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arXiv:1002.3638v1 [hep-th] 18 Feb 2010
2d Wall-Crossing, R-twisting, and a
Supersymmetric Index
Sergio Cecotti1∗, and Cumrun Vafa2†
1Scuola Internazionale Superiore di Studi Avanzati via Beirut 2-4 I-34100 Trieste, ITALY
2Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA
Abstract
Starting from N = 2 supersymmetric theories in 2 dimensions, we formulate a
novel time-dependent supersymmetric quantum theory where the R-charge is twisted
along the time. The invariance of the supersymmetric index under variations of the
action for these theories leads to two predictions: In the IR limit it predicts how
the degeneracy of BPS states change as we cross the walls of marginal stability. On
the other hand, its equivalence with the UV limit relates this to the spectrum of the
U(1) R-charges of the Ramond ground states at the conformal point. This leads to a
conceptually simple derivation of results previously derived using tt* geometry, now
based on time-dependent supersymmetric quantum mechanics.
February, 2010
∗e-mail: cecotti@sissa.it
†e-mail: vafa@physics.harvard.edu
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Contents
1 Introduction2
2The Basic Setup5
3A Time-Dependent SQM: Formulation of an Index7
3.1 The index Ik. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parisi–Sourlas supersymmetry . . . . . . . . . . . . . . . . . . . . . .
7
3.210
3.3 Time-dependent supersymmetry . . . . . . . . . . . . . . . . . . . . .12
3.4 The underlying time–dependent TFT . . . . . . . . . . . . . . . . . .13
4Evaluation of the Index 14
4.1 UV limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
4.2 IR limit: path–integral analysis . . . . . . . . . . . . . . . . . . . . .15
4.3Extensions to Two–dimensions and More General N = 2 Theories . .19
5Conclusion 21
A The case of collinear vacua22
A.1 The 1/n rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
B Evalutation in the Schroedinger picture: Links to tt* Geometry24
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1Introduction
Supersymmetric quantum field theories often lead to exactly computable quantities.
This arises from the fact that quantum corrections for many amplitudes can be
shown to vanish. A well known example of this is the Witten index which counts
the net number of ground states of the theory. With more supersymmetry there
are other quantities that can also be computed exactly. For example with N = 2
supersymmetry in d = 2 at the IR limit one can compute both the mass and the
spectrum of BPS states, or at the UV limit one can compute the spectrum of R-
charges of Ramond ground states exactly. In particular the masses of the BPS state
interpolating between vacua a and b is given by
mab= |Zab|
where Zab∈ C is the central charge of the supersymmetric algebra in the ab sector
(for an N = 2 Landau-Ginzburg theory Zab= Wa−Wbwhere Wadenotes the value
of the superpotential at the critical point corresponding to the a-th vacuum). The
number of such BPS states is given by an integer Nab (which can be positive or
negative, depending on the fermion number of the kink).
There is an interesting subtlety in this ‘exact’ prediction of BPS masses and
degeneracies: As we change the coupling constants of the theory it can happen
that the phases of central charges of BPS states can align. Passing through such a
configuration, the number of BPS states can jump. In particular suppose Zaband
Zbcalign. Then the number of BPS states in the ac sector jumps according to
Nac→ Nac± NabNbc
where the ± in the above formula depends on the orientation of the crossing of the
phases. This fact was derived in the case of LG theories by explicit computation [1]
and can be explained in full generality using the continuity of contribution of BPS
particles to certain computation as one crosses the wall [2].
On the other hand one can consider the UV limit of such theories, where the BPS
masses go to zero, and we obtain a conformal theory. One would naturally ask what
is the conformal theory imprint of the solitons and their degeneracies. It was shown
in [1] by studying the solutions to tt* equations and comparing the corresponding
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monodromy data from UV to IR that the spectrum of the N = 2 U(1) R-charges of
the Ramond ground states are captured by the soliton degeneracies1. In particular
for each sector ab consider the operators acting on the space of vacua given by
Sab= I + NabTab
(1.1)
where I is the identity operator, and Tab is the ‘upper/lower triangular’ operator
which takes the a-th vacuum to the b-th vacuum, and Nab is the net number of
solitons in that sector. One considers
M = T
??
ab
Sab
?
(1.2)
where the product is ‘T’ ordered, in the sense of ordered according to phases of Zab.
Then
eigenvalues of M = {exp(2πiqa)}
or, for k integer,
Ik= Tr Mk=
?
i
exp?2πikqa
?
(1.3)
where qaare the R-charges (i.e. N = 2 qa= qaL= qaRcharges) of the Ramond
ground states at the conformal point.2This relation can be used to derive the BPS
jumping phenomenon: The reason is that the RHS is fixed. Thus the LHS should
also remain fixed. However, as we change the couplings of the theory, the central
charges may change and when the phases of the central charges reorder, the fact
that the monodromy does not change implies that the BPS numbers should change
so that
SabSacSbc= SbcS′
acSab
where S,S′denote the corresponding operators before and after the wall-crossing
respectively. This leads to the degeneracy changing formula given above.3
1An alternative derivation of this result using D-branes was given in [5].
2For an N = 2 CFT in d = 2 to have deformations with mass gap all the ground states of the
Ramond sector should have equal left- and right-moving U(1) charges.
