M2-branes and AdS/CFT: A Review

We briefly review some of the important developments in the last decade in the theory of multiple M2-branes and AdS4/CFT3 correspondence. Taking the examples of the superconformal index, free energy on S^3 and entropy of charged black holes, we illustrate how the large N limit was studied and the correspondence was checked.


Introduction
In 1998, one day when I was a Ph.D. student, Prof. Eguchi came to me in the tea room and asked if I already read the paper by Maldacena about a new duality which is now known as AdS/CFT correspondence [1]. It was when another important paper by Witten [2] appeared. I had just finished my first paper [3] on emissions from D1-D5 black holes, which was a project suggest by him. Though he did not mean anything special by that small conversation, it remained in my memory because I felt like being treated as an independent researcher for the first time.
In 2009, I started working with Prof. Eguchi again at Yukawa Institute, where he was the Director at that time. We worked together, sometimes jointly with cosmologists, on organizing and running a series of conferences. That was a work requiring a different level of dedication to physics. I was influenced a lot from the eagerness with which he kept these activities running for many years, and also from the way he cared about the purpose and the real outcome for each of those events.
In 2019 we had a conference in Kyoto in memory of Prof. Eguchi. In this article, partly based on the talk given there, I will briefly review some of the important developments in the last decade in the theory of multiple M2-branes and AdS 4 /CFT 3 correspondence. I will illustrate how the large N limit was studied and the correspondence was checked by taking the superconformal index, free energy on S 3 and the entropy of charged black holes as examples.
Most of the discussions are restricted to the ABJM model [4] for N M2-branes probing the orbifold C 4 /Z k . In 3D N = 2 convention, it is a U (N ) k × U (N ) −k Chern-Simons theory with chiral multiplets A 1 , A 2 in the bifundamental and B 3 , B 4 in the anti-bifundamental representations and a superpotential The gauge field, scalar and the auxiliary field in the two U (N ) vectormultiplets will be denoted as (A µ , σ, D) and (Ã µ ,σ,D), respectively. The model should be dual to the quantum supergravity on AdS 4 × S 7 /Z k . The AdS 4 and S 7 /Z k have radii L and 2L, which are related to N and the 11D Newton constant G (11) via (2πℓ P ) 6 N = 384L 6 · vol(S 7 /Z k ), 16πG (11) = (2πℓ P ) 9 2π . (1.2) Definition. The 3D N = 6 superconformal symmetry of the ABJM model has conformal symmetry SO(2, 3) and R-symmetry SO(6) as bosonic subgroup. Let us denote by ǫ and j 3 the Cartan generators for SO(2) × SO(3) ⊂ SO (2,3), and h 1 , h 2 , h 3 for SO (6). Then one can find nilpotent supercharges Q and S satisfying and both commuting with h 1 , h 2 and ǫ + j 3 . The superconformal index is defined by the trace over the Hilbert space of radial quantization. Here x ≡ e −β , y 1 ≡ e −γ 1 , y 2 ≡ e −γ 2 . Note that it is independent of β ′ since it only receives contributions from the states annihilated by Q and S. The index can be computed as a path integral of the theory on S 1 × S 2 with the S 1 parametrized by Euclidean time τ ∼ τ + β + β ′ . The presence of j 3 and h i 's in the trace translates into twists in the periodicity of the fields. If one prefers to work with periodic fields, one can take account of them by turning on background SO(6) gauge fields and off-diagonal metric components.
