University of Amsterdam
MSc in Physics and Astronomy
Track: Theoretical Physics Master Thesis
Properties of holographic CFTs and the
assessment of SPOCFTs as holographic
Eloy de Jong, MASt BSc
Supervised by: Dr. Alejandra Castro Dr. Alexandre M.F. Belin
Second assessor: Prof. Dr. Erik P. Verlinde
Abstract
In the context of the ADS3/CF T2-conjecture one can ask whether, and if yes how, CFTs dual to
weakly coupled Einstein gravity (i.e. holographic CFTs) can be characterized. Holographic CFTs are expected to have large central charge and therefore, large N symmetric product orbifold CFT’s (SPOCFTs) are a prime candidate to be holographic. Various attempts to identify characteristic properties of holographic CFTs have been made and I present a self-contained overview of the derivation of these results and their relevance to the assessment of SPOCFTs as holographic CFTs. In particular, I discuss thermodynamic properties of SPOCFTs that are independent of the details of the seed theory, studied in [1], which include a universal growth of states for a particular range of conformal dimensions and universal behaviour of the free energy. Moreover, I study results from [2] that show that a CFT has an extended range of validity for the Cardy formula when the light states from the CFT’s spectrum satisfy a sparseness constraint. Additionally, one can study supersymmetric CFTs (SCFTs) dual to theories of supergravity. In particular, when N = (2,2) supersymmetry is present, one can consider the elliptic genus [3], a holomorphic partition function-like object that contains information about the energy and U(1) charge of states in the Hilbert space. I show how the coefficients of the elliptic genus of a holographic SCFT can be constrained so that the free energy behaves gravity-like in terms of β and µ, as in [4].
Contents
1 Introduction 1
2 Gravity thermodynamics in three dimensions 3
2.1 Solutions to the Einstein equations in three dimensions . . . 3
2.2 The gravitational partition function . . . 4
3 Conformal field theory in two dimensions 7 3.1 Free fermion . . . 7
3.2 Partition function CFT . . . 8
3.2.1 Free fermion partition function . . . 11
3.3 Cardy formula . . . 13
4 Symmetric product orbifolds CFTs 15 4.1 Permutation groups . . . 15
4.2 Symmetric product orbifold CFT partition function . . . 15
4.3 Modular invariance . . . 21
5 Supersymmetry 23 5.1 N = (1,1) supersymmetry . . . 24
5.2 N = (2,2) supersymmetry . . . 25
5.2.1 Spectral flow . . . 27
5.2.2 The elliptic genus . . . 28
6 Analysis of the spectrum of a CFT 31 6.1 Universality of symmetric product orbifold CFTs . . . 31
6.2 Constraint on low-energy spectrum of a CFT . . . 38
6.3 Constraining the elliptic genus . . . 43
7 Discussion 48
Acknowledgements 50
A Conformal field theory in two dimensions I A.1 Conformal transformations in two dimensions . . . I A.2 Fields under conformal transformations . . . II A.3 Mode expansions of fields . . . IV A.4 Free boson . . . V A.5 Operator product expansions in CFTs . . . VII A.6 Conformal transformation of the energy-momentum tensor . . . VIII B Decomposition of modular transformations X
1
Introduction
In 2000, Raphael Bousso and Joseph Polchinski proposed that there may be an astonishing 10500 different string theories, which describe as many different universes with their own constants of nature [5]. String theory is often hailed as a prime candidate to provide a unified theory that describes all fundamental forces of nature, and if the Bousso-Polchinksi number is of the correct order of magnitude, it will be an immense and daunting task to find the one theory that reproduces the universe as we know it.
A similar problem arises in the context of the conjecture that every two-dimensional conformal field theory (CFT), without gravity, is dual to a theory of quantum gravity in a three-dimensional asymptotically Anti-De-Sitter (ADS) spacetime, id est (i.e.) the ADS3/CF T2-conjecture. Duality
means that two such theories would be merely two different descriptions of the same physics, so that results in the one theory have a particular interpretation in the other. The gravity that we observe on and around earth is governed by the Einstein-Hilbert action and is weakly coupled, i.e. the value of the coupling constant is small, and the Einstein-Hilbert action depends only on the Ricci tensor and is thusly relatively simple. However, the space of gravity theories is much larger than just Einstein gravity, as one could couple various matter field to gravity or opt for exempli gratia e.g. a Brans-Dicke description [6]. Therefore, there is a priori no reason to expect that the action of a gravity theory dual to a generic CFT does not depend on other contractions of the Riemann tensor, or is even weakly coupled. One can thusly ask whether, and if yes how, CFTs dual to weakly coupled Einstein gravity (known as holographic CFTs) can be characterized.
One way to recognize a duality between theories is by studying the theories’ symmetries, for if two theories are truly dual, one would expect the symmetries that they demonstrate to be the same. Holographic CFTs, like all two-dimensional CFTs, demonstrate a Virasoro algebra with some central charge, which is explored in appendix A. When one studies theories of gravity in an asymptotically ADS3 spacetime, it turns out that the charges generating the asymptotic ADS3
symmetries also obey a Virasoro algebra, with the gravity central charge inversely proportional to the gravity coupling constant. We consider small gravity coupling constants corresponding to large gravity central charge, so we expect holographic CFTs to have large central charge, as well.
One way to obtain CFTs with arbitrarily large central charge is by considering an N -fold tensor product of a seed CFT with central charge c, yielding a new CFT with central charge N c. By letting N tend to infinity one can let the tensor product CFT’s central charge tend to infinity. However, other restrictions, which I go into briefly in chapter 6, on holographic CFTs rule out this relatively simple construction.
Another idea gaining traction is that symmetric product orbifolds have holographic properties, since taking an orbifold involves taking an N -fold tensor product, which multiplies the central charge by N . Symmetric product orbifold CFTs (SPOCFTs) are discussed in chapter 4 and holographic properties of SPOCFTs are studied in e.g. [7, 1].
Apart from focussing on SPOCFTs in particular as holographic CFTs, one can choose to con-sider a completely general CFT and attempt to impose constraints on its density of states, by demanding that characteristic weak gravity properties are reproduced. The authors of [2] use the
high energy density of states of weakly coupled Einstein gravity to derive such a constraint, against which candidate holographic CFTs can be tested.
Furthermore, one can study supersymmetric CFTs and duality to theories of quantum super-gravity. In some supersymmetric theories, one can define generalized partition function-like objects such as the elliptic genus, whose spectrum can again be constrained for holographic supersymmetric CFTs by looking at properties from the gravity theory. This is done in [4], in which the attempted reproduction of the gravity free energy leads to bounds on the numbers of states counted in the elliptic genus.
To derive and be able to interpret these constraints, one firstly needs a grasp of two-dimensional conformal field theory to be able to do analysis on the relevant CFTs. Secondly, one should understand how to construct permutation orbifolds and how a SPOCFT’s properties are different from the seed CFT’s. Lastly, some understanding of supersymmetry is required to be able to appreciate the meaning and properties of the elliptic genus and the states it counts. Even though this information is widely and freely available, see e.g. [8, 9, 10, 11, 12], it is usually not summarized as one self-contained piece of writing, which is what this work will set out to do. ADS/CFT holography is promising as a means to lift the veil on the secrets of quantum gravity, for some gravity computations are easier to complete in the dual CFT picture. Hence, identifying holographic CFTs would be a major step forward and it is important to collect preliminary theory and a discussion of ongoing research in one place.
This thesis aims to study various attempts to identify characteristic properties of holographic CFTs, in the context of the ADS3/CF T2-conjecture. Moreover, it will discuss how these methods
are relevant to the assessment of symmetric product orbifold CFTs as candidates for dual theories to weakly coupled Einstein gravity.
Chapter 2 briefly covers 3D gravity thermodynamics and chapters 3 − 5 are concerned with the theoretical preliminaries concerning 2D (supersymmetric) CFTs and orbifolds. Chapter 6 discusses relevant computations done in [1, 2, 4] in detail, which are then discussed in chapter 7. Additional clarifying calculations are relegated to the appendices.
2
Gravity thermodynamics in three dimensions
Since this thesis concerns ADS3/CF T2holography, we will begin by briefly discussing the properties
of gravity in three-dimensional ADS spaces that are relevant to the argument presented here. This will allow us to refer back to these properties whenever we need or wish to in the next chapters. The most important result is the free energy graph at the end of this chapter, which is important in the arguments given in chapter 6.
2.1 Solutions to the Einstein equations in three dimensions
Let us consider the Einstein equations in three dimensions with negative cosmological constant:
Gµν+ Λgµν = 8πTµν, Λ < 0. (2.1)
The most general solution to these equations is empty three-dimensional Anti-De-Sitter space (ADS3), which may be obtained from four-dimensional flat space with two timelike coordinates,
parametrised by global coordinates X0, X1, X2, X3, in which the metric is:
ds2 = −(dX0)2− (dX1)2+ (dX2)2+ (dX3)2. (2.2)
ADS3 is then obtained by restricting to the hyperboloid specified by the ADS-radius L:
XAXA= −L2, (2.3)
where the ADS-radius is related to the cosmological constant as Λ = −1L2 [13, 14]. We may
parametrise the subspace that satisfies (2.3) as follows:
X0 = L cosh ρ sin τ X1 = L cosh ρ cos τ X2 = L sinh ρ cos φ X3 = L sinh ρ sin φ.