3This formula was originally derived under the assumption that only two central charges align
at the same time. The argument in the present paper does not require that assumption, since it is
directly expressed in terms of the group element Sφ,φ′, namely the phase–ordered product of the
contributions from all BPS states having phases in the interval (φ′,φ), which will be interpreted
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The basic aim of this paper is to formulate a supersymmetric path-integral which
computes Ik, and use the invariance of the path-integral, upon deformations, to get
from the R-charge computation in the UV to the soliton counting problem in the IR.
The form of (1.2) is very suggestive: The phase ordered product should arise as a
time ordered product in the physical theory, as was recently proposed in the context
of 4d wall-crossing [4]. On the other hand the phase of the BPS states correspond,
at the conformal point, to the R-charge rotation. Since R-charge is conserved at the
conformal point, it suggests that we compute the path integral with the insertion of
exp(2πikR)
Indeed if we compute
Tr
?
(−1)Fg exp(−βH)
?
where g is a symmetry (commuting with H) and which also commutes with the
supersymmetry charges, then as argued in [8] it receives contributions only from the
ground states, and moreover it does not change under compact deformations of the
theory commuting with g. This motivates one to consider in our case
g = (−1)kFexp(2πikR)
The inclusion of (−1)kFis to make sure supercharges, which have ±1/2 charge under
R, commute with g. Indeed at the conformal point this would precisely give us Ik
defined above (noting that the Ramond ground states have zero fermion number
because F = qL− qR).
Ik= Tr
?
(−1)(k+1)Fexp(2πikR)exp(−βH)
?
However this definition does not quite work away from the conformal fixed points, be-
cause R is not a symmetry! Thus our main challenge is to define an index away from
the CFT fixed points which reduces to the above definition at the fixed points. The
fact that the time ordered expression we are after is making jumps at specific times
(corresponding to phases for which BPS masses exist) in the IR suggests that we are
dealing with a time-dependent evolution operator. In other words it is not going to
in section 4 as a quantum evolution kernel. This is important since the situations encountered in
N = 2 theories in 4d involve the case where infinitely many phases align at the same time.
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be a conventional supersymmetric theory whose Hamiltonian is time independent.
The idea that this can be done is motivated by the simple observation that, if we
consider deformations of a conformal theory, we can always promote the parameters
to be time-dependent, thus compensating for the violation of R-symmetry. The re-
sulting supersymmetry turns out to be a far–reaching time–dependent generalization
of the Parisi–Sourlas one [10–13].
The organization of this paper is as follows: In section 2 we formulate the basic
setup in the context of Landau-Ginzburg theories. In section 3 we formulate a novel
supersymmetric system which is time-depedent. In section 4 we evaluate the path-
integral in the UV and IR limits and recover the formula (1.3). In section 5 we
present our conclusions. In Appendix A we discuss some subtleties arising in the
cases where many classical vacua are alligned in the W–plane. In Appendix B we
discuss some connections with tt* geometry.
2 The Basic Setup
We will be interested in 2 dimensional theories with N = 2 supersymmetry with
isolated vacua and mass gap. In such theories one can consider Hilbert spaces Hab
corresponding to real line, where on the left we approach the vacuum a at infinity
and on the right the vacuum b. The supersymmetry algebra will in general have a
central charge in this sector represented by the complex number Zab. This means
that
Eab≥ |Zab|
where Eabis the energy eigenvalue in this sector. The BPS kinks in this sector will
saturate this bound. One can view the RG flow as acting on the central charges by
Zab→ λZab, where the IR limit corresponds to λ → ∞. We will often assume that
the theory comes from deformations of an N = 2 CFT by relevant deformations.
The conformal theory is then recovered in the limit λ → 0.
The constructions of this paper can be done in full generality only assuming this
structure. However, for simplicity of presentation, it will be convenient to illus-
trate the construction in the context of N = 2 Landau-Ginzburg theories. We will
comment on the construction in the general setup later in the paper.
LG theories have chiral superfields, which we denote by Xi, and the action for
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them involves D-term (integrating over the full superspace) and F-terms (integrating
over half the superspace). D-term is characterized by a real function K(Xi,X
the F-term by the holomorphic superpotential W(Xi). The action is given by
i), and
S =
?
d4θd2xK(Xi,X
i) +
??
d2θd2xW(Xi) + c.c.
?
.
For most of our discussion we take K =?
iXiX
i. Our results will not depend on
this assumption. The bosonic part of the action can be written as
?
d2x
?
∂Xi∂Xi+ |∂iW|2?
.
This, in particular, implies that the vacua are given by the critical points of W,
which we assume to be isolated, and denote them by Xi
the ab sector is given by
Zab= W(Xi
a. Then the central charge in
a) − W(Xi
b).
When W is a quasi-homogeneous function of Xi, i.e.,
W(λqiXi) = λW(Xi),
it is expected that (with a suitable choice of K) this theory corresponds to a conformal
theory [17,18] with central charge
ˆ c =
?
i
(1 − 2qi).
In this case the R-charge of the field Xiis qi. The chiral ring of this theory is
identified with
R = C[Xi]/{dW = 0},
and its elements are in one to one correspondence with the ground states of the
Ramond sector (whose R-charges are shifted by ˆ c/2). In particular this leads to
TrRamond Ground StatetR= t−ˆ c/2?
i
(1 − t1−qi)
(1 − tqi).
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In this context we can compute the supersymmetric index
Ik= Tr
?
(−1)(k+1)Fexp(2πikR)exp(−βH)
?
and find
Ik=
?
kqi∈Z
(1 − qi)
qi
.
The formulation of Ikas the dimension of a cohomological problem at the conformal
fixed point, and its relation to LG orbifolds was studied in [6].
