JHEP11(2019)080 Springer July 1, 2019 : October 31, 2019 : November 13, 2019 : ´ Energies, LPTHE, Received Accepted Published -invariant to a Callias index η , -invariant can be related to the η a Published for SISSA by https://doi.org/10.1007/JHEP11(2019)080 Dedicated to the memory of Michael Atiyah and Arnab Rudra a,b [email protected] , . 3 Diksha Jain 1905.05207 a,c,d The Authors. c Anomalies in Field and String Theories, Differential and Algebraic Geometry,
We show that the Atiyah-Patodi-Singer , -invariant, path integrals, and mock modularity [email protected] η E-mail: [email protected] Via Bonomea 265, Trieste 34136,Sorbonne Italy Universit´e,CNRS, Laboratoire de PhysiqueF-75005 Th´eoriqueet Hautes Paris, France CNRS, UMR 7589, LPTHE, Paris, F-75005 France Abdus Salam International Centre forStrada Costiera Theoretical Physics 11, (ICTP), Trieste 34151,Scuola Italy Internazionale Superiore di Studi Avanzati (SISSA), -invariant for the elliptic genus of a finite cigar is related to quantum modular forms ob- b c d a Open Access Article funded by SCOAP Keywords: Scattering Amplitudes, Sigma Models ArXiv ePrint: and compute it usingη localization of atained supersymmetric from path the completion integral. ofpath a We mock integral. show Jacobi form that which the we compute from the noncompact Abstract: temperature-dependent Witten indexthe of APS a theorem noncompact using theory scattering and theory. give a We relate new the proof of Atish Dabholkar, APS JHEP11(2019)080 (1.1) ] is defined by 4 36 i Michael Atiyah βH 33 ] and associated quantum modular − 2 e 22 F 16 1) − 27 ( 4 20 h 18 35 – 1 – 12 H 8 30 11 Tr -invariant 24 η 38 ) := -invariant 13 β ( η -invariant W 5 -invariant η η ], mock Jacobi forms [ 29 -dimensional spacetime, the Witten index [ Witten index of a noncompact theory. For a supersymmetric 1 1) superspace d , 37 1 -invariant [ η -invariant of a finite cigar -invariant and quantum modular forms η η “Any good theorem should have several proofs, the more the better.” -invariant and path integrals η ], and supersymmetric path integrals. The link between these three is provided by 3 temperature-dependent 4.3 The 5.1 Mock Jacobi5.2 forms Elliptic genus5.3 of an infinite The cigar 3.2 Scattering theory and the APS theorem 4.1 Supersymmetric4.2 worldpoint integral Callias index theorem and the 2.1 Compact Witten2.2 index Atiyah-Patodi-Singer 2.3 Elliptic genera and Jacobi forms 3.1 Noncompact Witten index Patodi-Singer forms [ the quantum field theory in a 1 Introduction In this note, we establish a connection between three distinct sets of ideas — the Atiyah- D Determinants and ultralocality A Conventions and (1 B Error function and incomplete GammaC function Scattering theory 5 Mock modularity and the 4 The 3 Witten index and the Contents 1 Introduction 2 Supersymmetry and index theorems JHEP11(2019)080 ] ) I 1 → ∞ . This β β obtained . One can + M ∼ N × c M R N × 0) limit using the heat → is the fermion number and ]. Evaluating the path inte- β 5 F also contains delta-function nor- with the square-integrable norm. c c M M ]. Only the zero energy states graded by 4 ]. 7 , – 2 – 6 by glueing it to a half-cylinder has a product form near the boundary is the Hamiltonian, -dimensional Euclidean base space Σ with periodic d M and give a suitable definition using the formalism as the target space. By construction, the solutions H M c M 3.1 with APS boundary conditions are in one-one correspon- -invariant can be seen as follows. Consider a compact mani- ) where the hat is added as a reminder that it corresponds η β ( . One can define Atiyah-Patodi-Singer boundary conditions [ M N c W is the inverse temperature, is a finite interval. It is useful to introduce a noncompact manifold -invariant and mock modularity and compute it using deformation invariance and with a boundary η contribute and consequently, the Witten index is a topological invariant. This is the I β F M The relation to the APS If the field space is noncompact and the spectrum is continuous, then the above ar- If the quantum field theory is compact in the sense that the spectrum of the Hamilto- is the Hilbert space of the theory. As usual, this trace can be related to a supersym- 1) − to the Diracdence operator with on the solutions toHowever, the the Dirac spectrum operator of onmalizable the the scattering Dirac states Hamiltonian with on bound continuous states energies, with in discrete addition energies. to We assume the that square-integrable the continuum is separated from the fold for the Dirac operator assuming that where by a trivial extension ofdefine the the manifold Witten index to the noncompact theory with longer topological but isdeformations that nevertheless do ‘semi-topological’ not in changeuation, the that temperature asymptotic. independence it and Note is deformationbecome that independent invariance clear in are of later. a logically general distinct any non-compact as will sit- of Gel’fand triplet.precisely In cancel general, the thebe fermionic bosonic temperature-dependent. density density We of of relate states states,Singer this in temperature and this dependence the continuum tolocalization noncompact the may of Atiyah-Patodi- Witten not the index supersymmetric can path integral. The temperature-dependent piece is no gument can fail becausea now continuum instead of of scattering aneeds states. discrete a indexed To framework sum, define toWe one the incorporate address