An elliptic triptych
Received: June
An elliptic triptych
Jan Troost 0 1
0 24 rue Lhomond , 75005 Paris , France
1 Ecole Normale Superieure, PSL Research University , Sorbonne Universites, UPMC, CNRS
We clarify three aspects of noncompact elliptic genera. Firstly, we give a path integral derivation of the elliptic genus of the cigar conformal eld theory from its nonlinear sigmamodel description. The result is a manifestly modular sum over a lattice. Secondly, we discuss supersymmetric quantum mechanics with a continuous spectrum. We regulate the theory and analyze the dependence on the temperature of the trace weighted by the fermion number. The dependence is dictated by the regulator. From a detailed analysis of the dependence on the infrared boundary conditions, we argue that in noncompact elliptic genera rightmoving supersymmetry combined with modular covariance is anomalous. Thirdly, we further clarify the relation between the at space elliptic genus and the in nite level limit of the cigar elliptic genus.
Conformal Field Models in String Theory; Conformal Field Theory; Super

symmetry and Duality
1 Introduction
2
The in nite cover of the orbifolded trumpet
The lattice sum
3
Supersymmetric quantum mechanics on a half line
3.1
Quantum mechanics on a half line
Quantum mechanics on the line
Quantum mechanics on the half line
3.2
Supersymmetric quantum mechanics on the half line
3.3 Infrared regulators and the weighted trace
Supersymmetric quantum mechanics on the line
Supersymmetric quantum mechanics on the half line
An interval
The free supersymmetric particle on the half line
The weighted trace
The application to the elliptic genus
4
at space limit conformal eld theory
3.4
A
4.1
4.2
4.3
3.1.1
3.1.2
3.2.1
3.2.2
3.2.3
3.3.1
3.3.2
5
Conclusion
1
Introduction
Flat space regulated
Twist two
A miniature
at space superconformal eld theory and the
in nite level limit of the cigar conformal eld theory using their elliptic genera.
2
The path integral lattice sum
In this section, we wish to obtain a simpler path integral understanding of the compact
formula for the elliptic genus of the cigar in terms of a lattice sum, derived in [8]. To that end,
we provide a new derivation of the elliptic genus of the cigar, through its supersymmetric
nonlinear sigmamodel description. The latter has the advantage of being parameterized
in terms of the physical degrees of freedom only.
2.1
The guises of the genus
The cigar elliptic genus
cig( ; ) = T rRR( 1)FL+FR e2 i QqL0 2c4 qL0 2c4
(2.1)
is a partition sum in the RamondRamond sector, weighted by left and rightmoving
fermion numbers FL;R, as well as twisted by the leftmoving Rcharge Q. It was computed
manifestly covariantly through a path integral over maps from the torus into the coset
SL(2; R)=U(1) target space [6]. The result obtained in [6, 9, 10] was
cig( ; ) = k
ds1;2
Z 1
0
X
m;w2Z
1(s1 + s2
1(s1 + s2
k
; )
k +k1 ; ) e2 i w
e k2 j(m+s2)+(w+s1) j2 ;
(2.2)
where the 1 functions arise from partition functions of fermions and bosons with twisted
boundary conditions on the torus, the integers m; w are winding numbers for the maps
from the torus onto the target space angular direction, and the angles s1;2 are holonomies
on the torus for the U(1) gauge eld used to gauge an elliptic isometry of SL(2; R). The
twist with respect to the leftmoving Rcharge is given by
. This modular Lagrangian
result was put into a Hamiltonian form in which the elliptic genus could be read directly
as a sum over rightmoving ground states plus an integral over the di erences of spectral
densities for the continuous spectrum of bosonic and fermionic rightmovers [6, 10] . The
di erence of spectral densities is determined by the asymptotic supercharge [
6, 11, 12
].
{ 2 {
In [8], a rewriting of the result (2.2) in terms of a lattice sum was obtained. The
resulting expression for the cigar elliptic genus is
a lattice Z + Z . Our goal in this section is to understand the formula (2.3) in a more
direct manner than through the route laid out in [6, 8{10]. We recall that a key step in
the derivation of the lattice sum (2.3) was to rst compute the elliptic genus of the in nite
HJEP10(27)8
cover of the Zk orbifold of the trumpet geometry [8, 13].
2.2
The in nite cover of the orbifolded trumpet
We start our calculation from the cigar geometry [14{16]
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
ds2 = 0k(d 2 + tanh2 d 2)
e
= e 0 = cosh ;
ds2 = 0 kd 2 +
coth2 d 2
e
= e 0 = sinh
1
k
where the angle
is identi ed modulo 2 . The range of the radial coordinate
is from
0 to 1. The metric and dilaton determine the couplings of a conformal twodimensional
nonlinear sigmamodel. The Tdual geometry is the Zk orbifold of the trumpet:
where the angle
is again identi ed modulo 2 . The trumpet geometry is singular at the
rim of the horn, at
= 0. The in nite cover of the orbifold of the trumpet is the geometry
in which we no longer impose any equivalence relation on the variable .
We perform the path integral on the cover as follows. Firstly, we consider the integral
over the zero modes and the oscillator modes separately. We suppose that the oscillator
contribution on the left is proportional to the free eld result
Zo1sc =
1
1=(4 2 2) has two sources. One can be viewed as the result of the spacetime covariant
integral over the radial momentum (at 0 = 1) while the second is the proper normalization
of the
zero mode volume integral (to be performed shortly). The rightmoving oscillators
cancel among each other.
We want to focus on the remaining integral over zero modes, which contains the crucial
information on the modularly completed AppellLerch sum [2]. The leftmoving fermionic
zero modes have been lifted by the Rcharge twist.