In general we will assume that W is a deformation away from quasi-homogeneous
form by relevant deformations, giving rise to isolated vacua. Rescaling W → λW
corresponds to the RG flow in this context, and the conformal point, where W is
quasi-homogeneous is the fixed point of this transformation.
In defining the relevant supersymmetric theory, we will be compactifying this
theory first on a circle. It turns out that the radius of the circle does not enter the
computation, and we can consider the case of the theory reduced to 1-dimension,
i.e. the quantum mechanics theory with 4 supercharges. Later in the paper we show
why our construction does not change when we go to the full 2d theory.
3A Time-Dependent SQM: Formulation of an In-
dex
3.1The index Ik
We consider the reduction of the LG theory to 1d. We first consider the quasi-
homogeneous case, where we have an N = 2 SQM system (4 supercharges) with an
R–symmetry with generator R, under which the supercharges have charges ±1
make this section self-contained, let us review the construction of the index Ikin this
context: Let k be an integer, and consider
2. To
Ik= Tr
?
e2πikR(−1)(k−1)Fe−βH?
. (3.1)
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This expression is an index receiving contributions only from the supersymmetric
vacua. Indeed, non–zero energy states appear in multiplets of R–charges (r,r +
1
2,r −1
2,r) and the factor e2πik R(−1)(k−1)Fis equal to
e2πikR(−1)(k−1)F=
?
±e2πikr
∓e2πikr
bosonic states
fermionic states
(3.2)
so that the total contribution of each non–trivial supermultiplet is zero. (The same
conclusion remains true even in presence of central charges in the supersymmetric
algebra).
Then
Ik=
?
vacua
(−1)(k−1)Fe2πikr. (3.3)
As in the last section we consider a LG model with superpotential W(Xi) which is
quasi–homogeneous in the chiral superfields Xiwith weights qi, that is, W(λqiXi) =
λW(Xi), with qi> 0.
The index Ikmay be represented by a path–integral with the boundary conditions
Xi(β) = e2πikqiXi(0)
ψi(β) = e2πikqiψi(0),
(3.4)
(3.5)
under which the superpotential is periodic
W(Xi(β)) = W(e2πikqiXi(0)) = e2πikW(Xi(0)) = W(Xi(0)).(3.6)
Using the periodicity of W, we may rewrite the action in the form
β
?
0
dt
?
˙Xi+ e−iα∂iW
??
˙X∗
i+ eiα∂iW
?
+ fermions,(3.7)
since the difference with the usual action is
2Re[eiα(W(β) − W(0))] = 0. (3.8)
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We may also rewrite the fields as
Xi(t) = e2πikqit/βYi(t)
ψi(t) = e2πikqit/βχi(t)
(3.9)
(3.10)
where Yi(t) and χi(t) are strictly periodic. In the new variables, the bosonic action
becomes
β
?
0
i
?
???˙Yi+2πikqi
β
Yi+ e−i(α+2πkt/β)∂iW(Yi)
???
2
dt(3.11)
and the fermionic part
β
?
0
dt
?
˜ χ¯i(dtχi+2πikqi
β
χi) + ˜ χi(dtχ¯i−2πikqi
β
χ¯i)+
+ ei(α+2πkt/β)(∂i∂jW(Y ))˜ χ¯iχ¯j+ e−i(α+2πkt/β)(∂¯i∂¯jW(¯Y ))˜ χiχj
?
.
(3.12)
Even though we were motivated to formulate the above path-integral starting
from the conformal case, where W is quasi-homogeneous, the path-integral we have
ended up in this formulation makes sense even when W is deformed by relevant
terms away from the quasi-homogeneous limit. We thus consider the above action
for arbitrary deformed W4.
The resulting path integral is an invariant index. This follows from the fact that
it has a supersymmetry, albeit not of the standard kind5. Rather, our path integral
is invariant under a generalized version of the Parisi–Sourlas supersymmetry [10–13].
The extension of the Parisi-Sourlas techniques to a time-dependent situation, which is
what we need for the present work, is novel. Let us briefly review how Parisi-Sourlas
supersymmetry works in the usual setting.
4Rewriting this action in terms of Xi,ψigives the usual LG Lagrangian with two differences: The
superpotential W is now time-dependent, and we have an additional term given by [−?dteiα ∂W
5The system is not invariant under time translations, so the square of a supersymmetric trans-
formation is not a translation in time.
∂t+
c.c.].
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3.2Parisi–Sourlas supersymmetry
Writing
hi=˙Yi+2πikqi
β
Yi+ e−i(α+2πkt/β)∂iW(Yi) (3.13)
the bosonic action (3.11) becomes simply
SB=
β
?
0
?
i
|hi|2dt, (3.14)
while the fermionic action is
SF=
β
?
0
?
˜ χj
˜ χ¯j
?
δh¯j
δYi
δhj
δYi
δh¯j
δY¯i
δhj
δY¯i
?
χi
χ¯i
?
dt.(3.15)
A Parisi–Sourlas supersymmetric system is defined by an action of the form
S = SB+ SF
(3.16)
where hi= hi[Yj] is any functional map from the original bosonic fields Yito the
Gaussian fields hi(this map is also known as the Nicolai map [11,14]).