has noncompact the this an Witten non-normalizable integral issue index over scattering in properly, states one section into the trace. or the Dirac indexone of can the target evaluate manifold. itkernel in Using expansion the temperature to prove much independence thegral simpler of Atiyah-Singer corresponding high the index to index, temperature theorem the ( [ another Witten derivation index of in the index this theorem high-temperature [ semiclassical limit gives follows from the observation thatand the states do with not nonzero energy contribute( come to in Bose-Fermi the pairs Wittencase, for index example, [ for aappropriate choice supersymmetric of sigma the model sigma withlimit model, can a the be compact Witten related index target to in space. some the of By zero the an temperature classic ( topological invariants such as the Euler character H metric Euclidean path integral overboundary a conditions for all fields in the Euclidean timenian direction. is discrete, then the Witten index is independent of the inverse temperature where JHEP11(2019)080 , ], ]. τ 2 .A 38 (1.2) – 35 4.2 without a c M -invariant has a num- η we relate the continuum c M . ) -invariant is nonzero precisely because ∞ ). There is a bulk contribution to the η ( ] , in relation to spectral flow in quan- ∞ . For a compact SCFT, by an argument ( c W τ 18 ], in the context of WRT invariants [ , c − W 17 34 ], mock modular forms and their cousins have and ¯ – 3 – limit, only the ground states corresponding to , (0) ] index theorem as we explain in section τ 33 ] for a recent review). Similarly, apart from their c W 23 , 15 equals . The elliptic genus of an SCFT is a generalization , ]. For a path integral on a manifold 21 τ by the relation 22 14 13 → ∞ M – = 2 N 8 β η ], and more recently in the description of symmetry-protected a function of 20 , 19 -invariant of η a priori ], in umbral moonshine [ ], in fermion fractionization [ 32 is that it becomes easier to obtain its path integral representation. Defining a – 16 c M 24 , 2 0 limit as in the compact case. However, now there is a left-over piece coming from Apart from its intrinsic importance in differential topology, the An advantage of mapping the APS index to the Witten index on the noncompact The relation to mock modularity arises similarly as a consequence of noncompactness as the target space, this argument fails. There is a ‘holomorphic anomaly’ because the → c tum chromodynamics [ phases of topological insulatorsintrinsic (see interest [ in numbercome theory to [ play an importantraphy role [ in the physics of quantum black holes and quantum holog- relating it to apath Callias-Bott-Seeley integral [ representation alsoto makes see the the modular connection invariance with manifest mock making modularity. it easier ber of interesting physics applications,anomalies for [ example, in the analysis of global gravitational path integral measure in aEven target for space with a awas very boundary achieved simple is relatively in system recently generalboundary, [ like even rather if complicated. a it particle isfacilitates noncompact, in computations one using a can supersymmetric use box, localization. the the canonical We measure. derive path The the integral path APS formulation integral result by a completion ofThe a holomorphic mock anomaly Jacobi is‘noncompact once form right-moving again Witten — index’. governed a by the new temperature mathematical dependence objectmanifold of introduced the in [ similar to the above,and it is is independent a of (weakly)M the holomorphic ‘right-moving Jacobi temperature’ form. andright-moving bosonic hence Once density of of again, states ¯ density for does of not a precisely states. noncompact cancel the SCFT In right-moving this with fermionic case, the elliptic genus is no longer a Jacobi form but is rather of the field manifold forwith a a superconformal complex field structure theoryof on parameter the a Witten base index space that Σexcitations. counts which the is It right-moving a ground is 2-torus states with arbitrary left-moving contribution to the This yields a new proofthe of noncompact the Witten APS index theorem. is temperature-dependent. The the square-integrable solutions of thefor Dirac the operator original contribute compact andAPS manifold hence index coming the from APS the index β integral of a local indexthe density contribution which of can the be continuum. computed Using in scattering the theory on ground states by a gap. In the JHEP11(2019)080 1 , a 2 T . See ]. We } 44 A – -invariant. , θ ]. It would η α 39 σ 46 { are real Grass- -invariant to an i A and to compute η ψ , in the Lorentzian or ]. 