(2.4)
One then obtains the pull-back of (2.2) to the hyperboloid in the standard way:
gµν = ∂XA ∂xµ ∂XB ∂xν ηAB, ds 2 = L2 − cosh ρ2dτ2+ dρ2+ sinh ρ2dφ2. (2.5)
In the parametrisation (2.4), we have implicitly chosen for ranges 0 ≤ τ, φ < 2π and 0 ≤ ρ < ∞. We can then construct closed timelike curves that have constant ρ, φ and varying τ . To avoid this, we may choose to consider (2.5) with −∞ < τ < ∞.
Finally, it is convenient to rewrite (2.5) in terms of coordinates r = L sinh ρ and t = Lτ . The metric then becomes:
ds2 = − 1 + r 2 L2dt 2+ 1 + r2 L2 −1 dr2+ r2dφ2. (2.6)
Let us study the hypersurface in the limit r −→ ∞. On this hypersurface, the metric is: ds2r−→∞= − r2 L2dt 2+ r2dφ2= r2 −dt2 L2 + dφ 2. (2.7)
We can compare this to a cylinder, of which the periodic coordinate is spatial and the coordinate along the central axis is temporal. Such a cylinder has the following metric:
ds2cylinder = −dt2+ r2dφ2. (2.8) By comparison then, we find that the boundary of ADS3 at r → ∞ is a cylinder with one temporal
and one spatial, periodic coordinate. This is convenient in light of the ADS/CFT conjecture, since this cylinder is a manifold on which we may define a CFT.
Another solution to the three-dimensional Einstein equations is given by the BTZ black hole [15], which is defined by the following metric in terms of its mass M and angular momentum J :
ds2 = −N2dt2+ N−2dr2+ r2(Nφdt + dφ)2, N2(r) = −M + r 2 L2 + J2 4r2 Nφ(r) = −J 2r2. (2.9)
The range of the coordinates is −∞ < t < ∞, 0 ≤ φ < 2π, 0 < r < ∞ excluding r = r± =
LM12 1± r 1− M LJ 2 2 12
, where an inner and outer horizon are located at r±. The geodesic structure
of this spacetime, which involves a region behind a horizon from which particles nor radiation can escape, as we expect from a black hole, is discussed in [16].
2.2 The gravitational partition function
The spacetimes discussed in the previous section are classical gravity solutions, i.e. they satisfy the equations of motion that can be derived from the Einstein-Hilbert action:
SEH =
1 16πGN
Z
d3x √g R − 2Λ, (2.10) i.e. the Einstein equations. When one studies quantum gravity, the partition function of the theory is a path integral over all metric field configurations, modulo diffeomorphisms, weighted by an action: Z(β) = Z Dg e−Sgravity[g], S gravity = SEH + SGHY + Sct SGHY = 1 8πGN Z ∂M d2x √hK (2.11)
we impose that tE ∼ tE + β, which is equivalent to fixing the temperature of the system to be
T = β1. The addition of the Gibbs-Hawking-York term SGHY is necessary to yield an action that
only depends on the metric and its first derivatives and depends on the extrinsic curvature K on the boundary and the induced metric [17, 18, 19]. It makes that one only needs to fix the induced metric on the boundary to have a well-defined variational principle [17]. Additionally, we must include the counterterm Sct, subtracting ”the extrinsic curvature of the boundary as measured in
flat space” [18], to prevent divergences.
Our inability to compute these types of path integrals lies at the core of many questions that remain about quantum theory today. To make this problem easier to handle, we may choose to assume that the gravity coupling constant GN is very small, so that the suppression of non-classical
solutions is large enough that we can do a saddle approximation of (2.11), summing over classical solutions gi:
Z(β) ≈X
gi
e−SEH[gi]= e−SEH[gADS3]+ e−SEH[gBT Z]. (2.12)
The classical solutions that we consider are ones that satisfy the Brown-Henneaux boundary condi-tions, specified in [20] and further discussed in [21] and [22]. These include the solutions described above, as well as an additional set of black holes corresponding to all elements of the group SL(2, Z) [23, 24, 13]. However, we will discard these additional solutions in this analysis, since only empty ADS (i.e. the metric described above with compactified temporal coordinate) and the BTZ black hole will be dominant in the saddle approximation, see e.g. chapter 6 of [25].
By plugging these two solutions into Sgravity, one finds that the contributions to the partition
function of empty ADS and the BTZ black hole are:
Sgravity[gBT Z] = −3L 24GN 4π2 β , Sgravity[gADS3] = −3L 24GN β, (2.13)
hence we approximate (2.11) as:
Z(β) ∼ Z0(β) = e 3L 24GN 4π2 β + e 3L 24GNβ = e12c 4π2 β + e12cβ. (2.14)
It is immediately clear that the first term dominates when β is small, while the second term dominates when β is large. At temperature T = 2π1 , both terms are equal. Since we are considering weakly coupled gravity, the exponents in the equation above are large and we expect one term or the other to dominate as soon as we move away from T = 2π1 . Furthermore, in the second line one defines c = 2G3L
N, which one finds after calculating the algebra of the charges of the asymptotic
currents that correspond to the asymptotic ADS symmetries of the spacetime. This algebra turns out to be a Virasoro algebra with central charge c: Virasoro algebras are discussed further in appendix A.
F (β) = −1 βlog Z 0 (β) ∼ −c 12 4π2 β2 β < 2π −c 12 β > 2π. (2.15)
Hence, the free energy is constant at low temperatures, until the tipping point where it diverges quadratically, which is schematically shown in figure 1. This phase transition is known as the Hawking-Page transition [26].
Furthermore, it is instructive to compute the entropy in different regimes:
S(β) = 1 − β ∂ ∂β log Z 0 (β) ∼ c 6 4π2 β β < 2π 0 β > 2π. (2.16)
These classical expressions will inevitably suffer from quantum corrections, for instance from the aforementioned SL(2, Z) black hole states. However, it is clear is that we have very low entropy at low temperature, while at high temperature the entropy diverges.
Figure 1: a schematic representation of the classical gravity free energy behaviour, in the approxi-mation discussed above.
3
Conformal field theory in two dimensions
Material concerning two-dimensional CFTs that is relevant for this chapter can be found in appendix A. This chapter serves to introduce the free fermion, which is relevant for the supersymmetry discussion in chapter 5, it sets up notation for the orbifold discussion in chapter 4, and it introduces the CFT partition function and the Cardy formula.
3.1 Free fermion
Keeping in mind that we will work with supersymmetric theories later on, which will involve fermionic fields, I want to introduce the free fermion right away. Consider the action:
SF =
i 8π
Z
dwd ¯w ψ(w, ¯w) ¯∂ψ(w, ¯w). (3.1) Here, ψ is a real Grassmannian field that thus satisfies ψ(w, ¯w)ψ(w0, ¯w0) = −ψ(w0, ¯w0)ψ(w, ¯w) and ψ∗= ψ. The i serves to yield the action a real number.
The equation of motion is ¯∂ψ(w, ¯w) and we are thus dealing with a holomorphic field. Further-more, we have the propagator:
hψ(w)ψ(w0)i = 1
(w − w0), (3.2)
and we have the energy-momentum tensor:
TF(w) =
1
2 : ψ(z)∂ψ(z) : . (3.3) It is straightforward to find that the field’s conformal dimensions are (hψ, ¯hψ) = (12, 0). We refer
to (A.15) to write: ψ(w) =X r ψrw−r− 1 2. (3.4)
There are multiple options for the range of the sum here, depending on whether we take the fermions periodic or anti-periodic when they are rotated around the origin of the plane. When they are anti-periodic as w → e2πiw and we say that they are in the Ramond sector. When they are periodic they are said to be in the Neveu-Schwarz sector:
ψ(w) =X r∈Z ψrw−r− 1 2 Ramond sector ψ(w) = X r∈Z+12 ψrw−r− 1 2 Neveu-Schwarz sector. (3.5)
The periodicity of these sectors depends on the coordinates, e.g. one can map the complex plane to an infinite cylinder using the conformal map, see also figure 2:
If we consider the field on the cylinder and exponentiate, we must take into account (A.8) to determine how the field transforms:
ψ(w) →∂w ∂z
12
ψ(w) = w12ψ(w), (3.7)
which tells us that Ramond sector fermions are anti-periodic on the plane, but periodic around the spatial circle on the cylinder, and vice versa for Neveu-Schwarz sector fermions.
3.2 Partition function CFT
This section serves to introduce the notion of partition functions in the context of two-dimensional CFTs, from which we will later determine the spectrum of a CFT, and which will be important in defining and calculating the elliptic genus.