Computation. The index can be evaluated with the help of SUSY localization. The path integral I ≡ D(fields)e −(action) is supersymmetric; namely there is a supercharge Q under which the measure D(fields) and the action are both invariant. As such, I is invariant under modification of the action by terms of the form 1 g 2 QΨ, with Ψ fermionic and Q 2 Ψ = 0. By choosing Ψ suitably and taking the weak coupling limit g 2 → 0, one can show the path integral is given exactly by the sum over contributions of saddle points, and that the contribution of each saddle point can be evaluated using Gaussian approximation. For the ABJM superconformal index, the saddle points are labeled by integers n i ,ñ i and periodic variables α i ,α i (i = 1, · · · , N ). They appear in the value of flux and temporal holonomy as follows, 3) The value of the action at this saddle point is This is multiplied by two "determinants" to make up the contribution of a given saddle point. Note, as it turns out, that both determinants are invariant under simultaneous shift of the 2N variables α i ,α i by the same amount. The integration over α i ,α i along this direction thus gives rise to a constraint i n i = iñ i . One of the determinants is the Faddeev-Popov determinant. The flux (n i ,ñ i ) generically breaks the gauge group The saddle point condition requires the holonomy to take values in this subgroup. Gauge-fixing the holonomy to be also diagonal gives rise to a factor 1 Sym ·∆ FP , where The other is the one-loop determinant arising from Gaussian integration over fluctuation of fields. It can be computed by KK reducing the free theory of fluctuations along S 2 . The resulting system can be regarded as a bunch of simple bosonic and fermionic harmonic oscillators with periodic Euclidean time. The determinant is its partition function where we used an abbreviation for the a-th bosonic or fermionic oscillator. The term (gauge) represents the gauge charge: for example it is α i −α j if the oscillator originates from the (i, j)-component of a bi-fundamental field. For later use, we rewrite it into a plethystic exponential where the Casimir energy ǫ 0 and the letter index f are defined by The KK reduction is performed using monopole harmonics (spherical harmonics for charged fields in the flux background). The quantities ǫ 0 and f therefore depend on the flux n i and n i as well, though not indicated explicitly. The contribution to the index from saddle points with flux (n i ,ñ i ) is thus given by where the quantity Sym is defined in (2.5), and the letter index f takes account of both the Faddeev-Popov and one-loop determinants.
f (x, y 1 , y 2 , e iα i , e iα i ) = − i =j The Casimir energy is given by The full superconformal index I(x, y 1 , y 2 , y 3 ) is given by the sum of (2.10) over different flux sectors with an additional weight y . The new fugacity parameter y 3 counts the KK momentum along the M-theory circle, i.e. Hopf fiber circle of S 7 /Z k .
The large N limit. A nice way [9,10] to treat the integral over the N + N variables α i ,α i in the limit is to express it in terms of the eigenvalue density functions ρ(α),ρ(α) and their Fourier modes, Note that ρ 0 =ρ 0 = N . As a simple exercise, let us rewrite the contribution of zero-flux sector I (0) using these variables. We find that the result is a simple Gaussian integral, which gives .
The evaluation of contributions of the sectors with nonzero flux is apparently much harder. The idea employed in [8] is to divide the integration variables α i ,α i into 3 groups. The first contains those α i orα i for which the corresponding flux (n i orñ i ) is positive, and the second contains those corresponding to negative flux. As long as one looks at the sectors carrying O(N 0 ) momentum along the M-theory circle, these two groups have O(N 0 ) variables. All the rest, corresponding to zero flux, are in the third group. In the large N limit one can apply the change of variables described in the previous paragraph to the third group, after which the integration measure becomes schematically as follows. (2.16) A nice observation of [8] is that the integral over ρ n ,ρ n is still Gaussian, and moreover the result takes the following factorized form.
This implies that the full index I takes factorized form, where I + , I − are positive and negative power series in y 3 , respectively. Though we are left with a finite-dimensional integral, the computation becomes increasingly complicated as the flux increases.

Free energy on S 3
Free energy measures the number of low-energy degrees of freedom. A supergravity analysis predicted [11] that the free energy for the system of N M2-branes should scale as N 3/2 at large N . This behavior was reproduced from the exact partition function of the ABJM model on S 3 . Generally, for a system of N M2-branes with near-horizon geometry AdS 4 × Y , the gravitational free energy is given by the classical action evaluated on the corresponding Euclidean background. Though it is naively infinite, after subtracting the power-law divergences by suitable counterterms [12,13,14] one can extract a finite positive value Here L is the radius of AdS 4 and G (4) is the effective 4D Newton constant. As a function of N and the volume of Y (normalized so that its metric satisfies R mn = 6g mn ), F becomes .
The path integral of the Euclidean ABJM model on S 3 was studied in [15]. It was shown that the saddle points are parametrized by 2N variables σ i ,σ i (i = 1, · · · , N ), and the scalar fields in the vectormultiplets take constant values at the saddle points. The partition function is given by the following integral, where we introduced g s ≡ 2πi/k.
Large N limit: traditional approach. A standard way to evaluate this integral is to use the idea of large N expansion [16]. Let us generalize the gauge group to U (N 1 ) × U (N 2 ) for a while and consider the limit N 1 , N 2 , k → ∞ with the 't Hooft couplings t 1 ≡ g s N 1 and t 2 ≡ g s N 2 kept fixed. The free energy then has an expansion of the form The planar contribution g −2 These equations are often interpreted as the condition for the equilibrium of forces acting on each eigenvalue. The forces between two σ's or twoσ i 's are repulsive, whereas σ i andσ j attract each other.
