SUSY Summary. Maximal SUGRA

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1 SUSY Suary This suary covers all interesting supersyetries which exist in spacetie diensions 11 d 3. In y notations,, n,... are Lorentz indices (in d diensions), a, b,... are Dirac or Majorana indices, α, β,... and α, β,... are left-handed and right-handed Weyl indices; N is the nuber of supersyetries (in units of the inial SUSY in the sae diension d); i, j = 1,..., N are flavor indices of the R syetry. Maxial SUGRA Maxial supersyetry in any diension d has 32 heritian supercharges. The only consistenf field theories with axial SUSY are pure supergravities; their particle spectru coprises one SUGRA ultiplet and nothing else. d = 11, N = 1, the other of axial supergravities. The supercharges Q a 32 (real). No R syetry. Bosonic fields: the etric g n and a 3-for C ln. Ferionic fields: one Majorana gravitino ψ. a d = 10, N = (2, 0), type IIB SUGRA. The supercharges Q i α 16 (real), i = 1, 2. R syetry R = SO(2). Bosonic fields: the etric g n, two real scalars (the dilaton φ and the axion a, often cobined into one coplex scalar field τ = a + ie φ ), two 2-for fields Bn i and one self-dual 4-for field D kln. Ferionic fields: two Majorana Weyl gravitini ψ α,i spinors χ α,i (right). (left) and two Majorana Weyl All other axial SUGRAs are diensional reductions of the d = 11 theory. d = 10, N = (1, 1), type IIA SUGRA. The supercharges Q α 16 (real) and Q α 16. No R syetry. Bosonic fields: the etric g n, a real scalar φ (the dilaton) a 1-for (vector) field A, a 2-for field B n and a 3-for field C ln. Ferionic fields: one Majorana (left+right MW) gravitino ψ a and one Majorana (left+right MW) spinor χ a. 1

2 d = 9, N = 2 SUGRA. The supercharges Q i a 16 (real), i = 1, 2. R syetry R = SO(2). Bosonic fields: the etric g n, three real scalars, three vectors, two 2-for fields and one 3-for field. Ferionic fields: two Majorana gravitini ψ a,i and four Majorana spinors. d = 8, N = 2 SUGRA. The supercharges Q i α 8 and Q α,i 8 (coplex plus h.c.), i = 1, 2. R syetry R = U(2). Bosonic fields: the etric g n, seven real scalars, six vectors, three 2-for fields and one 3-for field. Ferionic fields: two Weyl (left) gravitini ψ α,i heritian conjugates of opposite chirality. and six Weyl (right) spinors, plus d = 7, N = 2 SUGRA. The supercharges Q i a, Q a,i 8 (pseudoreal), i = 1, 2. R syetry R = Sp(2). Bosonic fields: the etric g n, 14 real scalars, ten vectors and five 2-for fields. Ferionic fields: two Dirac gravitini ψ a,i and eight Dirac spinors, plus heritian conjugates. d = 6, N = (2, 2) (AKA N = 4) SUGRA. The supercharges Q i α 4 and Q i a α 4 (both pseudoreal), i = 1, 2. R syetry R = Sp(2) Sp(2). Bosonic fields: the etric g n, 25 real scalars, 16 vectors and six 2-for fields. Ferionic fields: two Dirac (left+right Weyl) gravitini ψ a,i Weyl) spinors, plus h.c. d = 5, N = 4 SUGRA. and ten Dirac (left+right The supercharges Q i a, Q a,i 4 (pseudoreal), i = 1, R syetry R = Sp(4). Bosonic fields: the etric g n, 42 real scalars and 27 vectors. Ferionic fields: four Dirac gravitini ψ a,i and 24 Dirac spinors, plus h.c. 2

3 d = 4, N = 8 SUGRA. The supercharges Q i α 2 and Q α,i 2 (coplex plus h.c.), i = 1,..., 8. R syetry R = U(8). Bosonic fields: the etric g n, 70 real scalars and 28 vectors. Ferionic fields: eight Majorana gravitini ψ a,i d = 3, N = 16 SUGRA. and 56 Majorana spinors. The supercharges Q i a 2 (real), i = 1,..., 16. R syetry R = SO(16). Bosonic fields: 128 real scalars (the etric field is non-dynaical in d = 3). Ferionic fields: 128 Majorana spinors. Maxial Rigid SUSY Maxially supersyetric field theories without gravity are SSYM theories with 16 supercharges. The spectra of such theories are coprised of several vector superultiplets in the adjoint representation of soe gauge group G. In d = 10, the SSYM theories are chiral (and hence suffer fro the axial anoaly). The d < 10 SSYM theories are diensional reductions fro d = 10 and the chirality is lost. d = 10, N = 1 SSYM (the other of all SSYM ). Each vector ultiplet contains one vector field A and one Majorana Weyl spinor field ψ α. d = 9, N = 1 SSYM. Each vector ultiplet contains one vector field, one real scalar and one Majorana spinor. d = 8, N = 1 SSYM. Each vector ultiplet contains one vector field, two real scalars and one Majorana spinor. (Or equivalently, one vector, one coplex scalar and one Weyl spinor). R = SO(2) = U(1). The anoalies cancel for G = SO(32) or G = E 8 E 8 SSYM theories coupled to SUGRA. 3

