QMUL-PH-11-17

Quantum states to brane geometries

via fuzzy moduli spaces of giant gravitons

Jurgis Pasukonis^{1}^{1}1
and Sanjaye Ramgoolam^{2}^{2}2

Department of Physics

Queen Mary, University of London

Mile End Road

London E1 4NS UK

Abstract

Eighth-BPS local operators in SYM are dual to quantum states arising from the quantization of a moduli space of giant gravitons in . Earlier results on the quantization of this moduli space give a Hilbert space of multiple harmonic oscillators in 3 dimensions. We use these results, along with techniques from fuzzy geometry, to develop a map between quantum states and brane geometries. In particular there is a map between the oscillator states and points in a discretization of the base space in the toric fibration of the moduli space. We obtain a geometrical decomposition of the space of BPS states with labels consisting of representations along with Young diagrams and associated group theoretic multiplicities. Factorization properties in the counting of BPS states lead to predictions for BPS world-volume excitations of specific brane geometries. Some of our results suggest an intriguing complementarity between localisation in the moduli space of branes and localisation in space-time.

###### Contents

- 1 Introduction
- 2 Review of phase space and quantization
- 3 Fuzzy and giants as points on a simplex
- 4 A geometrical group theoretic labelling for eighth-BPS states
- 5 World-volume excitations of maximal giants
- 6 Excitations from partition function
- 7 Excitations from local quantization
- 8 World-volume excitations beyond maximal giants
- 9 Discussion
- 10 Summary and outlook
- A Symplectic form for perturbations of sphere giant

## 1 Introduction

The study of BPS states in the non-planar regime of four-dimensional , Yang-Mills theory (SYM) has been a very rich area of research, allowing investigations of the AdS/CFT correspondence [1, 2, 3] beyond the supergravity approximation. In the regime of energies of order , the states correspond to D3 brane geometries in the bulk, also known as “giant gravitons” [4]. For the case of half-BPS giant gravitons, the gauge theory duals were found in terms of operators associated with Young diagrams [5, 6]. The operators are related to states by the operator-state correspondence in the radial quantization of the conformal field theory. An elegant description of the moduli space of eighth-BPS giant gravitons was found in terms of holomorphic surfaces in [7]. The construction of gauge theory operators associated with this general class of giant graviton geometries has been a long-standing problem.

The correspondence between SYM operators and brane geometries for the half-BPS sector has been particularly illuminating, shedding light on the emergence of the background where the dual strings propagate. The giant gravitons in the half-BPS sector are systems of multiple spherical D3 branes, expanding either in (sphere giants) or in (AdS giants) [8, 9]. The Schur basis for multi-trace operators in the gauge theory constructed from one complex matrix are associated with Young diagrams of (with no more than rows). They offer natural candidates for duals of these brane geometries [5] [6]. Operators with order long columns (length order ) correspond to sphere giants while those with order long rows (length order ) correspond to AdS giants. This basic picture has been confirmed by constructing modifications of the Schur operators, which correspond to attaching strings to the branes [10, 11, 12, 13, 14, 15, 16]. The prescription can be described in terms of “restricted Schur” operators constructed using restrictions of symmetric group representations to their subgroups. It leads to evidence of integrability beyond the setting of the usual planar limit [17, 18, 19]. The open string excitations include vibrations of the branes, which have been studied from the world-volume perspective in [20].

The sector of quarter-BPS or eighth-BPS states, annihilated by 8 or 4 supercharges respectively, is far less understood. The eighth-BPS brane geometries extending in were constructed in [7]: the moduli space of these solutions is the moduli space of intersections of a four-dimensional holomorphic surface in , with . These configurations are more rich than those in half-BPS sector, containing intersecting branes, vibrating branes and other intricate surfaces. There has been significant progress in constructing quarter- or eighth-BPS operators in field theory [21, 22, 23, 24, 25]. However the problem of finding precise duals to these brane geometries is still unsolved.

