We provide a heuristic explanation for the emergence of worldsheet fermions
in the continuum limit of some matrix models. We also argue that turning on
Ramond-Ramond flux confines the fermionic degrees of freedom of the
Ramond-Neveu-Schwarz formalism.Comment: 7 pages, late

Kapustin, Anton

English

arXiv

We provide a heuristic explanation for the emergence of worldsheet

fermions in the continuum limit of some matrix models. We also argue

that turning on Ramond-Ramond flux confines the fermionic degrees

of freedom of the Ramond-Neveu-Schwarz formalism

Caltech Authors - Main

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A remark on worldsheet fermions and
double-scaled matrix models
Anton Kapustin
California Institute of Technology, Pasadena, CA 91125, U.S.A.
January 1, 2014
Abstract
We provide a heuristic explanation for the emergence of worldsheet
fermions in the continuum limit of some matrix models. We also argue
that turning on Ramond-Ramond flux confines the fermionic degrees
of freedom of the Ramond-Neveu-Schwarz formalism.
1
It has been realized recently that certain noncritical Type 0 superstring
backgrounds admit a nonperturbative description by means of double-scaled
matrix models [1, 2, 3]. Namely, Type 0B backgrounds with cˆ ≤ 1 are
described by two-cut Hermitian matrix models, while Type 0A backgrounds
with cˆ ≤ 1 are described by complex (and in general rectangular) matrix mod-
els. The evidence given in Refs. [1, 2, 3] is circumstantial: it has been shown
that many observables in the matrix and worldsheet descriptions agree. It
would be very interesting to find a direct argument for the emergence of
worldsheet fermions and worldsheet supersymmetry in the continuum limit
of these matrix models. On one hand, it is far from clear that this goal is feasi-
ble: after all, Type 0 strings are described by GSO-projected superconformal
field theories, which do not have any fermionic states in the spectrum. In
other words, one may regard worldsheet fermions and worldsheet supersym-
metry as an artefact of a particular continuum description of Type 0 strings,
and it is entirely possible that the matrix models discussed in Refs. [1, 2, 3]
discretize some other continuum description of the same string theories. On
the other hand, the RNS formalism is the only known continuum description
of noncritical superstrings, and it is natural to try to interpret the matrix
model fatgraphs in RNS terms. The main benefit of such an interpretation
would be a qualitative understanding of the effects of the Ramond-Ramond
(RR) flux on the worldsheet degrees of freedom in the RNS formalism. We
will argue below that turning on a RR flux leads to a confinement of the
fermionic degrees of freedom.
In this short note we present a heuristic picture of how the fermions
emerge from the matrix models. Previous discussions of this issue can be
found in Refs. [4, 5]. We will show that our picture explains some features
of the matrix-model/superstring duality in a natural way. At the very least,
this should be regarded as evidence that the matrix model fatgraphs provide
a discretization of the RNS worldsheet.
We will mostly focus on a special case: Type 0B superstring with cˆ = 0.
The corresponding matrix model is a large-N Hermitian matrix integral with
a symmetric double-well potential. The saddle point with equal numbers of
eigenvalues in the two minima of the potential corresponds, in the simplest
double-scaled limit, to the simplest Type 0B backgrounds: N = 1 super-
Liouville theory coupled to bc and βγ ghosts. The Liouville field and the
bc ghosts describe the metric on the Riemann surface; in the discretized
approach we expect them to be replaced by random fatgraphs. On the other
hand, local supersymmetry on a lattice is problematic, so we will treat the
1
superpartner of the Liouville field, the Mayorana fermion ψ, as well as the
βγ ghosts, as matter fields. On our very heuristic level of discussion, we
will ignore the βγ ghosts and try to “see” the Mayorana fermion ψ in the
structure of the fatgraphs of the Type 0B matrix model.
When constructing the diagrammatic expansion of the matrix model par-
tition function, it is important to expand about the correct vacuum. In the
case of the double-well potential V (x) with minima at x = a and x = −a,
this means that the eigenvalues must be localized near ±a, and (for vanishing
RR flux) the number of eigenvalues in the two minima must be the same.
To accomplish this it is natural to partially gauge-fix the U(N) symmetry
by requiring the matrix to be block-diagonal, with blocks of size M ×M ,
M = N/2 [6]. As usual, this leads to ghosts, which in the present con-
text are M ×M matrices C1, C2, B1, B2 with Grassmann-odd entries. The
gauge-fixed action is [6, 5]
S(X, Y,B, C) = Tr (V (X − a) + V (Y − a)− 2a(B1C1 −B2C2)
+B1XC1 − B2XC2 + Y B1C1 − Y B2C2) . (1)
Here X and Y are M × M matrices describing eigenvalues localized near
a and −a, respectively. The residual gauge symmetry is U(M) × U(M).
The diagrammatic expansion of the corresponding matrix integral leads to
fatgraphs with the following additional structure. There are several types of
vertices: the ones coming from V (X), which we color black, the ones coming
from V (Y ), which we color white, and four kinds of vertices involving the
BC matrix ghosts. X and Y propagators cannot connect vertices of different