By construction, a general Parisi–Sourlas system with Lagrangian6
L =1
2hihi+ ¯ χiδhj
δYi
χj,(3.17)
is invariant under the tautological supersymmetry
δYi= ¯ χiǫ
δχi= −hiǫ
(3.18)
(3.19)
δ¯ χi= 0,(3.20)
where ǫ is a complex Grassmannian parameter. Here we have written the action in
6For generality, we write the Lagrangian in the real notation corresponding to a general
N = 1 SQM model. If the underlying model has N = 2 susy everything gets complexified as
in eqns.(3.14)(3.15).
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the case of real fields — our case can be recovered from this by viewing the index
of the fields to also label the complex conjugate fields (see §3.3). In the usual time–
independent setting the Lagrangian (3.17) is actually Hermitian; in that case the
model is invariant under a second supersymmetry, namely the Hermitian conjugate
of the above one. For special forms of the functionals hi[Y ] the supersymmetry
enhances further.
In any Parisi–Sourlas system [10–13], the Gaussian integral over the fermions
produces precisely the Jacobian determinant7for the functional change of variables
Yi→ hi, and then the full path integral takes in the new fields the Gaussian form
?
[dhi]exp
?
−1
2
?β
0
?
i
h2
idt
?
. (3.21)
Thus the hi(t) are Gaussian fields with exact correlation functions
?hi(t)hj(t′)? = δijδ(t − t′),(3.22)
also known as “white noise” fields. Then the Nicolai map hi[Yj] = hiis interpreted
as a stochastic differential equation with a white noise source. For instance, the map
in eqn.(3.13) with k = 0 (the corresponding index, I0, being the usual Witten index)
is the standard Langevin equation for a ‘drift potential’ equal to 2Re(eiαW). In
appendix B we shall relate the Fokker–Planck equation associated to such a Langevin
equation to the tt* geometry of the corresponding N = 2 model.
In 1d all supersymmetric LG models are, in particular, Parisi–Sourlas systems.
The same is true in 2d, provided the LG model has N = 2 supersymmetry [12,13].
However, in these dimensions, there are many Parisi–Sourlas supersymmetric systems
which are not equivalent to the standard ones. The models relevant for the present
paper are a special instance of such more general supersymmetric systems.
In a Parisi–Sourlas supersymmetric model, the Witten index ∆ is interpreted [11]
as the degree of the stochastic map
Yi→ hi≡ hi[Yj],(3.23)
7Up to a crucial sign. See discussion below.
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that is, ∆ is the number of the periodic solutions Yi(t) of the equation
hi[Yi(t)] = hi(t),
for a given generic periodic function hi(t). Indeed, ∆ = Tr[(−1)Fexp(−βH)] is
equal to the path–integral with all fields periodic, and hence, after the integration of
the fermions, it is given by the Gaussian integral (3.21). Since the Gaussian integral
(computed over the space of the periodic functions) is 1, the value of the original
path integral is given just by the net number of times that the space of periodic paths
Yi(t) covers, under the map Yi(t) → hi(t), the space of the periodic functions in the
functional hi–space [11], which is what is meant by the degree of the functional map.
Mathematically what this means is that we have a map from the infinite dimensional
space of loops in CNto itself, given by h, and we simply compute the degree of this
map8(3.23).
The degree, being the analog of the Witten index for the present theory, is invari-
ant under the continuous deformations of the superpotential which do not change its
leading behavior at infinity.
3.3Time-dependent supersymmetry
From the above discussion, it is obvious that the path integral of a Parisi–Sourlas sys-
tem is supersymmetric even if the functional hi[Yj] has an explicit time–dependence,
provided this dependence is periodic with the same period β appearing in the bound-
ary conditions for the bosonic/fermionic fields. This applies, in particular, to our
model, defined by eqns.(3.13)(3.14)(3.15) where the fields Yi(t), χj(t) are now strictly
periodic of period β.
Explicitly, the action S = SB+ SF is invariant under the supersymmetry
δYi= χiǫδY ¯i= χ¯iǫ
δ˜ χ¯j= h∗
(3.24)
δ˜ χj= hjǫ
¯jǫ (3.25)
δχj= 0δχ¯j= 0. (3.26)
8In general there may be subtleties [15] in defining the degree of this infinite dimensional map.
However, if the ‘drift prepotential’ W is holomorphic, or if our system is a compact deformation of
such a model, we do not have to worry about such pathologies.
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where hj(resp. h∗
¯j) stands for the rhs of eqn.(3.13) (resp. its complex conjugate).
We are thus led to study this novel time-dependent supersymmetric quantum
mechanical system.
Then our index Ikis the Witten index for the generalized Parisi–Sourlas system
defined by the modified Langevin equation (3.13) or, equivalently, the degree of this
stochastic map. In particular, this implies that Ikis an integer, a property that is
not immediately obvious from its definition (3.1).
3.4The underlying time–dependent TFT
In this paper we have adopted the language of the ‘physical’ theory. However many
of the manipulations in section 4 below could have been done equally well in the
framework of the (time–dependent) ‘topological’ QM theory [16]. The topological
version of the Parisi–Sourlas model (3.17) is given by the Lagrangian
L = Fihi[Y ] + ¯ χiδhj
δYi
χj
(3.27)
where now Fiis an independent (auxiliary) field9. The Lagrangian (3.27) is invariant
under the nilpotent topological supersymmetry
δYi= ¯ χiǫ
δχi= −Fiǫ
δFi= 0(3.28)
δ¯ χi= 0. (3.29)
The functional integral over the auxiliary field Fiproduces a delta–function enforcing
the differential equation hi[Y ] = 0. That is, in the topological theory the stochas-
tic Langevin equation gets replaced by the corresponding deterministic differential
equation obtained by setting the “noise” to zero. Indeed, in the old days [16] this
topological version was seen as a trick to give a path integral representation to the
solutions of classical deterministic systems.
