An Index for Intersecting Branes in Matrix Models

An Index for Intersecting Branes in Matrix Models^{⋆This paper is a contribution to the
Special Issue on Deformations of Space-Time and its Symmetries.
The full collection is available at http://www.emis.de/journals/SIGMA/space-time.html}

Harold STEINACKER and Jochen ZAHN

H. Steinacker and J. Zahn

Fakultät für Physik, Universität Wien, Boltzmanngasse 5, 1090 Wien, Austria \Email,

Received September 17, 2013; Published online November 08, 2013

We introduce an index indicating the occurrence of chiral fermions at the intersection of branes in matrix models. This allows to discuss the stability of chiral fermions under perturbations of the branes.

matrix models; noncommutative geometry; chiral fermions

81R60; 81T75; 81T30

## 1 Introduction

The IKKT or IIB model [8] admits solutions which can be interpreted as branes embedded in a flat target space, cf. [11, 12] for recent reviews. Of particular interest is the case of intersecting branes, as these can give rise to chiral fermions living on the intersection [6], thus having the potential of yielding a physically viable model. The aim of this note is to show that the occurrence of chiral fermions can be rephrased as the non-vanishing of a certain index, which counts the number of zero modes of the Dirac operator, weighted with their chirality. Moreover, conditions ensuring the stability of the index under perturbations are given. This is also demonstrated in a concrete example.

Let us explain the relation of the index we propose with different indices discussed in the context of emergent geometries in matrix models. In [2], two-dimensional compact branes embedded in are studied. In our language, the intersection of such branes with a point is considered, and the index counts the difference of the number of positive and negative modes of the corresponding Dirac operator. The set of ’s where the index changes is then interpreted as the locus of the brane. Crucial differences to our setting are the odd dimension of the target space (so that there is no chirality operator and eigenvalues of the Dirac operator are not symmetric around ), and the restriction to finite matrices.

Another index for noncommutative branes was considered in [1]. The difference to our definition is the usage of another Dirac operator, the so-called Ginsparg–Wilson Dirac operator, which does not coincide with the Dirac operator appearing in the IKKT action.

The article is structured as follows: In the next section, we recall aspects of the matrix model framework and its effective (brane) geometry. We introduce the notion of intersecting branes, and introduce the index indicating the occurrence of chiral fermions. In Section 3, we present conditions guaranteeing the stability of the index under deformations, and discuss a concrete example. We conclude with a brief summary and an outlook.

## 2 Matrix models, intersecting branes, and chiral fermions

We briefly collect the essential ingredients of the matrix model framework and its effective geometry, referring to the recent review [12] for more details. The starting point is the maximally supersymmetric IKKT or IIB model [8], whose action is given by

Here the are Hermitian matrices, i.e., operators acting on a separable Hilbert space . The indices run from to , and will be raised or lowered with the invariant tensor of . Furthermore, is a matrix-valued Majorana Weyl spinor of , and is the Dirac operator, defined by

where are the -dimensional matrices.

Even though this will not be used explicitly, the picture to have in mind is that the matrix configurations describe embedded noncommutative branes. By this one means that the can be interpreted as quantized embedding functions [12] of a -dimensional submanifold . More precisely, there should be some quantization map which maps classical functions on to a noncommutative (matrix) algebra, such that commutators can be interpreted as quantized Poisson brackets. The are then the image of classical embedding functions under this map. For more details, we refer to [12].

If the matrices are of block-diagonal form

we speak of two intersecting branes. If we analogously split the fermions as

then the Dirac operator acts on the off-diagonal components as

We will consider the case when, in the semiclassical picture, the two branes are of the form

(1) |

where is even and is embedded in the subspace of generated by , and are embedded in the directions spanned by , . Furthermore, the symplectic form on is required to respect the split (1), i.e., it should vanish for one vector in and one in . In formal terms, this means that

(2) |

Here labels the indices , whereas labels . Furthermore,

(3) |

and, under this isomorphism, . This encodes the requirement that the two branes share a common -dimensional brane. Using the identification of and , we may write the Dirac operator as

We also split the chirality operator (note the different signs in and stemming from the signature of ),

and remark that it fulfills

and

We also note that the matrices may be represented as

(4) |

where the form the -dimensional Lorentzian Clifford algebra, is the corresponding chirality operator, and the form the -dimensional Euclidean Clifford algebra.

Given that the are represented on Hilbert spaces , the off-diagonal
fermions are elements of .
Due to the split (2), a general ansatz for solutions of
is^{1}^{1}1Note that the condition (2) is crucial here.
In [10], the same ansatz for is used, but (2) is not
fulfilled.
Hence, in that work, the ansatz is not general enough to find all solutions of the Dirac equation.

where we defined

Here we used (3) and the same factorization of the spinorial representation space as in (4). Using the operator norm, can be given the structure of a Banach space. Due to (4), have

where and are anticommuting operators on . In particular, non-zero eigenvalues of come in pairs , which are interchanged by , and whose eigenvectors may be combined to eigenvectors of of opposite chirality. It is then clear that given an eigenvector of with eigenvalue , the Dirac equation for becomes, cf. (2),

which does not admit chiral solutions unless .
Furthermore, given a zero mode , the chirality of w.r.t. determines , i.e., the -dimensional chirality of , by the total chirality constraint .
Hence, a -dimensional chiral fermion requires a zero eigenvector of with no
corresponding eigenvector of opposite chirality^{2}^{2}2Otherwise, their combination will in general
acquire a mass through quantum corrections.. Note that if we have a chiral fermion in the sector, then
the Majorana condition ensures that the sector contains the conjugate fermion with opposite chirality.