Let us start by considering a general quantum field theory, not necessarily conformally invariant, for which we can define following partition function:
Z = T rHe−βH, (3.8)
where the trace is taken over all states in the QFT’s Hilbert space, H is the QFT’s Hamiltonian operator and β is the inverse temperature, which is equal to the length of the temporal cycle after it is compactified. A theory on a two-dimensional plane can be mapped to an infinitely long cylinder using (3.6), the temporal direction of which can then be compactified. This turns the cylinder into a two-dimensional torus and it is this manifold on which we will define CFTs so that we can compute their partition functions.
For any QFT, we must then specify which torus it lives on. One way to define a torus is to consider R2, or alternatively C, and pick two linearly independent vectors. We then state that we will identify any two points that differ by any integer combination of these two vectors with one another, yielding a torus such as in figure 3.
Let us consider pairs of vectors ~v1, ~v2 and ~w1, ~w2. These describe two different tori in general,
but if the latter pair can be written as integer linear combinations of the former and vice versa,
Figure 2: illustration of conformal mapping from the cylinder to the plane, depicting constant time curves mapping to circles around the origin and a constant position curve to a ray.
then they do in fact describe the same torus. If we have such pairs, then a modular transformation is the transformation from one to the other. Let:
~
v = (~v1, ~v2), w = ( ~~ w1, ~w2). (3.9)
If these describe the same torus, then:
~v = A ~w = d c b a
! ~
w, a, b, c, d ∈ Z, (3.10)
since ~v1, ~v2 must be some linear combination of ~w1, ~w2. Multiplying by A−1 on both sides yields:
~ w = A−1~v = 1 ad − bc a −b −d d ! ~v, a, b, c, d ∈ Z. (3.11)
Since ~w1, ~w2 must also be a linear combination of ~v1, ~v2, the entries of A−1 should be integers, too.
Hence, we must impose ad − bc = 1, so that we may take the modular transformations to be the two-by-two integer valued matrices with unit determinant, i.e. the group SL(2, Z).
Let us then consider what this means in terms of two-dimensional CFTs. Conformal invariance of a theory gives us freedom to rotate and scale the periodicity vectors. We can then arrange for the situation that one of the vectors is the unit vector along the horizontal axis, so that the sum of the periodicity vectors is as in figure 4.
It is then sufficient to know only the ratio of the two vectors, which is universally denoted by the letter τ = ~v2
~v1 and known as the modular parameter of the torus. After rotation and rescaling,
we simply have τ = ~v2.
A modular transformation of the form of equation (3.10) changes the modular parameter as follows: τ = w~2 ~ w1 → τ0= b ~w1+ a ~w2 d ~w1+ c ~w2 = aτ + b cτ + d. (3.12) In the last line, we have divided all terms by ~w1. Hence, tori parametrised by τ and τ0 are really
the same torus. Moreover, it is easy to see from (3.12) that the transformations corresponding to the following matrices are the same:
d c b a ! , −1 0 0 −1 ! · d c b a ! , (3.13)
hence we must quotient out this Z2 symmetry to obtain the final modular group: SL(2, Z)/Z2.
This has far-reaching consequences for CFT partition functions: these will depend on the shape of the torus and thus on the modular parameter, but all modular transformations of the form (3.12) must leave the partition function invariant.
We will thus require from any CFT partition function defined on a torus that it satisfies the following property, known as modular invariance:
Z(τ ) = Z(τ0), τ0 = aτ + b cτ + d, d c b a ! ∈ SL(2, Z)/Z2 (3.14)
We will refer to SL(2, Z)/Z2 as the modular group. Note that:
d c b a ! ∈ SL(2, Z)/Z2 if and only if a b c d ! ∈ SL(2, Z)/Z2. (3.15)
It is more conventional in the literature to use the matrix on the right, for reasons that mostly concern aesthetics. The modular group is generated by the following elements:
U = 1 0 1 1 ! : τ 7→ τ τ + 1, T = 1 1 0 1 ! : τ 7→ τ + 1, (3.16) while an alternative basis of generators that is widely used is:
S = U T−1U = 0 −1 1 0 ! : τ 7→ −1 τ , T = 1 1 0 1 ! : τ 7→ τ + 1. (3.17) In terms of the second set of generators, one finds the first by checking that U = T ST . The issue of decomposing a general modular transformation of the form τ → aτ +bcτ +d into a sequence of T,S operations is discussed in appendix B.
Let us then find the explicit expression for the partition function. A torus parametrised by τ is taken to have cycles of length 2πτ1 and 2πτ2. We must find the equivalent of the operator e−βH,
which can be interpreted as taking a state and translating it in the temporal direction along a distance β, i.e. around the temporal cycle. On the torus, the equivalent operation is translating a state along the modular parameter. From the picture above, it is clear that a translation along the vector τ involves a spatial translation along 2πτ2, then a temporal one along 2πτ1. With the theory
in section A.1, one can show that the operator L0+ ¯L0 generates dilatations, while the operator
i(L0− ¯L0) generates rotations. These are interpreted as time translations and spatial translations
respectively and therefore the following expression is used for the partition function:
Z(τ, ¯τ ) = T rHe2πiτ1He2πiτ2P = T rHe2πiτ1(L0,cyl+ ¯L0,cyl)e2πiτ2i(L0,cyl− ¯L0,cyl)
= T rHqL0,cylq¯ ¯ L0,cyl = T r HqL0− c 24q¯L¯0− ¯ c 24 . (3.18)
In the last line, we have defined q = e2πiτ and taken into account the conformal transformation of the energy momentum tensor under the mapping from the plane to the cylinder, see appendix A.6. As was mentioned before, any CFT partition function on a torus is necessarily invariant under the transformations: Z(τ, ¯τ ) = Z(τ0, ¯τ0), τ0 = aτ + b cτ + d, ¯τ 0 = e¯τ + f g ¯τ + h, a b c d ! , e f g h ! ∈ SL(2, Z)/Z2. (3.19)
3.2.1 Free fermion partition function
To familiarize the reader with the language from the last section, and also in preparation for the orbifold discussion later on, let us consider the partition function of the theory that corresponds to the action (3.1). Let us consider the Neveu-Schwarz sector first, so that the fermionic modes are indexed by half-integers, see (3.5). This can be combined with (3.3) to obtain:
TF(w) = − 1 2 X r,s∈Z+12 (s + 1 2)w −r−s−2 : ψrψs: = 1 2 X r,s>−32 (s + 1 2)w −r−s−2ψ rψs− X r,s≤−32 (s + 1 2)w −r−s−2ψ sψr , (3.20)
where in the second line one takes into account that only ψs, s > −32 are annihilation operators.
One then uses (A.17) to obtain:
L0 =
X
r≥12
rψ−rψr, (3.21)
which is the zero-mode of the free fermion energy-momentum tensor. Let us then denote a general state from the Hilbert space as:
n−1 2 , n −3 2 , ... E = (ψ−1 2 ) n−1 2 (ψ−3 2 ) n−3 2 ... |0i , ni∈ {0, 1}, (3.22)
so that its eigenvalue with respect to L0 is:
L0 n−1 2 , n −3 2 , ... E =X r≥12 rn−r n−1 2 , n −3 2 , ... E , (3.23)
which can be easily checked by using the anticommutation properties of the fermionic modes, {ψr, ψs} = δr+s,0 and the fact that the vacuum |0i is annihilated by L0. We can then obtain the
following expression: T rN SqL0− c 24 = q −1 48 1 X n−1 2 =0 1 X n−3 2 =0 ... q P r≥ 1 2 rn−r = q−148 1 X n−1 2 =0 1 X n−3 2 =0 ... Y r≥12 qrn−r = q−481 ∞ Y n=0 (1 + qn+12) =: A
A , (3.24)where from the second to the third line one moves the sums past the product. In the context of the free fermion theory, one can use the square notation in the last line to specify traces such as the one above by their boundary conditions around the spatial and temporal cycles. The A in the box’ subscript refers to the anti-periodicity of the Neveu-Schwarz sector, while the A on its left refers to the anti-periodicity of the fermionic field around the temporal cycle. We can choose boundary conditions around the two cycles of the torus in three more ways, yielding the following results along the same lines as above:
P
A := T rN S(−1)FqL0− c 24 = q− 1 48 ∞ Y n=0 (1 + qn+12) A P := T rRqL0− c 24 = q− 1 48 ∞ Y n=0 (1 + qn+12) P P := T rR(−1)FqL0− c 24 = q− 1 48 ∞ Y n=0 (1 + qn+12) = 0. (3.25)A subscript P refers to the Ramond sector of the fermionic field, while a P on the left of the box signals an inclusion of the operator (−1)F in the trace. If it seems counter-intuitive that a (−1)F insertion corresponds to periodic temporal boundary conditions, this has to do with the inherent anti-periodicity of correlation functions of fermionic fields, see e.g. chapter 7 of [9].