Eighth-BPS operators in the sector of SYM are constructed
from holomorphic gauge invariant functions of three complex scalar matrices,
subject to the condition that they are annihilated by the
one-loop dilatation operator^{1}^{1}1There is a more general
class of eighth-BPS
operators involving fermionic highest weights which form the sector.
In this paper we will always work in the subsector and
eighth-BPS or simply BPS will refer to this. If desired the considerations can be restricted to the or quarter-BPS sector.. It is in principle possible to do this systematically [24] and calculate an orthogonal basis of eighth-BPS operators for any fixed charges in terms of representation theory. In practice, however, the procedure is computationally difficult due to calculation of Clebsch-Gordan coefficients, and only possible for charges of order . In order to study duals of brane geometries we need a basis at energies of with large , and no such basis has been explicitly constructed.

One description of the eighth-BPS sector in SYM is provided by the chiral ring, where the states built from three chiral scalars are identified up to F-terms. It was used in [26] to calculate the exact spectrum of eighth-BPS states. The number of states as a function of the three charges was found to agree with the counting of states of bosons in a 3D harmonic oscillator. The structure of the chiral ring is, however, not enough to calculate operator two-point functions which provide an inner product. The explicit construction of gauge invariant operators, which are annihilated by the one-loop dilatation operator, would allow the calculation of this inner product and would help find duals of the brane geometries.

The eighth-BPS spectrum can be constructed either by quantizing the moduli space of giant gravitons large in the directions [27] or alternatively by quantizing [28, 29] the giants which are large in the directions [7]. Our main interest in this paper will be the giants which are large in It was argued that the quantization of this moduli space is equivalent to the geometric quantization of a complex projective space. This physical moduli space is related to the moduli space of polynomials in three complex variables, also a projective space, through a non-trivial procedure. This procedure involves, among other things, the shrinking of holes in the moduli space of polynomials associated with polynomials whose zero set does not intersect the . The partition function over the resulting Hilbert space exactly matched the one counting elements of the chiral ring. This construction thus gives additional structure to the states in the chiral ring: it maps them to states in a Hilbert space, which, furthermore, has structure carried over from the moduli space of branes.

In this paper, we initiate a systematic study of the correspondence between quantum states in the geometric quantization of the physical moduli space and explicit eighth-BPS brane geometries. This can be viewed as an intermediate step in connecting quantum states associated to gauge theory operators (by the operator state correspondence) with the geometries.

Let us state that our main focus is not the construction of an overcomplete basis of coherent states associated with arbitrary points on the moduli space, but rather the association of subspaces of moduli space to a complete basis of orthogonal energy eigenstates. We expect that combining the constraints associated with orthogonality, completeness and symmetries can be a powerful guide in finding how gauge theory local operators map to branes in the dual space-time. In studying the map between geometric quantization states and geometries, we find that fuzzy geometry provides the ideal set of tools. A lot of work has been done on fuzzy projective spaces with a view to modeling fuzzy space in string theory or with a view to regulating continuum field theories. We are able to draw on and apply this existing literature to clarify how the oscillator basis corresponds to the geometry of the moduli space of branes.

This allows us to predict some physical properties of various brane configurations, such as the BPS open string spectrum: a check that has been crucial in the study of half-BPS states. The understanding of this Hilbert space lets us make predictions about the structure of Hilbert space of BPS operators in the gauge theory. In particular, we develop a new group-theoretic labelling of the states which relies on the decomposition of the moduli space of giant gravitons according to the degree of the polynomials appearing in the Mikhailov description. From the AdS/CFT correspondence, this geometric and group theoretic labelling will apply equally to the gauge theory construction of operators and can be expected to provide a valuable guide in this construction.

We start the paper in Section 2 by reviewing the eighth-BPS brane geometries found in [7] and their quantization according to [29].

In Section 3 we review some fuzzy geometry techniques and apply them to study the correspondence between Hilbert space and classical geometries. Once we have a space of states , we also have a space of operators forming the endomorphisms . In the case at hand, by considering the space of states coming from quantized projective spaces , the corresponding algebra of operators can be identified with a fuzzy , with fuzziness . We will also recall the structure of as a toric manifold with a fiber over a simplex in . By using fuzzy geometry constructions, we find that the states are uniformly spread out along the tori but localized at points in the simplex. This makes contact with recent literature on fuzzy projective spaces as models of space in string theory [30, 31, 32, 33, 34]. From this point of view, distinguished states lie at the corners (vertices) of the simplex.

