aa r X i v : . [ h e p - t h ] J un An Elliptic Triptych
Jan Troost
Laboratoire de Physique Th´eoriqueD´epartement de Physique de l’ENS´Ecole Normale Sup´erieurePSL Research University, Sorbonne Universit´es, UPMC, CNRSParis, France
Abstract:
We clarify three aspects of non-compact elliptic genera. Firstly, we give a pathintegral derivation of the elliptic genus of the cigar conformal field theory from its non-linearsigma-model description. The result is a manifestly modular sum over a lattice. Secondly,we discuss supersymmetric quantum mechanics with a continuous spectrum. We regulatethe theory and analyze the dependence on the temperature of the trace weighted by thefermion number. The dependence is dictated by the regulator. From a detailed analysis of thedependence on the infrared boundary conditions, we argue that in non-compact elliptic generaright-moving supersymmetry combined with modular covariance is anomalous. Thirdly, wefurther clarify the relation between the flat space elliptic genus and the infinite level limit ofthe cigar elliptic genus. ontents
Mock modular forms have an illustrious history in mathematics [1]. However, a systematicunderstanding of mock modular forms is recent [2] and evolving. Mock modular forms alsoappeared in physics in various guises [3–5]. A natural habitat for mock modular forms and theirnon-holomorphic modular completion was provided by the demonstration that they arise aselliptic genera of two-dimensional superconformal field theories with continuous spectrum [6].As such the completed forms appear also as duality covariant counterparts to black holeentropy counting functions [7].In this paper, we wish to clarify three aspects of non-compact elliptic genera. The firstcomment we make is on the compact form of the elliptic genus of the cigar derived by Eguchiand Sugawara in [8]. It is a modular covariant sum over lattice points which is an exponentiallyregulated Eisenstein series. Since it is manifestly modular covariant, one can wonder whetherit has a simple direct path integral derivation. We demonstrate that a path integration ofthe non-linear sigma-model description of the cigar provides such a derivation. The secondremark, in section 3, is based on an analysis of the temperature dependence of the weightedtrace
T r ( − F e − βH in supersymmetric quantum mechanics with a continuous spectrum. Uponregularization, the trace becomes β -dependent in a manner that hinges upon the choice ofregulator. We demonstrate this in detail, analyze the supersymmetric regulator and its pathintegral incarnation, and the role of infrared boundary conditions. We use it to lay barethe unresolvable tension between right-moving supersymmetry and modularity in the non-compact elliptic genus. In a third and final part, we clarify the relation between the flat space2uperconformal field theory and the infinite level limit of the cigar conformal field theory usingtheir elliptic genera. In this section, we wish to obtain a simpler path integral understanding of the compact formulafor the elliptic genus of the cigar in terms of a lattice sum, derived in [8]. To that end, weprovide a new derivation of the elliptic genus of the cigar, through its supersymmetric non-linear sigma-model description. The latter has the advantage of being parameterized in termsof the physical degrees of freedom only.
The cigar elliptic genus χ cig ( τ, α ) = T r RR ( − F L + F R e πiαQ q L − c ¯ q ¯ L − c (2.1)is a partition sum in the Ramond-Ramond sector, weighted by left- and right-moving fermionnumbers F L,R , as well as twisted by the left-moving R-charge Q . It was computed manifestlycovariantly 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 χ cig ( τ, α ) = k Z ds , X m,w ∈ Z θ ( s τ + s − α k +1 k , τ ) θ ( s τ + s − αk , τ ) e πiαw e − kπτ | ( m + s )+( w + s ) τ | , (2.2)where the θ functions arise from partition functions of fermions and bosons with twistedboundary conditions on the torus, the integers m, w are winding numbers for the maps fromthe torus onto the target space angular direction, and the angles s , are holonomies on thetorus for the U (1) gauge field used to gauge an elliptic isometry of SL (2 , R ). The twist withrespect to the left-moving R-charge is given by α . This modular Lagrangian result was putinto a Hamiltonian form in which the elliptic genus could be read directly as a sum overright-moving ground states plus an integral over the differences of spectral densities for thecontinuous spectrum of bosonic and fermionic right-movers [6, 10] . The difference of spectraldensities is determined by the asymptotic supercharge [6, 11, 12].In [8], a rewriting of the result (2.2) in terms of a lattice sum was obtained. The resultingexpression for the cigar elliptic genus is χ cig ( τ, α ) = θ ( α, τ )2 πη X m,w ∈ Z e − πkτ ( α + | m − wτ | +2 α ( m − w ¯ τ )) α + m − wτ . (2.3)This expression is also manifestly modular covariant, because it is written as a sum over alattice Z + Z τ . Our goal in this section is to understand the formula (2.3) in a more directmanner than through the route laid out in [6,8–10]. We recall that a key step in the derivationof the lattice sum (2.3) was to first compute the elliptic genus of the infinite cover of the Z k orbifold of the trumpet geometry [8, 13]. 