Only by replacing the basic point-particle components of the theory with extended one-dimensional objects (strings) is it given the necessary freedom to accommodate gravity [2, 3]. In trying to describe the known symmetries of the vacuum, the study of string theory has led to the discovery of a dramatic new class of fundamental symmetries known as dualities.
The holographic entropy bound
A striking solution to this problem, proposed by 't Hooft [15], is that nature obeys a holographic entropy bound, which states that the degrees of freedom available to a physical system occupying a volume V can be mapped to a physical theory that is defined to exist. strictly speaking on the boundary ∂V (see also. The maximum entropy of a system is thus limited by the number of degrees of freedom that can be mapped from the system's interior to its boundary.
Holography and string theory
During these studies, ’t Hooft observed that the finite extension of the Feynman graph, when 1/Nc is interpreted as the coupling strength, is topologically identical to the genus extension of the world sheet of generic interacting string theory. In addition, the sources for closed string states are studied, and the physics in the region just outside the brane is described by type IIB closed superstring theory in the AdS5 × S5 background geometry.
The Penrose limit
This will be a useful preliminary to our main goal of estimating corrections to the Penrose limit of GS action. The canonical momentumA also has a mode expansion, related to that ofxA by the free-field equation pA= ˙xA. The coefficient functions are best expressed in terms of harmonic oscillator sweep and sweep operators: 0.3.13).
The 1/J expansion and post-BMN physics
Concurrent with these studies, a new formalism for the dimensions of computer operators in gauge theory emerged. The work in the SO(6) sector was extended by Beisert and Staudacher, who formulated a Bethe ansatz for the full P SU(2,2|4) superconformal symmetry of the theory (under which the full dilation generator is invariant). 32].
Overview
Most of the information to be covered in this section first appeared in [28], although we will orient our review around a restatement of these results first presented in [26]. In the one-loop order in the 't Hooft coupling gY M2 Nc the action of the dilation operator based on eqn.
The complete supermultiplet
One-loop order
As a concrete example, the basis of three-impurity states of the K = 6 su(2) spin chain is. The eigenvalues of the spin chain (or the anomalous dimensions of the corresponding gauge theory operator) are then obtained as a power series in 1/K by expanding the Bethe roots in eq.
Two- and three-loop order
As before, we fit the spectral data to a power series in 1/K to read the leading scaling coefficients for the low-lying eigenvalues. It seems to us to provide useful additional evidence that the long-range Bethe ansatz for the su(2) sector of.
A closed su(1|1) subsector of su(2|3)
In [33], Beisert put the action of the Hamiltonian on the su(2|3) spin chain into three-ring order.4 In the notation [33], the action of the Hamiltonian on the ground states can be represented by expressions of special permutation operators denoted by. The su(1|1) Hamiltonian of the two-loop bootstrap space can be extracted in the same way (the position space version is too long to print here):. 2.2.6) Finally, the full Hamiltonian of three loops is for this subsector. These results imply the following predictions for the one-loop and two-loop scaling coefficients:
The sl(2) sector
The virial argument further tells us that higher strengths in the fields will determine higher strengths of K−1 in the expansion of the energy. For our present purposes it is sufficient to know that the Hamiltonian is expanded to terms of fourth order in the fields and this truncation of the Hamiltonian can be easily constructed by inspection: . In the sl(2) sector the highest weight is −1/2: the Dynkin diagram therefore has coefficient Vsl(2) = −1 and the Cartan.
Discussion
Comparison of the resulting interaction spectrum with corrections (in inverse powers of the R charge) to the dimensions of the corresponding gauge theory operators largely (but not completely) confirms expectations of AdS/CFT duality (see [26, 30] for discussion ). In Section 3.1 we introduce the problem by considering the bosonic sector of the theory alone. We comment on some interesting aspects of the theory that arise when it is restricted to the point-particle (or zero-mode) subsector.