In the large N limit, the eigenvalues {σ i } and {σ i } will form continuous distributions along some intervals C andC. Let ρ(x) andρ(x) be their densities. Moreover, let us define the resolvent by It turns out that the equations (3.6) translate into the following discontinuity relations for ω(z), which implies that f (z) ≡ e ω(z) + e 2z−ω(z) is an entire function. By combining it with the boundary condition at infinity one can determines ω(z) up to an arbitrary constant κ, The square-roots produce two branch cuts C andC + iπ, and the left of Fig. 1 shows their form when t 1 − t 2 > 0 for a suitable choice of κ. A useful fact is that the κ-derivative of the integral ω(z)dz is an elliptic integral, where we denoted e z ≡ u.
To find a relation between the planar free energy and 't Hooft couplings, we express them using contour integrals of ω(z)dz. First of all, one finds Figure 1: (Left) the branch cuts of ω(z) and the contours α,α, β, γ. (Right) a sketch of the numerical solution of (3.6) found in [17].
where α,α are the contours encircling C andC + iπ as shown in Fig.1. Also, notice that one can transport one σ eigenvalue from infinity to a point z * ∈ C by integrating 1 gs (z − ω(z)) along a contour γ shown in the figure. If the integral were not divergent, it would correspond to the change of the free energy under the shift of N 1 by one (or the shift of t 1 by g s ). It turns out that a simultaneous shift of t 1 , t 2 corresponds to a finite contour integral, where the contour β is as shown in the figure.
We now restrict to the ABJM model and set t 1 = t 2 ≡ t. For a large positive κ, the four branch points of the elliptic integral are approximately at with C runnning between the first two andC + iπ between the latter two. It is not difficult to extract, by evaluating the elliptic integrals and integrating with respect to κ, the leading large κ behavior which is in precise agreement with the prediction of supergravity.
The original work [16] and [18] also studied non-planar corrections (higher orders of perturbative series in g s ) and instanton corrections by making use of the connection of the integral (3.4) with the one for Chern-Simons theory on the lens space, which is in turn dual at large N to topological string theory on local P 1 × P 1 . The full perturbative series (3.5) was computed in [19] using holomorphic anomaly equation, and the result turned out to be given simply by Airy function.
Large N limit: another approach. A different method for evaluating the integral (3.4) was invented in [17], and it turned out very efficient for studying the large N limit with k fixed. It is partly based on the numerical result for the extrimization of F (σ i ,σ i ) which look like the right of Fig.1. It implied that the eigenvalue distribution is described by two functions ρ(x), y(x) in such a way that the following replacement works for arbitrary function ϕ(x) in the large N limit. By rewriting F (σ i ,σ i ) using this rule one finds it becomes a local functional of ρ(x) and y(x), where f (x) is a function of period 2π and f (x) = π 2 − x 2 for |x| ≤ π. The balance of the two terms in the right hand side requires α = 1 2 , which immediately implies the N 3/2 scaling of the free energy. The initial assumption that the distributions of Re(σ i ) and Re(σ i ) are described by the same function ρ(x) is also justified, because otherwise there would be terms of higher order in N remaining on the right hand side. It is now easy to extremize F with respect to ρ(x), y(x) under the condition dxρ(x) = 1. The result reads The value of F [ρ, y] at this extremum is F = √ 2π 3 k 1 2 N 3 2 , thus the supergravity result was correctly reproduced again.
Though this method is efficient, it is not very obvious how to go beyond the strict large N limit. Another powerful method, called "fermi gas" approach, to study the model systematically at large N with k kept fixed was introduced in [20]. It is based on a reformulation of the integral (3.4) as the partition function of a 1D gas of N non-interacting fermions with a non-trivial Hamiltonian. Combination of this approach with other methods from TBA and topological strings led to a very detailed understanding on the structure of non-perturbative corrections [21,22,23,24,25,26].
