String Theory II GEORGE SIOPSIS AND STUDENTS

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1 String Theory II GEORGE SIOPSIS AND STUDENTS Department of Physics and Astronomy The University of Tennessee Knoxville, TN U.S.A. Last update: 2006

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3 Contents 9 Low Energy Physics Type IIA Superstring Supergravity

4 UNIT 9 Low Energy Physics 9.1 Type IIA Superstring Sectors: (Ns+,NS+), (R+,NS+), (NS+,R-), (R+,R-). (NS+, NS+): massless states: A µν ψ µ 1/2 ψν 1/2 0; k A µν is decomposed into a scalar, antisymmetric field and traceless symmetric field: 8 8 = The scalar field represents the dilaton. We do not see it in Nature and it is believed to have settled into it s ground state value and not affect dynamics any further. We will set it to zero to avoid complications in an already complicated discussion (it should be set to a constant, but we can always tweak couplings, etc., so setting it to zero will be fine). Let B µν be the antisymmetric tensor and g µν the traceless symmetric tensor (graviton). The dynamics of g µν is described by the Einstein action S g = 1 4πG 10 d 10 x g R, where G 10 is the ten-dimensional Newton s constant. This can be derived from string theory tree-level amplitudes and loop amplitudes introduce corrections. B µν has field strength H µνρ = µ B νρ ν B µρ + ρ B µν. H µνρ is totally antisymmetric in it s indices. The action is given by S B = 1 d 10 x gh µνρ H µνρ, 8πG where indices are raised and lowered by g µν. (R+,R-): massless states: s; k s ; k, 8 8 of them. Recall ψ µ 0 s; k is also a ground state (annihilated by all ψµ r, r > 0). States decompose into C µ ψ µ 0 0 and C µνρψ [µ, 0 ψ[ν, 0 ψρ]] 0 0 where we antisymmetrize over all indices. There are 8 C µ (transverse µ) and 56 C µνρ (8+56 = 64)

5 58 UNIT 9: Low Energy Physics so they span the ground states. The action is given by S R = 1 d 10 x g (F µν F µν + 8πG F µνρσ F ) µνρσ 10 where F µν = µ C ν ν C µ is the field strength of C µ. F µνρσ = F µνρσ 1 4 (C µh νρσ + C ν H ρσµ + C ρ H σµν + C σ H µνρ ) where F µνρσ = µ C νρσ +... (add terms such that F µνρσ is completely antisymmetric) and is the field strength of C µνρ. There is one more contribution to the action that does not involve the metric (topological). This is a Chern-Simons term given by S CS = 1 8πG 10 d 10 xɛ µ1µ2...µ10 B µ1µ 2 F µ3µ 4µ 5µ 6 The total action is the sum of all the actions and is given by S = S g + S B + S R + S CS There is a fermionic counterpart which we will not discuss. 9.2 Supergravity Let us compare with supergravity (SUGRA). Unfortunately, SUGRA lives in 11 dimensions, yet it looks so much like the type-ii superstring, that is hard to ignore. It turns out that (modern wisdom holds) we really live in eleven dimensions and ten dimensional strings are really an eleven dimensional theory! What theory? Nobody knows... M-Theory. We have seen this problem with dimensions before. We compactified one dimension a la Kaluza-Klein, then let R 0 and the extra dimension would not go away. Same here. We will compactify the eleventh dimension to get 10D superstrings, but the eleventh dimension will remain lurking in the background. Action: S SUGRA 11 = 1 4πG πG 11 d 11 x G (R (11) 12 ) F MNQRF MNQR d 11 x ɛ M1M2...M11 A M1M 2M 3 F M4M 5M 6M 7 F M8M 9M 10M 11 where F MNQR is the field strength of A MNQ (F MNQR = M A NQR +...), where F is completely antisymmetric. The last term is a Chern-Simons term and is gauge invariant δa MNQ = M λ NQ +...

6 9.2 Supergravity 59 even though it doesn t look like it. We need to reduce the dimension from eleven to ten to compare with superstrings. We will do that a la Kaluza-Klein. Assume nothing depends on the eleventh coordinate and call it u. The metric: ds 2 = G MN (x µ )dx µ dx ν, M, N = 0, 1,..., 10, µ = 0, 1,.., 9 We may decompose the metric as such ds 2 = G µν dx µ dx ν + 2G µu dx µ du + G uu du 2 Let G u u = 1 for simplicity (fixes the size of the extra dimension, which can be rescaled later). Introduce vector A m u = G µu and metric g µν = G µν A µ A ν. Then we may write ds 2 = g µν dx µ dx ν + (du + A µ dx µ ) 2. The potential A MNQ may also be grouped into A µνρ and A µν = A µνu (no other components exist, because A MNQ is antisymmetric, so we can not have two u indices). So now the field content becomes (from the 10D perspective) g µν, A µ, A µν, A µνρ, very similar to the type-ii superstring. Futhermore, where F µν = µ A ν ν A µ. R (11) = R (10) 1 2 F µνf µν F MNQR F MNQR = F µνρ F µνρ + F µνρσ F µνρσ where F µνρ = µ A µρ +..., F µνρσ = F µνρσ A µ F νρσ +... and F µνρσ = µ A νρσ +... and 1 A 6 ɛm1m2...m11 M1M 2M 3 F M4M 5M 6M 7 F M8M 9M 10M 11 = ɛ µ1µ2...µ10 A µ1µ 2 F µ3µ 4µ 5µ 6 = ɛ µ1µ2...µ10 A µ1µ 2µ 3 F µ4µ 5µ 6 + total derivative The last equality can easily be verified by integrating by parts. Also G = g. If the eleventh dimension has length 2πR, then the action becomes S SUGRA 11 = 1 4πG 10 d 10 x g ( R (10) 1 2 F µνf µν 1 2 F µνρf µνρ 1 2 F µνρσ F µνρσ 1 2 ɛµ1µ2...µ10 A µ1µ 2 F µ3µ 4µ 5µ 6 ) where G 10 = 2πRG 11 is the 10D Newton s constant and we have rescaled A µν 1 2πR A µν, A µνρ 1 2πR A µνρ. This action is identical to the one obtained from type-iia superstring if we identify g µν g µν, A µ C µ, A µν B µν, A µνρ C µνρ.