4.3 τ 45 M -invariant and quantum = η 0 -dimensional quantum field σ (2.1) d ) σ or ( i t = ¯ θθF 0 . We review the definitions of mock 1 2 σ we review the supersymmetric quan- 5.2 2.1 ) + and σ and relate it to the Callias index. We use ( σ i 4.1 = ¯ θψ 1) supersymmetry which we describe below. It we define the noncompact Witten index using – 4 – 1 , σ ) + 3.1 Σ with real superspace coordinates σ s ( dimensional real field manifold i x n ) = -invariant of the finite cigar in section σ η ( . i we discuss the path integral representation of the and discuss the connection with the X we present the definition of the elliptic genus of an SCFT and 4 5.3 5.1 0. The Euclidean base space Σ in the three cases is a 2-torus , 2.3 1 are auxiliary fields. The components of the superfield can be thought be real super-fields with expansion , i } F ) = 2 d we apply localization to reduce the path integral for the . In section σ, θ ( i are the coordinates of 2 4.2 } 3.2 X for the conventions. We write i , and a point which we will refer to as the worldsheet, worldline, and worldpoint { x we describe the Atiyah-Patodi-Singer construction for a compact manifold with a 1 A { S Let The examples considered in this paper are simple but sufficiently nontrivial and il- The paper is organized as follows. In section 2.2 Indeed, this was posed by Atiyah a decade ago as a problem for the future [ 1 where mann fields and is convenient to usesection the superspace Euclidean version respectively. We will be interested in thetheories supersymmetric with path integrals for circle respectively. All our examples are obtainedof by reductions a of 1 Euclidean + Wick-rotated version 1 dimensional worldsheet with (1 even for the manifoldsalso that be do interesting not tosuperconformal have generalize product field the theories. construction form to We near will elliptic the return genera boundary to of these [ generic problems noncompact 2 in the future. Supersymmetry and index theorems consequence of noncompactness. Supersymmetricobtain methods a have path been integral used derivationmanifold, successfully of but to the to Atiyah-Singer our index knowledge,Using theorem no such for our derivation a formulation exists compact for inobtain target a terms manifold a with of more a a complete boundary. path noncompact integral Witten derivation index, of it the should APS be index theorem, possible for to example, the elliptic genus forJacobi the forms infinite in cigar section modular in forms section in section lustrative. Our results indicate that these interesting connections are a rather general its relation to Jacobithe forms. Gel’fand In triplet. section Thenin we section use thisIn formalism section to present aordinary proof super-integral of evaluated in the section these APS results theorem to compute the expect our results will have useful implications in thesetum diverse mechanics contexts. for indextion theory on aboundary. compact manifold In without section a boundary and in sec- and more generally in the context of elliptic genera of noncompact SCFTs [ JHEP11(2019)080 ] n 7 2 , (2.7) (2.5) (2.6) (2.2) (2.3) (2.4) 6 ≤ , 4 k a, b , and one ψ : ) F i ab ≤ δ ¯ ψ 1) j b j e − i F a e , + ) with 1 = j ¯ a X ψ i b ij i ( e g ψ h F is the superspace covari- k ( for easy comparison with action of ( ij used in the classic [ dt D dx 0 + 2 g 2 k α j Z ∂ M akb generated by the Hamiltonian X 1 4 ω t D and its components transform as i + k j + . ) k ψ i X F b X i j i x ¯ ) F D x dt F j F ) dψ α ψ Q → . + ∂ i X − i i ( i ¯ a ψ j jk ψ x ij = (¯ i ij ψ ψ α α g g / ∂ ∂ ab 1 2 ∇ k . The Lorentzian action is i α α ) in the target space. When the superpotential δX + Γ ∂ ¯ and the inverse vielbein i δ ψ x , x i 1 4 – 5 – iγ i ¯ γ i θ ( M i i a i ψ 2 + s as well as under the − e ψ i jk − α ψ − − j ∂ j σ d P ) dt ) ]. It is convenient to introduce an orthonormal basis x 2 1) supersymmetry generated by a real constant spinor = ¯ = ( = j dx ) is the superpotential, and j = associated with the Christoffel symbols. , α d i i i i ψ → − ψ i ∂ i X i i 48 Σ i dt , ( kb ¯ s ψ ¯ dx δx ψ δψ ψ ψ x δF a h Z α ) ( α 47 )( , ω 0 j x l ∇ ∂ ( ψ h 1 M ij ) in superfield components is given by ∂x πα k 2 i ij g 2 ¯ ψ ∂ g may be compact with or without boundary, or noncompact. As 2.2 − 1 2 ∂x )( generated by dt x 1 2 ( σ M σ ij 2 Z ) = − , using the vielbein g d i i l 1 2 X ∂ Σ ( F dx k Z i I i = a ∂ . It is also invariant under translations of e π 1 4 I ∂h 1 ∂x 2 M = + + − ) is the metric on a x e ( = = 1, the action ( ij 0 I g α The target space under which the superfield transforms as and translations of A To set the stage, we firstpath-integral consider derivation a superparticle of on compact thegiven by Atiyah-Singer index theorem. The Lorentzian action is can define the spin connection explained in the introduction,appropriate definition each of of the these Witten index. cases has2.1 to be Compact treated Witten differently index with an is defined using the Christoffel symbolsis Γ zero, eliminating thethe auxiliary Riemann fields curvature yield tensor the [ of familiar forms, quartic fermionic termas involving the tangent space indices. The metric can then be expressed as where the covariant derivative With target space H Moreover, it is invariant under the (1 where ant derivative on the (base)other superspace. normalizations in We the have literature. introduced The action is invariant under diffeomorphisms in the of as the coordinates of a supermanifold JHEP11(2019)080 2 is = π } b (2.8) (2.9) (2.12) (2.13) (2.14) (2.16) (2.10) , ψ a ψ { (2.15) -matrices with = 0 . γ a } a ψ b = would imply . ¯ = 0. However, since γ, γ ψ b a { + ψ } γ b iab a ω γ , ψ a and a is diagonal, the Dirac spinor , ψ ψ iab . π -dimensional { i γ 2 , ψ i j ω n 1 4 iδ . = 0 + i i = 1 (2.11) = 0 + } ˙ x x i 2 ] = F ) j ∂ ∂ , − = ˙ ∂σ γ 1) i (¯ , p ˙ = , i x i − with itself, = ˙ i i x ( x i ) by setting ] which uses slightly different conventions. [ , roughly as an average of the naive commutators. ∂L ! ψ ∂ ψ , D 2 ab − + Q, , n 49 δ 2.2 2 { √ Ψ Ψ := = . The canonical commutations imply the com- i