Let O[Y ] be an observable depending on the bosonic fields Yi. Integrating away
9Of course, the auxiliary fields Fimay be introduced also in the ‘physical’ theory to make the
supersymmetry in eqns.(3.24)–(3.26) to be nilpotent off–shell.
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the fermions and the auxiliary field
?O[Y ]?
topologic=
?
[dYj]det
?δhi
?
?
δYj
?
δ
?
hi[Y ]
?
O[Y ] =
=
?
[dhi]
det
δhi
δYj
?
????
???det
δhi
δYj
δ[hi] O?Y [h]?=
?
classical
solutions
±O[Yclas.]. (3.30)
In particular, the path–integral with periodic boundary conditions gives
?
periodic
[dYj]det
?δhi
δYj
?
=
= net #
?
classical periodic solutions of
the differential equation hi= 0
?
≡ ∆,(3.31)
where by the net number of solutions we mean the difference between the num-
ber of positively and negatively oriented solutions. More generally, the transition
amplitudes
?Y(f),t′|Y(i),t?
topological=
Yj(t′)=Y(f)
?
Yj(t)=Y(i)
[dYj]det
?δhi
δYj
?
(3.32)
are given by the net number of classical solutions to the differential equations hi[Y ] =
0 satisfying the corresponding boundary conditions. Some of the manipulations of
section 4 may be easily rephrased in this topological language.
4 Evaluation of the Index
In the previous section we formulated a path–integral representation of the index Ik.
As discussed, its value does not depend on small deformations of the superpotential.
In particular, varying the parameters which flow the theory from UV to the IR limit
is such a deformation.
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4.1UV limit
To study the UV limit, we decompose W(Yi) as
λW(Yi) = λ
?
W1(Yi) +
?
q<1
Wq(Yi)
?
, (4.1)
where the term Wq(Yi) is quasi–homogeneous of weight q. The deformation is ‘rele-
vant’ and hence does not change the behaviour at ∞ and the degree of the Nicolai
map. Under the redefinition Yi→ λqiYi, the superpotential becomes
W1(Yi) +
?
q<1
λ1−qWq(Yi). (4.2)
Therefore, as λ → 0, the theory becomes conformal and we get the result for the
quasi–homogeneous superpotential W1(Yi), and hence the index
Ik=
?
vacua
(−1)(k−1)fe2πikr.
4.2IR limit: path–integral analysis
The limit λ → ∞ corresponds to the IR limit and should give the same answer for
the index. The bosonic action is rewritten as
β
?
0
λ2???1
λ
dYi
dt
+2πikqi
βλYi+ e−i(α+2πkt/β)∂iW(Yi)
???
2
dt,(4.3)
and, in the limit λ → ∞, the path–integral should be saturated by the configurations
satisfying
1
λ dt
dYi
+2πikqi
βλYi+ e−i(α+2πkt/β)∂iW(Yi) = 0.(4.4)
Let us consider the (Euclidean) transition amplitudes
?Y′
i,t0+ t | Yi,t0− t? =
Y′
i, t0−t
?
Yit0+t
[d(fields)]e−SE
(4.5)
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from a configuration Yiat time t0−t to a configuration Y′
by the above path integral (of course, these are not the physical amplitudes for the
original quantum system).
iat t0+t which are defined
As λ → ∞ the rhs of eqn.(4.5) gets saturated by the solutions to eqn.(4.4) having
the right boundary conditions. We set τ = 2πkλ(t − t0)/β, ˆ α = α + 2πkt0/β, and
µ = β/(2πk). Then
dYi
dτ
+ iqi
λYi+ µe−i(τ/λ+ˆ α)∂iW(Yi) = 0. (4.6)
As λ → ∞, this equation becomes the one describing a BPS soliton of phase eiˆ α.
Writing Yi(τ) = e−iqiτ/λXi(τ),
dXi
dτ
+ µe−iˆ α?
q
ei(q−1)τ/λ∂iWq(Xi) = 0. (4.7)
One looks for an asymptotic solution as λ → ∞. If t in the lhs of (4.5) is much
smaller than β, so that τ/λ ≪ 1 everywhere along the paths contributing to the path
integral in the rhs of eqn.(4.5), the O(λ−1) corrections to the BPS soliton equation
(4.6) remain small as the fields go from one vacuum to the other. Measured in units
of τ, the small time 2t becomes of order O(λ), and hence infinitely long as λ → ∞,
so there is plenty of τ time to complete the transition from one asymptotic vacuum
to the other one interpolated by the given BPS soliton. Indeed, there is plenty of
rescaled time to accomodate a chain of BPS kinks, corresponding to multiple jumps
from one vacuum to the next one, provided all the involved solitons have the (same)
appropriate BPS phase. The saturating configuration differs from a classical vacuum
by a quantity of order O(e−Mτ) = O(e−Cλ(t−t0)). Therefore, as λ → ∞ the saturating
configuration is a vacuum, except for a time region of size O(1/λ) (measured in
‘physical’ time t) around the special times t0at which eiˆ α= eiαe2πikt0/βis the phase
of a BPS soliton.