Strings beyond the Penrose limit
This metric has the full SO(4,2)×SO(6) symmetry associated with AdS5 ×S5, but only the translational symmetries in t and φ and the SO(4)×SO(4) symmetry of the transverse coordinates remain manifest. This implies that we can use the structure of the Laplacian to correctly order operators in the string Hamiltonian. The HS5 equation is just a repackaging of the problem of finding the eigenvalues of the SO(6) Casimir invariant (another name for the scalar Laplacian onS5) and HAdS5 poses the corresponding problem for SO(4,2).
The left-invariant Cartan forms are defined in terms of the coset space representativeG by. This restricts the worldsheet fermions to lie in the 8s representation of SO(8) (and projects the 8c spinor out), reducing the number of independent components of the worldsheet spinor from 16 to 8. However, it will be shown that this new coordinate system induces correction terms for the spacetime curvature of the world sheet metric.
Curvature corrections to the Penrose limit
The advantage is that the results will ultimately be unambiguous (and free of ambiguities in normal order). This allows us to isolate the bosonic scaling contribution from the covariant derivative when combining different terms in the Lagrangian. In the pp-wave limit, keeping the world sheet metric flat in this light cone gauge is consistent with the equations of motion.
Quantization
The O(1/Rb2) correction to the Hamiltonian can also be expressed in terms of canonical variables. Finally, we found that this problem can be traced to the presence of second-class constraints involving ˙ψ†. Identification of these Dirac brackets with the quantum anticommutators of the fermionic fields in the theory naturally leads to additional O(1/Rb2) corrections to the energy spectrum.
Energy spectrum
Evaluating Fock space matrix elements of H BB
There is a parallel no-mixing phenomenon in gauge theory: two-impurity bosonic operators carrying spacetime vector indices do not mix with spacetime scalar bosonic operators carrying R charge vector indices. Due to operator-ordered ambiguities, two-impurity matrix elements can differ from HBB by contributions proportional to δADδBC, depending on the specific prescription chosen [26]. The zero-order term must vanish if the energy correction is to be disturbing in the gauge coupling.
Evaluating Fock space matrix elements of H FF
The next term in the expansion contributes one random constant (then BB term) and each higher term in the λ0 expansion basically contributes one additional random constant to this sector of the Hamiltonian. The index structure of the fermionic matrix elements is similar to that of its bosonic counterpart (3.5.6). We will now introduce some useful projection operators that will help us understand the selection rules implicit in the index structure of (3.5.7).
Evaluating Fock space matrix elements of H BF
The 128-dimensional subsector of space-time fermions is mixed with matrix elements of the same Hamiltonian taken between the states of the fermion array of the general form bα†n aA†−n|Ji. The key point is that the structure of the perturbative Hamiltonian implies certain relations between all normal ordering functions. A close inspection of the manner in which the conventional ordering functions contribute to the energies of the states in the two impurity sector shows that the states at the L = 0.8 levels are shifted only by NBB.
Diagonalizing the one-loop perturbation matrix
Since conformal invariance is part of the full symmetry group, states are organized into conformal manifolds built on conformal primaries. A supermultiplet will contain several conformal primaries that have the same value of ∆ and that transform into each other under overloads. Although the states in the degenerate multiplet all have the same J , they actually belong to different L levels in more than one supermultiplet.
Details of the one-loop diagonalization procedure
The identification of the representations associated with particular eigenvalues is easy to do based on multiplicity. The explicit realization of the two US(2) factors involved here is thus found. The important fact to note is that the Λ eigenvalues and their multiplicity are exactly as required for consistency with the full P SU(2,2|4) symmetry of the theory.
Gauge theory comparisons
The extended supermultiplet spectrum perfectly matches the full spectrum of one-loop string theory in Table 3.6 above.
Energy spectrum at all loops in λ 0
As explained above, N = 4 supersymmetry ensures that the dimensions of operators at other levels of the supermultiplet are obtained by making the substitution R → R + 2 − L/2 in the expression for the dimension of the L = 4 operator . Applying to this expression the same logic as applied to the result of the two-loop gauge theory (3.6.6), we obtain the following three-loop correction on the anomalous dimension of the general level of the supermultiplet of the two-impurity operator : . 3.6.9) We see that this expression is different from the third order contribution to the string result (3.6.5) for the corresponding quantity. However, our discussion of the question of normal order earlier in this chapter seems to rule out such freedom.