, = 0 − – 6 – } p b . . . γ M ab , δψ a − , = 2 δ , ψ i with γ a Ψ = for the Dirac representation 1 = ψ a Q γ { γ iψ } n ψ ] = 0 b i = i 2 , ψ i / , ψ = D a = ¯ δx H,Q γ = 0 ψ = [ a { ˙ . For a review see [ i using the vielbein. The conjugate variables are ψ ∂L F i D ∂ i ψ 1) i γ a -derivative. The nonvanishing canonical commutation relations are e − := t furnishes a Dirac representation of 2 , h = H, a = π Q H a = 0 = 2 ψ i } F , we have a constraint on the phase space and must use Dirac brackets instead of Poisson ψ . The chirality matrix ¯ j Q, Q iγ { is the worldline Hamiltonian. In the basis in which ¯ − − H = j apparently in conflict with the anticommutator of The worldline supersymmetry is now parametrized by a single Grassman parameter − ) on field space can be written as The naive anticommutator obtained from the Poisson bracket of ψ x 2 2 ab = δ 2 proportional to brackets to obtain the correct quantization where Ψ( which is the Dirac operatormutation on relations manifold The corresponding Noether supercharge is Upon quantization, we get can be identified with ( The Hilbert√ space where the dot refers to We have defined which can be obtained as a specialization of ( JHEP11(2019)080 τ −i E, ) that (2.22) (2.21) (2.17) (2.18) | := 2.15 is naturally i + . I (2.19) is the number ) − E, − | . n b and the Euclidean Q βH ψ ! − k iτ + 0 dτ − H . The Euclidean time dx exp β − ) (2.20) − = kab 0 0, then β t + H − ω ( and the index n
βH + W H and can thus be interpreted as − − b := E > + , dτ H →∞ dψ i n ! β ]) x L ab | = † 0 δ = lim Tr(exp of the Dirac operator on the manifold † L βH X, β L † I a [ − βH e →∞ S 0 ψ − β F LL − are discrete. It then follows from ( + with positive chirality and 1)
= lim exp j † − H L – 7 – = ( F dτ dim Ker | dx x i 1) LL ] exp ( − h − dτ dx ( L dX ) dx [ as usual by Wick rotation x in the Hilbert space H ( Tr t Z Z ij F g dim Ker = 1) →∞ ) = − ,H β − β dτ L ( respectively i.e. ) of this worldline theory has a path integral representation = lim ! = dim Ker † β 0 L L I 0 − W I 1.1 Z † H ), the Hamiltonian above can be identified with the Hamiltonian of come in Bose-Fermi pairs and cancel out of the trace. The Witten 0 L 1 2 with negative chirality, equivalently the number of zero modes of the
H † 2.15 and ] = = L + / D = dim Ker H ]. , so the Euclidean base space Σ is a circle of radius X, β denote the chiralities (not to be confused with the chirality of the Grassman [ β 5 I ) and ( , S is the number of zero modes of 4 ) on the worldsheet). The Dirac operator and the Hamiltonian take the form is a bosonic eigenstate with energy eigenvalue σ + ( i 2.14 n A + ψ The Witten index ( is then defined as: E, | with period is related to theaction Lorentzian is: time where the path integral is over superfield configurations that are periodic in Euclidean time zero eigenvalues of index in this caseinvariant [ receives contribution only from the ground states and is a topological For a compact manifold theif eigenvalues of is a fermionic eigenstate with the same energy eigenvalue. Hence, eigenstates with non- Positive and negative chirality spinors on theeigenstates field of space correspond the to operator positivebosonic and ( negative and fermionic states inidentified the with worldline the Hilbert Witten space index of the supersymmetric quantum mechanics. where of zero modes of Hamiltonians supersymmetric quantum mechanics. TheM index field Using ( where the JHEP11(2019)080 has a (2.24) (2.25) (2.27) (2.28) M introduced is a compact, M s N . ], assuming ). In local coordinates 1 where 1 mn is curved because the factor g N 2 M = − with the boundary located at defined by = I shown in red. , M } I ∂ m n matrices: n b γ b γ γ dy , . on the supermanifold m N × m 3 m b γ ) (2.26) m ) (2.23) dy { D b γ x B ) ( N m γ | ¯ γ α γ u + ¯ mn γ + M g u Z u – 8 – 0 on the interval ∂ + ∂ = ( ( near the boundary (figure = cancels against a similar factor in the fermionic measure with a collar u 2 u , ≤ γ I m g I γ γ u du M √ = mn x g = / n N × D 2 2 with a single boundary and 2 d -invariant − η ds N M := = } on . Manifold dx n } , γ 1 m − γ { n 2 is the boundary Dirac operator with M Figure 1 ] is induced from the supermeasure m D dX ,..., 2 m . , g b γ 1 ) is the topological index density given by the Dirac genus. √ ). One can now use the deformation invariance of the supersymmetric path x = ψ ( = 1 n 2.1 α B 2 in the bosonic measure d m ; g It is well-known that the supermeasure is flat even if the manifold := 3 √ m = 0, the metric takes the form y which satisfy the same Clifford algebra as the original of dψ where The Dirac operator near the boundary becomes It can be expressed as to impose the nonlocalproduct Atiyah-Patodi-Singer form boundary in conditions the ‘collar’ [ { region u Consider a compact manifold connected, oriented manifoldDirichlet with or no Neumann boundary. doreflection define at Usual the a local boundary, self-adjointchirality such boundary Dirac and local operator. conditions do boundary not conditions like However, mix allow because the one positive of to and the define negative the index. To preserve chirality, it is necessary where 2.2 Atiyah-Patodi-Singer The measure [ after ( integral to obtain JHEP11(2019)080 ). c M 2 (2.35) (2.32) (2.34) (2.29) . + -norma- R 2 L 0 (figure N × ≥ u . 0 0 . is the half line ) is positive on the semi-infinite obtained by gluing ) u λ > λ < + (2.30) (2.31) ( u . u ! ) ) ( R λ − 0 0 − + y y M λ + (0) (2.33) ∀ ∀ . For each mode we obtain ( ( Ψ Ψ Ψ λ λ λ Ψ B have the form e e E
