The first concrete example of the duality between gauge theory and string theory was proposed by Maldacena in reference [1]. The ABJM theory has two parameters: the rank of the gauge group N and the Chern-Simons level k.
The Free Theory
One can derive the conserved supercurrent using the Noether method, which involves varying the action while allowing to have arbitrary x-dependence. However, this is not a general theorem, so it is a good idea to check conformal symmetry explicitly, as we have done.
The SO(4) Theory
Of these, only the conformal transformation, obtained as the commutator of two η-transformations, is not a manifest symmetry of the action. Any apparent discrepancies in the minus sign are due to the different treatment of the Dirac matrices discussed earlier.
Parity Conservation
Then it is concluded that the Chern-Simons term is parity preserving.4 To conclude that the whole theory is parity preserving, there is another term that needs to be analyzed. The usual story is that the parity transformation (associated with the spatial inversion~x→ −~x) of a spinor is given byψ→γ0ψ.
The Search for Generalizations
It is well known that tabc can be considered as an invariant tensor describing the totally antisymmetric union of three seven-dimensional representations of the Lie group G2. These can then be used in the usual way to express the parts of the measure G2 as the seven-dimensional matrix Aab.
Relation to antide Sitter Gravity?
We thus believe that there is no reasonable interpretation of the BL theory as a three-dimensional theory of gravity. Also a proposal for the physical interpretation of the BLSO(4) theory in terms of M2-branes in M-theory at an M-fold singularity has been given in [30, 31].
The Basic Idea
Note that this theory has a noncompact gauge group whose Lie algebra is the semidirect sum of any regular Lie algebra of a compact Lie group G and dim(g) of abelian generators. In the case of M2-brane theory in the next section, conformal symmetry is expected to survive in quantum theory.).
Modifying the Lorentzian Metric BL-Theory
The supersymmetries of the new gauge fields must be detected in such a way that Lnew is invariant. We will find that the resulting supersymmetry algebra closes on shell when the new gauge symmetries are taken into account.
Discussion
Note also that the commutator closure requirement in the shell [δ(ε1), δ(ε2)]Ψ− gives the expected equation of motion for Ψ− since it is observed that the commutator receives a contribution from δ(η)Ψ− . In addition, [14] derived an apparent SU(4) invariant form of the scalar scale potential, which is sixth order in the scalar parts.
These expressions are not necessary to derive the equations of motion of the gauge fields, which are. Note that in the special case U(1)×U(1)Jµ=−Jˆµ, so the equations of motion imply Fµν= ˆFµν. The coefficients are chosen such that L4=L4a+L4b+L4c is the correct result required by supersymmetry, as shown in Appendix C.
In particular, for a solution where the scalar fields X A and X A are constant and the gauge fields vanish, the variations δ3ΨA and δ3ΨA should vanish. It is straightforward to verify the equivalence of equations (4.12) and (4.13) for this choice of the coefficient using the key identity. As already noted, NIA and NAI vanish when the scalar fields are diagonal matrices.
Conclusion
World-Sheet Formalism
Using the metric in equation (5.1), the bosonic part of the string Lagrangian in conformal gauge is given by . has two killing vectors and CP3 has three killing vectors, any solution to the bosonic equations of motion has at least ve conserved charges. The spectrum of bosonic uctuations around a classical solution can be calculated by expanding the bosonic Lagrangian in equation (5.4) to quadratic order in the uctuations and finding the normal modes of the resulting equations of motion. To calculate the spectrum of fermionic uctuations, we need only the quadratic part of the fermionic GS action for type IIA string theory.
We will now reformulate the fermionic Lagrangian in equation (5.9) in a form that allows us to calculate the fermionic uctuation frequencies in a simple way. In addition, if the classical solution satisfies. then the fermionic Lagrangian simply becomes ermi to LF. 5.15) Finally, if we consider the Fourier modeΘ (σ, τ) = ˜Θ exp (−iωτ+inσ), whereΘ˜ is a constant spinor, then the equations of motion for the fermionic uctuations are given by. Explicit calculations of the fermionic frequencies for the classical solutions studied in this chapter are described in sections 5.2.2 and 5.3.2.
Algebraic Curve Formalism
- Classical Algebraic Curve
- Semiclassical Quantization
From this formula, we see that q1(x)andq2(x) correspond to the AdS4 part of the algebraic curve, while q3(x),q4(x), andq5(x) correspond to the CP3 part of the algebraic curve. While Nnij is input to the calculation, ∆(y) will be determined in the process of determining the uctuations of quasi-momenta. The coefficients of the other terms on the right-hand side of equation (5.27) can be determined using arguments similar to those in [98].
Finally, when the spectral parameter approaches the location of one of the poles, the uctua-. After calculating the anomalous part of the energy shift, the uctuation frequency is given by The on-shell frequencies are then obtained by evaluating the o-shell frequencies at the location of the poles determined by solving equation (5.24), i.e. ωijn = Ωij xijn.
Summation Prescriptions
It should be noted that heavy and light frequencies are not as well defined in the worldsheet approach. In general, the only way to identify heavy and light frequencies in the world-sheet approach is to compare the world-sheet spectrum with the algebraic curve spectrum, i.e. a world-sheet frequency heavy/light is called if the corresponding algebraic frequency curve frequency is heavy/light. In [89] it was shown that equation (5.37) yields a one-loop correction corresponding to the all-loop Bethe ansatz when applied to the spectrum of the folded twisting string.