Discussion
In the boson space-time sector, the subspace of purely bosonic statesA†q aB†r aC†s |Ji is 512-dimensional. When each of the three mod indices (q, r, s) is different, the states with bi-fermionic excitations saA†q bα†r bβ†s |Ji are inequivalent under permutation of the mod indices and form a 1536-dimensional subsector. Hint bζ†s b†r bδ†q |Ji aC†s aD†r bδ†q |Ji aC†r aD†q bδ†s |Ji aC†r aD†s bδ†q |Ji hJ|bαqbβrbγs HFF HBF HBF HBF hJ|bαqaAraBs HBF HBB+HBF HBF HBF hJ|bαsaAqaBr HBF HBF HBB+HBF HBF hJ|bαraAsaBq HBF HBF HBF HBB+HBF.
For comparison, next to the projection of the spectrum of two impurities onto the same subspace (as shown in the chapter Table 4.6: The energy spectrum of three impurities in the projection of the pure SO(4) boson (left panel) and the energy spectrum of two impurities in the same projection (right panel) subsector is shown SO(4)S5 clearly originates from a singlet with two impurities (1,1;1,1) in the same SO(4) subgroup.We can perform a projection onto the subsector in Table 4.2 similar to Table 4.5. The results for both subspaces are presented in Table 4.8: Bifermion states with two impurities in table 4.8 are.
Assembling eigenvalues into supermultiplets
A full analysis of the agreement with gauge theory anomalous dimensions will have to be deferred to a later section: the dimensions of three-impurity gauge. The sector described above is often called a so(6)2 sector on the gauge theory side, referring to the subalgebra of the full superconformal algebra under which it is invariant. In the string theory, the subsectors analogous to the gauge theory sl(2) ensu(1|1) are constructed from respectively fully symmetric SO(4)AdS bosons and fully symmetric fermions of the same Π eigenvalue (see Chapter 3 or ref. [27]) .
The diagonal contributions from the pure fermionic HFF sector in blocks (2,2) and (3,3) of Table 4.11 look like. If we follow the irep SO(4)×SO(4) structure, we find that the traceless SO(4)S5 symmetric boson states arising from the su(2) closed subsector belong to. The states at level L = 4 in the second multiplet in Table 4.13 correspond to operators in the closed sector sl(2) of the gauge theory and the eigenvalue Λ =−7/3 [280B] represents the prediction for the one-loop anomalous dimension of this class of operators of the gauge theory.
Three-impurity spectrum: all orders in λ 0
In fact, the superconformal multiplet structure of the three-impurity problem is such that the energies/dimensions of all other irreps can be derived from those of the three protected irreps. Therefore, this method will give us exact expressions for all the energy levels of the three-impurity problem. The predicted closed sector eigenvalues (listed in table 4.14) match, with the accuracy of the calculation, entries in the list of numerical eigenvalues.
By invoking the angular momentum shift J → J+ 2−L/2 in the BMN limit, we can use the energy shift of the L = 4 level to recover the exact energy shifts of all other levels in the superconformal multiplets of Table 4.13. The energy shifts of the vector multiplet containing the shielded bosonic irrep SO(4)AdS at the L = 4 level are 4.2.21). We note again that the BMN energy of the original degenerate multiplet must be added to these results to obtain energies instead of energy shifts.
Gauge theory anomalous dimensions
This provides specific string theory predictions for large-K scaling of one-loop anomalous dimensions of the closed AdS sector. We now turn to the closed su(2) sector of gauge theory operators, corresponding to the symmetric-traceless bosonic SO(4)S5 sector in string theory. The large-K spectrum of the three-loop contribution is examined in Chapter 2; one again finds disagreement with string theory.).
Discussion
The SO(4) AdS (sl(2)) sector
The su(1|1) sector
Spectral decomposition
Discussion
Lax representation
Spectral comparison with gauge theory
Discussion