3 .2 The Infinite Cover of The Orbifolded Trumpet We start our calculation from the cigar geometry [14–16] ds = α ′ k ( dρ + tanh ρ dθ ) e Φ = e Φ / cosh ρ , (2.4)where the angle θ is identified modulo 2 π . The metric and dilaton determine the couplings ofa conformal two-dimensional non-linear sigma-model. The T-dual geometry is the Z k orbifoldof the trumpet: ds = α ′ ( kdρ + 1 k coth ρ dθ ) e Φ = e Φ / sinh ρ (2.5)where the angle θ is again identified modulo 2 π . The infinite cover of the orbifold of thetrumpet is the geometry in which we no longer impose any equivalence relation on the variable θ . We perform the path integral on the cover as follows. Firstly, we consider the integralover the zero modes and the oscillator modes separately. We suppose that the oscillatorcontribution on the left is proportional to the free field result Z ∞ osc = 14 π τ θ ( α, τ ) η , (2.6)for a left-moving fermion of R-charge 1 and two uncharged bosonic fields. The factor 1 / (4 π τ )is the result of the integral over momenta (at α ′ = 1). The right-moving oscillators cancelamong each other.We want to focus on the remaining integral over zero modes, which contains the crucialinformation on the modularly completed Appell-Lerch sum [2]. The left-moving fermionic zeromodes have been lifted by the R-charge twist. Thus, we can concentrate on the integrationover the bosonic zero modes as well as the right-moving fermionic zero modes, with measure dρdθd ˜ ψ ρ d ˜ ψ θ . (2.7)The square root of the determinant in the diffeomorphism invariant measures has canceledbetween the bosons and the fermions. The relevant action is the N = (1 ,
1) supersymmetricextension of the non-linear sigma-model on the curved target space. The term in the actionthat lifts the right moving fermion zero modes is [17] S lift = 14 π Z d z G µν ˜ ψ µ Γ νρσ ∂X ρ ˜ ψ σ (2.8)and more specifically, the term proportional to the Christoffel connection symbolsΓ θθρ = − Γ ρθθ = 12 ∂ ρ G θθ . (2.9) See e.g. formula (12.3.27) in [17]. S lift = 14 π Z d z ˜ ψ θ ˜ ψ ρ ∂ ρ G θθ ∂θ . (2.10)We can descend this term once from the exponential in order to absorb the right-moving zeromodes and obtain a non-zero result.We wish to introduce a twist in the worldsheet time direction for the target space angulardirection θ because we insert a R-charge twist operator in the elliptic genus, and the field θ ischarged under the R-symmetry [6, 8–10]. We thus must twist θ ( σ + 2 πτ , σ + 2 πτ ) = θ ( σ , σ ) + 2 πα , (2.11)and we still have θ ( σ + 2 π ) = θ ( σ ). Since we study the infinite cover of the Z k orbifold ofthe trumpet, there are no winding sectors. We thus obtain the classical configuration θ cl = σ α/τ . (2.12)We plug this classical solution (2.12) into the action for the infinite order orbifold of thetrumpet, and descend a single insertion of (2.10) to lift the right-moving zero mode, use theChristoffel connection (2.9) and then find the zero mode integral Z ∞ = 2 πN ∞ Z ∞ dρ α∂ ρ ( − πk coth ρ ) e − πα kτ coth ρ = 2 πN ∞ τ α e − πα kτ . (2.13)We have represented the integral over the variable θ by a factor of 2 πN ∞ where we think of N ∞ as the order of the cover, which goes to infinity. Putting this together with the oscillatorfactor (2.6) we proposed previously, we find Z ∞ = N ∞ θ ( α, τ ) η πα e − πα kτ . (2.14)This precisely agrees with the elliptic genus of the infinite cover of the orbifolded trumpetcalculated in [8]. Our next step is the path integral incarnation of the procedure of the derivation of the latticesum formula in [8]. We undo the infinite order orbifold of the cigar, i.e. we undo the infiniteorder cover of the orbifolded trumpet. This will reproduce the lattice sum elliptic genusformula.There are two changes that we need to carefully track. The first one is that since the field θ becomes an angular variable with period 2 π , we must sum over the world sheet windingsectors. Thus, we introduce the identifications θ ( σ + 2 πτ , σ + 2 πτ ) = θ ( σ , σ ) + 2 π ( α + m ) θ ( σ + 2 π, σ ) = θ ( σ , σ ) + 2 πw , (2.15) The factor N ∞ is absorbed in the definition of Z ∞ in [8, 13]. θ cl = σ w + σ ( m + α − wτ ) /τ = − i τ ( z ( m + α − w ¯ τ ) − ¯ z ( m + α − wτ )) . (2.16)We then have the classical contribution to the action ∂θ cl ¯ ∂θ cl = 14 τ ( m + α − w ¯ τ )( m + α − wτ )= 14 τ ( | λ | + α ( λ + ¯ λ + α )) (2.17)where λ = m − wτ . After tracking normalization factors, one finds that the action acquiresanother overall factor of 4 πτ /k (see e.g. [27]).The second effect we must take into account is that the left-moving R-charge correspondsto the left-moving momentum of the angle field. When we introduce a winding number w , wemust properly take into account the contribution of the winding number to the left-movingmomentum. This amounts to adding a factor of e − πiαw/k to a contribution arising fromwinding number w . (Recall that the radius is R /α ′ = 1 /k .) We rewrite e − πiαw/k = e α ( λ − ¯ λ ) πkτ (2.18)which leads to a total contribution to the exponent equal to − πkτ ( | λ | + α ( λ + ¯ λ ) + α + α ( − λ + ¯ λ )) = − πkτ ( | λ | + 2 α ¯ λ + α ) . (2.19)The denominator in the final expression is obtained from a factor ( λ + α )(¯ λ + α ) in thedenominator that arises from the exponent (2.17) in the generalized zero mode integral (2.13)on the one hand, and a factor of ¯ λ + α in the numerator from the z -derivative of the angularvariable θ on the