]. One sets the exponentially growing E λ < λ > ) ) )Ψ √ u u E where √ . Since 50 ( ( M √ ∀ ∀ λ u + λ − λ + c ) = M ∓ R Ψ Ψ ) = = u ( u ( λ λ ! λ + – 9 – X X (0) = 0 (0) = 0 λ − N × − + Ψ + λ − λ Ψ Ψ Ψ Ψ Ψ ) = ) = λ λ λ ) = exp ( λ u ! + ( (0) = 0 (0) = 0 u, y u, y 0 + L is a trivial extension of ( ( λ λ + λ − d † − + du Ψ d Ψ Ψ (0) + (0) + 0 du c L M Ψ Ψ λ + λ −
− Ψ Ψ du du d d − c M are the complete set of eigenmodes of . The noncompact } ) y ( λ e { Figure 2 For the remaining half, one uses Robin boundary conditions is consistent with themode APS to boundary zero condition which amounts [ to Dirichlet boundary condition for half the modes: One can ask whichlizable of solutions these on solutions the cancylinder, noncompact be the manifold extended solutions are to normalizable square-integrable if or the argument of the exponent is negative. This To motivate the APSobtained boundary by gluing conditions a semi-infinite consider cylinder Near a the noncompact boundary, ‘trivial’ the extension zero energy solutions on where The eigenfunctions can be written as The eigenvalue equation for the Dirac operator near the boundary takes the form JHEP11(2019)080 η on B (2.39) (2.40) (2.41) (2.42) (2.37) is the Atiyah- η 0 0 (2.38) is given by -normalizability for = 0 2 N L λ λ > λ < -function regularization ). By construction, im- ζ . -invariant is given by c . η . 2.32 M ) . ). ) β λ s ( | b η β ] is the λ 2.36 1 ) (2.36) | 1 p h − sgn( | s 2 λ with boundary . | . 6=0 ) η X λ ) s = 0, so the ( dββ ( M λ 1 2 s = ∞ 0 APS ) erfc − η +1 Z sgn( λ ) s λ | ) 0 x is equivalent to requiring ( 6=0 λ – 10 – π → | , then X λ 2 α +1 ) s sgn( s B √ M = s 6=0 M = lim 6=0 X λ Γ( Z η 3.15 X λ η = ) = ) = I s is the number of zero modes of the boundary operator. s ) := ( ( β ) is analytic near h ( s b ( η APS APS 1 for η η − APS η ) = 1) = ) = regular dependent for for λ λ λ ] that sgn( sgn( sgn( 50 -invariant and η ) is Atiyah-Singer term present also in the compact case, x be the set of eigenvalues of ( α } λ { M R This infinite sum can be regularized in many ways. A natural regularization that arises The APS index theorem states that the index of Dirac operator with APS boundary . Let One can prove [ It is expected that the answerthat is are independent of implicit the in regularization the up definition to local of counter-terms a path integral. The two regularization schemes are related by a Mellin transform Another regularization used in the original APS paper [ but the answer depends on theis direction also in present which one in approaches the zero. original This APS sign formula ambiguity infrom equation the ( path integral derivation is ( The case of a zero eigenvalue can be treated by slightly deforming the boundary operator N here sgn is the sign function which is defined as where Patodi-Singer is a measure ofnumber the spectral of asymmetry modes which with is positive equal and to the negative regularized eigenvalues difference of in the the boundary operator posing the APS boundary conditionthe on solutions of Dirac equation on the noncompact extension conditions on the compact Riemannian manifold which are consistent with supersymmetry as one sees from ( JHEP11(2019)080 c 0 ˜ L and and − and ˜ (2.45) (2.43) (2.47) 0 w P 2. The c L = for the weight . P ) ) (2.44) w Z , πiz 2) supersymme- Z ’ and ‘elliptic in ) WZW model. 2 , τ R , one can identity e ∈ ) of weight , 2 . , the elliptic genus ˜ iτ SL(2 J := τ, z ( ∈ + πizJ φ , y 2 1 ) has (2 ! e τ 4 would change by ) πiτ to cause level crossing. 2.2 ˜ c η = 2 24 c d a b e theta function for index. We use − B τ ) (2.46) 0
˜ L with λ, µ m := m ( τ, z ¯ τ q ( ( πi 2 m,` ) − ); ϑ which is ‘modular in e ) ) z τ τ, z c τ, z ( 24 ( , the elliptic genus is a weak Jacobi form ( ` φ c − φ h ) 0 -functions i.e. 2 and Z L λz ϑ d ( ) is the index m mn τ + 2 +2 – 11 – / cτ πiτ τ corresponds to the level of the SL(2 can be identified with the right-moving fermion ) has the following consequence. As one varies the πimcz +2 τ, z X Z 2 2 ` 2 ( k e λ ∈ ˜ y e J ( as the chemical potentials for the operators ` J m was used for weight and 2.36 m,l w + are the R-symmetry generators. In our normalization 4 ) ˜ ϑ J k / πim 6. Recall that a Jacobi form ˜ d πz 2 J 2 ) = ) can pass through a zero and in ( 1) c/ − + e η − B mn τ, z ( = and ( cτ and 2 +2 ] where φ ` H ) = J it transforms as 2 m Tr ( 1 µ = ( q z Z πτ 1 as expected for an integer. It also shows that neither the index + ) = ∈ . ∓ d X for the cigar coset, n is K¨ahler,then the worldsheet action in ( J λτ τ, z + z generates time translations on the cylindrical worldsheet, ( 4.3 + are the left and right-moving Virasoro generators respectively, ) M commutes with the right-moving fermion number χ c ) := cτ and 0 +˜ , 24 J c ˜ L b τ, z ( d P τ, z ( ( + − + φ = 0 and index , the eigenvalues of 0 and ) is a modular form and m,` and think of 2 ˜ N cτ aτ w L -invariant are strictly topological and can change under smooth deformations. τ ϑ 0 is a holomorphic function of both ( η 2 + ` L 0 h φ m πτ L 2) superconformal field theory with left-moving and right-moving super Virasoro al- For a compact SCFT with central charge The factor of half in front of , Our convention differs from [ = 4 respectively. In the path integral representation, the presence of additional insertions = 2 ’. Thus, under modular transformation it transforms as where here because in section A Jacobi form can be expanded in terms of z and under translations of has the effect ofcharged twisting under