This suggests that only heavy modes with even mode number should contribute to the one-loop correction. The coecientsAandB can then be uniquely determined by requiring that the integral approach to this formula be reduced to equation (5.37) in the large-κ limit, to ensure that this summation prescription gives a one-loop correction to the folded twisting rope- energy corresponding to the all -loop Bethe ansatz. One of the advantages of the new summation prescription in equation (5.38) compared to the one in equation (5.35) is that it gives more well-covered results for one-loop corrections.
Point-Particle
- Classical Solution and Dual Operator
- Point-Particle Spectrum from the World-Sheet
- Point-Particle Algebraic Curve
- Excitation Spectrum
- One-Loop Correction to Energy
One of them has the dispersion relation in the above equation, and the other four have the dispersion relation in equation (5.41). This can be seen by extending the Virasor constraints in equation (5.8) to linear order in the perturbations. In order to calculate the spectrum of fermionic fluctuations with respect to the point particle solution given by equation (5.39), we only need to know the Vielbein recoil and spin coupling in the background of this classical solution.
In this section we calculate the algebraic curve for the classical solution given in equation (5.40). If we remember that κ=J and plug these results into equation (5.22), we find that the classical quasi-momenta are. Note that this ansatz satisfies the inversion symmetry in equation (5.26) and has pole structure consistent with equation (5.28).
Spinning String
- Classical Solution and Dual Operator
- Spinning String Spectrum from the World-Sheet
- Spinning String Algebraic Curve
- Excitation Spectrum
- One-Loop Correction to the Energy
As a result, both the standard summation prescription in equation (5.36) and the new summation prescription in equation (5.38) provide a vanishing one-loop correction to the energy. The higher order terms in the expansion of the classical string energy in equation (5.50) correspond to O λ4/J3. In this section, we calculate the spectrum of fermionic uctuations about the spinning string solution in equation (5.48).
For the classical solution in equation (5.49) one finds that the relation in equation (5.18) is given by In the previous section we found that when Ω (J, n, m) = Ωold, AC(J, n, m), the one-loop correction is divergent, but for the other three cases in equation (5.72), it is convergent. Note that the first term in equation (5.77) came from the expansion of the classical distribution relation for the spin string to first order in the BMN parameterλ/J2 and then the substitution λ→2λ2.
Comparison with Bethe Ansatz
Conclusions
More importantly, if we calculate one-loop corrections by adding the algebraic curve frequencies using the standard summation prescription used in AdS5×S5, we get a linear divergence. This is inconsistent with supersymmetry, because we expect the one-loop correction to vanish for the point particle and to be nonzero but nite for the spinning string. We propose a new summation prescription in equation (5.38) that produces exactly these results when applied to both the algebraic curve spectrum and the worldsheet spectrum.
In particular, it also gives a single-loop correction to the folded spinning cord, which matches the all-loop Bethe ansatz. For example, we found that our summation recipe generally produced more accurate results for single-loop corrections. It would also be interesting to evaluate the single-loop correction to the spinning string energy in a way that does not rely on summation rules.
Spin(8) Dirac Algebra
Ten-dimensional Dirac Algebra
We will now take the supersymmetric variations of the Lagrangian by separating it into five terms: A, B, C, D and the Chern-Simons term. 2ic1c4abcdbef gψaΓIJΓKLMφIcφJdφKeφLfφMg −6ic2abcdbef gψaΓKφIcφJdφKeφJfφIg= 0,. We analyze only half of the articles, since the other half is only their adjunct.
The other half of the terms of the variation of the action, which are additional to those dealt with here, are similarly repealed. Using ∂µε(x) =γµηandγµγργµ=−γρ this gives a variation of the action that almost cancels except for a few terms. These remaining terms can be canceled by including an extra variation of the spinor fields.
Having determined L4, we are now able to determine δ3Ψ by computing the schematic structure terms tr(ΨADXBXCXD), tr(ΨAXBDXCXD) and tr(ΨAXBXCDXD) arising from varying the gauge parts in the kinetic term X and varying the spinor elds in L4. Verifying that this current is conserved as a consequence of the equations of motion is quite tedious. The first term in this expression is canceled by the variation δ0ΨA of the spin kinetic term.
After all, this is not a logical consequence of the other symmetries that have been verified. Scalars have mass dimension 1/2 and transform in the fundamental representation of the R-symmetric group SU(4). For form operators. the two-loop dilation operator is given by. where λ = N/k, P is the permutation operator and T is the tracking operator [55].
CP 3
Fluxes
- Field content of the BL SO(4) theory
- Index notation for the BL SO(4) theory
- Labels for the uctuations of the AdS 4 × CP 3 algebraic curve
- Relations between heavy and light o-shell frequencies
- O-shell frequencies for uctuations about the point-particle solution
- Spectrum of uctuations about the point-particle solution computed using the world-
- O-shell frequencies for the uctuations about the spinning string solution
- Spectrum of uctuations about the spinning string solution computed using the world-
- Notation for spinning string frequencies
- Large- n limit of Ω (J , n, m) + Ω (J , n, −m)
Tseytlin, Semiclassical Strings and AdS/CFT, in String Theory: From Gauge Interactions to Cosmology (L. Baulieu, J. de Boer, B. Pioline, & E. Rabinovici, eds.), p. Stelle, T-Duality in the Green-Schwarz formalism, and the massless/massive duality map IIA, Nucl.
Classication of the supersymmetry variations of the BL SO(4) Lagrangian
