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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

1Introduction2

2 The Basic Setup5

3A Time-Dependent SQM: Formulation of an Index7

3.1The index Ik. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Parisi–Sourlas supersymmetry . . . . . . . . . . . . . . . . . . . . . .

7

3.210

3.3Time-dependent supersymmetry . . . . . . . . . . . . . . . . . . . . .12

3.4The underlying time–dependent TFT . . . . . . . . . . . . . . . . . .13

4Evaluation of the Index14

4.1UV limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2IR limit: path–integral analysis . . . . . . . . . . . . . . . . . . . . .15

4.3Extensions to Two–dimensions and More General N = 2 Theories . . 19

5 Conclusion21

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.

3 A 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.

4Evaluation 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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