Thus, we can concentrate on the integration over the bosonic zero modes as well as the rightmoving fermionic zero modes, with measure
d d d ~ d ~ :
{ 3 {
The square root of the determinant in the di eomorphism invariant measures has canceled
between the bosons and the fermions. The relevant action is the N = (1; 1) supersymmetric
extension of the nonlinear sigmamodel on the curved target space.1 The term in the action
that lifts the right moving fermion zero modes is [
17
]
and more speci cally, the term proportional to the Christo el connection symbols
This leads to a term in the action equal to
Slift =
Slift =
=
zero modes and obtain a nonzero result.
We wish to introduce a twist in the worldsheet time direction for the target space
angular direction
because we insert a Rcharge twist operator in the elliptic genus, and
the eld
is charged under the Rsymmetry [6, 8{10]. We thus must twist
the Christo el connection (2.9) and then nd the zero mode integral
1See e.g. formula (12.3.27) in [
17
].
2The factor N1 is absorbed in the de nition of Z1 in [8, 13].
{ 4 {
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
We have represented the integral over the variable
by a factor of 2 N1 where we think of
N1 as the order of the cover, which goes to in nity. The singular geometry at
= 0 makes
for a vanishing contribution from this region. Putting this together with the oscillator
factor (2.6) we proposed previously, we nd
Z1 = N1
This precisely agrees with the elliptic genus of the in nite cover of the orbifolded trumpet
calculated in [8].2
Our next step is the path integral incarnation of the procedure of the derivation of the
lattice sum formula in [8]. We undo the in nite order orbifold of the cigar, i.e. we undo the
in nite order cover of the orbifolded trumpet. This will reproduce the lattice sum elliptic
genus formula.
There are two changes that we need to carefully track. The rst one is that since the
eld
becomes an angular variable with period 2 , we must sum over the world sheet
winding sectors. Thus, we introduce the identi cations
which lead to the classical solutions
We then have the classical contribution to the action
where
= m
w . After tracking normalization factors, one nds that the action acquires
another overall factor of 4
2=k (see e.g. [27]).
The second e ect we must take into account is that the leftmoving Rcharge
corresponds to the leftmoving momentum of the angle
eld. When we introduce a winding
number w, we must properly take into account the contribution of the winding number to
the leftmoving momentum. This amounts to adding a factor of e 2 i w=k to a contribution
arising from winding number w. (Recall that the radius is R2= 0 = 1=k.) We rewrite
e 2 i w=k = e (
) k 2
which leads to a total contribution to the exponent equal to
k 2
(j j2 + ( + ) +
2 + (
+ )) =
k 2
(j j2 + 2
+
2) :
The denominator in the nal expression is obtained from a factor ( + )( + ) in the
denominator that arises from the exponent (2.17) in the generalized zero mode integral (2.13)
on the one hand, and a factor of
+
in the numerator from the zderivative of the angular
variable
on the other hand (arising from the zero mode lifting term (2.10)). Multiplying
these, we nd the nal formula
which is the compact lattice sum form [8] of the cigar elliptic genus. We have given a
direct derivation of the lattice sum form, using the nonlinear sigma model description.
This concludes the rst panel of our triptych.
3
Supersymmetric quantum mechanics on a half line
In this section, we wish to render the fact that the nonholomorphic term in noncompact
elliptic genera arises from a contribution due to the continuum of the rightmoving
supersymmetric quantum mechanics [6] even more manifest. For that purpose, we discuss
to what extent the rightmoving supersymmetric quantum mechanics can be regularized
in a supersymmetric invariant way, or a modular covariant manner, but not both. That
fact leads to the holomorphic anomaly [6]. The plan of this section is to rst review how
boundary conditions in ordinary quantum mechanics show up in its path integral
formulation. We then extend this insight to supersymmetric quantum mechanics. We illustrate the
essence of the phenomenon in the simplest of systems. We end with a discussion of how the
regulator of the noncompact elliptic genus cannot be both modular and supersymmetric,
which leads to an anomaly.
3.1
Quantum mechanics on a half line
We are used to path integrals that map spaces with boundaries into closed manifolds.
Less frequently, we are confronted with path integrals from closed spaces to spaces with
boundaries. It is the latter case that we study in the following in the very simple setting
of quantum mechanics.
In particular, we discuss quantum mechanics on a half line, its path integral
formulation, and pay particular attention to the path integral incarnation of the boundary
conditions. The easiest way to proceed will be to relate the problem to quantum mechanics
on the whole real line. What follows is a review of the results derived in e.g. [18{20], albeit
from an original perspective.
3.1.1
Quantum mechanics on the line
Firstly, we rapidly review quantum mechanics on the real line. We work with a Hilbert
space which consists of quadratically integrable functions on the line parameterized by a
coordinate x. We have a Hamiltonian operator H of the form
where V (x) is a potential. We can de ne a Feynman amplitude to go from an initial
position xi to a nal position xf in time t through the path integral
H =
A(xi; xf ; t) =
Z x(t)=xf
x(0)=xi
dx eiS[x] ;
S =
Z t
0
dt0
x_
2
2
V (x) :
{ 6 {
(3.1)
(3.2)
(3.3)
where the action is equal to
The Schrodinger equation for the wavefunction of the particle reads
and we work with normalized wavefunctions . We can also write the amplitude in terms
of an integral over energy eigenstates E:
Z
and the amplitude satis es the function completeness relation at t = 0, as well as the
Schrodinger equation (3.4) in the initial and nal position variables xi and xf .
Quantum mechanics on the half line
The subtleties of quantum mechanics on the open real half line x
0 have been understood
for a long time [21]. Boundary conditions compatible with unitarity have been classi ed.
The path integral formulation for quantum mechanics on the half line has resurfaced several
times over the last decades [18{20], and is also wellunderstood. We review what is known.