In conclusion, in the limit λ → ∞, for almost all times t, t′, with t − t′≪ β,
one may effectively replace the quantum amplitude (4.5) with a finite n × n matrix
(where n is the number of supersymmetric vacua)
?Yi(a),t′|Yi(b),t? → (St′,t)a
b= δab+ (Nt′,t)a
b
(4.8)
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where the integer (Nt′,t)a
connecting vacuum b to vacuum a though a sequence of intermediate vacua of the
form
a ≡ a0→ a1→ a2→ ··· → am≡ b,
bcounts with signs the number of different BPS kink chains
(4.9)
which satisfy the condition of well–ordered central charge phases10
eiαe2πikt/β≤ phase(Wa1− Wa) ≤ phase(W2− W1) ≤ ···
··· ≤ phase(Wb− Wm−1) ≤ eiαe2πikt′/β, (4.10)
where each vacuum transition ai→ ai+1is triggered by a BPS state of the appropriate
phase.
In the special case that the time interval t′− t is small enough that the angular
sector
?eiαe2πikt/β, eiαe2πikt′/β?
contains only one BPS phase1
pair11of vacua ba, the integer (Nt′,t)a
solitons connecting the two vacua a, b, thus reproducing eqn.(1.1). However, our
present discussion, in terms of the integral–valued time–evolution kernel St′,t, is more
general than eqn.(1.1) and holds for any number of BPS rays and also for vacua which
are aligned in the W–plane.
(4.11)
2log(Zba/Zba), and this phase is associated to a unique
bis just plus or minus the number of BPS
The ambiguity in the sign of (Nt′,t)a
fermionic determinants around a given saturating configuration. Supersymmetry
guarantees that these two determinants are equal in absolute value, but not neces-
sarily in sign12(the path integral is essentially a Witten index which must be an
integer, but may be a negative integer). We shall formalize the correct sign rule for
(Nt′,t)a
breflects the relative signs of the bosonic and
bmomentarily.
The time–evolution kernels satisfy a group–law, namely
(St′,t)a
b= (St′,t′′)a
c(St′′,t)c
b
for t′′∈ (t,t′).(4.12)
10The symbol ≤ stands for the natural order of increasing phases.
11This is the generic situation for 2d LG models.
12The bosonic determinant is the square root of the square of the fermionic one, so it is always
positive, while the fermionic one may, in principle, be negative.
17
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Then, even if our IR computation was reliable for time intervals |t′− t| ≪ β, the
result can be extended to arbitrary long time–intervals by defining
Stm,t0= Stm,tm−1Stm−1,tm−2··· S1,t0
where0 < t1< t2< ··· < tm,ti+1− ti≪ β.
(4.13)
The discussion between eqn.(4.8) and eqn.(4.10) applies to arbitrary time intervals.
In particular, (as λ → ∞) all amplitudes are given by integer numbers.
(Nt′,t)a
bis a strictly triangular matrix (in a suitable basis) provided the time
interval (t,t′) satisfies
0 < (t′− t) <β
2k.
Indeed, if the phase of Zbabelongs to the corresponding angular sector (4.11), so that
(Nt′,t)a
angular sector, and then (Nt′,t)b
bis possibly non–zero, the phase of Zab= −Zbabelongs to the complementary
a= 0.
Consider the amplitude translated in time by β/2k,
St′+β/2k,t+β/2k.
It counts BPS states with phases in the opposite angular sector with respect to
the one contributing to St′,t, that is, it counts the states in the conjugate sectors.
The absolute number of solitons in the ab sector is the same as in the sector ba,
|Nab| = |Nba|. However, their signs may be different. We want to argue that, indeed,
they have opposite signs. For simplicity, we focus on the generic situation in which
there is no vacuum alignement, and all amplitudes St′,tmay be written as a time–
order product of elementary ones of the form in eqn.(1.1)
St′,t= T??
S(i)
?
∈ SL(n,Z),
where S(i) = 1 + N(i)with N(i)a nilpotent matrix of rank 1. If the signs of the
N(i)’s of conjugate sectors were the same, a translation in time by β/2k will act on
the elementary amplitudes by simply transposing them. Under deformation to a
non–generic configuration, this would correspond to the rule
St′+β/2k,t+β/2k= ST
t′,t.
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Page 20
But this equation is inconsistent with the group law, since transposition is not an
automorphism of SL(n,Z). What is a group automorphism is transposition and
inverse combined, S ?→ (ST)−1(the Cartan involution), which acts on the elementary
factors as
S(i)= 1 + N(i)?→ 1 − (N(i))T≡ (S−1
(i))T, (4.14)
that is, invertes the sign of the elementary soliton multiplicities, Nab= −Nba. Then,
in full generality,
St′+β/2k,t+β/2k=?S−1
This equality fixes the rule for the signs (up to convention–dependent choices). A
different argument for the sign rule will be presented in appendix B.
t′,t)T. (4.15)
Let S = Sβ/2k,0be the amplitude for the largest time interval such that it is still
a triangular matrix. From eqn.(4.15) we have
Sβ,0= (S−1)TS(S−1)TS ···(S−1)TS =?(S−1)TS)k, (4.16)
and then
Ik= tr??(ST)−1S?k?. (4.17)
This is, of course, the well–known formula [1].
4.3Extensions to Two–dimensions and More General N = 2
Theories
We have mainly focused on the one dimensional formulation of the index.
nothing essential changes in going to the main case of interest, which is d = 2. One
quantizes the system in a rectangular torus of sizes L, β with the boundary conditions
But
Xi(L,t) = Xi(0,t),Xi(x,β) = e2πikqiXi(x,0). (4.18)
The N = 2 2d Landau–Ginzburg models may still be written as Parisi–Sourlas
supersymmetric systems [12,13]. Eqn.(3.7) is replaced by (z = x + it)
?
torus
d2z
?
