � ) )] . h � (6.55) iv) The N � 4 (1, 4, 3) case. We have: Q x g g x gi j j i i ij ijk k ij ijk k( ; , ; ) ( , � ; � � )� � � � � � � �� � � � � , ( ; , ; ) ( ; �, ; � ).Q x g x gj j j j4 � � � �� (6.56) The most general invariant Lagrangian L of dimension d � 2 is given by L x x g x g g i i i i i ijk i j k � � � � � � � �� � � � � � � � ( )[ � � � ] ( )[ 2 2 1 2 ] [ ].� �� � � � � � � � x ijk i j k6 (6.57) It is worth recalling that N � 4 is associated, as we have discussed, to the algebra of the quaternions. This is why in cases (4, 4), (3, 4, 1) and (1, 4, 3) the invariant actions can be written by making use of the quaternionic tensors ij and �ijk. In the (2, 4, 2) case two fields are dressed to be auxiliary fields and this spoils the quaternionic covariance property. 7 Octonions and N � 8 sigma-models Just as the N � 4 supersymmetry is related with the alge- bra of quaternions, the N � 8 supersymmetry is related with the algebra of the octonions. More specifically, it can be proven that the N � 8 supersymmetry can be produced from the lifting of the Cl(0, 7) Clifford algebra to Cl(0, 9). On the other hand, it is well-known, as we have discussed before, that the seven 8×8 antisymmetric gamma matrices of Cl(0, 7) can be recovered by the left-action of the imaginary octonions on the octonionic space. As a result, the entries of the seven antisymmetric gamma-matrices of Cl(0, 7) can be expressed in terms of the totally antisymmetric octonionic structure con- stants Cijk’s. The non-vanishing Cijk’s are given by C C C C C C C123 147 165 246 257 354 367 1� � � � � � � (7.58) The non-vanishing octonionic structure constants are associ- ated with the seven lines of the Fano projective plane, the smallest example of a finite projective geometry, see [24]. The N � 8 supersymmetry transformations of the various irreps can, as a consequence, be expressed in terms of octonionic structure constants. This is in particular true for the dressed (1, 8, 7) multiplet, admitting seven fields which are “dressed” to become auxiliary fields. This is an example of a multiplet which preserves the octonionic structure since the seven dressed fields are related to the seven imaginary octonions. We have that the supersymmetry transformations are given by Q x g g x C g Ci j j i i ij ijk k ij ijk k( ; , ; ) ( , � ; � � )� � � � � �� � � � � , ( ; , ; ) ( ; �, ; � ).Q x g x gj j j j8 � � � �� (7.59) for i j k, , , ,�1 7� . The strategy for constructing the most gen- eral N � 8 off-shell invariant action of the (1, 8, 7) multi- plet makes use of the octonionic covariantization principle. When restricted to an N � 4 subalgebra, the invariant actions should have the form of the N � 4 (1, 4, 3) action (6.57). This restriction can be made in seven non-equivalent ways (the seven lines of the Fano plane). The general N � 8 action should be expressed in terms of octonionic structure con- stants. With respect to (6.57), an extra-term could in principle be present. It is given by � dt x Cijk i j k l� � � � �( ) and is con- structed in terms of the octonionic tensor of rank 4 C Cijkl ijklmnp mnp� 1 6 � (7.60) (where �ijklmnp is the seven-index totally antisymmetric ten- sor). Please note that the rank-4 tensor is obviously vanishing when restricting to the quaternionic subspace. One immedi- © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 67 Acta Polytechnica Vol. 48 No. 2/2008 ately verifies that the term � dt x Cijk i j k l� � � � �( ) breaks the N � 8 supersymmetries and cannot enter the invariant action. As concerns the other terms, starting from the general action (with i j k, , , ,�1 7� ) �S t x x g x g C g i i i i i ijk i � � � � � � � � d �� � � � � � ( )[ � � � ] ( )[ 2 2 1 2 � j k ijk i j k x C � � � � � ] ( ) [ ]� �� 6 (7.61) we can prove that the invariance under the Qi generator ( , , )�1 7� is broken by terms which, after integration by parts, contain at least a second derivative �� . We obtain, e.g., a non-vanishing term of the type � ��dt C gijk i k l � � � 2 . In order to guarantee the full N � 8 invariance (the invariance under Q8 is automatically guaranteed) we have therefore to set �� � � �x 0, leaving a linear function in x. As a result, the most general N � 8 off-shell invariant action of the (1, 8, 7) multi- plet is given by �S t ax b x g a g C g i i i i i ijk i j � � � � � � � � d ( )[ � � � ] [ 2 2 1 2 �� � � � � � � �k ] . (7.62) We can express this result in the following terms: the associa- tion of the N � 8 supersymmetry with the octonions implies that the octonionic structure constants enter as coupling con- stants in the N � 8 invariant actions. The situation w.r.t. the other N � 8 multiplets is more complicated. This is due to the fact that the dressing of some of the bosonic fields to auxiliary fields does not respect octonionic covariance. The construc- tion of the invariant actions can however be performed along similar lines, the octonionic structure constants being re- placed by the “dressed” structure constants. The procedure for a generic irrep is more involved than in the(1, 8, 7) case. The full list of invariant actions for the N � 8 irreps is cur- rently being written. The results will be reported elsewhere. The method proposed is quite interesting because it allows us in principle to construct the most general invariant actions. It is worth mentioning that various groups, using N � 8 super- field formalism, are still working on the problem of con- structing the most general invariant actions. Let us close this section by pointing out that the only sign of the octonions is through their structure constants entering as parameters in the (7.62) N � 8 off-shell invariant action. (7.62) is an ordinary action, in terms of ordinary associative bosonic and fermionic fields closing an ordinary N � 8 super- symmetry algebra. 8 Non-equivalent representations with the same field content The irreducible representations of the N-extended super- symmetry algebra are nicely presented in terms of N-colored graphs with arrows (we will explain below how to draw the graphs). The existence of irreducible representations admit- ting the same field content, but non-equivalent graphs was pointed out in [25]. In [15] the non-equivalent graphs associ- ated to irreducible representations up to N � 8 were classified. We discuss here both construction of [15] and also its main re- sults. Since it can be quite easily proved that non-equivalent graphs are not encountered for N � 4, it is sufficient to dis- cuss the irreducible representations of N � 5 6 7 8, , , , which are obtained through a dressing of the N � 8 length-2 root multi- plet of type (8, 8) (see the previous discussion). Inequivalent graphs (see [15]) are described by the so-called connectivity of the irreps. Connectivity can be understood as follows. For the class of irreducible representations under consideration any given field of dimension d is mapped, under a supersymmetry transformation, either i) to a field of dimension d � 12 belonging to the multiplet (or to its opposite, the sign of the transformation being irrelevant for our purposes) or, ii) to the time-derivative of a field of dimension d � 12. If the given field belongs to an irrep of the N-extended one-dimensional supersymmetry algebra, therefore k N� of its transformations are of type i), while the N k� remaining ones are of type ii). Let us now specialize our discussion to a length-3 irrep (the interesting case for us). Its field content is given by (n n n n1 1, , � ), while the set of its fields is expressed by ( ; ; )x gi j k� , with i n�1 1, ,� , j n�1, ,� , k n n� �1 1, ,� . The xi’s are 0-dimensional fields (the �j are 1 2 -dimensional fields and gk are 1-dimensional fields, respectively). The connectivity associated to the given multiplet is defined in terms of the �g symbol. It encodes the following information. The n 1 2 -dimen- sional fields �j are partitioned in the subsets of mr fields admitting kr supersymmetry transformations of type i) (kr can take the 0 value). We have m nrr� � . The �g symbol is expressed as � g k km m� � �1 