the boundary conditions along the timeof direction weight for theindex fields generates the space translations,number. and Since can be thought ofβ as a right-moving Witten index.J Writing where are the central charges, and H 2.3 Elliptic generaIf and the Jacobi target space forms try. We assume that thea theory is (2 Weyl-invariant. The nonlineargebras. sigma model The then elliptic defines genus of an SCFT is defined by metric on index then changes by nor the They are nevertheless semi-topologicaldata near in the the boundary sense is changed they to change alter the only spectrum if of the asymptotic JHEP11(2019)080 . J and, (3.1) (3.2) β without boundary. c M ): ∞ ( . W . ) (0) ∞ is discrete and is paired by supersymmetry. W ( 0 W ˜ L ) = – 12 – of a noncompact manifold := ∞ ( I c W W -invariant . As a result, the Witten index is independent of . The holomorphic elliptic genus thus counts right-moving ] physically by bosonizing the U(1) R-symmetry current it is given instead by the completion of mock Jacobi form. η H β Consequently, it is invariant under the mapping class group 5 52 without boundary or with a boundary and APS boundary 5 ] of the left-moving superconformal field theory, and the theta M . This is essentially the same argument we used to show that the 51 τ of the Dirac operator equals has a boundary, but there is an obvious difficulty. In general, path I ) has a path integral representation, which is the starting point to M 1.1 0 limit using the high-temperature expansion of heat kernels. → β ) which is the group of global diffeomorphisms of the torus worldsheet modulo Weyl Z , For a compact target space without a boundary, the index of a supersymmetric quan- For a non-compact target space, this argument fails. Therefore, the noncompact elliptic For a compact SCFT, the spectrum of We consider nonchiral theories so there is no possibility of gravitational anomalies. 5 integral formulation is much morecannot subtle use the for canonical a measure.the target computation For space this of with reason, the a it WittenIt boundary is index will because convenient also to one lead map to the a problem ‘spectral to theoretic’ reformulation of the APS theorem. simpler tum mechanics ( obtain a derivation of thelocalization AS theorem when using localization. One would like to similarly apply vectors span the Hilbertin space particular, This equality is the essentialPatodi-Singer step index in theorems because the one proofs can of then both evaluate the the Witten Atiyah-Singer index and in the the Atiyah- much For a compact manifold conditions, the Dirac Hamiltonian is self-adjoint. It has a discrete spectrum and its eigen- 3 Witten index and the At zero temperature, onlydefinition, the the index ground states contribute to the Witten index. Thus, by ground states with arbitrary left-moving oscillators. genus need not bethat holomorphic. it However, must it nevertheless isform. have clear modular As from and we its explain elliptic path in transformation section integral properties representation of a Jacobi from the spectral flowexpansion [ can be understood [ Hence, only the right-moving groundgenus states is contribute independent to of the elliptic ¯ Witten genus index and is the independent elliptic of The path integral is diffeomorphismregulator invariant when such regulated as covariantly a using a shorttheory covariant proper on time the flat cutoff. worldsheet. ItSL(2 is also Weyl invariant fortransformations. a Similarly, the conformal elliptic field transformation properties of the elliptic genus follow The modular invariance of the elliptic genus follows from its path integral representation. JHEP11(2019)080 . is H c M H (3.4) . The ), it is on 2 β p ( b H = c W H is ‘trace class’ : R b H β ; (3.3) − e ∞ F is expected to contain . < 1) ) b . The Hilbert space H − x ( × i ∈ S ψ S ) ψ x ( such that ∗ i} φ with the Hamiltonian dx ψ {| R Z ∞ ∀ | has no normalizable eigenvectors, so the with the unit norm. This is essential for := < i H i . This, however, is not always possible. A H ψ φ | | H is the set on ψ – 13 – ψ h h × H S if ) of square-integrable wave functions on with R × dx, ( 2 i} , and the conjugate Schwartz space L ψ i ∈ S S φ {| | -normalizable solution of the extended Dirac Hamiltonian ]. An advantage of this formalism is that one can discuss the spectral = 2 L 54 H , ). 53 ]. The notion of the Schwartz space is motivated by the fact that it is left 1.1 ) at zero temperature, which admits a simpler path integral representation. 54 , ∞ ( 53 c W , there is a For a quantum particle on a real line, the Gel’fand triplet consists of a Hilbert space The first Von Neumann axiom states that every physical system is represented (up A natural formalism for this purpose is provided by ‘rigged Hilbert space’ or ‘Gel’fand M -normalizable states. It is not clear then that the operator ( , the Schwartz space 2 The Gel’fand tripletDirac provides and a offers rigorous avalues way way [ to to discuss define the the spectral theory bra of and operators ket with formulation continuous eigen- of isomorphic to the space The Schwartz space is the spacefall of off. infinitely The differentiable ‘test conjugate functions’ Schwartz with space exponential with such more generalrepresent physical physical situations, observables it by is operatorsusing necessary defined a to on Gel’fand relax a triplet domain the rather in second than a axiom on rigged a and Hilbert domainH