The halfline has a boundary, and we must have that the probability current vanishes
at the boundary. This is guaranteed by the Robin boundary conditions
(3.6)
When the constant c is zero, we have a Neumann boundary condition and when it is
in nite, the boundary condition is in e ect Dirichlet,
(0) = 0. Suppose we are given a
Hamiltonian H of the form (3.1) with a potential V (x) on the half line x > 0. We can
extend the quantum mechanics on the half line to the whole real line by extending the
potential in an even fashion, declaring that V ( x) = V (x). It is important to note that
this constraint leaves the potential to take any value at the origin x = 0. We can then think
of the quantum mechanics on the half line as a folded version of the quantum mechanics
on the real line.3 The even quantum mechanics that we constructed on the real line has
a global symmetry group Z2. We can divide the quantum mechanics problem on the real
line, including its Hilbert space, by the Z2 operation, and
nd a wellde ned quantum
mechanics problem on the half line, which is the original problem we wished to discuss.
An advantage of this way of thinking is that the measure for quantum mechanics on
the whole line is canonical. It leads to the Green's function (3.5). Since the quantum
mechanics that we constructed has a global Z2 symmetry, we can classify eigenfunctions
in terms of the representation they form under the Z2 symmetry, namely, we can classify
them into even and odd eigenfunctions of the Hamiltonian. We then obtain the whole
line Green's function in the form that separates the even and odd energy eigenfunction
contributions
Z
3In string theory, one would say that we think of the half line as an orbifold of the real line.
{ 7 {
(3.4)
(3.5)
1
2
1
2
(A(xi; xf ; t)
A(xi; xf ; t)) =
E;o(xi) E;o(xf ) ;
(3.8)
Z
dEe iEt
is wellde ned on the halfline and satis es Dirichlet boundary conditions. We divide by
a factor of two since we are projecting onto Z2 invariant states. From the path integral
perspective, the subtraction corresponds to a di erence over paths that go from xi to xf
and that go from xi to
xf , on the whole real line, with the canonical measure (divided
by two). This prescription generates a measure on the half line which avoids the origin,
since we subtract all paths that cross to the other side [18, 19].4 If we represent the Z2
action oppositely on the odd wavefunctions, we arrive at the Green's function that satis es
Neumann boundary conditions:
A 21 ;N (xi; xf ; t) =
(A(xi; xf ; t) + A(xi; xf ; t)) =
E;e(xi) E;e(xf ) :
(3.9)
Z
dEe iEt
In this second option, we add paths to the nal positions xf and
xf with their whole line
weights (divided by two). This path integral represents a sum over paths that re ect an
even or an odd number of times o the origin x = 0, and in particular, allows the particle
to reach the end of the half line.
We clearly see that the naive folding operation projects the states of the quantum
mechanics onto those states that are even, or those that are odd.5 However, concentrating
on these two possibilities only fails to fully exploit the loop hole that the even potential
V (x) allows, which is an arbitrary value V (0) at the
xed point x = 0 of the folding
operation.6
We can make use of this freedom by taking as the total potential an even
potential V (x), zero at x = 0, complemented with a function:
The Green's function
A 12 ;D(xi; xf ; t) =
(3.10)
(3.11)
We take the wavefunction on the whole line to be even and continuous, with a discontinuous
rst derivative at the origin. When we consider the onesided derivative at zero, we nd
that the wavefunction satis es the Robin boundary condition [19]
We have gone from a purely even continuous and di erentiable wavefunction on the real
line that satis es the Neumann boundary condition (at c = 0) to an even wavefunction
that satis es mixed Robin boundary conditions, by in uencing the wavefunction near zero
with a deltafunction interaction.7 It is intuitively clear, and argued in detail in [19] that
4This is a common manipulation in probability theory.
5These are states in the untwisted sector of an orbifold, projected onto invariants under the gauged
discrete symmetry.
strongly constrained by consistency.
6In string theory orbifolds, the xed point hosts extra degrees of freedom which in that case are very
7The even wavefunction on the side x > 0 corresponds to the linear combination
(x) / ( E;e(x) +
c E;o(x)) in terms of even and odd solutions to the problem on the real line without the deltafunction
interaction [19]. It is an invariant under the Z2 action with discontinuous derivative at the origin.
{ 8 {
it is harder to push an initial problem with Dirichlet boundary conditions at the origin
towards a mixed boundary condition problem. In order to achieve this, one needs a very
deep well [19]. For later purposes, we note in particular that an ordinary deltafunction
insertion at the origin will not in uence an initial Dirichlet boundary value problem.
As an intuitive picture, we can imagine that the deltafunction is generated by
possible extra degrees of freedom that are localized at the origin, and whose interaction with
the quantum mechanical degree of freedom we concentrate on induces the deltafunction
potential localized at the origin.
Thus far, we brie y reviewed the results of [18, 19] on path integrals on the half line
and discussed how they are consistent with folding. Next, we render these techniques
HJEP10(27)8
compatible with supersymmetry.
3.2
Supersymmetric quantum mechanics on the half line
In this section, we extend our perspective on quantum mechanics on the half line to a
quantum mechanical model with supersymmetry. We again start from a quantum mechanics on
the whole of the real line, with extra fermionic degrees of freedom and supersymmetry. In
a second stage, we fold the quantum mechanics onto the half line in a manner consistent
with supersymmetry.
3.2.1
Supersymmetric quantum mechanics on the line
We discuss the supersymmetric system with Euclidean action (see e.g. [22])
SE =
W 2
where W 0(x) = @xW (x). The action permits two supersymmetries with in nitesimal
variations
(3.12)
(3.13)
(3.14)
(3.15)
When we quantize the fermionic degrees of freedom, we tensor the space of quadratically
integrable functions with a two component system. We call one component bosonic and
the other fermionic. The two components have the Hamiltonians [22]8
x =
=
+
W ) :
H
= p2 + W 2
W 0 :
p =
We introduced the operator
8We follow standard conventions for supersymmetric quantum mechanics in this section. These di er
by a factor of two from the standard conventions for quantum mechanics used in section 3.1.
and can represent the supercharges by
Q = (p + iW )
Qy = (p
iW )
When we trace over the fermionic degrees of freedom, we need to compute the fermionic
determinant with antiperiodic boundary conditions. It evaluates to [22]
HJEP10(27)8
f
Zanti per(x) =
Z
W 0) ) = cosh
;
(3.17)
after regularization. This is the path integral counterpart to the calculation of the
Hamiltonians (3.14).