other hand (arising from the zero mode lifting term (2.10)). Multiplyingthese, we find the final formula χ cig ( τ, α ) = θ ( α, τ )2 πη X m,w ∈ Z e − πkτ ( α + | m − wτ | +2 α ( m − w ¯ τ )) α + m − wτ , (2.20)which is the compact lattice sum form [8] of the cigar elliptic genus. We have given a di-rect derivation of the lattice sum form, using the non-linear sigma model description. Thisconcludes the first panel of our triptych. In this section, we wish to render the fact that the non-holomorphic term in non-compactelliptic genera arises from a contribution due to the continuum of the right-moving super-symmetric quantum mechanics [6] even more manifest. For that purpose, we discuss to whatextent the right-moving supersymmetric quantum mechanics can be regularized in a super-symmetric invariant way, or a modular covariant manner, but not both. That fact leads to the6olomorphic anomaly [6]. The plan of this section is to first review how boundary conditionsin ordinary quantum mechanics show up in its path integral formulation. We then extend thisinsight to supersymmetric quantum mechanics. We illustrate the essence of the phenomenonin the simplest of systems. We end with a discussion of how the regulator of the non-compactelliptic genus cannot be both modular and supersymmetric, which leads to an anomaly.
We are used to path integrals that map spaces with boundaries into closed manifolds. Lessfrequently, 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 quantummechanics.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. Theeasiest way to proceed will be to relate the problem to quantum mechanics on the whole realline. What follows is a review of the results derived in e.g. [18–20], albeit from an originalperspective.
Firstly, we rapidly review quantum mechanics on the real line. We work with a Hilbert spacewhich consists of quadratically integrable functions on the line parameterized by a coordinate x . We have a Hamiltonian operator H of the form H = − ∂ x + V ( x ) , (3.1)where V ( x ) is a potential. We can define a Feynman amplitude to go from an initial position x i to a final position x f in time t through the path integral A ( x i , x f , t ) = Z x ( t )= x f x (0)= x i dx e iS [ x ] , (3.2)where the action is equal to S = Z t dt ′ ( ˙ x − V ( x )) . (3.3)The Schr¨odinger equation for the wave-function of the particle reads i∂ t Ψ = H Ψ , (3.4)and we work with normalized wave-functions Ψ. We can also write the amplitude in terms ofan integral over energy eigenstates Ψ E : A ( x i , x f , t ) = Z dEe − iEt Ψ E ( x i )Ψ E ( x f ) , (3.5)and the amplitude satisfies the δ -function completeness relation at t = 0, as well as theSchr¨odinger equation (3.4) in the initial and final position variables x i and x f .7 .1.2 Quantum Mechanics on the Half Line The subtleties of quantum mechanics on the open real half line x ≥ ∂ x Ψ(0) = c Ψ(0) . (3.6)When the constant c is zero, we have a Neumann boundary condition and when it is infinite,the boundary condition is in effect Dirichlet, Ψ(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 quantummechanics 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 potentialto take any value at the origin x = 0. We can then think of the quantum mechanics on thehalf line as a folded version of the quantum mechanics on the real line. The even quantummechanics that we constructed on the real line has a global symmetry group Z . We candivide the quantum mechanics problem on the real line, including its Hilbert space, by the Z operation, and find a well-defined quantum mechanics problem on the half line, which is theoriginal problem we wished to discuss.An advantage of this way of thinking is that the measure for quantum mechanics on thewhole line is canonical. It leads to the Green’s function (3.5). Since the quantum mechanicsthat we constructed has a global Z symmetry, we can classify eigenfunctions in terms of therepresentation they form under the Z symmetry, namely, we can classify them into even andodd eigenfunctions of the Hamiltonian. We then obtain the whole line Green’s function in theform that separates the even and odd energy eigenfunction contributions A ( x i , x f , t ) = Z dEe − iEt (Ψ E,e ( x i )Ψ E,e ( x f ) + Ψ E,o ( x i )Ψ E,o ( x f )) . (3.7)The Green’s function A ,D ( x i , x f , t ) = 12 ( A ( x i , x f , t ) − A ( x i , − x f , t )) = Z dEe − iEt Ψ E,o ( x i )Ψ E,o ( x f ) , (3.8)is well-defined on the half-line and satisfies Dirichlet boundary conditions. We divide bya factor of two since we are projecting onto Z invariant states. From the path integralperspective, the subtraction corresponds to a difference over paths that go from x i to x f and that go from x i to − x f , on the whole real line, with the canonical measure (divided bytwo). This prescription generates a measure on the half line which avoids the origin, since wesubtract all paths that cross to the other side [18,19]. If we represent the Z action oppositelyon the odd wave-functions, we arrive at the Green’s function that satisfies Neumann boundaryconditions: A ,N ( x i , x f , t ) = 12 ( A ( x i , x f , t ) + A ( x i , − x f , t )) = Z dEe − iEt Ψ E,e ( x i )Ψ E,e ( x f ) . (3.9) In string theory, one would say that we think of the half line as an orbifold of the real line. This is a common manipulation in probability theory.