By our assumptions, is a Dirac operator on the intersection of branes with Riemannian signature, so we may expect it to have discrete spectrum (in Section 3.1, this is shown to be the case in a concrete example). The above discussion then motivates the following definition of an index for the Dirac operator :

Here is some closed curve that encircles the origin and does not intersect an eigenvalue of , and is the orthogonal projector on the eigenspaces whose eigenvalues are encircled by . As discussed above, nonzero eigenvalues of occur in pairs of opposite chirality, so the definition is independent of the choice of . The index counts the number of 0 eigenmodes, weighted with their chirality. This index can also be written in the form

for generic , which is analogous to the usual definition of the index on compact Riemannian spaces, cf. [4, Theorem 3.50].

The motivation for introducing an index to describe chirality is that it takes discrete values, so by continuity, one would expect it to be constant under deformations of the branes. In the next section, we will discuss criteria which indeed ensure this.

## 3 Deformation stability of chiral modes

Let us begin by recalling a notion from perturbation theory. Let be a closed, in general unbounded operator on a Banach space. Then is -bounded, if , and there are positive constants , such that

holds for all . A straightforward consequence of [9, Theorem IV.3.18] is now the following: {proposition} Let have discrete spectrum. Given a closed curve in that encircles a finite part of the spectrum, we define the projector on the corresponding eigenspaces. Given an -bounded operator , the map is norm-continuous in a small enough neighborhood of .

Now fix some . By the above proposition and the fact that takes discrete values, one easily obtains precise conditions that ensure the invariance of the index under perturbations of the : {proposition} Let be self-adjoint and the corresponding Dirac operator. Assume that and are bounded. Then there is a neighborhood of such that

for all . {remark} If are finite-dimensional, then has finite even dimension. It is then no longer necessary to restrict the trace to a finite number of eigenvalues, so one can dispose of the projector in the definition of . It follows that for finite-dimensional representation spaces (corresponding to compact branes), the chirality index always vanishes.

### 3.1 An example

Up to now, the discussion was generic, in particular independent of the commutation relations of the . Let us now consider the concrete example of intersecting Moyal planes (recall that a Moyal plane is defined by canonical commutation relations , with a real antisymmetric matrix). Take , and let span two 2-dimensional orthogonal Moyal planes, i.e.,

where and are the canonical position and momentum operators on . As shown in [6] (and also below), the index of this configuration is . It is easy to see that

(5) |

where

This operator acts on , where we use that

As the first four terms on the r.h.s. of (5) form a positive definite quadratic form,
it follows from the above proposition that the index is invariant under perturbations for small enough , if the are bounded or
linear (corresponding to intersections at angles^{3}^{3}3For intersections at angles in the context of
string compactifications, cf. [3, 5, 7].) in
the (or a sum of such contributions).
In order to see this explicitly, let us consider the case where the are linear in the .
As an example, consider

For the square of the Dirac operator, one obtains

Here acts on , while acts on the spinorial representation space . To have a zero eigenvector of requires a pair of eigenvectors of and which add up to zero. Let us compute the lowest eigenvalue of . We use the ansatz

The eigenvalue equation then leads to

the eigenvalue being given by . It is straightforward to find the eigenvalue . For the eigenvalues of the spinorial part , one finds

Hence, there is exactly one way to cancel the eigenvector of , i.e., there is one eigenvector of with eigenvalue (the higher eigenvalues of can obviously not lead to further zero eigenvalues). One can also explicitly check that it has positive chirality. Analogously, one can treat the dimensional intersection of a - and an -dimensional brane, and similar configurations [6].

An example of intersecting branes with a vanishing index is provided by a degenerate intersection of two quantum planes, such as

In this case the part of that is quadratic in the coordinates of the quantum plane is given by

which is not a positive definite quadratic form. In particular, the condition of being bounded is not fulfilled for rotations of this plane. One easily checks that the index for this configuration vanishes. This underlines the necessity of spanning the full in order to get chiral fermions, as already pointed out in [6].

## 4 Summary and outlook

We presented a definition of an index describing the occurrence of chiral fermions on intersecting branes
in matrix models and discussed the stability of this index under perturbations.
In particular, this implies the existence of chiral fermions for branes intersecting at angles.
The drawback of our approach is that it requires strong restrictions on the embedding, in
particular (2).
It is for example not applicable for situations in which (in the semiclassical picture) the brane
is not flat.
One possibility to treat this case could be to work in the semiclassical limit, or to use a modified
chirality operator, like^{4}^{4}4This particular operator has the disadvantage that it does in general not
anticommute with the Dirac operator, but it may be useful nevertheless.

for a -dimensional brane. We plan to come back to this issue in future work.

As noted in Remark 3, the index always vanishes for intersections of compact fuzzy spaces
.
This raises an apparent paradox, since the results on chiral fermions on intersections should apply at
least approximately for each intersection.
What happens is that pairs of “almost-localized” fermionic near-zero modes arise on the intersections
, such that for each ‘‘effectively’’ chiral fermion localized on
some intersection, there is another fermion with opposite chirality at some other
intersection^{5}^{5}5This is verified in numerical simulations.. This means that if, e.g., the chiral
fermions of the standard model arise from some intersections such as in [6],
there are additional sectors with fermions of opposite chirality localized at different intersections.
The approximate localization on different intersections suggests that these unwanted sectors could be
effectively hidden or removed in some way.
A natural strategy to achieve this is to give up the product ansatz (2), as proposed
in [10], and as realized, e.g., by solutions with split
noncommutativity [13].
These are interesting directions for further research.

### Acknowledgments

This work was supported by the Austrian Science Fund (FWF) under the contract P24713.

[1]Referencesref

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