3.2.2
Supersymmetric quantum mechanics on the half line
We study the supersymmetric quantum mechanics on the half line by folding the
supersymmetric quantum mechanics on the whole line. We wish for the folding Z2 symmetry
to preserve supersymmetry. Since the particle position x is odd under the Z2 action (as
is its derivative with respect to time, since we choose world line time to be invariant), we
demand that the superpotential W (x) is odd under parity, and that the fermionic variables
and
are odd as well. See equation (3.13). Thus, we have the Z2 action
(x; ;
) ! ( x;
;
) ;
and the superpotential W is odd. For the moment, we consider the superpotential to be
continuous, and therefore zero at zero.
We project onto states invariant under the Z2 action (3.18). Thus, in any path integral,
we will insert a projection operator PZ2 that consists of
1
2
PZ2 =
(1 + P ( 1)F )
where P is the parity operator that maps P : x !
themselves. When we trace over the fermionic degrees of freedom with a ( 1)F insertion,
we must impose periodic boundary conditions on the fermions. The fermionic determinant
x and ( 1)F maps fermions to minus
in this case evaluates to [22]
Zfper(x) =
Z
W 0) ) = sinh
;
(3.20)
Z T
0
d
W 0(x)
2
which leads to the same Hamiltonians (3.14) for the two component system, and when
we compare to equation (3.17) we
nd a minus sign up front in the path integral over
the second component. As a consequence, for the rst component of the two component
system, from the insertion of the projection operator PZ2 in equation (3.19), we will obtain
a path integral measure
while for the second component, we obtain a path integral measure
1
2
1
2
Z xf
xi
Z xf
xi
Z
Z
xi
xi
xf
xf
dx +
dx ;
dx
dx :
Thus, from the discussion in subsection 3.1, the upper component, which we will call
fermionic and indicate with a minus sign, will satisfy a Neumannn boundary condition
at zero, while the bosonic component will satisfy the Dirichlet boundary condition. We
carefully crafted our setup to be consistent with supersymmetry, and must therefore expect
the boundary conditions we obtain to be consistent with supersymmetry as well. Indeed,
the operator Q maps the derivative of the fermionic wavefunction to the bosonic
wavefunction (when evaluated at the boundary, and using W (0) = 0). Thus, the operator Q
maps the boundary conditions into one another.9
The next case we wish to study is when the superpotential is wellde ned on the
halfline for x > 0, and approximates a nonzero constant as we tend towards x = 0. Since the
superpotential is odd on the line, the distributional derivative of the superpotential will be
a deltafunction with coe cient twice the limit of the superpotential as it tends towards
zero. If we call the latter value W0, then we have the equation
W 0(0) = 2W0 (x) :
The derivative of the superpotential arises as a term in the component Hamiltonians (3.14).
The function interaction at the origin will result in a change in the Neumann (but not
the Dirichlet) boundary conditions, as we saw in subsection 3.1. If we follow through the
consequences, we nd that the supersymmetric quantum mechanics on the half line that
we obtain by folding now satis es the boundary conditions
(3.21)
(3.22)
(3.23)
(3.24)
+(0) = 0
(0) :
These boundary conditions are consistent with supersymmetry.
3.2.3
An interval
We have used the folding technique to obtain a supersymmetric or ordinary quantum
mechanics problem on a half line. We can use the same technique to generate quantum
mechanics problems on an interval. We perform a second folding by the re ection symmetry
x ! 2L
x where L is the length of the desired interval. The fermions also transform with
9Note that the choice of action of ( 1)F on the two components (assigning to one component a plus
sign) broke the symmetry between Q and Qy in this discussion. In other words, the opposite assignment
would have resulted in the operator Qy mapping one boundary condition into the other.
a minus sign under the second Z2 generator. Again, we can render the superpotential odd
under the second ip, take into account a possible deltafunction potential on the second
end of the interval, and
nd boundary conditions consistent with supersymmetry on both
ends. Our application of these ideas lies in regulating a weighted trace, and we proceed
immediately to apply them in that particular context.
Infrared regulators and the weighted trace
We wish to discuss the trace
Z( ) = T r( 1)F e
H
over the Hilbert space of states, weighted with a sign ( 1)F corresponding to their fermion
number F . It is wellknown that this weighted trace is equal to the supersymmetric
(Witten) index when the spectrum of the supersymmetric quantum mechanics is discrete [
23
].
It then reduces to the index which equals the number of bosonic minus the number of
fermionic ground states.10
When the spectrum of the supersymmetric quantum mechanics is continuous, the
situation is considerably more complicated (see e.g. [
11, 24, 25
]), and the debate in the
literature on this quantity may not have culminated in a clear pedagogical summary. We
attempt to improve the state of a airs in this subsection. The origin of the di culties is
that the trace over a continuum of states is an illde ned concept. An in nite set of states
contributing a
nite amount gives rise to a divergent sum. A proper de nition requires
a regulator. An infrared regulator will reduce the continuum to a discretuum and render
the trace
nite. The alternating sum can remain
nite in the limit where we remove the
regulator. There has been a discussion on whether and how the resulting weighted trace
Z( ) depends on the inverse temperature , and on the infrared regulator. To understand
the main issues at stake, and to draw
rm conclusions, it is su cient to consider the
example of a free supersymmetric particle on the half line.
3.3.1
The free supersymmetric particle on the half line
Let us consider a supersymmetric quantum mechanics, based on the superpotential which
is equal to a constant for x > 0, namely W (x > 0) = W0.
We obtain the half line supersymmetric quantum mechanics by folding the problem on the whole line, and induce supersymmetric boundary conditions at the end of the half line. We recall the Hamiltonians (3.25)
HJEP10(27)8
with boundary conditions
H
2 W0 (x) ;
We can then solve for the wavefunctions on the half line. The solutions for energy E =
p2 + W02 are given by re ecting waves. The phase shift is set by the boundary condition.