8n this second option, we add paths to the final positions x f and − x f with their whole lineweights (divided by two). This path integral represents a sum over paths that reflect an evenor an odd number of times off the origin x = 0, and in particular, allows the particle to reachthe end of the half line.We clearly see that the naive folding operation projects the states of the quantum me-chanics onto those states that are even, or those that are odd. However, concentrating onthese 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 fixed point x = 0 of the folding operation. Wecan make use of this freedom by taking as the total potential an even potential V ( x ), zero at x = 0, complemented with a δ -function: H c ( x ) = − ∂ x / V ( x ) + c δ ( x ) . (3.10)We take the wave-function on the whole line to be even and continuous, with a discontinuousfirst derivative at the origin. When we consider the one-sided derivative at zero, we find thatthe wave-function satisfies the Robin boundary condition [19] ∂ x Ψ(0 + ) = c Ψ(0) . (3.11)We have gone from a purely even continuous and differentiable wave-function on the real linethat satisfies the Neumann boundary condition (at c = 0) to an even wave-function thatsatisfies mixed Robin boundary conditions, by influencing the wave-function near zero with adelta-function interaction. It is intuitively clear, and argued in detail in [19] that it is harderto push an initial problem with Dirichlet boundary conditions at the origin towards a mixedboundary condition problem. In order to achieve this, one needs a very deep well [19]. Forlater purposes, we note in particular that an ordinary delta-function insertion at the originwill not influence an initial Dirichlet boundary value problem.As an intuitive picture, we can imagine that the delta-function is generated by possibleextra degrees of freedom that are localized at the origin, and whose interaction with thequantum mechanical degree of freedom we concentrate on induces the delta-function potentiallocalized at the origin.Thus far, we briefly reviewed the results of [18, 19] on path integrals on the half line anddiscussed how they are consistent with folding. Next, we render these techniques compatiblewith supersymmetry.
In this section, we extend our perspective on quantum mechanics on the half line to a quantummechanical model with supersymmetry. We again start from a quantum mechanics on thewhole of the real line, with extra fermionic degrees of freedom and supersymmetry. In a These are states in the untwisted sector of an orbifold, projected onto invariants under the gauged discretesymmetry. In string theory orbifolds, the fixed point hosts extra degrees of freedom which in that case are verystrongly constrained by consistency. The even wave-function on the side x > x ) ∝ (Φ E,e ( x ) + c Φ E,o ( x )) in terms of even and odd solutions to the problem on the real line without the delta-functioninteraction [19]. It is an invariant under the Z action with discontinuous derivative at the origin. We discuss the supersymmetric system with Euclidean action (see e.g. [22]) S E = Z t dτ ( 12 ∂ τ x + 12 W − ψ ∗ ( ∂ τ − W ′ ) ψ ) , (3.12)where W ′ ( x ) = ∂ x W ( x ). The action permits two supersymmetries with infinitesimal variations δx = ǫ ∗ ψ + ψ ∗ ǫδψ ∗ = − ǫ ∗ ( ∂ τ x + W ) δψ = ǫ ( ∂ τ x − W ) . (3.13)When we quantize the fermionic degrees of freedom, we tensor the space of quadraticallyintegrable functions with a two component system. We call one component bosonic and theother fermionic. The two components have the Hamiltonians [22] H ± = p + W ∓ W ′ . (3.14)We introduced the operator p = − i∂ x (3.15)and can represent the supercharges by Q = ( p + iW ) (cid:18) (cid:19) Q † = ( p − iW ) (cid:18) (cid:19) . (3.16)When we trace over the fermionic degrees of freedom, we need to compute the fermionicdeterminant with anti-periodic boundary conditions. It evaluates to [22] Z anti − perf ( x ) = Z dψdψ ∗ anti − per exp( ψ ∗ ( ∂ τ − W ′ ) ψ ) = cosh( Z t dτ W ′ ( x )2 ) , (3.17)after regularization. This is the path integral counterpart to the calculation of the Hamilto-nians (3.14). We study the supersymmetric quantum mechanics on the half line by folding the supersym-metric quantum mechanics on the whole line. We wish for the folding Z symmetry to preservesupersymmetry. Since the particle position x is odd under the Z action (as is its derivative We follow standard conventions for supersymmetric quantum mechanics in this section. These differ by afactor of two from the standard conventions for quantum mechanics used in section 3.1. W ( x ) is odd under parity, and that the fermionic variables ψ and ψ ∗ are oddas well. See equation (3.13). Thus, we have the Z action( x, ψ, ψ ∗ ) → ( − x, − ψ, − ψ ∗ ) , (3.18)and the superpotential W is odd. For the moment, we consider the superpotential to becontinuous, and therefore zero at zero.We project onto states invariant under the Z action (3.18). Thus, in any path integral,we will insert a projection operator P Z that consists of P Z = 12 (1 + P ( − F ) (3.19)where P is the parity operator that maps P : x → − x and ( − F maps fermions to minusthemselves. When we trace over the fermionic degrees of freedom with a ( − F insertion, wemust impose periodic boundary conditions on the fermions. The fermionic determinant inthis case evaluates to [22] Z perf ( x ) = Z dψdψ ∗ per exp( ψ ∗ ( ∂ τ − W ′ ) ψ ) = sinh( Z T dτ W ′ ( x )2 ) , (3.20)which leads to the same Hamiltonians (3.14) for the two component system, and when wecompare to equation (3.17) we find a minus