In Section 4, we use the connection between BPS states and quantization of projective spaces, to give a group theoretic labelling of the states in terms of representations along with a Young diagram of . States with specified are generically not unique, but all additional multiplicities are described in terms of other group theory data such as Littlewood-Richardson numbers. This is one of our main new results. We discuss the geometry of the above labelling of states in terms of fibration structures of projective spaces and of the simplices in the base of their toric fibrations.

We study the physics of the states at the corners of the base simplex in Section 5. We find that the polynomial equation in the Mikhailov description of these giants is actually a monomial equation, simply setting to zero a monomial. For these corner states, the subtleties of the map between the physical moduli space and the moduli space of Mikhailov polynomials can be obviated by using symmetries. These branes share the property of being static with the familiar maximal giants of the half-BPS sector, although they are not the most general static configurations, these being general homogeneous polynomials. We interpret these corner states described by monomial equations as composites of maximal half-BPS giants with angular momentum in different directions, along similar lines to [35] Hence we will refer to these corner states as maximal giants. We consider the states in the Hilbert space which are near the states for these maximal brane geometries, and disentangle them into a tensor product of bulk closed string excitations and world-volume open string excitations. The spectrum of world-volume excitations is consistent with the interpretation of the brane as a composite of half-BPS maximal giants. It gives specific predictions which should be testable by construction of operators in the gauge theory or by world-volume calculations for branes in the bulk space-time.

In Section 6, we display the tensor product structure of open and closed string excitations in the form of factorization properties of the partition function. A very useful strategy in order to exhibit this in the simplest way is to focus on states which are near the stringy exclusion principle [36] cut-off, i.e states which exist in the Hilbert space for rank , but not for rank . The phenomenon of the stringy exclusion principle [36] and its explanation by the growth of a brane [4] is a remarkable example of how classical geometry explains the disappearance of specific quantum states as the rank of the gauge group is changed. This is in fact one of the key ingredients in the map between Young diagram operators for half-BPS in the gauge theory and brane geometries [5, 6]. It is therefore no surprise that the stringy exclusion principle continues to be illuminating in the eighth-BPS sector.

In Section 7 we consider Hilbert space of “nearby states” directly by going to the Mikhailov’s polynomials, without assuming that the global structure of the physical moduli space is given by projective spaces as argued by [29]. This is done by conducting a local analysis of the symplectic form near the points on the moduli space, corresponding to geometries of interest. We do the case of perturbations around a single giant graviton. We find agreement with the discussion in Sections 5 and 6.

In Section 8, we extend the discussion of the correspondence between states and geometries beyond the maximal branes. This is substantially more subtle, but allows some geometrical understanding of the multiplicities encountered in the analysis of the world-volume excitations of the maximal branes.

In Section 9, we discuss the implications of our results for the construction of gauge theory operators. We observe that some of our results can be interpreted, at a qualitative level, in terms of a complementarity between localization in space-time and localization of branes in space-time. We also consider implications of the lessons we have learned for the broader discussion of states and geometries in the context of bulk deformations of AdS spacetime and black hole physics.

## 2 Review of phase space and quantization

In this section we will review how the phase space of eighth-BPS Mikhailov’s solutions in is described by . This phase space can be geometrically quantized to give a Hilbert space isomorphic to bosons in a three-dimensional harmonic oscillator. The material in this section is largely based on [29] and we refer the reader there for the more complete treatment.

We first describe the moduli space of giant graviton solutions. The 3-brane action gives a symplectic form on this space, which gives it the structure of a phase space. We describe the symplectic form and then use the geometric quantization prescription [37] to build the Hilbert space and operators.

The starting point is the following construction by Mikhailov [7]. We consider D3 branes wrapping surfaces in which preserve 1/8 of supersymmetries (eighth-BPS). Mikhailov showed that all such surfaces can be constructed by taking holomorphic functions in

(2.1) |

and intersecting the four-dimensional surface with the unit five-sphere embedded in . The intersection is generically a three-dimensional surface in on which we wrap the D3 brane. More precisely, the shape of the D3 brane is a time-dependent solution given by polynomial

(2.2) |

That is, the time evolution keeps the shape of the D3 brane fixed, and it just rotates with a phase factor in all coordinates.

For the simplest example take the polynomial

(2.3) |

Then the surface is or time dependent and intersection with is

(2.4) |

This defines a with radius , which is the original half-BPS giant graviton of [4].