space in a Hilbert space. corresponds to asimple self-adjoint counterexample operator is on aself-adjoint free operator particle corresponding on to aset line of eigenvalues ofexpects this the operator free is particle empty. to On have the continuous energy other with hand, a on sensible physical classical grounds limit. one To deal present context. to a phase) bythe Born a interpretation vector because in thephysical total a system probability Hilbert must of outcomes be space of unity. measurements The for second any axiom requires that every physical observable generalizes ( triplet’ which generalizes the VonHilbert Neumann formulation space of [ quantum mechanicstheory based of on operators a with anot continuous be spectrum square-integrable. with ‘generalized’ We eigenvectors which review may some of the relevant concepts as they apply in the delta-function normalizable scattering states with aL continuous spectrum in addition toin the the conventional sensescattering because states. it may Thus, notnecessary even have to a before first convergent give trying trace a to proper after develop definition including for the the it path in integral the canonical for formulation that correctly The APS boundary conditionson imply that for everyOne solution can, of therefore, the aim Diracindex Hamiltonian to express theOne Dirac immediate index problem in with terms this idea of is the that noncompact the Witten spectrum of 3.1 Noncompact Witten index JHEP11(2019)080 ]. . (3.8) (3.5) (3.6) (3.7) S reside. 55 , 53 ipx e . One can has the same × can be expanded H S ⊂ H H ). This extension of and may not be nor- H ( H D . The conjugate Schwartz . which may contain both the x ) and plane waves x Sp ( × with generalized eigenfunctions by the equation δ i ∈ S i ∈ S × ψ by ψ × S × i ∈ S H E . × | i ∈ S defined on a domain 2 φ H 2 | d dx H i ∀ | i ∀ | , φ − | E i | = E ψ – 14 – | h Hψ × E h E acting on H = = = × i i i can be Fourier-expanded in terms of plane waves. We φ H E | E | | × H × may lie outside the Hilbert space × H i H | H ) is much larger than the domain | E ψ | × . In any lab with a finite extent, one can never experimentally ψ h h } H 2 ( p as a differential operator: { D H need not have finite inner product with themselves and hence may not × . This is the content of the Gel’fand-Maurin spectral theorem [ S i} provides a complete basis in the sense that any state in E {| i} is where objects like the Dirac delta distribution E {| × and eigenvalues S } In the present situation, we are interested in global properties that depend sensitively Normally, one can gloss over these niceties essentially because of the locality. A particle For the example of a free particle discussed earlier, the operator Consider now a self-adjoint Hamiltonian ipx e periodic or Dirichlet boundarycomputations. conditions in a large box, as oneon indeed the finds boundary in conditions. textbook Forwhile example, preserving one supersymmetry. cannot impose The Dirichlet Gel’fand boundary triplet condition provides an appropriate formulation states with continuous energies. on an infinite line isthat an measurements extreme of idealization local in a quantitiesexperiment universe such which as in may scattering be a cross-sections finite. in labof a One should particle expects the physics not universe. be affected One by should boundary arrive conditions at imposed the at same the physical end conclusions whether one uses realize an exact planeeigenfunction. wave but only Nevertheless, a wavea the packet square-integrable that plane function is waves in sufficientlydenote form close the to total a space the of complete energy square-integrable generalized basis bound eigenvectors of states in with discrete the energies sense as well that as nonnormalizable scattering However, the domain the Hamiltonian is diagonalizable{ in the larger space the set in terms of formal expression as should be interpreted in terms of the overlap with test functions: A ‘generalized eigenvector’ malizable. This means that it cannot be prepared in any experimental setup. Nevertheless, With this definition, the eigenvalue equation for space The elements of be square-integrable, but they have finite overlap with ‘testdefine functions’ the belonging conjugate to Hamiltonian invariant by unbounded operators like the position operator JHEP11(2019)080 . − c M H (3.9) (3.10) (3.11) (3.13) (3.12) ) on and between ). Since β ( ∞ + ( H c W c W . Under suitable x difference . 2 ). ) y β . 4 − 4.19 x ( − → ∞ i b H dx α β exp − Z M e Z F . πβ ) 1 πβ 4 1) = 1 4 : ∞ − √ α ( ( √ c h M c = M c W are separated from the ground states by a gap, = when there is no ambiguity to unclutter the notation. Z i – 15 – i SP Tr = y × | x Sp | I H βH βH ) := (0) = − − β e | ( e c W | x -normalizable ground states contribute to x h 2 c W h L instead of dx ) = H β Z ; x, y ( with a boundary in terms of a noncompact Witten index 0, one can evaluate the Witten index by using the short proper time and use it for applying localization using ( K M → 4.1 with APS boundary conditions, we conclude β over which one traces now has a precise meaning, it is not clear that the M 6 Sp on H Within this framework, one can now express the index of the Dirac operator on the In what follows, we will use 6 . As a result, the volume divergent contribution cancels in the supertrace. In the path Atiyah-Singer term but now evaluated over space In the limit of expansion of the