sign up front in the path integral over the secondcomponent. As a consequence, for the first component of the two component system, fromthe insertion of the projection operator P Z in equation (3.19), we will obtain a path integralmeasure 12 ( Z x f x i dx + Z − x f x i dx ) , (3.21)while for the second component, we obtain a path integral measure12 ( Z x f x i dx − Z − x f x i dx ) . (3.22)Thus, from the discussion in subsection 3.1, the upper component, which we will call fermionicand indicate with a minus sign, will satisfy a Neumannn boundary condition at zero, while thebosonic component will satisfy the Dirichlet boundary condition. We carefully crafted our set-up to be consistent with supersymmetry, and must therefore expect the boundary conditionswe obtain to be consistent with supersymmetry as well. Indeed, the operator Q maps thederivative of the fermionic wave-function to the bosonic wave-function (when evaluated at theboundary, and using W (0) = 0). Thus, the operator Q maps the boundary conditions intoone another. The next case we wish to study is when the superpotential is well-defined on the half-line for x >
0, and approximates a non-zero constant as we tend towards x = 0. Since the Note that the choice of action of ( − F on the two components (assigning to one component a plus sign)broke the symmetry between Q and Q † in this discussion. In other words, the opposite assignment would haveresulted in the operator Q † mapping one boundary condition into the other. W , then we have the equation W ′ (0) = 2 W δ ( x ) . (3.23)The derivative of the superpotential arises as a term in the component Hamiltonians (3.14).The δ -function interaction at the origin will result in a change in the Neumann (but notthe Dirichlet) boundary conditions, as we saw in subsection 3.1. If we follow through theconsequences, we find that the supersymmetric quantum mechanics on the half line that weobtain by folding now satisfies the boundary conditionsΨ + (0) = 0 ∂ x Ψ − (0) = W Ψ − (0) . (3.24)These boundary conditions are consistent with supersymmetry. We have used the folding technique to obtain a supersymmetric or ordinary quantum mechan-ics problem on a half line. We can use the same technique to generate quantum mechanicsproblems on an interval. We perform a second folding by the reflection symmetry x → L − x where L is the length of the desired interval. The fermions also transform with a minus signunder the second Z generator. Again, we can render the superpotential odd under the secondflip, take into account a possible delta-function potential on the second end of the interval,and find boundary conditions consistent with supersymmetry on both ends. Our applicationof these ideas lies in regulating a weighted trace, and we proceed immediately to apply themin that particular context. We wish to discuss the trace Z ( β ) = T r ( − F e − βH (3.25)over the Hilbert space of states, weighted with a sign ( − F corresponding to their fermionnumber F . It is well-known that this weighted trace is equal to the supersymmetric (Witten)index when the spectrum of the supersymmetric quantum mechanics is discrete [23]. It thenreduces to the index which equals the number of bosonic minus the number of fermionicground states. When the spectrum of the supersymmetric quantum mechanics is continuous, the situationis considerably more complicated (see e.g. [11, 24, 25]), and the debate in the literature onthis quantity may not have culminated in a clear pedagogical summary. We attempt toimprove the state of affairs in this subsection. The origin of the difficulties is that the traceover a continuum of states is an ill-defined concept. An infinite set of states contributing afinite amount gives rise to a divergent sum. A proper definition requires a regulator. An We use the name weighted trace because we will soon encounter contexts in which it is not an index. Z ( β ) depends on the inversetemperature β , and on the infrared regulator. To understand the main issues at stake, and todraw firm conclusions, it is sufficient to consider the example of a free supersymmetric particleon the half line. Let us consider a supersymmetric quantum mechanics, based on the superpotential which isequal to a constant for x >
0, namely W ( x >
0) = W . We obtain the half line supersymmetricquantum mechanics by folding the problem on the whole line, and induce supersymmetricboundary conditions at the end of the half line. We recall the Hamiltonians H ± = p + W ∓ W δ ( x ) , (3.26)with boundary conditions ∂ x Ψ − = W Ψ − Ψ + (0) = 0 . (3.27)We can then solve for the wave-functions on the half line. The solutions for energy E = p + W are given by reflecting waves. The phase shift is set by the boundary condition. We have thewave-functions on the half line x ≥ + ( x ) = c + ( e ipx − e − ipx ) , Ψ − ( x ) = c − ( e ipx + ip − W ip + W e − ipx ) . (3.28)We find that the supercharge Q maps the wave-function Ψ − into Ψ + if we identify c − ( p + iW ) = c + . Thus, we have computed the space of eigenfunctions for bosons and fermions andhow they are related. Our intermediate goal is to evaluate the weighted trace Z ( β ) in this model. To evaluate thetrace, we need an infrared regulator. Moreover, the weighted trace depends on the infraredregulator, as we will demonstrate. In any case, we need to introduce an infrared regulatorto make the trace well-defined. We cut off the space at large x = x IR . We need to imposeboundary conditions at this second end, at x IR . As a result, the spectrum becomes discrete,and we will be able to perform