10We use the name weighted trace because we will soon encounter contexts in which it is not an index.
= W0
+(0) = 0 :
(3.26)
(3.27)
We nd that the supercharge Q maps the wavefunction
into
+ if we identify c (p +
iW0) = c+. Thus, we have computed the space of eigenfunctions for bosons and fermions
and how they are related.
Our intermediate goal is to evaluate the weighted trace Z( ) in this model. To evaluate
the trace, we need an infrared regulator. Moreover, the weighted trace depends on the
infrared regulator, as we will demonstrate. In any case, we need to introduce an infrared
regulator to make the trace wellde ned. We cut o the space at large x = xIR. We need to
impose boundary conditions at this second end, at xIR. As a result, the spectrum becomes
discrete, and we will be able to perform the trace over states weighted by the corresponding
fermion number. We consider two regulators in detail.
In a rst regularization, we construct the supersymmetric quantum mechanics on the
interval as we described previously. The result will be a Hamiltonian
H
We have the wavefunctions on the half line x
e ipx) ;
(x) = c
eipx +
ip
b(0) = 0
b(xIR) = 0 :
eipnxIR
e ipnxIR = 0 ;
pn =
n
xIR
(3.28)
(3.29)
(3.30)
(3.31)
(3.32)
HJEP10(27)8
The reason that the boundary condition on both sides is the same despite the sign ip in the
function coe cient in (3.29) is because we are evaluating either the derivative with a left
or a right approach to the singular point. Because the Z2
Z2 folding procedures commute
with supersymmetry, the infrared regulated model preserves supersymmetry. Explicitly,
we have a spectrum determined by the infrared boundary condition
which implies
where n is an integer. All states are twofold degenerate. The state with the lowest energy
has energy equal to E = W02. The weighted trace reduces to a supersymmetric index and
the Witten index is equal to zero.
A second regularization of the weighted trace proceeds as follows. We rather put
Dirichlet boundary conditions at the infrared cuto
xIR for both component wavefunctions. We
can intuitively argue that we expect a normalizable wavefunction to drop o
at in nity,
and that the Dirichlet boundary condition is a good approximation to this expectation.
It has the added advantage of not introducing extra degrees of freedom at the end point
which we imagine to be responsible for a deltafunction potential. The disadvantage is that
this infrared regulator breaks supersymmetry. The regulated weighted trace will now sum
over bosonic and fermionic states determined by the respective conditions (see (3.28))
eipfn0 xIR +
eipbnxIR
ipfn0
ipfn0 + W0
W0
eipbnxIR = 0 ;
e ipfn0 xIR = 0 :
We de ne the phase shift
of the fermionic wavefunction. Then the solutions to the bosonic and fermionic boundary
conditions are
As the infrared cuto is taken larger, the number of states per small dp interval will
grow, to
nally reach the continuum we started out with. To measure this growth, we can
compute the bosonic and fermionic densities of states
b(p) =
f (p) =
dn
dp
dn0
dp
=
=
xIR
1
2
2xIR +
d (p)
dp
:
Thus, when we approximate the weighted trace at large infrared cuto by the appropriate
integral formula, we nd [11]
T r( 1)F e
H =
dp( b(p)
f (p))e
E(p)
ei (p) =
ip
W0
p
bn =
n
xIR
2pfn0 xIR + (pfn0 ) = 2
n0 +
:
1
2
(3.33)
(3.34)
(3.35)
(3.36)
(3.37)
(3.38)
where the di erence of densities of states is given by
f (p) =
0(p)
=
=
b(p)
1 d
2 i dp
log
ip
ip + W0 =
W0
1
2
1
This second way of regularizing shows that the boundary condition we impose at the
infrared end of our interval is crucial in determining the end result. When we put, as
we did in the rst case, a boundary condition consistent with supersymmetry, then the
di erence of spectral densities is zero for all values of the cuto , and therefore also in
the limit of in nite cuto . When we put identical boundary conditions for fermions and
bosons at the infrared endpoint, then the spectral densities di er by the phase shift in the
continuum problem. It should now be clear that one can choose another mix of boundary
conditions that will lead to yet another outcome for the spectral measure. Before a choice
of regulator, the weighted trace is illde ned. The
nal result depends on the regulator
choice, even after we remove the regulator. We have illustrated this e ect in two cases,
but there is an in nite number of choices, and the dependence of the
nal result Z( )
is determined by the choice of regulator. We should rather think of the weighted trace
Z( ; regulator) as a function of both the inverse temperature
and the regulator.
The rst regulator is interesting, since it preserves supersymmetry. The second
regulator, with identical boundary conditions for bosons and fermions is also interesting, it
turns out. Although we computed the spectral density in our particular model of the free
particle on a half line, the
nal result is universal in an appropriate sense. The relative
phase shift of bosons and fermions at large xIR is determined by the asymptotic form of the
supercharge Q alone. This can be seen from the fact that the fermionic wave function in
the infrared is determined by the bosonic wave function in the infrared and the asymptotic
supercharge. Thus, only the asymptotic value of the superpotential limx!1 W (x) = W0,
which we assume to be constant, will enter the phase shift and spectral density formula [11].
Thus, the result for the dependent weighted trace is universal, given the regularization
procedure. Both the universality and the caveat are crucial.
The
nal result for our free particle on the half line with Dirichlet infrared regulator
becomes [11]
Z( ; Dirichlet) =
e
(p2+W02)
=
1
2
dp
1
1
1
Of course, we recuperated the standard wisdom that any
supersymmetric regulator makes the weighted trace into a supersymmetric Witten index which is
independent. However, another choice of infrared regulator can give rise to a dependent
weighted trace, and the dependence is dictated by the regulator.
It is quite striking that there are applications of supersymmetric quantum mechanics on
a half line in which the infrared regulator is dictated by another symmetry of an overarching,
higher dimensional model. In such circumstances, the weighted trace and its dependence
become wellde ned and useful concepts.