21 2 � (8.63) As an example, the N � 7 (6, 8, 2) multiplet admits con- nectivity � g � �6 22 1 (see (9.68)). This means that there are two types of fields �j. Six of them are mapped, under super- symmetry transformations, into the two auxiliary fields gk. The two remaining fields �j are only mapped into a single auxiliary field. An analogous symbol, x�, can be introduced. It describes the supersymmetry transformations of the xi fields into the �j 68 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 fields. This symbol is, however, always trivial. An N-irrep with ( , , )n n n n1 1� field content always produces x n N� � 1 . Let us now discuss how to compute the connectivities. (8, 8) involves 8 bosonic and 8 fermionic fields entering a column vector (the bosonic fields are accommodated in the upper part). The 8 supersymmetry operators �Q i (i �1 8, , )� in the (8, 8) N � 8 irrep are given by the matrices � , � ,Q H Q Hj j j � � � � � � �� � � � � � � 0 0 0 08 8 8 � � 1 1 (8.64) where the �j matrices ( j �1 7, ,� ) are the 8×8 generators of the Cl(0, 7) Clifford algebra and H i t � dd is the Hamiltonian. The Cl(0, 7) Clifford irrep is uniquely defined up to similarity transformations and an overall sign flipping [22]. Without loss of generality we can unambiguously fix the �j matrices to be given as in the Appendix. Each �j matrix (and the 18 iden- tity) possesses 8 non-vanishing entries, one in each column and one in each row. The whole set of non-vanishing entries of the eight (A.1) matrices fills the entire 8 8 64� � squares of a “chessboard”. The chessboard appears in the upper right block of (8.64). The length-3 and length-4 N � 5 6 7 8, , , irreps (no irrep with length l % 4 exists for N � 9, see [14]) are acted upon by the Qi’s supersymmetry transformations, obtained from the original �Q i operators through a dressing, � �Q Q DQ Di i i� � �1, (8.65) realized by a diagonal dressing matrix D. It should be noted that only the subset of “regular” dressed operators Qi (i.e., having no 1 H or higher poles in its entries) act on the new irre- ducible multiplet. Apart from the self-dual (4, 8, 4) N � 5 6, irreps, without loss of generality, for our purpose of comput- ing the irrep connectivities, the diagonal dressing matrix D which produces an irrep with ( , , )n n n n1 1� fields content can be chosen to have its non-vanishing diagonal entries given by pq qd , with dq �1 for q n�1 1, ,� and q n n� � 1 2, ,� , while d Hq � for q n n� �1 1, ,� . Any permutation of the first n entries produces a dressing which is equivalent, for comput- ing both the field content and the �g connectivity, to D. The only exceptions correspond to the N � 5 (4, 8, 4) and N � 6 (4, 8, 4) irreps. Besides the diagonal matrix D as above, non- -equivalent irreps can be obtained by a diagonal dressing �D with diagonal entries pq qd� , with � �d Hq for q � 4 6 7 8, , , and � �dq 1 for the remaining values of q. Similarly, the ( , , , )n n n n n n1 2 1 2� � length-4 multiplets are acted upon by the Qi operators dressed by D, whose non- -vanishing diagonal entries are now given by pq qd , with dq �1 for q n�1 1, ,� and q n n n� � �2 1 22 , ,� , while d Hq � for q n n n� � �1 21 2, ,� . The N � 5 6 7 8, , , length-2 (8, 8) irreps are unique (for the given value of N), see [26]. It is also easily recognized that all N � 8 length-3 irreps of a given field content produce the same value of �g con- nectivity (8.63). As concerns the length-3 N � 5 6 7, , irreps the situation is as follows. Let us consider the irreps with ( , , )k k8 8 � field content. Their supersymmetry transforma- tions are defined by picking an N & 8 subset from the com- plete set of 8 dressed Qi operators. It is easily recognized that for N � 7, no matter which supersymmetry operator is dis- carded, any choice of the seven operators produces the same value for the �g connectivity. Irreps with different connectivity can therefore only be found for N � 5 6, . The 8 6 28 � � � � � choices of N � 6 operators fall into two classes, denoted as A and