Let us consider how these boundary conditions behave under the action of the modular group: Under S : τ → −1τ , the boundary conditions are interchanged between cycles, as can be deduced from τ moving from numerator to denominator. Under T : τ → τ +1, the spatial cycle is unaffected, but boundary condition on the temporal cycle is multiplied by the spatial boundary condition. In summary:
T : g
h − → gh h T−1 : g h − → gh−1 h U : g h − → g gh S : g h − → h−1 g . (3.26)Concretely, this means that the trace P
P
is unaffected by the actions of the modular group, while the other three traces are transformed into one another. Hence, a fully modular invariant of a free fermion necessarily includes all these sectors.
3.3 Cardy formula
For modular invariant partition functions of CFTs, one can derive an asymptotic expression for the entropy [27]. Later on, we will compare this formula and its range of validity to the entropy of Einstein gravity. Let us consider a CFT C that is characterised by a modular invariant partition function: ZC(β) = ∞ X m, ¯m=0 d(m, ¯m)qm−24cq¯m−¯ ¯ c 24. (3.27)
We may define 2πiτ = −β + iµ, so that we relate the periodicity of the torus to the temperature and angular potential of the theory. The partition function then becomes:
ZC(τ, ¯τ ) = ∞ X m, ¯m=0 d(m, ¯m)e−β(m+ ¯m−c+¯24c)eiµ(m− ¯m− c−¯c 24) = eβc+¯24c ∞ X m, ¯m=0 d(m, ¯m)e−β(m+ ¯m), (3.28)
where we have set µ = 0 in the second line. In the low temperature limit when β is large, the m, ¯m = 0 term will be dominant over all others, assuming that the CFT’s spectrum is discrete. This is the vacuum contribution. Assuming that there is one vacuum state, we thus find:
lim β→∞ZC(β) = e βc+¯24c lim β→∞FC(β) = − c + ¯c 24 lim β→∞SC(β) = 0. (3.29)
On the other hand, the modular invariance assumption tells us that Z(τ ) = Z(−1τ ), which can be shown to be equivalent to Z(β) = Z(4πβ2), hence we may deduce that the high temperature, low
β limits of the above thermodynamic quantities are: lim β→0ZC(β) = e 4π2 β c+¯c 24 lim β→0FC(β) = − 4π2 β2 c + ¯c 24 lim β→0SC(β) = 2π2 β c + ¯c 24 . (3.30)
Furthermore, we may obtain an asymptotic expression for the entropy as a function of energy, known as the Cardy formula [27]. Using E = m + ¯m, we can write (3.28) as:
ZC(β) = eβ c+¯c 24 ∞ X E=0 ρ(E)e−βE= e4π2β c+¯c 24 ∞ X E=0 ρ(E)e−4π2β E, (3.31)
and to obtain the density of states ρ(E), we may do an inverse Laplace transform of the partition function in the high energy limit, which we solve using a saddle approximation, taking c = ¯c:
ρ(E) = 1 2π Z ∞ −∞ dβ0 e(iβ0)EZ(iβ0) = 1 2π Z ∞ −∞ dβ0 e(iβ0)Ee 4π2 iβ0 c 12 ∼ 1 2πe (iβ0)E e 4π2 iβ0 c 12 β0=β∗=−i q π2c 3E = 1 2πe 2π q cE 3 , (3.32)
where in the second line we used equation (3.30) for the partition function when β → 0, and in the third line we determined the maximum of the exponent with respect to β0. This maximum features a −i out front so that the argument of Z(iβ0) is positive. We can then deduce that at high temperature, the entropy behaves as:
S(E) ∼ log ρ(E) ∼ 2π r
cE
3 . (3.33)
Note that the saddle approximation (3.32) assumes a high-temperature limit, β 1, hence β∗ 1, so that the range of validity of the CFT Cardy formula is:
E c. (3.34)
Note that at this point, we have two formulas for the entropy at high energy or temperature, namely in (3.30) and (3.33). The relevant formula is determined by whether one considers E fixed or β fixed, i.e. on whether one considers a microcanonical or canonical ensemble.
4
Symmetric product orbifolds CFTs
The research that this thesis is concerned with relies heavily on the concepts of symmetric product orbifolds and orbifolded CFTs and this section serves as an introduction to these concepts, or as a reminder for readers already familiar with the notion. Symmetric product orbifolds are discussed in [28, 29, 30, 31].
4.1 Permutation groups
This subsection serves merely to remind the reader of properties of permutation groups. Let Sn be
the permutation group acting on the set X = {1, ..., n}. A cycle of length m is an element of Sn
that is written as {x1...xm}, m ≤ n, xi ∈ X, which moves xi to where xi+1 is in a cyclic manner.
Two cycles {x1...xm} and {y1...yl}, xi, yj ∈ X, m, l ≤ n are disjoint if and only if xi 6= yj∀i, j.
Disjoint cycles commute.
We may write every element of Snuniquely as the product of disjoint cycles - the disjoint report.
If g is a disjoint report, let ji be the number of cycles of length i that appear in it. The jis are
collected in ’signature’ j. If we work neatly, then:
n
X
i=1
iji = n,
so for instance, we should denote the identity element as (1)(2)...(n), i.e. a product of n cycles of length 1. In practice, we will omit cycles of length 1.
Generally for a group G acting on a set X, the stabiliser of an element x ∈ X is:
Stabx = {g ∈ G|gx = x}. (4.1)
4.2 Symmetric product orbifold CFT partition function
Consider a seed CFT C on a two-dimensional torus specified by modular parameter τ , with a modular invariant partition function Z(τ ). Around both the spatial and temporal cycle, boundary conditions for all fields must be defined: the easiest boundary conditions would simply be periodicity around both cycles. We may take an N -fold tensor product of this theory, C⊗N, so that one effectively has N CFTs living on seperate tori, see figure 5. We must impose boundary conditions for all N tori, and again the easiest thing is to impose periodicity around all cycles. This yields partition function Z(τ )N, which we can write as:
Z = I
I
. (4.2)
The identity elements should be interpreted as elements of the permutation group SN and represent
an operation that is done upon going around one of the cycles. In this case, when letting a field go around either cycle of torus i, i ∈ {1, ..., N }, one applies the identity element to the number i. Because this does not change i, the field is simply identified with itself, so the different factors of the tensor product behave completely independently. A field’s boundary conditions are thus:
φi(x, t + 2πτ ) = φi(x, t), φi(x + 2π, τ ) = φi(x, t), (4.3)
where the subscript refers to which copy of the seed theory it is in.
One can define many other boundary conditions for the tensor product CFT. For instance, one could use the element (12) ∈ SN, N ≥ 2, which swaps number 1 and 2, to impose (12)
I
. In that case, one must act with (12) when going around the temporal cycle of a torus, which effectively means that a field on torus 1 is identified with the same field on torus 2 after going around and vice versa. Effectively, the temporal cycle for these fields becomes twice as long.
The tensor product theory possesses an inherent SN symmetry. After all, ’swapping’ one factor
of the tensor product with another factor does not affect the theory as a whole. Let G then be a subgroup of SN. We define the N -fold permutation orbifold CFT of C by G by its partition
function [10]: ZC/G(N )(τ ) = 1 |G| X hg=gh g
h , (4.4)where the sum is over ordered pairs of elements in G, but only pairs whose elements commute, so that when a field is transported around both cycles, the order in which this happens does not matter. The permutation orbifold CFT is called a symmetric product orbifold CFT when G = SN.
The partition function of the permutation orbifold CFT is a sum over ordered commuting pairs of elements of G, (g, h), g, h ∈ G. The element g(h) specifies boundary conditions in the temporal (spatial) direction, e.g. when we have a 3-fold tensor product and g = (1)(23) and h = (1)(2)(3), then any field φi(x, t), where the subscript again refers to which copy of the seed theory it is in,
obeys:
φ1(x, t + 2πτ ) = φ1(x, t), φ1(x + 2π, τ ) = φ1(x, t)
φ2(x, t + 2πτ ) = φ3(x, t), φ2(x + 2π, τ ) = φ2(x, t)
φ3(x, t + 2πτ ) = φ2(x, t), φ3(x + 2π, τ ) = φ3(x, t).
(4.5)
If the element h is not equal to the identity element, one speaks of a twisted sector of the permutation orbifold CFT.
According to [32], we can then obtain the partition function of the permutation orbifold CFT using the following formula:
ZC/G(N )(τ ) = 1 |G| X gh=hg Y ξ∈O(g,h) Z(τξ), (4.6)
where the product is over the set of orbits of the subgroup generated by the elements g and h [32]. For instance, if N = 2 and g = h = (1)(2) = I, then the orbits generated by this subgroup are simply the elements of the set X themselves.
Finally, τξ is defined as in [32]:
τξ=
µξτ + κξ
λξ
, (4.7)
whose variables are defined in [28] and [32] in terms of the ordered pair of commuting elements x, y and one of the orbits ξ of the subgroup of which x, y are generators:
• all g-orbits contained in ξ have the same length. This length is λξ.
• µξ is the number of g-orbits contained in ξ. One should check that λξµξ gives the number of
elements in ξ.