color, but they can terminate at “ghost” vertices. Also, the ghost number
is conserved, so the ghost propagators (which can be represented by double-
lines, just as the X and Y propagators) form closed loops. These loops
separate regions with only black (X) vertices from regions with only white
(Y ) vertices. Thus each Feynman diagram is a random fatgraph discretizing
a Riemann surface plus some loops on it which separate black and white
domains. This picture was recently discussed in Ref. [5], where a similar
description of the Type 0A complex matrix model was also given.
Recall now that the Mayorana fermion can be discretized by means of
the Ising model. More precisely, the continuum limit of the Ising model with
periodic boundary conditions at the critical temperature is described by a
massless Mayorana fermion with a Type 0B GSO projection. The necessity
of the Type 0B GSO projection can be seen intuitively as follows [7]. Each
2
state in the Ising model can be represented by a collection of domains on
the lattice where the spins are pointing “up”. In the complement of these
domains the spins are pointing “down”. The fermion loops are essentially
the boundaries of these domains. The change from the spin variables to
the fermionic variables is possible because knowing the boundaries of the
domains completely specifies the spin configuration. One potential problem
with this argument is that on a Riemann surface of nonvanishing genus there
exist closed loops which are homologically nontrivial and therefore are not
boundaries of domains. Contributions of such loop configurations must van-
ish or be removed “by hand”. The Type 0B projection achives precisely this.
For example, in the case of a torus, imagine a loop which winds once around
the a-cycle, which we will regard as the “time” direction. By cutting the
torus along the b-cycle, we see that this contribution to the partition func-
tion comes from a state with a single fermion. GSO projection removes this
configuration from consideration.
In view of the above, it is tempting to conjecture that the worldsheet
fermion ψ is the fermionization of the black and white domains on the fat-
graphs of the Type 0B matrix model. That is, the BC ghost loops can be
identified, in the continuum limit, with ψ loops.
We stress that at present this is merely a suggestive analogy, because the
Ising model coupled to gravity is certainly not equivalent to the noncritical 0B
superstring. In the remainder of this note we will give a couple of examples
showing that our heuristic picture has some explanatory power, so perhaps
it is more than a mere analogy. A proper derivation of the RNS worldsheet
theory from the matrix model fatgraphs remains to be found; it would have
to explain where the βγ ghosts come from. Hopefully, this can be achieved
by studying in detail the difference between the fatgraphs of the 0B matrix
model and the Ising model coupled to gravity.
It was argued in Ref. [3] that the response of the Type 0B string to RR
flux is qualitatively different depending on the sign of the super-Liouville
coupling constant µ. This coupling multiplies the term
ψ+ψ−e
φ (2)
in the continuum action. For µ > 0 every term in the perturbative expansion
of the free energy is analytic in the RR flux q, while for µ < 0 half-integral
powers of q appear. Thus only for µ > 0 is expansion in the number of RR
insertions well-defined. In the case of 0B string with cˆ = 0 the RR vertex
3
operator in question has the form
V = e−
ϕ+ϕ¯
2 σ,
where ϕ and ϕ¯ are the right and left-moving bosonized superconformal ghosts,
and σ is one of the two twist operators for ψ (the other one being removed
by the GSO projection).
A similar distinction arises in the Ising model picture. Recalling that eφ
is the conformal scale of the worldsheet metric, we see that the term Eq. (2)
is the mass term for the fermion ψ. Thus the super-Liouville coupling µ cor-
responds to the deviation of the Ising temperature from criticality. Ignoring
the βγ ghosts, V ∼ σ is simply a twist operator for the fermion ψ. There
are actually two fermionic twist operators, σ and σ˜. In the Ising language,
σ is the spin operator which lives on the vertices of the lattice, while σ˜ is
the so-called disorder operator which lives on the vertices of the dual lattice.
The disorder operator is defined as follows [8]. One draws a contour L on
the dual lattice which begins at the point where we wish to insert a disorder
operator and runs off to infinity (or to the insertion point of another disorder
operator). Then one flips the sign of the Ising coupling constant on each bond
of the original lattice intersected by L. It is easy to see that the choice of
L is immaterial. It is also easy to see that in the ordered (low-temperature)
phase the disorder operator creates a defect line. Thus the correlation func-
tion of two disorder operators in the low-temperature phase will be of order
e−ℓ/a where ℓ is the distance between the insertion points and a is the lattice
cut-off. In other words, in the low-temperature phase the disorder operator
is completely screened at macroscopic scales. On the other hand, in the high-
temperature phase it is the spin operator which has exponentially decreasing
correlators. In fact, the Kramers-Wannier duality exchanges the high and
low-temperature phases and the spin and disorder operators. If we consider