B, which can, potentially, produce ( , , )k k8 8 � irreps with dif- ferent connectivity. Similarly, the 8 5 56 � � � � � choices of N � 5 operators fall into two A and B classes which can, poten- tially, produce irreps of different connectivity. For some given ( , , )k k8 8 � irrep, the value of �g connectivity computed in both N � 5 (as well as N � 6) classes can actually coincide. In the next Section we will show when this feature does indeed happen. To be specific, we present a list of representatives of the supersymmetry operators for each N and in each N � 5 6, A, B class. We have, with diagonal dressing D, N N N A N B N A N B � � � � � � 8 7 6 6 5 5 ( ) ( ) ( ) ( ) case case case case � � � � � � Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 , , , , , , , , , , , , , 1 3 4 5 6 7 1 2 3 4 5 6 3 4 5 6 7 2 , , , , , , , , , , , , , , , Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q3 4 5 6, , , (8.66) and, with diagonal dressing �D for the (4, 8, 4) irreps, N A N A Q Q Q Q Q Q Q Q Q � � � � � � 6 5 1 3 4 5 6 7 3 4 ( ) ( ) , , , , , , , case case 5 6 7, ,Q Q (8.67) We are now in a position to compute the connectivities of the irreps (the results are furnished in the next Section). Quite literally, the computations can be performed by filling a chess- board with pawns representing the allowed configurations. 9 Classification of the irrep connectivities In this Section we report the results of the computation of the allowed connectivities for the N � 5 6 7, , length-3 irreps. It turns out that the only values of N � 8 allowing the existence of multiplets with the same field content but non-equivalent connectivities are N � 5 and N � 6. The results concerning the allowed �g connectivities of the length-3 irreps are reported in the following table (the A, �A , B cases of N � 5 6, are specified) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 69 Acta Polytechnica Vol. 48 No. 2/2008 It is useful to explicitly present, in at least one pair of ex- amples, the supersymmetry transformations (depending on the �i global fermionic parameters) for multiplets admitting different connectivities and the same field content. We write below a pair of N � 5 irreps (the (4, 8, 4)A and the (4, 8, 4)B multiplets) differing by connectivity. It is also convenient to vi- sualize them graphically. The graphical presentation at the end of this Section is given as follows. Three rows of (from bot- tom to up) 4, 8 and 4 dots are associated with the xi, �j and gk fields, respectively. Supersymmetry transformations are rep- resented by lines of 5 different colors (since N � 5). Solid lines are associated to transformations with a positive sign, and dashed lines to transformations with a negative sign. It is eas- ily recognized that in the type A graph there are 4 �j points with four colored lines connecting them to the gk points, while the 4 remaining �j points admit a single line connecting them to the gk points. In the type B graph we have 4 �j points with three colored lines and the 4 remaining �j points with two col- ored lines connecting them to the gk points. The supersymmetry transformations are explicitly given by i) The N � 5 (4, 8, 4)A transformations: � � � � � � � � � � � � � � � � � � x x 1 2 3 4 5 3 6 1 7 5 8 2 2 4 3 5 4 6 5 � � � � � � � � � 7 1 8 3 2 1 1 5 5 6 4 7 3 8 4 2 2 5 5 � � � � � � � � � � � � � � � � � � � � � � � � � � � x x � � � � � � � � � � � � � 1 6 3 7 4 8 1 2 3 4 1 3 2 1 3 5 4 2 � � � � � � � � � � i x g g g g� i x g g g g i x g g g � � � � � � � � � � 2 4 3 1 4 2 5 3 1 4 3 2 1 1 1 5 2 4 � � � � � � � � � � 3 3 4 4 2 2 5 1 1 2 3 3 4 4 5 4 1 3 � � � � � � � � � � � � � � � � � � g i x g g g g i x i � � � � � � � � x i x i x g i x i x i x i 2 1 3 5 4 2 3 6 3 1 4 2 5 3 1 � � � � � � � � � � � � � � � � � � � � x g i x i x i x i x g i 4 2 4 7 1 1 5 2 4 3 3 4 2 1 8 � � � � � � � � � � � � � � � �5 1 1 2 3 3 4 4 2 2 1 4 1 3 2 � � � � � � x i x i x i x g g i i i � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1 3 5 4 2 7 2 3 1 4 2 5 3 1 � � � � � � � � � � � � � � i