We now analyze the phase space^{2}^{2}2Note that surface defines a point in *phase space* rather than just configuration space, because it determines both position and velocity. This is a result of the BPS condition. See, for example, [20].
of such eighth-BPS giants in . Let us first consider the space of holomorphic surfaces in given by^{3}^{3}3We will often abbreviate as , nevertheless, these are always polynomials of three complex coordinates.
.
The points in are labelled by coefficients . In fact, the coefficients are projective coordinates , because multiplying them by a common factor keeps the surface unchanged. It is convenient to regularize the infinite-dimensional space by considering a *finite*-dimensional subspace where only a subset of coefficients are allowed to be non-zero. If is the number of elements in , then we get a space spanned by complex projective coordinates, that is, topologically . For example, we could take for which is the space of linear polynomials (see (2.19) in the next section), topologically . In the end the full can be defined as a limit , where is a sequence which includes ever more monomials . For example, could be all coefficients that multiply monomials of degree up to . The important aspect of this construction is that at every step we are dealing with a complex projective space . The limiting case is .

Next, the intersection of each with is a surface which defines the shape of a D3 brane and therefore labels a point in the phase space . That is, there is a map

(2.5) |

The regularized subspace is mapped to , which is a finite-dimensional subspace of . It is argued that is also . One problem that has to be dealt with is that the map is many-to-one, that is, different polynomials can lead to the same intersection . In fact, it was shown in [29] that two polynomials and have the same intersection with if and only if

(2.6) |

where and do not intersect . Therefore, in order to get the space from , we need to identify

(2.7) |

with any that does not intersect . Note that all polynomials that do not intersect are themselves identified with a single polynomial , which is the vacuum point () in the phase space. It was also shown in [29] that these identifications can be performed smoothly and the resulting space is indeed still . Let us denote the projective coordinates on by , with indices running over the same set . The map then takes the form of functions

(2.8) |

They should be such that whenever points and should be identified.

Let us now turn to the discussion of the symplectic form on the phase space , which is necessary for quantization. The starting point is the world-volume action on a single D3 brane with no world-volume field strength or fermions:

(2.9) |

Here is the induced metric, and is the four-form background gauge field, such that field strength is proportional to volume form. The symplectic form can then be written as

(2.10) |

Now the metric and the induced metric is taken on a unit radius ( is related to by rescaling). This symplectic form is defined on the phase space of *all* configurations of a D3 brane, supersymmetric or not. Space is, of course, much larger than the supersymmetric subspace . The “coordinates” on are fields , whereas is parametrized by “collective coordinates” . In any case, we have a map and the pullback of (2.10) defines a symplectic form on or . In fact, since we have a map we can also take a pullback of on the space of holomorphic polynomials . This pullback will inevitably be degenerate and have singularities, but it can nevertheless be convenient for explicit calculations.

Crucially, it was argued in [29], that not only is topologically , but also that the symplectic form is globally well defined, closed, and in the same cohomology class as .
This implies it is always possible to find such coordinates that the pullback of (2.10) becomes *proportional to the Fubini-Study form*, with coefficient :

(2.11) |

Here and we use shorthand for *inhomogeneous* coordinates on . For example in the patch the index runs over remaining tuples in .

Once we have the phase space manifold as with Fubini-Study form as the symplectic form, the geometric quantization is well known. The Hilbert space is spanned by wavefunctions, which are holomorphic polynomials of the projective coordinates of degree

(2.12) |

or equivalently polynomials of inhomogeneous coordinates of degree up to (if we take e.g. in the patch). It is important to note how enters the definition of Hilbert space purely through setting the scale of , which controls the effective Planck constant or the area in phase space that a single quantum state occupies. As we increase , the area occupied by a state decreases, and we get more states in .

Finally, we need to discuss the conserved charges in the system. There is a natural symmetry acting on the coordinates which preserves the shape of . The Cartan subgroup rotating each coordinate by a phase will give three commuting charges that we can use to label the states. The action induces transformation

(2.13) |

on , as seen from (2.1). Now we also need to use the fact argued in [29] that the map can be done in a invariant way, so that the action on the final phase space coordinates is also . That means we have three vector fields on generated by

(2.14) |

We have used the abbreviation

(2.15) |

Upon geometric quantization these become operators on the Hilbert space

(2.16) |

So that the charge of each excitation is simply under each of the respective , and the total charge of a state is the sum of all excitation charges. Note that the time evolution is given by an overall , generated by Hamiltonian

(2.17) |

This also reflects the BPS condition. Given the charge assignments we can write a partition function over the Hilbert space (2.12)

(2.18) |

The notation denotes the coefficient of , which enforces the degree of wavefunction. This matches the partition function over the chiral ring in , and so reproduces the correct supersymmetric spectrum from quantizing giant gravitons.