4 d = 7, N = 1 SSYM. Each vector ultiplet contains one vector field, three real scalars and one Dirac spinor. R = Sp(1) = SU(2) = SO(3). d = 6, N = (1, 1) (AKA N = 2) SSYM. Each vector ultiplet contains one vector field, four real scalars and one Dirac spinor (i.e., Weyl left+right spinors). R = Sp(1) Sp(1) = SO(4). d = 5, N = 2 SSYM. Each vector ultiplet contains one vector field, five real scalars and two Dirac spinors. R = Sp(2) = SO(5). d = 4, N = 4 SSYM. Each vector ultiplet contains one vector field, six real scalars and four Majorana spinors. R = SU(4) = SO(6). Despite quantu effects, the N = 4 SSYM theories in d = 4 are superconforally invariant. d = 3, N = 8 SSYM. Each vector ultiplet contains one vector field, seven real scalars and eight Majorana spinors, or equivalently 8 real scalars and 8 Majorana spinors. R = SO(8). Tensor Theories with 16 Supercharges In addition to all these SSYM theories, in d = 6 there also tensor theories with chiral N = (2, 0) SUSY. The basic ultiplet of this SUSY is tensor rather than vector; it coprises five real scalar fields φ I, two left Weyl spinors ψi α and a self-dual 2-for field B n (the tensor). However, its diensional reduction to d < 6 is equivalent to a vector ultiplet. Our present knowledge of interacting N = (2, 0) tensor theories is rather liited; ostly, it follows fro diensional oxidation of SSYM theories fro d = 5 to d = 6. In d = 3 a vector field is dual to a scalar. 4

5 Theories with 8 Supercharges Rigid theories with 8 supercharges allow for two kind of superultiplets, naely vector and hyper. The vector ultiplets for an adjoint repr. of soe gauge group G; the hyperultiplets ay belong to any coplete representations. Again, all such sypersyetric gauge theories are diensional reduction fro the highest diension which allows for 8 supercharges, naely d = 6. Consequently, in d = 6 we have chirality (and hence axial anoaly to contend with) but the d < 6 theories are non-chiral. d = 6, N = (1, 0) (AKA N = 1). A vector ultiplet coprises a vector field A and a right Weyl spinor field ψ α. A hyper ultiplet coprises a left Weyl spinor χ α and four real scalars. R = SU(2). d = 5, N = 1. A vector ultiplet coprises a vector, a real scalar and a Dirac spinor. ultiplet coprises a Dirac spinor and four real scalars. R = SU(2). A hyper d = 4, N = 2. A vector ultiplet coprises a vector, two real scalars and two Majorana spinors. A hyper ultiplet coprises two Majorana spinor and four real scalars. R = SU(2) U(1). d = 3, N = 4. A vector ultiplet coprises a vector, three real scalar and four Majorana spinors, or equivalently four real scalars and four Majorana spinors. A hyper ultiplet also coprises four real scalars and four Majorana spinors. However, the R syetry R = SU(2) SU(2) distinguishes between the vector and the hyper ultiplets. In addition, the d = 6, N = (1, 0) SUSY has tensor superultiplets, each coprising one real scalar φ, one left Majorana spinor χ α and one self-dual 2-for field B n. The tensor ultiplets are always neutral with respect to the gauge syetry. The diensional reduction of a tensor ultiplet to d < 6 is equivalent to a vector ultiplet. 5

6 Theories with 4 Supercharges Supersyetric theories with only 4 supercharges (existing in d 4) allow for Yukawa and scalar forces independent of the gauge couplings. d = 4, N = 1 SUSY. A chiral ultiplet coprises a left Weyl spinor χ α and a coplex scalar field φ. Its heritian conjugate coprised of the right Weyl spinor χ α and the conjugate scalar φ is a distinct anti-chiral ultiplet. A vector ultiplet coprises a vector field A and a Weyl spinor ψ α with its heritian conjugate ψ α. R = U(1) d = 3, N = 2 SUSY is the diensional reduction fro (d = 4, N = 1). In d = 3, the vector, the chiral and the anti-chiral ultiplets becoe equivalent on shell; each coprises two real scalar fields φ i and one Majorana spinor χ a. R = SO(2) = U(1). Theories with 2 Supercharges d = 3, N = 1 SUSY. On shell, there is only one kind of a superultiplet; it contains a real scalar field and a Majorana spinor. There is no R syetry. 6