3.4
The application to the elliptic genus
In the calculation of the cigar elliptic genus (2.1), there is a weighted trace over the
rightmoving supersymmetric quantum mechanics. For each sector labeled by the rightmoving
momentum m on the asymptotic circle of the cigar, there is a supersymmetric quantum
mechanics with superpotential W that asymptotes to W0 = m [
12
]. The point is now that,
as we saw, each of the rightmoving supersymmetric quantum mechanics labeled by the
rightmoving momentum can be cuto
supersymmetrically using a function potential
with coe cient depending on the rightmoving momentum m. The resulting elliptic genus
would be equal to the mock modular AppellLerch sum. The cuto
depending on the
rightmoving momentum is not modular covariant though. The rightmoving momentum
is a combination of a winding number of torus maps, and the Poisson dual of the other
winding number of torus maps, and as a result does not transform modular covariantly. The
second alternative (and the one generically preferred in the context of a twodimensional
theory of gravity in which we wish to preserve large di eomorphisms as a symmetry group)
is to have a Dirichlet cuto for all these supersymmetric quantum mechanics labeled by
the rightmoving momentum. This choice is covariant under modular transformations,
but is not supersymmetric, as we have shown. The result of the second regularization is
a modular completion of the mock modular form. We have thus shown that an anomaly
arises in the combination of rightmoving supersymmetry and modular covariance.
Let us comment on how generic the anomaly is. In cases where the asymptotic
supercharge relates a radial direction to a circle (as is the case in conical geometries), we expect
the supercharge to depend on radial and angular momentum. This momentum dependence
entails a di erence in the density of states as well. In such a rather generic setup, there is
an anomaly.11
Our analysis of supersymmetric quantum mechanics is interesting in itself. It also
provides the technical details of the reasoning in [6, 10], and thus produces a second panel
in our elliptic triptych. Moreover, our technical tinkering paints the background to
continuum contributions to indices, or rather their continuous counterparts in twodimensional
theories [28] as well as in fourdimensional theories with eight supercharges [29, 30]. In
particular, it clari es both the regulator dependence as well as the universality of the results
on weighted traces in the presence of supersymmetry and a continuum.
4
A
at space limit conformal
eld theory
In [
26
], we studied the in nite level limit of the cigar elliptic genus. In this limit, the target
space is attened. One is tempted to interpret the resulting conformal eld theory as a at
space supersymmetric conformal eld theory at central charge c = 3. Still, the theory has
features that distinguish it from a mundane at space theory. In this third panel, we add
remarks to the discussion provided in [
26
], to which we also refer for further context.
4.1
Flat space regulated
Firstly, we consider a at space conformal eld theory on R2, with two free bosonic scalar
elds, and two free Majorana fermions, for a total central charge of c = 3, and with
N = (2; 2) supersymmetry.
We consider the RamondRamond sector of the left and
rightmoving fermions.
11Exceptionally, the integral over the density of states can still be zero. An example is provided by the
cigar conformal eld theory at level one.
The ordinary bosonic partition function is divergent. There is an overall volume factor
arising from the integral over bosonic zero modes which makes the partition function
illde ned.
We can regulate the divergence in various ways.
One regulator would be to compactify the target space on a torus of volume V , and then take the radii of the torus to in nity. The result is that the partition function approximates (see e.g. [27])
V
0
ZV =
where V = 0 represents the volume divergence. Alternatively, we can compute the partition
function through zetafunction regularization and the rst Kronecker limit formula. See
HJEP10(27)8
e.g. [31]. The result is identical. If we regulate the bosons in this manner, and leave the
nite fermionic partition function unaltered, both the rightmoving fermions and the
leftmoving fermions will provide a zero mode in the RamondRamond sector partition sum.
Thus, we will nd that the regulated supersymmetric Witten index is zero for all nite
values of the volume regulator V . The limit of the supersymmetric index will be zero
under these circumstances.
A di erent way of regularizing is to twist the phase of the complex boson Z = X1+iX2.
In the path integral calculation of the complex boson partition function, this is implemented
in a modular covariant way by demanding that the eld con gurations we integrate over
pick up a phase as we go around a cycle of the torus. The phase is a character of the Z
homotopy group of the torus. If we parameterize the phases by e2 ium+2 ivw (for winding
numbers m; w on the two cycles of the torus), the result can be obtained either as the
RaySinger analytic torsion [32] (to the power minus two) or by using the second Kronecker
2
limit formula. The modular invariant result is
Ztwist = e
2
(Im( ))2 1( ; )
2
;
(4.1)
(4.2)
where
= u
v is the complexi ed twist. Near zero twist, there is a second order
divergence that is proportional to j j
2
j j
4 in accord with equation (4.1). The twist regulator
breaks the translation invariance in spacetime and preserves the rotational invariance.
In fact, it uses the rotation invariance to twist the angular direction and to remove all
bosonic zero modes. (The idea is generic in that one can use twists by global symmetries
to lift divergences in numerous contexts.) If we leave the fermions undisturbed, we again
have the fermionic zero modes that give rise to a zero elliptic genus for the full conformal
eld theory.
The twist regulator suggests an interesting alternative. We can twist the bosons and
preserve world sheet supersymmetry at the same time. The (tangent indexed) fermions
naturally transform under the SO(2) rotating the two spacetime directions, and if we twist
with respect to the complete action of the spacetime rotations, we twist the fermions as
well. In that case, we nd a partition function that equals one
Ztwist = e
2
(Im( ))2 1( ; ) 2
e
2
(Im( ))2 1( ; )
2
= 1 :
(4.3)
The two fermionic zero modes have canceled the quadratic volume divergence. The
supersymmetric partition function (or Witten index) is now equal to one for all values of the
twist, and therefore equals one in the limit where we remove the twist.
Again, as in section 3, we see that the nal result is regulator dependent (as is in nity
times zero).
We have two regulators that preserve world sheet supersymmetry as well
modular invariance, and they give rise to index equal to zero, or to one.