A comment needs to be made on the validity of the D3 world-volume action (2.9). It certainly is a good description for large branes of energy , but not for small ones with high curvature. However, the spectrum of BPS gravitons at energies is still correctly reproduced by derived from , and that part of the spectrum comes precisely from very small D3 branes, where should not be valid. This may be a result of the fact that the full symplectic form, corrected for small branes, is still in the same cohomology class as and also invariant.

### 2.1 Example: single half-BPS giant

In order to illustrate various concepts in this section, let us quickly go through an example of linear polynomials. It will also serve as a starting point for further calculations in this paper. Take , then is the space of hyperplanes

(2.19) |

We abbreviate . For inhomogeneous coordinates we set .

Intersection with yields an of radius

(2.20) |

– the same as in (2.4), only -rotated. The energy and momenta of this solution are:

(2.21) |

As typical, the map is not one-to-one, the region does not intersect and so maps to a single point: the vacuum. Good coordinates on as can be constructed by rescaling:

(2.22) |

As explained in detail in [29], this smoothly contracts “the hole” at to a point .

The symplectic form, which in this case can be calculated explicitly using (2.10), takes the following form in coordinates:

(2.23) |

as long as . In coordinates this becomes

(2.24) |

precisely times Fubini-Study form on , with perfectly good behavior at .

This can now be geometrically quantized to a Hilbert space of wavefunctions

(2.25) |

or in terms of only inhomogeneous coordinates (setting )

(2.26) |

The momenta are^{4}^{4}4
Remember (2.16) e.g.

(2.27) |

and total energy

(2.28) |

Note the maximum energy of a state in this is , that of a maximal sphere giant, corresponding to in (2.21).

Finally, let us emphasize one point which will be important later on: (2.24) is written in inhomogeneous coordinates where corresponds to the vacuum point with . But we can equally well take a different coordinate patch in , for example where and parametrize the point. The new inhomogeneous coordinates are expressed in terms of the old ones as

(2.29) |

The symplectic form (2.24) has the same form in terms of . But now the point corresponds to , , which is the maximal giant arising from polynomial

(2.30) |

We can choose to write the wavefunctions in terms of these coordinates

(2.31) |

which is, of course, still the same Hilbert space as in (2.25), with a map . One difference, though, is that now the vacuum has , and the excitations can have negative charges:

(2.32) |

Physically and quanta keep the giant energy the same, just rotate it in , while takes the giant away from maximal by decreasing energy and .

## 3 Fuzzy and giants as points on a simplex

The Hilbert space arising from geometric quantization of
is closely related to *fuzzy* or non-commutative .
We will review this relation and use it to show how
the holomorphic basis of the Hilbert space, is related to
a discretization of the base in a description of
as a toric fibration over a simplex in [30, 31, 32, 33, 34, 38, 39].
The wavefunctions are localized at points on the toric base
and spread out in the torus fibers.
With the fuzzy
technology in hand, we demonstrate this nice geometrical character
of the states, using
elementary calculations of expectation values of
Lie algebra elements and their products, evaluated on states of .
From this point of view, we find that the corners of
the simplex, where the tori degenerate, correspond to distinguished
states. We will return to these states in section (5).
We will show that they correspond to maximal giants, where
the Mikhailov polynomials become monomials.

In the case of half-BPS giant gravitons, this will allow us to relate the Young diagram labels which arise in the construction of corresponding operators in the dual SYM theory, to the coordinates of points in the discretized toric base, which as we will explain is a simplex in . The Young diagram labels have a physical interpretation in terms of brane multiplicities for branes of different angular momenta.

This shows that fuzzy geometry can be a powerful tool in providing a precise connection between quantum states and localization, with its complementary non-locality due to quantum uncertainty, in the moduli space of solutions.

The geometry of the discretized simplex will also play an important role in subsequent sections, where we will develop the states-geometries connection further.