∂zXi+ e−iα∂iW
??
∂zX∗
i+ eiα∂iW
?
+ fermions,(4.19)
19
Page 21
which again is equivalent, in the quasi–homogeneous case, to the standard action.
Indeed, the action (4.19) differs from the usual one by
i
2
?
rectangle
?
∂(eiαW) ∧ dz − ∂(e−iαW) ∧ dz
?
=
=i
2
?
∂(rectangle)
?
eiαW dz − e−iαW dz
?
= 0, (4.20)
since W is still periodic: W(x,0) = W(x,β) and W(0,t) = W(L,t).
Again, one can introduce strictly periodic fields Yi, χi, getting a time–dependent
superpotential
W(Yi,t) = e2πikt/βW(Yi).
W(Yi,t) is then deformed away from the quasi–homogeneous case, by adding
relevant operators to W(Yi). One ends up with the generalized Parisi–Sourlas super-
symmetric system defined by the stochastic partial differential equations
∂zYi+πik
β
Yi+ λe−iαe2πk(z−z)/β∂iW = hi(z, ¯ z), (4.21)
to which the discussion in section 3 applies word for word. Again, in the limit
λ → ∞, the amplitudes are given by a time ordered product of integral matrices
whose elementary entries are the multiplicities of the BPS particles with the given
phase. In fact, the matrices are numerically the same as in the corresponding 1d
models13.
Also one can formulate the action in a more general setup than the LG case.
All we needed for the definition of the action was to introduce a connection which
couples to the R-charges at the conformal AR
d2θ → d2θe2πikt/β. In this way we can formulate the supersymmetric path-integral.
Furthermore one can easily show that the UV and IR computations yield the R-
charges of the ground states in the UV, and the corresponding solitons in the IR.
We thus have a general proof of the main result for general massive deformations of
0= 2πk/β. Moreover, we replace the
13This construction can be extended to the case of N = 1 supersymmetric theories in 4d. These
theories can have BPS domain walls, and the considerations in this paper relate their spectrum to
the R-charges at the UV fixed points.
20
Page 22
N = 2 theories in 2 dimensions.
From another point of view, what we have done is to construct a quantum (time–
dependent) system whose Schroedinger equation corresponds to the Lax equations
whose compatibility conditions are the tt* equations for the given N = 2 model; all
of these structures exist for any N = 2 theory and not only in the LG case.
5Conclusion
In this paper we have seen how one can twist the path-integral of a supersymmetric
theory with R-symmetry. It was known that this object exists at the conformal point.
The novelty here is that we have found a way to extend R-twisting of the path-integral
even after we deform the action by relevant terms that break R-symmetry. As a by-
product we have found a simple picture of how the 2d wall-crossing in the N = 2
theories work, and why it is related to R-charges of the corresponding conformal
theory.
The ideas in this paper should apply to many other cases. For example for d = 4
theory with N = 2 supersymmetry, a similar picture should hold, which gives another
derivation of Kontsevich-Soibelman wall-crossing result [7], very much in line with
the string theoretic derivation proposed in [4]. Indeed similar ideas seem to hold
there which relates the d = 4 wall-crossing to the 2d case [19].
One could also consider other CFT’s where R-symmetry is non-abelian, say
SU(2). In such a case one can consider more interesting twistings along a higher
dimensional geometry than a circle. It would be very interesting to study potential
implications of such twists for conformal field theories and their deformations.
Acknowledgements
We would like to thank Andy Neitzke for valuable discussions. The research of CV
was supported in part by NSF grant PHY-0244821.
21
Page 23
A The case of collinear vacua
The arguments in the main body of the paper apply even in models having a plurality
of distinct solitonic sectors ab with the same phase of the central charge Zab. These
(non generic) models have some particular property, which are universal for a given
vacuum alignment geometry in the complex Z–plane. In this appendix we discuss
these properties in the light of the arguments of the present paper. The conclusions
are confirmed by the analyis of the explicit models whose tt* equations can be solved
in closed form.
A.1The 1/n rule
The case of three aligned vacua was already discussed in appendix B of [1]. There
it is shown that, if we have three vacua with critical values W1,W2,W3 lying (in
this order) on a straight line in the W–plane, and there is one physical BPS soliton
connecting vacua 1,2 and one connecting vacua 2,3, but no soliton connecting the
extremal vacua 1,3, then the leading IR behaviour of the index
Q13= Tr13
?
(−1)FF e−βH?
,
would look like as we had ±1
puzzling, since the number of particles should be integral. However, this strange
result is actually required by the integrality of the spectrum. As shown in the present
paper, what actually must be integral are the group elements St′,t∈ SL(n,Z). As
stressed by Kontsevich and Soibelman [7], and as it is manifest from our eqn.(B.6),
the ‘soliton multiplicities’ µab, as defined by the IR asymptotics of the index Qab,
belong to the corresponding Lie algebra sl(n,Q). Of course, we pass from the Lie
algebra to the group by taking the exponential; we shall write
2worth of BPS solitons of mass |Z13|. At first this looks
St′,t= exp?Lt′,t
?,(A.1)
where Lt′,t ∈ sl(n,Q) represents the Q–index ‘multiplicities’ µ for the ab sectors
having phases in the angular sector associated to the time interval (t,t′).