the trace over states weighted by the corresponding fermionnumber. We consider two regulators in detail.In a first regularization, we construct the supersymmetric quantum mechanics on theinterval as we described previously. The result will be a Hamiltonian H ∓ = p + W ± W δ ( x ) ∓ W δ ( x − x IR ) , (3.29)13nd boundary conditions ∂ x Ψ f (0 + ) = W Ψ f Ψ b (0) = 0 ∂ x Ψ f ( x − IR ) = W Ψ f Ψ b ( x IR ) = 0 . (3.30)The reason that the boundary condition on both sides is the same despite the sign flip in the δ function coefficient in (3.29) is because we are evaluating either the derivative with a left ora right approach to the singular point. Because the Z × Z folding procedures commute withsupersymmetry, the infrared regulated model preserves supersymmetry. Explicitly, we have aspectrum determined by the infrared boundary condition e ip n x IR − e − ip n x IR = 0 , (3.31)which implies p n = πnx IR (3.32)where n is an integer. All states are two-fold degenerate. The state with the lowest energyhas energy equal to E = W . The weighted trace reduces to a supersymmetric index and theWitten index is equal to zero.A second regularization of the weighted trace proceeds as follows. We rather put Dirichletboundary conditions at the infrared cut-off x IR for both component wave-functions. We canintuitively argue that we expect a normalizable wave-function to drop off at infinity, andthat the Dirichlet boundary condition is a good approximation to this expectation. It hasthe added advantage of not introducing extra degrees of freedom at the end point which weimagine to be responsible for a delta-function potential. The disadvantage is that this infraredregulator breaks supersymmetry. The regulated weighted trace will now sum over bosonic andfermionic states determined by the respective conditions (see (3.28)) e ip bn x IR − e ip bn x IR = 0 ,e ip fn ′ x IR + ip fn ′ − W ip fn ′ + W e − ip fn ′ x IR = 0 . (3.33)We define the phase shift e iδ ( p ) = ip + W ip − W (3.34)of the fermionic wave-function. Then the solutions to the bosonic and fermionic boundaryconditions are p bn = πnx IR p fn ′ x IR + δ ( p fn ′ ) = 2 π ( n ′ + 12 ) . (3.35)As the infrared cut-off is taken larger, the number of states per small dp interval will grow,to finally reach the continuum we started out with. To measure this growth, we can compute14he bosonic and fermionic densities of states ρ b ( p ) = dndp = x IR πρ f ( p ) = dn ′ dp = 12 π (2 x IR + dδ ( p ) dp ) . (3.36)Thus, when we approximate the weighted trace at large infrared cut-off by the appropriateintegral formula, we find [11] T r ( − F e − βH = Z ∞ dp ( ρ b ( p ) − ρ f ( p )) e − βE ( p ) (3.37)where the difference of densities of states is given by∆ ρ = ρ b ( p ) − ρ f ( p ) = 12 π δ ′ ( p )= 12 πi ddp log ip + W ip − W = 12 π ( 1 ip + W − ip − W ) . (3.38)This second way of regularizing shows that the boundary condition we impose at theinfrared end of our interval is crucial in determining the end result. When we put, as we didin the first case, a boundary condition consistent with supersymmetry, then the difference ofspectral densities is zero for all values of the cut-off, and therefore also in the limit of infinitecut-off. When we put identical boundary conditions for fermions and bosons at the infraredendpoint, then the spectral densities differ by the phase shift in the continuum problem. Itshould now be clear that one can choose another mix of boundary conditions that will leadto yet another outcome for the spectral measure. Before a choice of regulator, the weightedtrace is ill-defined. The final result depends on the regulator choice, even after we removethe regulator. We have illustrated this effect in two cases, but there is an infinite number ofchoices, and the β -dependence of the final result Z ( β ) is determined by the choice of regulator.We should rather think of the weighted trace Z ( β, regulator) as a function of both the inversetemperature β and the regulator.The first 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 ona half line, the final result is universal in an appropriate sense. The relative phase shift ofbosons and fermions at large x IR 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 infraredis determined by the bosonic wave function in the infrared and the asymptotic supercharge.Thus, only the asymptotic value of the superpotential lim x →∞ W ( x ) = W , which we assumeto be constant, will enter the phase shift and spectral density formula [11]. Thus, the resultfor the β -dependent weighted trace is universal, given the regularization procedure. Both theuniversality and the caveat are crucial.The final result for our free particle on the half line with Dirichlet infrared regulator15ecomes [11] Z ( β, Dirichlet) = Z ∞ dp π ( 1 ip + W − ip − W ) e − β ( p + W ) = Z + ∞−∞ dp π ip + W e − β ( p + W ) . (3.39) Conclusion