• κξ is a nonnegative number, smaller than λξ. It is the smallest number such that hµξg
−κξ
belongs to the stabiliser of all elements of ξ. Put differently, it is the smallest number such that hµξ and gκξ act identically on all elements of ξ.
Let us adopt the following notation for the partition function of a symmetric product orbifold CFT (SPOCFT): ZC/S(N )
N = ZC/N. An alternative formula for the untwisted sector of the partition
function of a SPOCFT is [7]: ZC/Nu (τ ) = 1 |Sn| X j AjZ(τ )j1...Z(nτ )jn. (4.8)
The sum here is effectively over different signatures that disjoint reports may have, i.e. over their composition in terms of cycles of different lengths. Aj is the number of elements in Sn whose
disjoint reports have signature j. In the case of S2, the possible disjoint reports are:
j1= (2, 0) j2 = (0, 1). (4.9) The identity element is the only element with signature j1, the ’swap’ is the only element with signature j2. Hence, A1= A2= 1. This information is sufficient to use (4.8) to compute:
ZC/2u (τ ) = 1 2 Z(τ )
2+ Z(2τ ), (4.10)
which can serve both as a check of the result of the full partition function, and as a way to obtain the (un)twisted sector partition function seperately.
What follows now is an explicit computation of the partition function of 2-fold and 3-fold SPOCFTs. Doing these calculations will help the reader grasp the essence of the action of orbifold-ing, whilst the final result helps interpret the SPOCFT in terms of the seed theory.
In the 2-fold case, where we are interested in finding ZC/2, we identify four commuting ordered
pairs of elements: {I, I}, {I, (12)}, {(12), I}, {(12), (12)}. Let us look at these in more detail: • {I, I}: The orbits of this subgroup are simply the elements 1 and 2. The length of these orbits
is λξ, which is 1. Thus κξ = 0 Their number is µξ = 2. Hence it follows from (4.7) that for
both orbits, τξ = τ and the corresponding term contributing to (4.6) is Z(τ )2.
• {I, (12)}: this subgroup only generates one orbit: ξ = {1, 2}. Here, g = I, so the length of the g-orbits is 1, their number is 2. Hence κξ= 0. The corresponding term is thus Z(2τ ).
• {(12), I}: We have the same orbit as above. However, here g = (12) and thus λξ= 1, µξ= 2
and κξ= 0. Term: Z(τ2).
• Finally, {(12), (12)}: Again the orbit is ξ = {1, 2}., so λξ= 1 and µξ= 2 hold again. However,
since here g = h, κξ= 1 and the term is Z(τ +12 ).
In all cases, we find indeed that the product lξµξequals the number of elements in ξ. Gathering
all pieces gives:
ZC/2 = 1 2 Z(τ )2+ Z(2τ ) + Z(τ 2) + Z( τ + 1 2 ) . (4.11)
Later on I will show explicitly that this expression is modular invariant. We immediately see that we can retrieve the untwisted part that we computed above, which allows us to read off the twisted part, as well.
(4.11) is a sum of the seed theory’s partition function for different values of the modular pa-rameter. The SPOCFT’s partition function thusly includes a contribution from the two-fold tensor product (the first term), as well as multiple contributions from the seed theory, as if it lived on a torus specified by a different modular parameter than originally.
In the 3-fold case, we may obtain the untwisted sector partition function through the same method as above. The elements of S3 are I, (12), (13), (23), (123) and (132). The occuring
signatures of disjoint reports are thus:
j1= (3, 0, 0) j2= (1, 1, 0) j1= (0, 0, 1). (4.12) Furthermore, Aj1 = 1, Aj2 = 3 and Aj1 = 1. It then follows from (4.8) that:
ZC/3u = 1 6
Table 1: commutation information I (12) (13) (23) (123) (132) I Y Y Y Y Y Y (12) Y N N N N (13) Y N N N (23) Y N N (123) Y Y (132) Y
We must then consider the following ordered pairs of elements:
• {I, I}: This subgroup forms three orbits or length 1, and yields a term Z(τ )3 by similar
arguments as before.
• {I, (12)}: now, there are two orbits: ξ1 = {1, 2} and ξ2 = {3}. We have g = I, thus λξ1 = 1,
µξ1 = 2 κξ1 = 0, λξ2 = 1, µξ2 = 1 κξ2 = 0. This yields a term Z(2τ )Z(τ ).
• {(12), I}: the orbits are the same, but because g = (12) here, we get λξ1 = 2, µξ1 = 1 κξ1 = 0,
λξ2 = 1, µξ2 = 1 κξ2 = 0. This term becomes Z(τ )Z(
τ 2).
• {I, (123)}: there is only one orbit ξ = {1, 2, 3}. λξ= 1, µξ= 3 κξ = 0. Term: Z(3τ ).
• {(123), I}: same orbit, but the values of λξ and µξ switch. Hence Z(τ3).
• {(12), (12)}: again, this gives orbits ξ1 = {1, 2} and ξ2 = {3}. The new feature here is that
κξ= 1, yielding Z(τ +12 )Z(τ ).
• {(123), (123)}: this yields Z(τ +13 ). • {(123), (132)} this yields Z(τ +2
3 ).
The other commuting pairs all give similar expressions, yielding final expression:
ZC/3 = 1 6 Z(τ )3+ 3Z(2τ )Z(τ ) + 3Z(τ 2)Z(τ ) + 2Z(3τ ) + 2Z(τ 3) + 3Z( τ + 1 2 )Z(τ ) + 2Z( τ + 1 3 ) + 2Z( τ + 2 3 ) , (4.14)
where as before, we recognize the untwisted sector partition function and we are this able to identify the twisted sector part. Again, one can identify the terms in this partition function as the partition functions of (tensor products of) the seed theory, on tori of different modular parameters than the original.
It is actually simpler to work out a generating function for the partition functions of all N -fold SPOCFTs of C [7]:
Z = X
N ≥0
This generating function may be interpreted similarly to a grand canonical ensemble (GCE) parti-tion funcparti-tion from thermodynamics, which may be written as a sum over states denoted by i:
ZGCE = X i e−β(Ei−µNi) = ∞ X N =0 X i|Ni=N e−β(Ei−µN )= ∞ X N =0 eβµN X i|Ni=N e−βEi = ∞ X N =0 pN X i|Ni=N e−βEi = ∞ X N =0 pNZN. (4.16)
Here, we have defined p = eβµ, which is referred to as the fugacity in the context of statistical physics. Then, ZN in the last line is the canonical ensemble partition function of a system with N
particles, which we may obtain from ZGEC by:
ZN = 1 N ! dN dpNZGCE p=0. (4.17)
In this sense, ZGCE generates all canonical ensemble partition functions. Equivalently, knowing it
in (4.15) amounts to knowing any ZC/N, since we can write the equivalent of (4.17):
ZC/N(τ, ¯τ ) = 1 N ! dN dpNZ p=0. (4.18)
In the case of symmetric orbifolds, it is given in terms of Hecke operators TL [29, 33]:
TLf (τ, ¯τ ) = X d|L d−1 X b=0 f (Lτ + bd d2 , L¯τ + bd d2 ). (4.19)
Note that T1Z(τ ) = Z(τ ). In terms of these Hecke operators, the generating function is:
Z = expn X L>0 pL LTLZC o . (4.20)
We will check this explicitly that this reproduces (4.11) and (4.14). Let N = 2, so that we must take two derivatives of Z:
d dpZ = Z X L>0 pL−1TLZ(τ ) d2 dp2Z = Z X L>0 pL−1TLZ(τ ) 2 + ZX L>0 (L − 1)pL−2TLZ(τ ), (4.21)
ZC/2(τ ) 1 2 d2 dp2Z p=0= 1 2 T1Z(τ ) 2 +X d|2 d−1 X b=0 Z(Lτ + bd d2 ) = 1 2 Z(τ )2+ Z(2τ + 0 1 ) + Z( 2τ + 0 4 ) + Z( 2τ + 2 4 ) = ZC/2(τ ), (4.22)
since we find the same expression as (4.11). To check ZC/3(τ ), we compute:
d3 dp3Z = Z X L>0 pL−1TLZ(τ ) 3 + 2Z X L>0 pL−1TLZ(τ ) X K>0 (K − 1)pK−2TKZ(τ ) + Z X L>0 pL−1TLZ(τ ) X K>0 (K − 1)pK−2TKZ(τ ) + X L>0 (L − 1)(L − 2)pL−3TLZ(τ ) . (4.23)
The only new sum is the last one, which has a nonzero term when L = 3. From the above we then find: 1 6 d3 dp3Z = 1 6 Z(τ )3+ 3Z(τ ) Z(2τ ) + Z(τ 2) + Z( τ + 1 2 ) + Z(3τ ) + Z(3τ + 0 9 ) + Z( 3τ + 3 9 ) + Z( 3τ + 6 9 ) = ZC/3, (4.24)
for it is the same expression as (4.14).
4.3 Modular invariance
Having computed ZC/2(τ ) and ZC/3(τ ), it is easy to explicitly check that these expressions are
modular invariant. This will be of importance later, e.g. because it guarantees that the Cardy formula is valid after orbifolding.