Going beyond the half-BPS sector to the eighth-BPS sector, a complete group theoretic basis including representation labels, along with Young diagram labels, will be developed in Section 4.

### 3.1 Fuzzy from operators on Hilbert space of giant states

The homogeneous coordinates for projective space are . A complete basis for rational functions is provided by

(3.1) |

where . The denominator ensures that these functions are invariant under scaling by a complex number . These functions span the function space for , which we will denote as . This decomposes into irreducible representations of as

(3.2) |

where transforms as an irreducible representation corresponding to the Young diagram with columns of length and columns of length of , which we denote as .

As we reviewed in Section 2, the geometric quantization of giant gravitons for with units of flux, in a sector of polynomials of dimension , leads to a quantization of the moduli space which produces a Hilbert space of holomorphic polynomials of degree . This Hilbert space consists of polynomials of degree in the homogeneous coordinates . It can be viewed as the -fold symmetric tensor product of the fundamental of

(3.3) |

This is also isomorphic to a Hilbert space of oscillators

(3.4) |

with the constraint . Note also that this counting of oscillator states is equivalent to counting of Young diagrams of . The oscillator states characterized by can be mapped to the Young diagrams with rows of length .

The dimension is

(3.5) |

Given this Hilbert space, it is natural to consider the algebra of operators, i.e the endomorphism algebra . The decomposition into representations of is

(3.6) |

where transforms as the irreducible representation described above. A basis for is given by operators

(3.7) |

or in oscillator language

(3.8) |

The indices on the oscillators range from to , and

(3.9) |

It is clear that the operators in (3.8) have a cut-off at , since polynomials of degree will be annihilated by more than derivatives. The matrix algebra provides a finite dimensional approximation to with an invariant cutoff at . The case has been much studied as a model for fuzzy brane world-volumes, fuzzy spatial directions for Kaluza-Klein reduction in F-theory and elsewhere and as a model for quantum field theory with a UV cutoff that preserves spatial symmetries. Here we are finding the fuzzy in the set-up of quantizing a moduli space of giant gravitons.

The algebra is generated by operators

(3.10) |

or in oscillator language

(3.11) |

These form a basis for the algebra of

(3.12) |

The traceless generators

(3.13) |

form the Lie algebra of .

Using (3.9) along with the constraint

(3.14) |

we may also obtain the relations

(3.15) |

We also have

(3.16) |

If we define

(3.17) |

we have relations

(3.18) |

These relations simplify at large . The generate a commutative algebra. We have

(3.19) |

There is a homomorphism from these generators of to given by

(3.20) |

where . The homomorphism property is easily established by verifying that the functions on the RHS of (3.20) obey relations (3.19). At finite , the algebra is a fuzzy deformation of . This can be made precise by using the map to define a star product on the classical algebra [30] [39] [31].

#### 3.1.1 Toric geometry of from the Lie algebra embedding

The coordinates give a description of as embedded in which is the Lie algebra of . This is an example of a general construction of co-adjoint orbits [30]. Another aspect of the geometry of will be of interest to us, namely the fact that it is a toric variety. Let us describe this in the cases of , which generalizes to the general case.

Given the homogeneous coordinates , we can impose the equivalence by first setting

(3.21) |

and then modding out by a phase . This shows that is the base space of a fibration of with fiber.

Consider the case where we have . Lets us recall the toric description [40]. Keeping in mind that , we can consider the quadrant parametrized by coordinates . The allowed values of fall inside a triangle with vertices . For each chosen point inside the triangle, there is, in the , a of phases given by collapses on the vertical axis (), the one parametrized by on the horizontal axis (), and the combination collapses on the line . See Figure 1. . The cycle parametrized by

This generalizes straightforwardly to . The toric description has a base space which is a generalized tetrahedron or simplex in (for more on simplices see [41]). There is a fiber related to angles of (modulo the overall ).

The identification

(3.22) |

from (3.20) shows that the diagonal are equal to the coordinates used to parametrize the toric base. The off-diagonal are sensitive to the angles. Their magnitudes are completely determined once the diagonal generators are known since

(3.23) |

We can write

(3.24) |

Hence, the off-diagonal elements of the Lie algebra are associated with the angular variables of the toric description and the diagonal ones with the base space.

#### 3.1.2 Giants: points on toric base and delocalized on fiber

For a state described by the monomial