We analyze how the above remarks in uence our reading of the in nite level limit of the
cigar elliptic genus [
26
]. First o , we further twist the leftmoving fermions (only) by their
leftmoving Rcharge, and wind up with the modular invariant at space partition sum
Ztwist two =
(Im( + ))2
e
e
2
(Im( ))2
2
1( + ; )
:
This chiral partition function su ers from a chiral anomaly. We have again decided (for
now) on a modular invariant choice of phase. The regulating twist
has canceled the
rightmoving zero mode against the antiholomorphic pole due to the in nite volume. The
leftmoving Rcharge twist
(when nonequivalent to zero) has reintroduced the holomorphic
pole in
, also associated to the divergent volume. When we take the limit
! 0, we
therefore again
nd an in nite result.
Once more, there are various ways to regularize the expression. One straightforward
way to obtain the result in [
26
] is to perform a modular covariant minimal subtraction.
We expand the expression (4.4) near = 0, and subtract the pole. Given the dictum of a modular covariant transformation rule for the constant term (e.g. the desired modular covariant transformation rule for the elliptic genus [33]) one then obtains the result [26]
Zms;cov =
2
3
2 2
11( ; )
3
:
We note that the elliptic properties are lost in the large level limit since the periodicity
grows with the level. This contrasts sharply with the (alternatively regulated) zero result
for the at space elliptic genus which exhibits a very large symmetry group. 12
The cigar elliptic genus manages to regulate the pole at
= 0 in a more subtle manner
than the covariant minimal subtraction advocated above [6]. It goes as follows. One
introduces an extra circle. Then, one couples the circle to the angular direction of the
plane (or the cigar), and gauges a U(1) such as to identify the two circular directions. The
net e ect on the toroidal partition function is to incorporate the twist
into a modular
covariant holonomy integral. The integral over the angle of the twist kills the divergent
holomorphic pole, and renders the
nal result
nite. The result is identical to the one
obtained by covariant minimal subtraction (see [
26
] for the detailed derivation of this
statement).
12The lack of periodicity makes the partition function hard to interpret as an in nite set of Dirac indices.
To recuperate such an interpretation as an in nite set of Dirac indices, one reverts to the holomorphically
regulated expression, i.e. the mock modular part of the partition function.
(4.4)
(4.5)
Finally, we wish to assemble a miniature triptych. Firstly, we revisit the path integral
approach of section 2 and apply it to
at space. We Tdualize at space, consider the
in nite covering, and nd instead of the zero mode factor (2.13)
Z01; at = 2 N1
e R2 2
2
;
r 2) e r 2
2
2
Z1; at(R) = N1
where we have introduced an infrared cuto
R on the radial integral. Thus, we nd for
the in nite cover of the Tdual of at space the infrared regulated elliptic genus
(4.6)
(4.7)
(4.8)
For at space then, we nd the same lattice sum (see equation (2.20)) as for the cigar
elliptic genus, with the level k replaced by the infrared cuto
R2.
Our second panel, in section 3, makes it manifest that we have implicitly used the same
boundary conditions for bosons and fermions, since we considered a single measure, a hard
infrared cuto
R, and no deltafunction insertion. Hence we
nd the antiholomorphic
dependence in our result (4.7). Furthermore, our discussion in this section agrees with
the fact that if we take the limit R ! 1 term by term, neglecting the exponential factor
in (4.7), then we nd a divergent result. Indeed, the lattice sum will be divergent.
Finally, we note that (at R = 1) the genus can be regulated in the manner of the
Weierstrass function (which is the regulated lattice sum of 1= ). If we take that ad hoc
route, the result can be made holomorphic and nonmodular, and equal to only the rst
term in (4.5), using the formula
( ; )
G2( ) =
1( ; )
;
where G2 is the second Eisenstein series (and multiplying in the prefactor 1( ; )= 3)).
On the other hand, if we infrared regulate with a radial cuto
as in (4.7), or using the
cigar model in the large level limit, we obtain the modular covariant, nonholomorphic
result (4.5) which equals the exponentially regulated Eisenstein series as proven in [
8, 26
].
This nal miniature illustrates how our conceptual triptych folds together seamlessly.
5
Conclusion
Our aim in this paper was to further explain conceptual features of completed mock
modular noncompact elliptic genera [6] with elementary means. Using the supersymmetric
cigar conformal eld theory as an example, we provided a simple path integral derivation
of the lattice sum formula [8] for the completed mock modular form. We derived the elliptic
genus from the nonlinear sigmamodel.13 We also laid bare the unresolvable tension
between rightmoving supersymmetry and modular covariance in de ning the weighted trace
13Other derivations are based on the coset conformal eld theory or the gauged linear sigmamodel [
34, 35
]
descriptions.
with an infrared regulator, and we analyzed the quirks of the identi cation of the large
level limit of the cigar model [
26
] with a at space conformal eld theory.
We believe these conceptual pointers provide a looking glass with which to revisit higher
dimensional elliptic genera, including the K3, the ALE [36] and the higher dimensional
linear dilaton space genera [37]. The ubiquitous possibility to factor the appropriate powers
of 1= 3 bodes well for this enterprise. For fourdimensional examples, for instance, we
expect the doubling of the number of rightmoving zero modes to be correlated to an
elliptic Weierstrass } factor in the result, et cetera. It will be interesting to study these
generalizations.
Acknowledgments
Many thanks to Sujay Ashok, Costas Bachas, Amit Giveon, Dan Israel, Sunny Itzhaki,
Sameer Murthy, Boris Pioline, Giuseppe Policastro, Ashoke Sen and Yuji Sugawara
for useful discussions over the years, including at the workshop on Mock Modular
Forms and Physics in Chennai, India in 2014. I acknowledge support from the grant
ANR13BS050001.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
(2002), arXiv:0807.4834 [INSPIRE].
Lett. B 196 (1987) 75 [INSPIRE].
[INSPIRE].
651 [arXiv:0906.1767] [INSPIRE].