Generalization. The check of AdS/CFT via comparison of free energy on S 3 can be generalized to the cases with less SUSY. Explicit formula is known for the free energy of general N ≥ 2 Chern-Simons matter theories [27,28]. For N M2-branes at the tip of some Calabi-Yau 4-fold cone X, the worldvolume dynamics is described by U (N ) p CS-matter theories with matters satisfying certain condition. By adopting an ansatz similar to (3.14), one can show that the free energy for such theories scale as N 3/2 and obtain a local functional of ρ(x) and y 1 (x), · · · , y p (x) [29].
A new issue arises from the fact that, for general N = 2 theories of vector and chiral multiplets, the Lagrangian on S 3 at the starting point has arbitrariness in the assignment of R-charges to chiral multiplets. By extremizing the functional of ρ(x), y a (x) one therefore ends up with a function of the matter R-charges. As was proposed in [27,29] and proved in [30], the correct assignment corresponding to the R-charge of N = 2 superconformal symmetry is the one which maximizes the free energy. As an illustrative exercise, let us break the SUSY of the ABJM model to N = 2 by assigning arbitrary R-charges ∆ 1 , ∆ 2 , ∆ 3 , ∆ 4 to the chiral fields A 1 , A 2 , B 3 , B 4 , with a constraint 4 a=1 ∆ a = 2 . (3.17) We also turn on the R-charge ∆ m for the momopole operator T carrying a unit flux. By deriving the free energy functional and extremizing it with respect to ρ(x), y(x), one obtains It has a flat direction under which (∆ 1 , · · · , ∆ 4 , ∆ m ) shifts by (δ, δ, −δ, −δ, kδ), which is a reflection of the fact that the operators A 1 , A 2 , B 3 , B 4 , T carry U (1) gauge charges corresponding to Tr(A µ −Ã µ ). By extremizing with respect to the other non-flat directions with the constraint (3.17) one recovers F = √ 2π 3 k 1 2 N 3 2 again. On the gravity side, we need to compute the volume of the 7D Sasaki-Einstein space Y which is the base of the cone X. If X is toric, there is a useful technique to compute the volume of Y as a solution to a minimization problem [31]. By definition X has U (1) 4 symmetry, and one can regard X as a T 4 fibration over a convex polyhedral cone C inside R 4 . Its Kähler form is given by with x i the coordinages on the base and ϕ i ∼ ϕ i + 2π on the fiber. We denote by v a the inward-pointing normal vector to the a-th facet of C. CY condition implies one may assume v a0 = 1 for all a. Note that its components v ai are all integers since v a also specifies the 1-cycle of T 4 which shrinks above the a-th facet. There is a distinguished isometry i b i ∂ ϕ i , called Reeb vector, which is paired up with the radial vector field under a chosen complex structure of X. As was shown in [31], b contains some information on the (Kähler but not necessarily Ricci-flat) metric of X which is actually enough to determine the volume of Y and all of its 5-cycles. Consider a hyperplane which intersects C to make a finite polytope ∆ b . Then Y is Sassaki-Einstein when b is chosen to minimize vol(∆ b ) under a condition b 0 = 4. As an example, for X = C 4 /Z k one can take The parametrization of b by ∆'s can be obtained by matching the R-charges of M5-branes wrapping various 5-cycles with those of gauge invariant operators in ABJM. One then finds where we used (3.17). Note that the above result reproduces (3.18) via (3.2) before extremization. Generalization of this correspondence was studied in [32,33], though it is not simple because the numbers of parameters on the gauge and gravity sides do not agree in general.

Entropy of charged black holes
According to AdS/CFT, any classical solution with AdS asymptotics should be described as an ensemble of states in the corresponding CFT. Construction of black holes in AdS spacetime was known to be considerably harder than those in flat spacetime, but an analytic solution for asymptotically AdS 4 static BPS black holes with magnetic (and electric) charges was found in [34]. A natural question is whether the dual CFTs correctly accounts for their entropy as the degeneracy of states.