To this end, we remind ourselves that, by assumption, Z(τ ) is modular invariant. In particular, it is invariant under the reparametrisations:
T : τ −→ τ + 1 S : τ −→ −1 τ .
Since these transformations generate the entire modular group, it is sufficient to check that ZS2(τ )
and ZS3(τ ) are invariant under these transformations. We have:
ZC/2(τ ) T −→ ZC/2(τ + 1) = Z(τ + 1)2+ Z(2(τ + 1)) + Z(τ + 1 2 ) + Z( τ 2) = Z(τ )2+ Z(2τ ) + Z(τ + 1 2 ) + Z( τ 2) = ZC/2(τ ). (4.25)
in (3.25) under modular transformations. A similar calculation gives: ZC/2(τ ) S −→ ZC/2(−1 τ ) = Z( −1 τ ) 2+ Z(−2 τ ) + Z( −1 2τ ) + Z( τ − 1 2τ ) = Z(τ )2+ Z(τ 2) + Z(2τ ) + Z(ST 2Sτ − 1 2τ ) = ZC/2(τ ). (4.26)
Note that here, a twisted and untwisted sector part have swapped roles. Explicitly, the four operations in the second-to-last line yield:
τ − 1 2τ S −→ 2τ 1 − τ T2 −→ 2 1 − τ S −→ τ − 1 2 . (4.27)
To check that (4.14) is modular invariant, we rewrite it:
6ZC/3= 3Z(τ ) Z(τ )2+ Z(2τ ) + Z(τ 2) + Z( τ + 1 2 ) − 2Z(τ )3+ 2Z(3τ ) + 2Z(τ 3) + 2Z( τ + 1 3 ) + 2Z( τ + 2 3 ) = 3Z(τ )ZC/2(τ ) − 2Z(τ )3+ 2Z(3τ ) + 2Z( τ 3) + 2Z( τ + 1 3 ) + 2Z( τ + 2 3 ). (4.28)
From the discussion above, we know that the first two terms in the last line are modular invariant separately, so we only need analyse the last four, which goes to:
T −→ 2Z(3(τ + 1)) + 2Z(τ + 1 3 ) + 2Z( τ + 2 3 ) + 2Z( τ + 3 3 ) = 2Z(3τ ) + 2Z(τ + 1 3 ) + 2Z( τ + 2 3 ) + 2Z( τ 3), (4.29)
where we only needed to apply one T−1 transformation to the last term. Similarly:
S −→ 2Z(−3 τ ) + 2Z( −1 3τ ) + 2Z( τ − 1 3τ ) + 2Z( 2τ − 1 3τ ) = 2Z(τ 3) + 2Z(3τ ) + 2Z(ST 3Sτ − 1 3τ ) + 2Z(ST −3 ST−12τ − 1 3τ ) = 2Z(τ 3) + 2Z(3τ ) + 2Z( τ + 2 3 ) + 2Z( τ + 1 3 ). (4.30)
The modular invariance of (4.4) for general N, G can be shown using the change of boundary conditions under modular transformations derived in appendix B [10]:
5
Supersymmetry
A supersymmetric theory necessarily includes both bosonic and fermionic degrees of freedom, i.e. its Hilbert space can be written as a direct sum of spaces of bosonic and fermionic states:
H = HBMHF. (5.1)
Moreover, there must be an operator Q that maps Q : HB/F → HF /B, such that Q2 = 0 and {Q, Q} = 2H, to render the theory supersymmetric. It is easy to show that this implies [H, Q] = 0 and the charge Q is thus conserved. Additionally, every state must have positive energy, except potentially a vacuum state, which is then annihilated by Q.
The above implies that the map Q : HB/F → HF /B is one-to-one. Namely, take two
non-vacuum states |ψi , |ψ0i ∈ HBand map them both to HF using the map Q. This map is one-to-one
if no linear combination of these states is zero, i.e. if the following is satisfied:
0 6= αQ |ψi + βQ ψ0
= Q α |ψi + β ψ0 , (5.2) which holds since |ψi , |ψ0i are linearly independent by assumption and they are non-vacuum, hence not annihilated by Q. It thusly follows that for every bosonic state at any non-zero energy, there is a fermionic state at the same energy level, illustrated in figure 5.
However, this matching does not necessarily happen for vacuum states. This motivated Witten in [34] to define the following trace, referred to as the Witten index:
T rH(−1)Fe−βH = nE=0B − nE=0F , (5.3)
where the equality holds if the operator (−1)F satisfies:
(−1)F|bosonic statei = |bosonic statei (−1)F |fermionic statei = − |fermionic statei .
(5.4)
An example of an operator that satisfies the above relation is eiπJ0, which will be discussed in
Figure 6: matching of bosonic/fermionic energy levels in a supersymmetric theory. The solid dots could be bosonic states, the open dots fermionic states.
section 5.2.
5.1 N = (1,1) supersymmetry
Since we will later consider supersymmetric CFTs dual to supergravity theories, let us briefly spell out the most important aspect of the simplest supersymmetric CFT in two dimensions. This theory features one bosonic and one fermionic field and is specified by the following action:
SSU SY,1=
1 4π
Z
d2w ∂X ¯∂X + iψ ¯∂ψ + iθ∂θ. (5.5) The fermions were already discussed in chapter 2, the bosonic field is discussed in appendix A: the field X is the free boson and the field ψ is the holomorphic free fermion, while the field θ is equivalent to ψ only anti-holomorphic, hence dependent on ¯z. In summary, we have the following conformal primary fields:
j(w) = i∂X(w), ¯j( ¯w) = ¯∂iX( ¯w), ψ(w), θ( ¯w), (5.6) and by combining these fields into one theory a new symmetry arises. Consider the following global variations:
δX = iψ, δψ = −X. (5.7) The parameter is a Grassman variable. The action then transforms as follows:
4πδSSU SY,1=
Z
d2z ∂(iψ) ¯∂X + ∂X ¯∂(iψ) + i(−∂X) ¯∂ψ + iψ ¯∂(−∂X) =
Z
d2z i∂ψ ¯∂X + i∂X ¯∂ψ − i∂X ¯∂ψ + iψ∂ ¯∂X = 0.
(5.8)
From the first to the second line, the last term switches sign because is moved past ψ. In the second line, the second term cancels the third and the first cancels the fourth, after one integration by parts. We may calculate the corresponding current, e.g. by letting the parameter depend on z, ¯z, to find that it is:
G(w) =: jψ : (w). (5.9)
Similarly, there is a supersymmetry involving the anti-holomorphic fields with a corresponding anti-holomorphic current:
hence we refer to this theory as a N = (1, 1) supersymmetric conformal field theory (SCFT), where the ones refer to the number of holomorphic and anti-holomorphic supersymmetric currents of conformal weight 32.
The modes of the supersymmetric currents and of the energy-momentum tensor obey the N = (1, 1) superconformal algebra, see e.g. [8]:
[Lm, Ln] = (m − n)Lm+n+ c 12(m 3−, )δ m+n [Lm, Gr] = ( m 2 − r)Gm+r {Gr, Gs} = 2Lr+s+ c 3(r 2−1 4)δr+s. (5.12)
Depending on whether we would consider fermions in the Neveu-Schwarz or the Ramond sector, the G modes will have integer or half-integer labels. The algebra (5.12) holds in either case.
5.2 N = (2,2) supersymmetry
A more ’advanced’ theory of supersymmetry, with a higher degree of supersymmetry, has the following action: SSU SY,2= 1 4π Z d2w ¯∂X∂X + iΨ ¯∂ ¯Ψ + iΘ∂ ¯Θ. (5.13) The first of the fields X, Ψ, Θ is bosonic, while the latter two are fermionic and all three are complex fields, meaning that they have two real degrees of freedom captured in one field in the following manner:
X = X1+ iX2, Ψ = Ψ1+ iΨ2, Θ = Θ1+ iΘ2. (5.14) It must be stressed that a bar over a partial derivative means ¯∂ = ∂w¯, while a bar over a field
means complex conjugation, i.e. ¯Ψ = Ψ1− iΨ2.
Since the number of holomorphic degrees of freedom has doubled, it may not be surprising that the number of supersymmetries has doubled, too. Consider the following global variations:
δX = Ψ, δ ¯X = − ¯Ψ δΨ = −i∂X, δ ¯Ψ = i∂ ¯X
∈ R.
(5.15)
It can be straightforwardly checked that this leaves the Lagrangian invariant up to a total derivative, and that the corresponding current is:
G1(w) =: Ψ∂ ¯X : (w)− : ∂X ¯Ψ : (w). (5.16)
the above current is a supersymmetric one. Additionally, we may take imaginary and consider the following global variations:
δX = Ψ, δ ¯X = ¯Ψ δΨ = i∂X, δ ¯Ψ = i∂X
∈ iR.