i g i i i i 4 2 8 3 1 1 5 2 4 3 3 4 2 5 4 � � � � � � � i g i i i i i g � � � � � � � � � � � � � � � � � � � � � � � �i i i i i� � � � � � � � � �5 1 1 2 3 3 4 4 2 6� � � � � (9.69) ii) The N � 5 (4, 8, 4)B transformations: � � � � � � � � � � � � � � � � � x x 1 5 2 2 3 4 5 3 6 1 7 2 5 1 2 4 3 5 4 � � � � � � � � � � � � � � � � � � � � � � � � � � � 6 1 8 3 2 1 5 4 1 5 4 7 3 8 4 2 2 5 3 � � � � � � � � � � x x � � � � � � � � � � � � � � � � � � � � � � 1 6 3 7 4 8 1 5 2 2 3 4 1 3 2 1 3i x i x g g g� � 2 5 1 2 4 3 1 4 2 1 4 3 2 1 5 4 1 � � � � � � � � i x i x g g g i x i x � � � � � � � � � � � � � g g g i x i x g g g i 1 4 3 3 4 4 2 2 5 3 1 2 3 3 4 4 5 � � � � � � � � � � � � � � � � � � � � 4 1 3 2 1 3 5 2 2 3 6 3 1 4 2 1 � � � � � � x i x i x g g i x i x i � � � � � � � � � � � � � � � x g g i x i x i x g g 4 5 1 2 4 7 1 1 4 3 3 4 2 1 5 4 8 � � � � � � � � � � � � � � � � � � � � i x i x i x g g� � � � �1 2 3 3 4 4 2 2 5 3� � �� � � � (9.70) 70 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 lenght-3 N � 7 N � 6 N �5 (9.68) (7, 8, 1) 7 11 0� 6 21 0� 5 31 0� (6, 8, 2) 6 22 1� 6 22 0� (A) 4 42 1� (B) 4 2 22 1 0� � (A) 2 62 1� (B) (5, 8, 3) 5 33 2� 4 2 23 2 1� � (A) 2 63 2� (B) 4 3 13 1 0� � (A) 1 5 23 2 1� � (B) (4, 8, 4) 4 44 3� 4 44 2� (A 2 4 24 3 2� � ( �A ) 83 (B) 4 44 1� (A) 1 3 3 14 3 2 1� � � ( �A ) 4 43 2� (B) (3, 8, 5) 3 55 4� 2 2 45 4 3� � (A) 6 24 3� (B) 1 3 45 4 2� � (A) 2 5 14 3 2� � (B) (2, 8, 6) 2 66 2� 2 66 4� (A) 4 45 4� (B) 2 2 45 4 3� � (A) 6 24 3� (B) (1, 8, 7) 1 77 6� 2 66 5� 3 55 4� Fig. 1: Graph of the N � 5 (4, 8, 4) multiplet of 4 44 1� connectiv- ity (type A) � � � � � � � � � � � � g i i i i i g i 1 4 1 3 2 1 3 5 6 2 7 2 3 1 � � � � � � � � � � � � � � � � � � � � � � i i i i g i i i � � � � � � � � � � � � � 4 2 1 4 5 5 2 8 3 1 1 4 3 3 � � � � � � � � � � � � � � � � � � � � � � � � � � 4 2 5 5 8 4 1 2 3 3 4 4 2 6 � � � � � � � � i i g i i i i i� �5 7 � 10 Tensoring irreducible representations: their fusion algebras and the associated graphs The tensor product of linear irreducible representations can be decomposed into their irreducible constituents. This decomposition contains useful information in the construc- tion of bilinear (in general, multilinear) terms entering a supersymmetric invariant action. We recall that the auxiliary fields in a given representation transform as a total derivative (a time derivative in one dimension). Useful information con- cerning the decomposition of the tensor products of the irre- ducible representations can be encoded in the so-called fusion algebra of the irreps and their supersymmetric vacua. The notion of a fusion algebra of the supersymmetric vacua of the N-extended one dimensional supersymmetry, introduced in [14], is constructed by analogy with the fusion algebra for ra- tional conformal field theories. Fusion algebras can also be nicely presented in terms of their associated graphs. We ex- plicitly present here the N �1 and N � 2 fusion graphs (with two subcases for each N, according to whether or not the irreps are distinguished w.r.t. their bosonic/fermionic statis- tics). Let us discuss here how to present the [14] results in graphical form. The irreps correspond to points. Nij k oriented lines (with arrows) connect the [j] and the [k] irrep if the de- composition [ ] [ ] [ ]i j N kij k� � holds. The arrows are dropped from the lines if the [j] and [k] irreps can be interchanged. The [i] irrep should correspond to a generator of the fusion alge- bra. This means that the whole set of N Nl lj k� fusion matrices is produced as the sum of powers of the N Ni ij k� fusion matrix. Let us discuss explicitly the N � 2 case. We obtain the fol- lowing list of four irreps (if we discriminate their statistics): [ ] ( , ) ; [ ] ( , , ) ; [ ] ( , ) ; [ ] ( , , 1 2 2 2 1 2 1 3 2 2 4 1 2 � � � � Bos Bos Fer 1)Fer (10.71) The corresponding N � 2 fusion algebra is realized in terms of four 4×4, mutually commuting, matrices given by N X N N 1 2 4 1 2 1 0 0 2 0 2 1 0 1 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 � � � � � � � � � � � ; 0 2 0 2 0 2 1 0 1 2 0 2 0 2 1 2 1 0 0 2 0 2 3 � � � � � � � � � � � � � � Y N ; � � � Z. (10.72) The fusion algebra admits three distinct elements, X, Y, Z and one generator (we can choose either X or Z), due to the relations Y X X Z X X X� � � � � � 1 8 2 1 4 6 43 3 2( ), ( ). (10.73) The vector space spanned by X, Y, Z is closed under multiplication X Z ZX X Y Z XY Y YZ Y 2 2 2 2 4 � � � � � � � � , (10.74) This fusion algebra corresponds to the “smiling face” graph below. We obtain the following four tables for the fusion graphs of the N �1 and N � 2 supersymmetric quantum me- chanics irreps. The “A” cases below correspond to ignore the statistics (bosonic/fermionic) of the given irreps. In the “B” cases, the number of fundamental irreps is doubled w.r.t. the previous ones, in order to take the statistics of the irreps into account. We have © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 71 Acta Polytechnica Vol. 48 No. 2/2008 Fig. 2: Graph of the N � 5 (4, 8, 4) multiplet of 4 43 2� connectiv- ity (typeB ) 11 Conclusions Supersymmetric quantum mechanics is a fascinating sub- ject with several open problems. The potentially most inter- esting one concern the construction of off-shell invariant actions with the dimension of a kinetic term for large values of N (let us say N % 8). They could provide a hint towards an off-shell formulation of higher dimensional supergravity and M-theory. Other important topics concern the nature of the non-linear realizations of the supersymmetry and their con- nection with linear representations. We have here presented the rich mathematics underlying the linear irreducible repre- sentations realized on a finite number of time-dependent fields. We have shown how to use this information to construct supersymmetric invariant one-dimensional sigma models. We have seen that behind supersymmetric quantum mechan- ics there exists an interlacing of several mathematical struc- tures, Clifford algebras, division algebras, graph theory. Fur- ther mathematical structures seem to enter the picture (Cayley-Dickson algebras, exceptional Lie algebras, etc.). The theory of supersymmetric quantum mechanics is rich in sur- prises and seems to lie at the crossroads of various mathemati- cal disciplines. We have just given a taste of it here. Acknowledgments I am grateful to the organizers of the Advanced Summer School for the opportunity they gave me to present these results. I am pleased to thank my collaborators Zhanna Kuznetsova and Moises Rojas. These lectures were mostly based on our joint work. References [1] Witten, E.: Nucl. Phys., Vol. B 188 (1981), p. 513. [2] Akulov, V., Pashnev, A.: Teor. Mat. Fiz. Vol. 56 (1983), p. 344; Fubini, S., Rabinovici, E.: Nucl. Phys., Vol. B 245 (1984), p. 17; Ivanov, E., Krivonos, S., Lechtenfeld, O.: JHEP 0303, 2003, p. 014; Bellucci, S., Ivanov, E., Krivonos, S., Lechtenfeld, O.: Nucl. Phys., Vol. B 684 (2004), p. 321. [3] Claus, P., Derix, M., Kallosh, R., Kumar, J., Townsend, P. K., Van Proeyen, A.: Phys. Rev. Lett., Vol. 81 (1998) p. 4553; de Azcarraga, J. A., Izquierdo, J. M., Perez- -Bueno, J. C., Townsend, P. K.: Phys. Rev., Vol. D 59 (1999), p. 084015; Michelson, J., Strominger, A.: JHEP 9909, 1999, p. 005. [4] Britto-Pacumio, R., Michelson, J., Strominger, A., Volovich, A.: “Lectures on Superconformal Quantum Mechanics and Multi-Black Hole Moduli Spaces”, hep-th/9911066. [5] Ivanov, E. A., Krivonos, S. O., Pashnev, A. I.: Class. Quan- tum Grav., Vol. 8 (1991), p. 19. [6] Donets, E. E., Pashnev, A. I., Rosales, J. J., Tsulaia, M. M.: Phys. Rev., Vol. D 61 (2000), p. 43512. [7] Rittenberg, V., Yankielowicz, S.: Ann. Phys., Vol. 162 (1985), p. 273; Claudson, M., Halpern, M. B.: Nucl. 72 Acta Polytechnica Vol. 48 No. 2/2008 Fig. 4: Fusion graph of the N � 1 superalgebra (B case, 2 irreps, bosons/fermions distinction) Fig. 5: Fusion graph of the N � 2 superalgebra (A case, 2 irreps, no bosons/fermions distinction) Fig. 6: Fusion graph of the N � 2 superalgebra (B case, 