Black hole solutions and their entropy. The black hole solutions were found in 4D N = 2 supergravity with n abelian vectormultiplets and gauging [35]. The bosonic fields in this theory are the metric g µν (x), (n + 1) gauge fields A Λ µ (x) (Λ = 0, · · · , n) and n complex scalars z i (x) (i = 1, · · · , n) which parametrize a special Kähler manifold M. There is a rank 2n + 2 holomorphic vector bundle over M, and the Kähler potential of M is expressed in terms of its section Ω ≡ (X Λ (z), F Λ (z)) and its conjugateΩ ≡ (X Λ (z),F Λ (z)) as where Ω,Ω ≡ F ΛX Λ − X ΛF Λ is the duality-invariant bilinear product. The covariant derivatives of Ω,Ω with respect to z i ,zī are and the Kähler metric on M is The condition Ω, ∇ i Ω = 0 implies the existence of the prepotential F(X), which is a homogeneous function of degree 2 in X Λ satisfying F Λ (z) = ∂F ∂X Λ (X(z)). It also implies there is a symmetric matrix N ΛΣ (z,z) such that (4.4) The first few terms in the supergravity action [36] reads where F ±Λ is the imaginary (anti-)self-dual part of the field strengths F Λ = dA Λ satisfying * F ±Λ = ±iF ±Λ . We define the magnetic and electric charges q ≡ (q Λ , q Λ ) of spherically symmetric solutions by The duality group Sp(2n + 2, R) rotates the vectors Ω and q in the same way. Static extremal black holes with flat asymptotics was studied in [36]. It was found that, for spherically symmetric solutions of the form ds 2 = e 2U (r) dt 2 − e −2U (r) dr 2 + r 2 (dθ 2 + sin 2 θdϕ 2 ) , z i = z i (r), (4.7) the BPS condition can be cast into a flow equation, Here g i (z,z) is the inverse metric on M and Z(z,z) ≡ e K/2 Ω, q is the central charge of the black hole with charge q. These imply that the value of the scalars z i at the horizon r = 0 should extremize |Z(z,z)|, and also that the entropy of the black hole is given by at its extremum, which is therefore a function of the charge q only.
To discuss black holes with AdS asymptotics, one needs to move to gauged supergravity. In 4D N = 2 supergravity, gauging amounts to assigning U (1) n+1 charges to fields according to their SU (2) R charges. We denote the couplings as g ≡ (g Λ , g Λ ), though the discussions of concrete theories are often restricted to those with g Λ = 0.
In order to explain the mechanism of gauging, we think of adding n H hypermultiplets whose scalars y m (m = 1, · · · , 4n H ) parametrize a quaternionic space M H . There is a principal SU (2) bundle over M H with connection V a = V a m (y)dy m (a = 1, 2, 3) such that the triplet of Kähler forms of M H is proportional to its curvature 2-form. All the fields with SU (2) R charges are then coupled to V a . For example, the SUSY transformation rule for gravitino ψ A µ (x) reads (4.10) If M H has a U (1) n+1 isometry and we want to gauge it, we covariantize the derivatives where k m Λ (y) is the Killing vector field on M H for the Λ-th U (1). At the same time, an analogue of the U (1) n+1 hyperKähler moment map P a Λ (y) takes part in the modification This procedure works even for the case with empty M H and constant P a Λ (y) = g Λ δ a3 , and thus couples the U (1) n+1 gauge fields to fields with SU (2) R charges. The gravitino SUSY transformation rule now becomes (4.13) Note that the coupling g Λ determines the quantization rule of the charges, (4.14) A peculiar feature of the black hole solutions of [34] with magnetic charge is that, with a spherical symmetric ansatz, the gauge field A Λ and spin connection ω ab both take the form ∼ cos θdϕ so that the second and the third terms in (4.13) cancel each other. This occurs when the twisting condition is satisfied. The cancellation among contributions from various connections is reminiscent of the construction of topologically twisted theories. The observation of this fact led to the identification of the dual description [37,38]. Black hole solutions with g Λ q Λ = 0 are also known but they require some rotation to be free of naked singularity [39,40].
For BPS black hole solutions in gauged supergravity, there is an attractor flow equation similar to (4.8) which allows one to obtain the black hole entropy without working out the metric explicitly [41]. It implies that the value of z i at the horizon extremizes 16) and the value of R at the extremum gives the horizon radius. Thanks to the homogeneity of the RHS, one can think of extremizing the numerator as a function of X Λ keeping the denominator fixed.
As an example, let us consider N = 8 maximal gauged supergravity truncated to N = 2 which is relevant to the ABJM model at k = 1. Its prepotential and couplings are given by where L is the AdS 4 radius. In terms of integer charges the twisting condition (4.15) becomes Λ n Λ = 2, and the entropy is given by extremized as a function of X Λ under a constraint Λ X Λ = 2π. Note that the extremum value has to be real and positive in order for the solution to have a smooth horizon. This puts an independent condition on (n Λ , e Λ ).