(5.17)
This is another set of supersymmetric variations with corresponding supersymmetric current:
G2(w) =: Ψ∂ ¯X : (w)+ : ¯Ψ∂X : (w), (5.18)
which we may combine with (5.16) to conveniently write:
G+(w) = i √ 2 G1(w) + G2(w) = i √ 2 : Ψ∂ ¯X : G−(w) = i √ 2 G1(w) − G2(w) = i √ 2 : ¯Ψ∂X : . (5.19)
There are equivalent symmetries between the anti-holomorphic bosons and fermions with corre-sponding supersymmetric currents:
¯ G1( ¯w) =: Θ ¯∂ ¯X : ( ¯w)− : ¯∂X ¯Θ : ( ¯w) ¯ G2( ¯w) =: Θ ¯∂ ¯X : ( ¯w)+ : ¯Θ ¯∂X : ( ¯w) ¯ G+( ¯w) = i √ 2 ¯ G1( ¯w) + ¯G2( ¯w) = i √ 2 : Θ ¯∂ ¯X : ¯ G−( ¯w) = i √ 2 ¯ G1( ¯w) − ¯G2( ¯w) = i √ 2 : ¯Θ ¯∂X :, (5.20)
whose conformal dimensions of these currents turn out to be:
(hG+, ¯hG+) = (hG−, ¯hG−) = ( 3 2, 0) (hG¯+, ¯hG¯+) = (hG¯−, ¯hG¯−) = (0,3 2). (5.21)
Therefore, we say that this theory is a N = (2,2) SCFT, where we are again counting the numbers of (anti-)holomorphic supersymmetric currents with the right conformal dimension.
Furthermore, the three complex fields all have U (1) symmetries, stemming from rotating the real part into the complex part and vice versa. Firstly, there is the U (1) symmetry of the bosonic
the above current is not, either. We thus exclude it from the analysis of this superconformal algebra and only consider the following U (1) variations:
δΨ1 = υΨ2, δΨ2 = −υΨ1
δΘ1 = υΘ2, δΘ2 = −υΘ1,
(5.23)
where υ is a global infinitesimal bosonic parameter. The following currents correspond to these symmetries:
J (w) =: ¯ΨΨ : (w), J ( ¯¯w) =: ¯ΘΘ : ( ¯w) (5.24) These conserved currents assign quantum numbers to states in the Hilbert space of this theory, so that states are not just labeled by their conformal dimensions but also by a U(1) charge. Expanding the energy-momentum tensor and the currents in modes and computing commutation relations, we arrive at the N = (2,2) superconformal algebra:
[Lm, Ln] = (m − n)Lm+n+ c 12(m 3− m)δ m+n [Lm, Jn] = −nJm+n [Lm, G±r] = ( m 2 − r)G ± m+r [Jm, Jn] = c 3mδm+n [Jm, G±r] = ±G ± m+r {G+ r, G−s} = 2Lr+s+ (r − s)Jr+s+ c 3(r 2−1 4)δr+s {G+r, G+s} = {G−r, G−s} = 0. (5.25)
The second line tells us that the current J has conformal dimension one. Of course, there is another anti-holomorphic copy of the same algebra.
5.2.1 Spectral flow
Spectral flow constitutes a continuous class of automorphisms of the algebra (5.25) and is thusly inherent to N = (2, 2) supersymmetric theories. Spectral flow is characterised by one parameter η [8]: Ln→ L0n= Ln+ ηJn+ η2 6 cδn Jn→ Jn0 = Jn+ c 3δn G±r → G0±r = G±r±η. (5.26)
{G0+r , G0−s } = {G+r+η, G−s−η} = 2Lr+s+ r − s + 2ηJr+s+ c 3 r 2+ 2rη + η2−1 4δr+s = 2Lr+s+ 2ηJr+s+ 2 η2 6 cδr+s+ r − sJr+s+ (r − s) c 3ηδr+s+ c 3 r 2−1 4δr+s = 2L0r+s+ (r − s)Jr+s0 + c 3(r 2−1 4)δr+s, (5.27)
where we have used 2rδr+s= (r − s)δr+s from the second to the third line. The other relations of
(5.25) can be checked in similar fashion.
In particular, when the parameter η = 12 and one applies spectral flow, moding of the super-symmetric currents is changed from integer to half-integer or vice versa. Hence, such flows map the Ramond sector to the Neveu-Schwarz sector of a CFT and vice versa.
5.2.2 The elliptic genus
Due to the presence of the U (1) currents in N = (2,2) SCFTs, we may extend the notion of the partition function to include information about this charge. In particular, with the definition of the Witten index in mind, the elliptic genus for such theories is defined as [3]:
ZEG(τ, z) = T rR,R(−1)F + ¯FqL0− c 24q¯L¯0− ¯ c 24yJ0 , (5.28)
which features the zero-mode of the holomorphic U (1) current and the variable y := e2πiz. Most notably, the dependence on ¯τ is absent. This can be understood in terms of figure 5. The anti-holomorphic factor in the trace above is (−1)Fq¯L¯0, and the anti-holomorphic supersymmetries
(5.20) guarantee that at every non-vacuum energy level, i.e. every ¯L0 eigenvalue, there are as many
bosonic as fermionic states. These cancel one another in the trace due to the (−1)F insertion, hence the only anti-holomorphic contribution comes from the vacuum and is therefore independent of ¯τ [3].
It was proposed in [35] that the elliptic genus of a N = (2, 2) supersymmetric CFT should satisfy particular transformation properties under modular transformations of the parameters z, τ and under spectral flow. Let us consider such a CFT with right- and left-moving central charge both equal to c = ¯c = 6m. The generators of the modular group induce the following transformations of the elliptic genus [35]:
ZEG(τ + 1, z) = ZEG(τ, z), ZEG( −1 τ , z τ) = e 2πimz2τ Z EG(τ, z). (5.29)
Such functions have the following expansion: φ(τ, z) = X n∈Z X r2<4nm c(n, r)e2πi(nτ +rz), (5.31) where one takes n to be the eigenvalue of the operator L0− 24c. Moreover, we may call a Jacobi
form weak if it allows an expansion of the form:
φ(τ, z) =X
n≥0
X
r2−4mn≤m2
c(n, r)qnyr. (5.32) Physical considerations lead one to expect to deal with weak Jacobi forms rather than Jacobi forms. This is due to the fact that a coefficient c(n, r), n < 0 counts states with holomorphic conformal dimension h < −c24, when n is interpreted as the eigenvalue of the operator L0− 24c. Such states
are absent in unitary CFTs [9, 10]. Finally, the expansion of a weak Jacobi form can be arranged differently as in [23, 36]: ZEG(τ, z) = X r∈Z/2mZ X n≥0 c(n, r)qn−r24kΘk,r(τ, z), Θk,r(τ, z) = X k=rmod2m q4mk2yk (5.33)
Let us consider the action SSU SY,2 and calculate the corresponding elliptic genus. A vacuum
of this theory is a tensor product of three vacua, one for bosonic modes and one for holomorphic and anti-holomorphic fermionic modes each. Additionally, since we consider the Ramond sectors for the elliptic genus, the vacua for the fermionic modes have eigenvalue ±1 with respect to the operator (−1)F or (−1)F¯, so that there are four vacuum states:
|0i±,± = |0iB⊗ |0iF,±⊗ |0iF ,±¯ , (5.34)
where on both sides of the equation the first ± denotes the (−1)F eigenvalue and the second ±
the (−1)F¯ eigenvalue. There are two real holomorphic bosonic degrees of freedom and two real anti-holomorphic bosonic degrees of freedom, and as many fermionic ones, see table 2. These can all be decomposed into modes that can be used to build states on top of a vacuum.
Table 2: modes of a free N = (2, 2) supersymmetric theory
Field Bosonic/fermionic (Anti-)holomorphic Number of excitations L0 eigenvalue U (1) charge
∂X B H mi, m0i, i ∈ Z≤1 -i 0
¯
∂X B A µi, µ0i, i ∈ Z≤1 -i 0
Ψ F H ni, n0i, i ∈ Z≤1 -i 1
Θ F A νi, νi0, i ∈ Z≤1 -i -1
Schematically then, one can write a general state in the Hilbert space as:
|ψi =
..., mi, ..., m0i, ..., µi, ..., µ0i, ..., ni, ..., n0i, ..., νi, ..., νi0, ...
±,±, (5.35)
where the subscripts signal which fermionic vacua are used. Consider then any combination of modes built on the vacuum |0i++. The contribution of this state to the elliptic genus depends
on its eigenvalues with respect to L0, ¯L0, J0, and the number of fermionic modes tells whether its
eigenvalue with respect to (−1)F + ¯F is positive or negative one. Furthermore, we can build the same number of modes on the vacuum |0i+−. The L0, ¯L0, J0 eigenvalues will be identical, but the
(−1)F + ¯F will be opposite since (−1)F¯|0i ¯
F ,−= − |0iF ,−¯ . Similarly, the |0i−+and |0i−− states will
give equal and opposite contributions. Since this happens independently of the modes built on the vacuum, the elliptic genus is identically zero for the free theory.