Of course, we recuperated the standard wisdom that any supersymmetric regulator makesthe weighted trace into a supersymmetric Witten index which is β -independent. However,another choice of infrared regulator can give rise to a β -dependent weighted trace, and the β -dependence is dictated by the regulator.It is quite striking that there are applications of supersymmetric quantum mechanics on ahalf 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 β -dependencebecome well-defined and useful concepts. In the calculation of the cigar elliptic genus (2.1), there is a weighted trace over the right-moving supersymmetric quantum mechanics. For each sector labeled by the right-movingmomentum ¯ m on the asymptotic circle of the cigar, there is a supersymmetric quantum me-chanics with superpotential W that asymptotes to W = ¯ m [12]. The point is now that, as wesaw, each of the right-moving supersymmetric quantum mechanics labeled by the right-movingmomentum can be cut-off supersymmetrically using a δ -function potential with coefficient de-pending on the right-moving momentum ¯ m . The resulting elliptic genus would be equal tothe mock modular Appell-Lerch sum. The cut-off depending on the right-moving momentumis not modular covariant though. The right-moving momentum is a combination of a windingnumber 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 onegenerically preferred in the context of a two-dimensional theory of gravity in which we wish topreserve large diffeomorphisms as a symmetry group) is to have a Dirichlet cut-off for all thesesupersymmetric quantum mechanics labeled by the right-moving momentum. This choice is covariant under modular transformations, but is not supersymmetric, as we have shown. Theresult of the second regularization is a modular completion of the mock modular form. Wehave thus shown that an anomaly arises in the combination of right-moving supersymmetryand modular covariance.Our analysis of supersymmetric quantum mechanics is interesting in itself. It also providesthe technical details of the reasoning in [6,10], and thus produces a second panel in our elliptictriptych. Moreover, our technical tinkering paints the background to continuum contributionsto indices, or rather their continuous counterparts in two-dimensional theories [28] as well asin four-dimensional theories with eight supercharges [29, 30]. In particular, it clarifies boththe regulator dependence as well as the universality of the results on weighted traces in thepresence of supersymmetry and a continuum.16 A Flat Space Limit Conformal Field Theory
In [26], we studied the infinite level limit of the cigar elliptic genus. In this limit, the targetspace is flattened. One is tempted to interpret the resulting conformal field theory as a flatspace supersymmetric conformal field theory at central charge c = 3. Still, the theory hasfeatures that distinguish it from a mundane flat space theory. In this third panel, we addremarks to the discussion provided in [26], to which we also refer for further context. Firstly, we consider a flat space conformal field theory on R , with two free bosonic scalarfields, and two free Majorana fermions, for a total central charge of c = 3, and with N =(2 ,
2) supersymmetry. We consider the Ramond-Ramond sector of the left- and right-movingfermions.The ordinary bosonic partition function is divergent. There is an overall volume factorarising from the integral over bosonic zero modes which makes the partition function ill-defined. We can regulate the divergence in various ways. One regulator would be to compactifythe target space on a torus of volume V , and then take the radii of the torus to infinity. Theresult is that the partition function approximates (see e.g. [27]) Z V = Vα ′ (4 π τ ) − | η | − , (4.1)where V /α ′ represents the volume divergence. Alternatively, we can compute the partitionfunction through zeta-function regularization and the first Kronecker limit formula. See e.g.[31]. The result is identical. If we regulate the bosons in this manner, and leave the finitefermionic partition function unaltered, both the right-moving fermions and the left-movingfermions will provide a zero mode in the Ramond-Ramond sector partition sum. Thus, we willfind that the regulated supersymmetric Witten index is zero for all finite values of the volumeregulator V . The limit of the supersymmetric index will be zero under these circumstances.A different way of regularizing is to twist the phase of the complex boson Z = X + iX .In the path integral calculation of the complex boson partition function, this is implementedin a modular covariant way by demanding that the field configurations we integrate over pickup a phase as we go around a cycle of the torus. The phase is a character of the Z homotopygroup of the torus. If we parameterize the phases by e πium +2 πivw (for winding numbers m, w on the two cycles of the torus), the result can be obtained either as the Ray-Singer analytictorsion [32] (to the power minus two) or by using the second Kronecker limit formula. Themodular invariant result is Z twist = | e − π (Im( β ))2 τ θ ( β, τ ) η | − , (4.2)where β = u − vτ is the complexified twist. Near zero twist, there is a second order divergencethat is proportional to | β | − | η | − in accord with equation (4.1). The twist regulator breaks thetranslation invariance in space-time and preserves the rotational invariance. In fact, it usesthe 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 innumerous contexts.) If we leave the fermions undisturbed, we again have the fermionic zeromodes that give rise to a zero elliptic genus for the full conformal field theory.17he twist regulator suggests an interesting alternative. We can twist the bosons and pre-serve world sheet supersymmetry at the same time. The (tangent indexed) fermions naturallytransform under the SO (2) rotating the two space-time directions, and if we twist with respectto the complete action of the space-time rotations, we twist the fermions as well. In that case,we find a partition function that equals one Z twist = | e − π (Im( β ))2 τ θ ( β, τ ) η | × | e − π (Im( β ))2 τ θ ( β, τ ) η | − = 1 . (4.3)The two fermionic zero modes have canceled the quadratic volume divergence. The super-symmetric 