Ising spins on a compact surface, Kramers-Wannier duality is broken. For
example, with periodic boundary conditions the state corresponding to the
disorder operator is not in the spectrum of the critical Ising model, for any
µ. This truncation is equivalent to the 0B GSO projection on the Mayorana
fermion. As explained above, the deviation of the Ising temperature from
criticality is proportional to µ. Thus the spin operator σ is screened for one
sign of µ and has a long-range order for the other sign.
To determine which phase of the Ising model corresponds to which sign
of µ, we recall [3] that µ > 0 corresponds to the situation when the resolvent
4
of the matrix model has two cuts, while for µ < 0 they merge into a single
cut. Since the cuts of the resolvent indicate the location of the eigenvalues of
the matrix, we conclude that for µ > 0 the eigenvalues are well localized near
the minima of the potential, and therefore the fatgraphs are in the ordered
(low-temperature) phase.
To provide more evidence for this identification, consider turning on the
RR flux for µ > 0. On one hand, the RR vertex operator is basically the
twist operator for the fermion, which we have identified with the spin oper-
ator of the Ising model. Adding a spin operator to the Ising action amounts
to turning on a magnetic field. It is well-known that this perturbation is
relevant and gives in the contunuum a rather complicated (but for µ = 0 still
integrable [9]) massive field theory. The Mayorana fermion is not among the
elementary excitations of this theory, because for it is confined by the mag-
netic field. Indeed, the magnetic field favors spins of a particular orientation,
which means that in the ground state all spins have the same orientation.
The contribution of a “droplet” of spins with the opposite orientation to
the partition function is suppressed by e−hA, where A is the area of the
droplet and h is the magnetic field. This means that the contribution of a
fermion loop will be exponentially suppressed by the area of the domain that
it bounds, i.e. fermions are confined.
On the other hand, in the matrix model turning on the RR flux is achieved
by making the numbers of eigenvalues localized in the two minima unequal [2,
3]. In order to have M1 eigenvalues in one minimum of the potential and M2
eigenvalues in the other minimum, one gauge-fixes the matrix so that it is
block-diagonal with blocks of size M1×M1 and M2×M2 [6, 5]. The gauged-
fixed action is given by the same equation Eq. (1), where now X and Y
have sizes M1×M1 and M2×M2, respectively, the B-ghosts are rectangular
matrices of size M2 ×M1, and the C-ghosts are rectangular matrices of size
M1 ×M2. The RR flux is proportional to M1 −M2. Suppose M1 > M2.
Then, according to the usual t’Hooft counting rules, each face of a black
domain will contribute eM1 to the partition function, while each face of a
white domain will contribute eM2 . Thus the contribution of white domains
will be suppresed by a relative factor of order e−(M1−M2)A, where A is the area
of the white domains. Therefore for M1 6= M2 the BC ghosts are confined,
in agreement with our identification of ψ loops with the BC loops. This also
confirms that the two-cut (µ > 0) phase of the matrix model corresponds to
the ordered (low-temperature) phase of the Ising spins.
We end this note with the following remark. In the limit of large RR
5
flux it appears that Type 0B cˆ = 0 string theory reduces to a noncritical
bosonic string. Indeed, one of the cuts in the eigenvalue density recedes to
infinity in this limit, so that only one cut remains near the maximum of the
potential, and this situation should be describable in the continuum by a
bosonic string. Our heuristic picture provides the following explanation for
this. As one turns on the RR flux, while keeping the string coupling fixed, the
fermions become confined by the worldsheet magnetic field and decouple. We
speculate that more generally turning on RR flux confines fermionic degrees
of freedom in the RNS formalism, leading to a bosonic string in the limit of
large RR flux.
The author would like to thank Jongwon Park for useful discussions and
Juan Maldacena for comments. This work was supported in part by the DOE
grant DE-FG03-92-ER40701.
References
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tinec and N. Seiberg, “A new hat for the c = 1 matrix model,”
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[3] I. R. Klebanov, J. Maldacena and N. Seiberg, “Unitary and complex
matrix models as 1-d type 0 strings,” arXiv:hep-th/0309168.
[4] A. Kapustin and A. Murugan, “Fatgraph expansion for noncritical su-
perstrings,” arXiv:hep-th/0404238.
[5] D. Gaiotto, L. Rastelli and T. Takayanagi, “Minimal superstrings and
loop gas models,” arXiv:hep-th/0410121.
[6] R. Dijkgraaf, S. Gukov, V. A. Kazakov and C. Vafa, “Perturbative
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[7] A. M. Polyakov, “Gauge fields and Strings”, Harwood (1987).
6
[8] L. P. Kadanoff and H. Ceva, “Determination Of An Operator Algebra
For The Two-Dimensional Ising Model,” Phys. Rev. B 3, 3918 (1971).
[9] A. B. Zamolodchikov, “Integrals Of Motion And S Matrix Of The
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7


A remark on worldsheet fermions and double-scaled matrix models

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