4 irreps, bosons/fermions distinction), “the smiling face”. From left to right the four points correspond to the [2] – [1] – [3] – [4] irreps, respectively. The lines are generated by the N X1 � fusion matrix, see (10.72) Fig. 3: Fusion graph of the N � 1 superalgebra (A case, 1 irrep, no bosons/fermions distinction) Phys., Vol. B 250 (1985), p. 689; Flume, R.: Ann. Phys., Vol. 164 (1985), p. 189. [8] Gates Jr., S. J., Linch, W. D., Phillips, J.: hep-th/0211034; Gates Jr., S. J., Linch III, W. D., Phillips, J., Rana, L.: Grav. Cosmol., Vol. 8 (2002), p. 96. [9] de Crombrugghe, M., Rittenberg, V.: Ann. Phys., Vol. 151 (1983), p. 99. [10] Baake,M., Reinicke, M., Rittenberg, V.: J. Math. Phys., Vol. 26 (1985), p. 1070. [11] Gates Jr., S. J., Rana, L.: Phys. Lett., Vol. B 352 (1995), p. 50; ibid. Vol. B 369 (1996), p. 262. [12] Bellucci, S., Ivanov, E., Krivonos, S., Lechtenfeld, O.: Nucl. Phys., Vol. B 699 (2004), p. 226. [13] Pashnev, A., Toppan, F.: J. Math. Phys., Vol. 42 (2001), p. 5257 (also hep-th/0010135). [14] Kuznetsova, Z., Rojas, M., Toppan, F.: JHEP (2006), p. 098 (also hep-th/0511274). © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 73 Acta Polytechnica Vol. 48 No. 2/2008 Appendix We present here for completeness the set (unique up to similarity transformations and an overall sign flipping) of the seven 8×8 gamma matrices �i which generate the Cl( , )0 7 Clifford algebra. The seven gamma matrices, together with the 8-dimen- sional identity 18, are used in constructing the N � 5 6 7 8, , , supersymmetry irreps, as explained in the main text. �1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 � � � 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 2 � � � � � � � � � � � � � � � �� 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 � � � 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 3 � � � � � � � � � � � � � � �� 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 � � � 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 4 � � � � � � � � � � � � � � �� 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 � � � � 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 5 � � � � � � � � � � � � � �� 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 � � � � 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 6 � � � � � � � � � � � � � � � � 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 � � � � � � � � � � � � � � � � � � � �7 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 � � � � � � � � � � � � � � � �18 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 � � � � � � � � � � � � � (A.1) [15] Kuznetsova, Z., Toppan, F.: Mod. Phys. Lett., Vol. A 23 (2008), p. 37 (also hep-th/0701225). [16] Bellucci, S., Krivonos, S.: hep-th/0602199. [17] Toppan, F.: Nucl. Phys. B (Proc. Suppl.) 102 &103 (2001), p. 270. [18] Carrion, H. L., Rojas, M. Toppan, F.: JHEP 0304 (2003), p. 040. [19] Wess, J., Bagger, J.: Supersymmetry and Supergravity, 2nd ed., Princeton Un. Press (1992). [20] Atiyah, M. F., Bott, R., Shapiro, A.: Topology (Suppl. 1), Vol. 3 (1964), p. 3. [21] Porteous, I. R.: Clifford Algebras and the Classical Groups, Cambridge Un. Press, 1995. [22] Okubo, S.: J. Math. Phys., Vol. 32 (1991), p. 1657; ibid. (1991), p. 1669. [23] Faux, M., Gates Jr., S. J.: Phys. Rev. D 71 (2005) 065002. [24] Baez, J.: The Octonions, math.RA/0105155. [25] Doran, C. F., Faux, M. G., Gates Jr., S. J., Hubsch, T., Iga, K. M., Landweber, G. D.: hep-th/0611060. [26] Toppan, F.: hep-th/0612276. Francesco Toppan e-mail: toppan@cbpf.br CBPF Rua Dr. Xavier Sigaud 150 cep 22290-180, Rio de Janeiro (RJ), Brazil 74 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 Table of Contents Lectures on Classical Integrable Systems and Gauge Field Theories 3 M. Olshanetsky 25 Years of Quantum Groups: from Definition to Classification 23 A. Stolin Correlation Functions for Lattice Integrable Models 27 F. Smirnov 3öLectures on Noncommutative Geometry 34 A. Sitarz Extended Supersymmetries in One Dimension 56 F. 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