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 final result is regulator dependent (as is infinitytimes zero). We have two regulators that preserve world sheet supersymmetry as well modularinvariance, and they give rise to index equal to zero, or to one. We analyze how the above remarks influence our reading of the infinite level limit of thecigar elliptic genus [26]. First off, we further twist the left-moving fermions (only) by theirleft-moving R-charge, and wind up with the modular invariant flat space partition sum Z twisttwo = | e − π (Im( α + β ))2 τ θ ( α + β, τ ) e − π (Im( β ))2 τ θ ( β, τ ) | . (4.4)This chiral partition function suffers from a chiral anomaly. We have again decided (for now)on a modular invariant choice of phase. The regulating twist β has canceled the right-movingzero mode against the anti-holomorphic pole due to the infinite volume. The left-movingR-charge twist α (when non-equivalent to zero) has reintroduced the holomorphic pole in β ,also associated to the divergent volume. When we take the limit β →
0, we therefore againfind an infinite result.Once more, there are various ways to regularize the expression. One straightforward way toobtain the result in [26] is to perform a modular covariant minimal subtraction. We expand theexpression (4.4) near β = 0, and subtract the pole. Given the dictum of a modular covarianttransformation rule for the constant term (e.g. the desired modular covariant transformationrule for the elliptic genus [33]) one then obtains the result [26] Z ms,cov = − π ∂ α θ ( α, τ ) η − α τ θ ( α, τ ) η . (4.5)The cigar elliptic genus manages to regulate the pole at β = 0 in a more subtle manner thanthe covariant minimal subtraction advocated above [6]. It goes as follows. One introducesan extra circle. Then, one couples the circle to the angular direction of the plane (or thecigar), and gauges a U (1) such as to identify the two circular directions. The net effect on thetoroidal partition function is to incorporate the twist β into a modular covariant holonomyintegral. The integral over the angle of the twist kills the divergent holomorphic pole, andrenders the final result finite. The result is identical to the one obtained by covariant minimalsubtraction (see [26] for the detailed derivation of this statement).18 .3 A Miniature Finally, we wish to assemble a miniature triptych. Firstly, we revisit the path integral approachof section 2 and apply it to flat space. We T-dualize flat space, consider the infinite covering,and find instead of the zero mode factor (2.13) Z ∞ ,flat = 2 πN ∞ Z R dr∂ r ( − πr − ) αe − r − πτ α = 2 πN ∞ τ α e − πR τ α , (4.6)where we have introduced an infrared cut-off R on the radial integral. Thus, we find for theinfinite cover of the T-dual of flat space the infrared regulated elliptic genus Z ∞ ,flat ( R ) = N ∞ θ ( α, τ ) η πα e − πα R τ . (4.7)For flat space then, we find the same lattice sum (see equation (2.20)) as for the cigar ellipticgenus, with the level k replaced by the infrared cut-off R .Our second panel, in section 3, makes it manifest that we have implicitly used the sameboundary conditions for bosons and fermions, since we considered a single measure, a hardinfrared cut-off R , and no delta-function insertion. Hence we find the anti-holomorphic ¯ τ dependence in our result (4.7). Furthermore, our discussion in this section agrees with thefact that if we take the limit R → ∞ term by term, neglecting the exponential factor in (4.7),then we find a divergent result. Indeed, the lattice sum will be divergent.Finally, we note that (at R = ∞ ) the genus can be regulated in the manner of theWeierstrass ζ -function (which is the regulated lattice sum of 1 /α ). If we take that ad hocroute, the result can be made holomorphic and non-modular, and equal to only the first termin (4.5), using the formula ζ ( α, τ ) − G ( τ ) α = ∂ α θ ( α, τ ) θ ( α, τ ) , (4.8)where G is the second Eisenstein series (and multiplying in the prefactor θ ( α, τ ) /η )). Onthe other hand, if we infrared regulate with a radial cut-off as in (4.7), or using the cigar modelin the large level limit, we obtain the modular covariant, non-holomorphic result (4.5) whichequals the exponentially regulated Eisenstein series as proven in [8, 26]. This final miniatureillustrates how our conceptual triptych folds together seamlessly. Our aim in this paper was to further explain conceptual features of completed mock modularnon-compact elliptic genera [6] with elementary means. Using the supersymmetric cigar con-formal field theory as an example, we provided a simple path integral derivation of the latticesum formula [8] for the completed mock modular form. We derived the elliptic genus from thenon-linear sigma-model . We also laid bare the unresolvable tension between right-moving Other derivations are based on the coset conformal field theory or the gauged linear sigma-model [34, 35]descriptions. K
3, the ALE [36] and the higher dimensional lineardilaton space genera [37]. The ubiquitous possibility to factor the appropriate powers of θ /η bodes well for this enterprise. For four-dimensional examples, for instance, we expect thedoubling of the number of right-moving 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, SameerMurthy, Boris Pioline, Giuseppe Policastro, Ashoke Sen and Yuji Sugawara for useful discus-sions over the years, including at the workshop on Mock Modular Forms and Physics inChennai, India in 2014. I acknowledge support from the grant ANR-13-BS05-0001.
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