An elliptic triptych

Journal of High Energy Physics, Oct 2017

We clarify three aspects of non-compact elliptic genera. Firstly, we give a path integral derivation of the elliptic genus of the cigar conformal field theory from its non-linear sigma-model 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 right-moving supersymmetry combined with modular covariance is anomalous. Thirdly, we further clarify the relation between the flat space elliptic genus and the infinite level limit of the cigar elliptic genus.

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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 non-compact elliptic genera. Firstly, we give a path integral derivation of the elliptic genus of the cigar conformal eld theory from its non-linear sigma-model 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 right-moving 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 non-linear sigma-model 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 Ramond-Ramond sector, weighted by left- and right-moving fermion numbers FL;R, as well as twisted by the left-moving R-charge 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 left-moving R-charge 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 right-moving ground states plus an integral over the di erences of spectral densities for the continuous spectrum of bosonic and fermionic right-movers [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 two-dimensional non-linear sigma-model. The T-dual 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 space-time 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 right-moving 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 Appell-Lerch sum [2]. The left-moving fermionic zero modes have been lifted by the R-charge twist. Thus, we can concentrate on the integration over the bosonic zero modes as well as the right-moving 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 non-linear sigma-model 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 non-zero result. We wish to introduce a twist in the worldsheet time direction for the target space angular direction because we insert a R-charge twist operator in the elliptic genus, and the eld is charged under the R-symmetry [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 left-moving R-charge corresponds to the left-moving 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 left-moving 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 z-derivative 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 non-linear 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 non-holomorphic term in non-compact elliptic genera arises from a contribution due to the continuum of the right-moving supersymmetric quantum mechanics [6] even more manifest. For that purpose, we discuss to what extent the right-moving 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 non-compact 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 wave-function of the particle reads and we work with normalized wave-functions . 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 well-understood. We review what is known. The half-line 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 well-de 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 well-de ned on the half-line 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 wave-functions, 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 wave-function on the whole line to be even and continuous, with a discontinuous rst derivative at the origin. When we consider the one-sided derivative at zero, we nd that the wave-function satis es the Robin boundary condition [19] We have gone from a purely even continuous and di erentiable wave-function on the real line that satis es the Neumann boundary condition (at c = 0) to an even wave-function that satis es mixed Robin boundary conditions, by in uencing the wave-function near zero with a delta-function 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 wave-function 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 delta-function 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 delta-function insertion at the origin will not in uence an initial Dirichlet boundary value problem. As an intuitive picture, we can imagine that the delta-function 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 delta-function 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 anti-periodic 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 set-up 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 wave-function 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 well-de ned on the halfline for x > 0, and approximates a non-zero constant as we tend towards x = 0. Since the superpotential is odd on the line, the distributional derivative of the superpotential will be a delta-function 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 delta-function 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 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 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 ill-de 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 wave-functions 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 wave-function 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 well-de 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 wave-functions 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 two-fold 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 cut-o xIR for both component wave-functions. We can intuitively argue that we expect a normalizable wave-function 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 delta-function 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 wave-function. Then the solutions to the bosonic and fermionic boundary conditions are As the infrared cut-o 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 cut-o 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 cut-o , and therefore also in the limit of in nite cut-o . 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 ill-de 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 well-de 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 right-moving 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 right-moving supersymmetric quantum mechanics labeled by the right-moving momentum can be cut-o supersymmetrically using a -function potential with coe cient depending on the right-moving momentum m. The resulting elliptic genus would be equal to the mock modular Appell-Lerch sum. The cut-o depending on the right-moving momentum is not modular covariant though. The right-moving 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 two-dimensional theory of gravity in which we wish to preserve large di eomorphisms as a symmetry group) is to have a Dirichlet cut-o for all these supersymmetric quantum mechanics labeled by the right-moving 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 right-moving 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 set-up, 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 two-dimensional theories [28] as well as in four-dimensional 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 Ramond-Ramond sector of the left- and right-moving 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 zeta-function 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 right-moving fermions and the leftmoving fermions will provide a zero mode in the Ramond-Ramond 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 space-time 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 space-time directions, and if we twist with respect to the complete action of the space-time 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 left-moving fermions (only) by their left-moving R-charge, 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 anti-holomorphic pole due to the in nite volume. The leftmoving R-charge twist (when non-equivalent 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 T-dualize 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 cut-o R on the radial integral. Thus, we nd for the in nite cover of the T-dual 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 cut-o 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 cut-o R, and no delta-function insertion. Hence we nd the anti-holomorphic 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 non-modular, 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 cut-o as in (4.7), or using the cigar model in the large level limit, we obtain the modular covariant, non-holomorphic 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 non-compact 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 non-linear sigma-model.13 We also laid bare the unresolvable tension between right-moving supersymmetry and modular covariance in de ning the weighted trace 13Other derivations are based on the coset conformal eld theory or the gauged linear sigma-model [ 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 four-dimensional examples, for instance, we expect the doubling of the number of right-moving zero modes to be correlated to an elliptic Weierstrass } factor in the result, et cetera. 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Jan Troost. An elliptic triptych, Journal of High Energy Physics, 2017, 78, DOI: 10.1007/JHEP10(2017)078