Microscopic theory. The microscopic theory for these black holes is a 3D N = 2 supersymmetric theory on S 2 × R with a topological twist by a unit background U (1) R flux through S 2 . The matter R-charge has to be integer due to Dirac quantization, but there are infinite choices for its assignments if the theory has flavor symmetry. The path integral of the theory with periodic time (i.e. on S 1 × S 2 ) is called the twisted index [42]. For the theories with conserved flavor charges J a , one can turn on the constant σ and A τ components of the corresponding vector multiplet in the background. In Hamiltonian description, the twisted index computes the trace over the Hilbert space H, It is a function of the complexified flat connections ∆ a ≡ β(A a τ + iσ a ) only, because the supercharge Q satisfies Q 2 = H − a σ a J a .
In the case of the ABJM model, the twist is labeled by the R-charges n 1 , · · · , n 4 ∈ Z of the chiral multiplets A 1 , A 2 , B 3 , B 4 obeying a constraint a n a = 2. The model has U (1) 3 flavor symmetry generated by J a − J 4 (a = 1, 2, 3), where J a phase-rotates the a-th chiral multiplet. It is therefore convenient to regard I as a function of flat connections ∆ 1 , · · · , ∆ 4 obeying a constraint ∆ 1 + ∆ 2 + ∆ 3 + ∆ 4 = 0 mod 2πZ. (4.21) Thanks to SUSY localization, the path integral can be reduced to an integral over saddle points labeled by integers m i ,m i and periodic variables u i ,ũ i (i = 1, · · · , N ). They appear in the value of vectormultiplet field at the saddle point as follows, The system also has fermionic zeromodes ξ i ,ξ i which are paired with u * i ,ũ * i under the supersymmetry. Consequently, after localization one is left with an integral over u's along some contour, not over the cylinder. With x i ≡ e iu i ,x i ≡ e iũ i and y a ≡ e i∆a one can express the index as follows. (4.23) The contour integral is performed following the Jeffrey-Kirwan residue prescription [43], which was first used in the study of SUSY localized path integrals in 2D [44,45] and 1D [46]. It goes roughly as follows. For each pole (intersections of singular hyperplanes) p of the integrand, there is a matter field responsible for each of the hyperplanes. Label p by the charges Q p = { q 1 , q 2 , · · · } of those matters under the Cartan of the gauge group. (In the present problem there are additional singularities at x i ,x i = 0 or ∞. They are labeled according to the Chern-Simons couplings [42].) The JK-residue prescription begins by choosing a reference charge η arbitrarily. Then one decides whether to pick up the residue of a pole p according to whether the cone spanned by the charge vectors in Q p includes η or not. The end result is independent of the initial choice of η.
For the index of the ABJM model (4.23), there is a suitable choice of η such that one only has to evaluate the residue of the pole at x i =x i = 0. Then the terms in I with m i very large (orm i negatively very large) can be discarded because there would not be a pole at x i = 0 (orx i = 0). As a result, one only has to sum over m i ≤ M andm i ≥ −M for some M , which can be performed easily before integrating over x i ,x i . One is then left with an integral over x i ,x i , and the integrand has poles at the solution of a Bethe ansatz like equations (4.24) These are actually the equations for the extremum of the potential, Here Li n (x) = k≥1 x k k n is the polylogarithm function and ε a = (+1, +1, −1, −1). c i ,c i are integers which arise from the multi-valuedness of log function.
The extremization of W was studied in [37]. It was found that by using an ansatz similar to (3.14), (4.26) one can rewrite W into a local functional of ρ(x) andỹ(x) − y(x), and moreover the local functional takes the same form as the one for the free energy (of the N = 2 deformed theory with general R-charge assignments) on S 3 . It turned out that the variational problem has a consistent solution only for a ∆ a = 2π, and the value of W and I at the solution are given (for k = 1) by ln I(n a ; ∆ a ) = i 4 a=1 n a ∂ W ∂∆ a , W = iN −ie a ∆ a + in a ∂ W ∂∆ a , (4.28) where the RHS should be extremized as a function of ∆ a with a constraint a ∆ a = 2π. In view of (3.1), we see that the black hole entropy (4.19) has been reproduced precisely by the microscopic theory.
Ever since I became a student of Prof. Eguchi, I used to feel tense every time we had discussions of physics. It always led me to commit to physics more seriously and helped me grow. The same feeling comes back still now whenever I remember him.
I am truly grateful for being a student of Prof. Tohru Eguchi.