In the case of the three aligned vacua, if their common phase is in the small time
22
Page 24
interval (t,t′), the rules of eqns.(4.9)(4.10) give
(St′,t)12= (St′,t)23= (St′,t)13= 1, (A.2)
where the first two entries correspond to single BPS soliton chains and the last one
to a BPS chain of length 2. Now
exp
?01
1
2
100
000
?
=
?111
011
001
?
≡ St′,t, (A.3)
and (Lt′,t)13is actually required to be1
we computed in section 4.
2in order to reproduce the integral amplitude
This analysis is easily estended to an arbitrary number of aligned vacua. Assume
we have n + 1 adiacent BPS solitons (ak,ak+1), all having multiplicity 1, which are
perfectly aligned and sorted in ascending order along the BPS ray,
|Za0an| = |Za0a1| + |Za1a2| + ··· + |Zan−1an|, (A.4)
while there is no physical BPS soliton connecting two vacua which are not next
neighborhood in the Z–plane. Then the leading IR asymptotics of (Q)a0anis as
there was ±1
nBPS ‘solitons’ of the total mass |Za0an|, that is (Lt′,t)a0an= ±1
To see this, we write T for the upper–triangular (n + 1) × (n + 1) matrix
n.
Tab= δb,a+1,Tn+1= 0.(A.5)
By assumption, the non–zero soliton multiplicities are Na,a+1 = 1. However, the
arguments of section 4 give
?St′,t
?
ab=
?
1 if a ≥ b
0otherwise.
⇔ Lt′,t= log(1 − T)−1= 1 + T +1
⇔ St′,t= (1 − T)−1= 1 + T + T2+ ··· + Tn
2T2+1
3T3+ ··· +1
nTn. (A.6)
which gives
(Lt′,t)ai,aj=
1
j − i
j > i,(A.7)
23
Page 25
which implies the claim. Notice that (S−1
AnLie algebra, in correspondence with the fact that this is the BPS spectrum of
a model which flows in the UV to the An minimal model. The extension of this
analysis to the case of inifnitely many collinear vacua and its application to 4d,
N = 2 supersymmetric theories will be discussed in [19].
t′,t)T+ S−1
t′,tis the Cartan matrix for the
B Evalutation in the Schroedinger picture: Links
to tt* Geometry
As a further check of the analysis in this paper (including the signs assignements)
and to clarify its relation with the tt* geometry, we perform the same computation
in the Schroedinger picture.
The time–evolution kernel St,t′, satisfies the (Euclidean) time-dependent Schroedinger
equation14
?d
As λ → ∞, the system remains most of the time near the ground states so, in the
adiabatic approximation, one would write the evolution kernel as
dt+ H(t)
?
St,t′ = 0, (B.1)
St,t′ = Ψa(t)A(t,t′)abΨb(t′)∗, (B.2)
where Ψa(t) is the a–th vacuum wave–function for the model with a superpotential
with a (frozen) phase exp(2iπkt/β) and the coefficients A(t,t′)abare suitable func-
tions of t and t′. However, we must recall that, even adiabatically freezing the phase
of W, we do not get the usual Schroedinger equation since, in that limit, our action
differs from the standard one by a surface term
S = Sstandard+ 2λRe?e2iπkt/βW?
14One may prefer to call it the Fokker–Planck equation associated to the (time–dependent)
stochastic equation (3.13).
f− 2λRe?e2iπkt/βW?
i.(B.3)
24
Page 26
Performing the compensating transformation on the states15, we obtain
St,t′ =
?
e−2λRe?
e2iπkt/βWa?
Ψa(t)
?
A(t,t′)ab
?
e2λRe?
e2iπkt′/βWb
?
Ψb(t′)∗
?
. (B.4)
In the adiabatic approximation, which is valid as λ → ∞, the Schroedinger
equation (B.1) reduces to a linear differential equation for the coefficient matrix
A(t,t′)ab. To get the equation satisfied by this matrix, we observe that the tt*
equations imply
d
dtΨa=2πik
β
QabΨb+ ···
(B.5)
where Qabis the Q–index [2] and +··· stands for a state orthogonal to all vacua.
The Schroedinger equation then reduces to
β
2πik
d
dtAab= QacAcb+ 2λIm
?
e2πikτ/λβWa
?
Aab
(B.6)
which is eqn.(4.11) of ref. [1] with16
x = exp?2πikt/β?
β = λ.
(B.7)
(B.8)
Then the IR monodromy computed with our present SQM system is the same as
the monodromy computed by the tt* connection. In particular, in the extreme IR
limit, the reduced time evolution kernel A(t,t′)abis an integral matrix taking values
in the Lie groups discussed in section 6 of [3]. This implies that the sign rule is
as in eqn.(4.14). Indeed, Qab is an element of the Lie algebra so(n), and hence
the elementary BPS multiplicity matrix Nab, which is proportional to the leading
IR contribution to Qab, must be real antisymmetric. In the language of Kontsevich–
Soibelman [7] this property corresponds to the symmetry under the Cartan involution
15This is the usual transformation mapping the Euclidean–time Schroerdinger equation into the
forward Fokker–Planck equation.
16As written, eqn.(B.6) differs from eqn.(4.11) of ref. [1] by exponentially small terms in the limit
λ → ∞. However, a more precise asymptotic analysis will match the subleading corrections too.
To avoid misunderstanding, we stress that the β of the present paper and the β of ref. [1] should
not be identified.
25
Page 27
of SL(n,Z),
g ?→ (g−1)T.
26
Page 28
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