[arXiv:1004.3649] [INSPIRE].
[1] D. Zagier, Ramanujan's mock theta functions and their applications d'apres Zwegers and
BringmannOno, Seminaire Bourbaki 986 (2007).
[2] S. Zwegers, Mock theta functions, Ph.D. thesis, Utrecht University, Utrecht, The Netherlands
[3] T. Eguchi and A. Taormina, Unitary representations of N = 4 superconformal algebra, Phys.
[4] A.M. Semikhatov, A. Taormina and I.Yu. Tipunin, Higher level Appell functions, modular
transformations and characters, Commun. Math. Phys. 255 (2005) 469 [math/0311314]
[5] J. Manschot, Stability and duality in N = 2 supergravity, Commun. Math. Phys. 299 (2010)
[6] J. Troost, The noncompact elliptic genus: Mock or modular, JHEP 06 (2010) 104
[7] A. Dabholkar, S. Murthy and D. Zagier, Quantum black holes, wall crossing and Mock
modular forms, arXiv:1208.4074 [INSPIRE].
JHEP 11 (2014) 156 [arXiv:1407.7721] [INSPIRE].
[8] T. Eguchi and Y. Sugawara, Compact formulas for the completed Mock modular forms,
[9] T. Eguchi and Y. Sugawara, Nonholomorphic modular forms and SL(2; R)=U(1)
superconformal eld theory, JHEP 03 (2011) 107 [arXiv:1012.5721] [INSPIRE].
[arXiv:1101.1059] [INSPIRE].
Nucl. Phys. B 246 (1984) 253 [INSPIRE].
096 [arXiv:1302.1045] [INSPIRE].
[11] R. Akhoury and A. Comtet, Anomalous behavior of the Witten index: exactly soluble models,
HJEP10(27)8
Phys. B 359 (1991) 581 [INSPIRE].
theory, Mod. Phys. Lett. A 6 (1991) 1685 [INSPIRE].
[15] G. Mandal, A.M. Sengupta and S.R. Wadia, Classical solutions of twodimensional string
selfadjointness, Elsevier, Germany (1975).
251 (1995) 267 [hepth/9405029] [INSPIRE].
138 (1984) 389.
B 173 (1986) 327 [INSPIRE].
(2014) 160 [arXiv:1401.3104] [INSPIRE].
University Press, Cambridge U.K. (1998).
[20] C. Grosche, Delta function perturbations and boundary problems by path integration, Annalen
[21] M. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis,
[22] F. Cooper, A. Khare and U. Sukhatme, Supersymmetry and quantum mechanics, Phys. Rept.
genera, JHEP 08 (2014) 087 [arXiv:1404.7396] [INSPIRE].
[29] S. Alexandrov, G.W. Moore, A. Neitzke and B. Pioline, R3 index for fourdimensional N = 2
eld theories, Phys. Rev. Lett. 114 (2015) 121601 [arXiv:1406.2360] [INSPIRE].
[30] B. Pioline, Wallcrossing made smooth, JHEP 04 (2015) 092 [arXiv:1501.01643] [INSPIRE].
(1997).
HJEP10(27)8
[arXiv:1311.0918] [INSPIRE].
linear models, JHEP 02 (2015) 110 [arXiv:1406.6342] [INSPIRE].
[12] S.K. Ashok , S. Nampuri and J. Troost , Counting strings, wound and bound , JHEP 04 ( 2013 ) [13] Y. Sugawara , Comments on nonholomorphic modular forms and noncompact superconformal eld theories , JHEP 01 ( 2012 ) 098 [arXiv: 1109 .3365] [INSPIRE].
[14] S. Elitzur , A. Forge and E. Rabinovici, Some global aspects of string compacti cations , Nucl.
[16] E. Witten , On string theory and black holes , Phys. Rev. D 44 ( 1991 ) 314 [INSPIRE].
[17] J. Polchinski , String theory. Volume 2 : superstring theory and beyond , Cambridge University [18] T.E. Clark , R. Meniko and D.H. Sharp , Quantum mechanics on the half line using path [19] E. Farhi and S. Gutmann , The functional integral on the half line , Int. J. Mod. Phys. A 5 Press , Cambridge U.K. ( 1999 ).
integrals , Phys. Rev. D 22 ( 1980 ) 3012 [INSPIRE].
[23] E. Witten , Constraints on supersymmetry breaking, Nucl. Phys. B 202 ( 1982 ) 253 [INSPIRE].
[24] A.J. Niemi and L.C.R. Wijewardhana , Fractionization of the Witten index , Phys. Lett. B [25] N.A. Alves , H. Aratyn and A.H. Zimerman , Beta dependence of the Witten index , Phys. Lett.
[26] A. Giveon , N. Itzhaki and J. Troost, Lessons on black holes from the elliptic genus , JHEP 04 [27] J. Polchinski , String theory. Volume 1 : An introduction to the bosonic string , Cambridge [28] S.K. Ashok , E. Dell'Aquila and J. Troost , Higher poles and crossing phenomena from twisted [31] P. Di Francesco , P. Mathieu and D. Senechal , Conformal eld theory, Apringer, Germany [32] D.B. Ray and I.M. Singer , Analytic torsion for complex manifolds , Annals Math . 98 ( 1973 ) [33] T. Kawai , Y. Yamada and S.K. Yang , Elliptic genera and N = 2 superconformal eld theory, Nucl . Phys. B 414 ( 1994 ) 191 [ hep th/9306096] [INSPIRE].
[34] S. Murthy , A holomorphic anomaly in the elliptic genus , JHEP 06 ( 2014 ) 165 [35] S.K. Ashok , N. Doroud and J. Troost , Localization and real Jacobi forms , JHEP 04 ( 2014 ) [36] J.A. Harvey , S. Lee and S. Murthy , Elliptic genera of ALE and ALF manifolds from gauged [37] S.K. Ashok and J. Troost , Elliptic genera and real Jacobi forms , JHEP 01 ( 2014 ) 082