6
Analysis of the spectrum of a CFT
Now we have covered the relevant theory about 3D gravity thermodynamics, (supersymmetric) 2D CFTs and SPOCFTs, let us cut to the chase and look into attempts to characterise holographic CFTs.
If one believes strongly in the ADS/CFT conjecture [37], then any CFT in two dimensions is dual to some theory of quantum gravity in a three-dimensional asymptotically ADS spacetime: this gravity theory is then said to live “in the bulk”. The conjecture has not been proven, but many hints as to its validity exist, in three as well as in a general number of dimensions. For instance, the fact that the algebra of the charges generating the asymptotic ADS symmetries of an asymptotially ADS3spacetime is a Virasoro algebra hints towards some duality with 2D CFTs. Furthermore, the
fact that the entropy of a black hole scales with the area of its horizon, as opposed to its volume, suggests the number of degrees of freedom of a black hole in D dimensions is equal to the degrees of freedom of some QFT in D − 1 dimensions.
This chapter discusses the work done in [1, 2, 4] on universal properties of symmetric orbifold CFTs, and on constraints on the spectra of CFTs, motivated by arguments from the gravity side of the duality.
Let us consider which theories may have a weakly coupled Einstein gravity dual. As we have seen in chapter 2, weakly coupled Einstein gravity in three dimensions is universally characterised by the Hawking-Page transition at T = 2π1 and one would expect any CFT dual to it to exhibit a similar phase transition. Furthermore, we defined a central charge in the gravity theory in chapter 2 as:
c = 2L 3GN
. (6.1)
The gravity duals that we are interested are weakly coupled and have GN 1, which corresponds
to a very large central charge, which is what we will look for in a CFT. As was briefly mentioned in chapter 1, a simple way to build such CFTs is to take the N -fold tensor product of a seed theory C with central charge c, yielding a theory with central charge N c, then considering the limit where N goes to infinity. However, this construction as such is unsatisfactory as the number of states at a given energy level E would diverge as N is taken to infinity. After all, any state in the seed CFT of energy E can be excited in any of the copies of the tensor product.
Hence, in a search for theories with finite numbers of states at given energy levels, one considers SPOCFTs instead of ordinary tensor products, as described in chapter 4, projecting out states that are invariant under the orbifold action and creating new twisted sectors [38], creating a theory whose spectrum is very different from that of the seed theory. It is hoped that such a construction reproduces the properties of a three-dimensional gravity theory.
6.1 Universality of symmetric product orbifold CFTs
This subsection focuses on work done in [1] and chapter two of said work in particular. Consider some seed CFT with modular invariant partition function Z(τ ) and its SPOCFT with partition
function ZC/N(τ ). Let us write the partition function of the seed theory as follows, mirroring the notation in [1]: ZC = X m, ¯m∈I d(m, ¯m)qmq¯m¯ = q−c24q¯ −¯c 24 X m, ¯m∈ ˜I ˜ d(m, ¯m)qmq¯m¯. (6.2)
By construction, the set I contains m ≥ −c24, ¯m ≥ −¯24c, while the set ˜I contains m, ¯m ≥ 0. Recall the generating function of the SPOCFTs’ partition functions and its expression in terms of the Hecke operators from section 4.2:
Z = expn X L>0 pL LTLZC o , (6.3) TLf (τ, ¯τ ) = X d|L d−1 X b=0 f (Lτ + bd d2 , L¯τ + bd d2 ). (6.4)
Assuming that the seed CFT has one unique vaccum, we may write the action of a Hecke operator on the partition function as:
TLZC(τ, ¯τ ) = X d|L d−1 X b=0 Z(Lτ + bd d2 , L¯τ + bd d2 ) =X d|L d−1 X b=0 expn− 2πi c 24 Lτ + bd d2 o expn2πi ¯c 24 L¯τ + bd d2 o X m, ¯m∈ ˜I ˜ d(m, ¯m) expn2πimLτ + bd d2 o expn− 2πimL¯τ + bd d2 o . (6.5)
We see that the d = 1, b = 0, m = ¯m = 0 contribution to the sum gives q−cL24q¯− ¯ cL 24, hence we may write: TLZC(τ, ¯τ ) = q− cL 24q¯− ¯ cL 24 1 + X m, ¯m>0 ˜ dTL(m, ¯m)q mq¯m¯. (6.6)
As one sees that the term within brackets features qmd2L terms, we may instead sum over m0 = Lm.
Assuming that the spectrum is discrete and the first non-vacuum state is at m = m0, the range of
the sum will be m, ¯m ≥ m0L. The author of [1] then defines ˜p = pq
−c 24q¯
−¯c
log X K≥0 xK = log 1 1 − x = log1 − log 1 − x =X L>0 1 Lx L, (6.8)
where in the last line we also use the Taylor expansion of the logarithm. Substituting (6.6) in (6.3) and using the equality above, we arrive at:
Z = X K≥0 ˜ pKexpn X L>0 1 Lp˜ L X m, ¯m≥m0 ˜ dTL(m, ¯m)q mq¯m¯o. (6.9)
Keeping in mind that we should obtain the partition function of the N -fold SPOCFT by taking N derivatives with respect to ˜p, one can do an attempt to find the number ˜dN(m, ¯m), which counts
the number of states at level (m, ¯m) in the N-fold SPOCFT. It is clear that this number is the coefficient of the ˜pNqmq¯m¯ in the generating function. We may then restrict the first sum above to an upper range of N , keeping only part of Z, and pull out a factor ˜pN to obtain:
Z 3 N X K≥0 ˜ pK−Np˜Nexpn X L>0 1 Lp˜ L X m, ¯m≥m0 ˜ dTL(m, ¯m)q mq¯m¯o = ˜pN N X K≥0 ˜ p−Kexpn X L>0 1 Lp˜ L X m, ¯m≥m0 ˜ dTL(m, ¯m)q mq¯m¯o. (6.10)
We may thus alternatively look for the ˜p0qmq¯m¯ contribution of: XN K≥0 ˜ p−Kexpn X L>0 1 Lp˜ L X m, ¯m≥m0 ˜ dTL(m, ¯m)q mq¯m¯o. (6.11)
To do so, we may attempt to only consider the linear approximation of the exponential, which would yield: Z 3 ˜pN XN K≥0 ˜ p−K 1 +X L>0 1 Lp˜ L X m, ¯m≥m0 ˜ dTL(m, ¯m)q mq¯m¯ + ... = ˜pN N X K≥0 ˜ p−K+ N X K≥0 X L>0 1 Lp˜ L−K X m, ¯m≥m0 ˜ dTL(m, ¯m)q mq¯m¯ + ... . (6.12)
It is then clear that in this approximation, we would obtain:
dN(m, ¯m) = N X L=1 1 Ld˜TL(m, ¯m). (6.13)
We now do two things: firstly, we use the fact that TLZ is a modular form and its decomposition
(6.6) to justify an approximation of ˜dL(m, ¯m) in terms of the Cardy formula: the validity of this
approximation is discussed in [2]. Secondly, since we are interested in taking the limit N → ∞, we may approximate the sum above by an integral, such that we obtain:
˜ dN(m, ¯m) = Z N 1 dL ˜dTL(m, ¯m) = Z N 1 dL exp n 2π r cL 6 (m − cL 24) o exp n 2π r cL 6 ( ¯m − cL 24) o . (6.14)
We may attempt to do this integral using a saddle approximation: it can be easily checked that the maximum of the argument of the exponential is:
L∗ = 24m ¯m
c(m + ¯m). (6.15)
We may then substitute this value of L in the exponential to obtain, in the limit N → ∞:
lim N →∞ ˜ dN(m, ¯m) = exp n 2π r 4m ¯m m + ¯m m − m ¯m m + ¯m + 2π r 4m ¯m m + ¯m m −¯ m ¯m m + ¯m o = expn2π r 4m ¯m m + ¯m m2 m + ¯m + 2π r 4m ¯m m + ¯m ¯ m2 m + ¯m o = e4π √ m ¯m. (6.16)
It is stressed in [1] that the above limit holds for c, ¯c m, ¯m N , where c, ¯c are the seed theory’s central charges. The latter inequality is trivially true when N is truly taken to infinity, but it is useful for the interpretation of the result. The fact that m, ¯m N means that the above result is a statement about states that are, in holographic terms, lighter than black holes. After all, we saw in section 2.2 that the energy of black holes goes as −c12∗ 4πβ2− 1, where the energy is shifted so that the vacuum energy vanishes and c∗ is used to avoid confusion with the seed theory central charge c. It is clear that the energy associated to black hole states depends linearly on c∗, which depends linearly on N when an N -fold SPOCFT is considered. Hence, states for which m, ¯m N holds states that are lighter than black holes. On the other hand, the fact that c, ¯c m, ¯m tells us that these states must be considerably heavier than the vacuum state, since c, ¯c are constant and assumed to be finite, O(1) real numbers and therefore relatively close to the vacuum energy. Hence, we deduce that (6.16) is a statement about heavier non-black hole states of the SPOCFT.