heat kernels to obtain a local expression. It must correspond to the original manifold Assuming that the continuum statesat in zero temperature onlythese the states are in one-one correspondence with the ground states in the original Hilbert over the ‘fermionic zeroconditions, modes’ the in addition fermionic to Berezinregion the integration on ‘bosonic localizes the zero-mode’ the realvanishing bosonic line contribution integral to from to yield thepoint a asymptotic a in compact infinity finite section in answer. field In space. particular, We the elaborate path on integral this receives two heat kernels correspondingrespectively. to If the there bosonic isthe and two a fermionic Hamiltonians gap differ Hamiltonians betweenR from the each ground other states onlyintegral and over representation, the a this scattering region corresponds states, with to then compact the fact support that in the supertrace involves integrals One might worry thatnately, the the Witten noncompact index is Witten a index supertrace is or equivalently also a trace similarly over divergent. the Fortu- trace thus defined actuallywell-defined converges. For example, for a free particle, the heat kernel is However, if we try to define a trace then there is the usual ‘volume’ divergence: ditions to ‘compactify’ space.Witten index With by these preliminaries, one can define the noncompact On the facespectrum of it, this definition is still not completely satisfactory. Even though the so that one can discuss the scattering states without the need to put any boundary con- JHEP11(2019)080 - β M (3.17) (3.16) (3.14) can be 7 3.2 ]. Since 1 -invariant as η ) can be viewed ) and the contin- 3.14 H ( -invariant. Moreover, ) of the Hamiltonian b η i H b Sp H β − thus measures the failure of e F . N 1) without boundary. as was suggested in [ − )) (3.15) ( ). This provides a natural regular- α 3 (0)) h M ∞ ( c M W s Z 3.14 Tr SP c W − to be temperature independent. ) + − c ) i W ∞ ( (0) = b β H ( β c W W − c W – 16 – 1 2 ). Therefore the Witten index admits a spectral e 0 expansion is local and gives the Atiyah-Singer F H ( 1) → s (0) that makes this connection with the bulk Atiyah- (0) + ( − β (0) = ) := 2( ( c Sp W c h W β ( c W b b η = ). With these identifications, equation ( Tr SP I ) the ] in a special example. . We can therefore write 2.39 56 + ) = by gluing its copy as in figure 3.13 R β ( . The doubled compact manifold 0 reduces to the bracket in ( . It is convenient to consider a regularized quantity M -invariant of the boundary manifold -invariant. We note that the spectrum Sp( c W η η → N × 3.2 β ) more manifest and easier to relate it to a path integral. One can simply Figure 3 3.13 M ) equals the is a direct sum of the discrete spectrum of bound states To prove the APS theorem, we would like to show that the term in the parentheses c 3.14 This was noticed earlier in [ M 7 on uum spectrum of scatteringdecomposition states index density. In summary, in ( There is a simpler waySinger to term compute ( double the manifold to is a manifold without aby boundary, there the is reasoning no before contribution from ( the and fermionic density ofcancel states precisely. in The this continuumthe may Witten not index be of exactly the equal noncompact and manifold need3.2 not Scattering theory and the APS theorem which in the limit ization described earlier in ( as the statement of the APSviewed theorem; as the a discussion rederivation above of togetherdependent the with because APS section at result. finite The temperature noncompact the Witten scattering states index also is contribute. in general The bosonic The term in thewill parantheses show is in no section longer zero and is in fact related to the where the second equality followson from the the half-cylinder fact that the topological index density vanishes JHEP11(2019)080 η k ), we . The (3.21) (3.23) (3.19) (3.20) (3.22) (3.18) } λ 3.15 { i . We can use iku ∞ of the boundary − -invariant: e η λ ) ) we obtain ) in the asymptotic λ + ). To prove ( 2.32 < u < 2.32 ik ∞ ( ( . ) ) with eigenvalues λ − λ λ i δ c W ) 2 N i + − ) k ) into ( i i k e ( ( ik ik λ − . ) with 0 on + ( ( iku iku δ ) βE 3.19 − − B β − − ) ) − ( iku e k k 2.24 ( ( ) b η e i = ) λ λ ) + − k ( λ k = iδ iδ λ λ + − ( λ δ δ + e e i 2 2 δ + λ − i i ) over scattering states can be expressed as b ) from scattering theory, which we review h H + + ρ e e . After all, they contain all the information β ik and hence with d − − dk − c iku iku 3.18 e M ) I ( C.13 e e 1 π F k h – 17 – h h ( , λ − 1) λ + λ − λ + ) c has the form ( c c ρ − λ ) = between the phase shifts is determined entirely by h ( = k h ∼ ∼ ( ] without making the connection to APS index theorem and + c E M i which in turn can be related to the difference in phase dk λ − ) ) s 60 √ ik ρ u u – Tr SP ( ( iku Z c − M 2 57 can be expressed in terms of eigenvalues − − ( invariant and scattering theory was observed earlier in special cases λk λk − + e ) λ ψ ψ η k X c λ + on = M . Generically, it would be impossible to compute any of them ( δ difference 2 2 8 u λ + i λ + λ − ρ e c c ) are the density of bosonic and fermionic states of the theory for + k ). Here, we assume that the boundary operator has no zero eigenvalue. ( iku λ − e ρ . This implies h 2.32 2 E λ √ + -matrix. The exact form of the scattering states similarly has a complicated λ + , we can relate the difference in the density of states to the difference in phase ) and 2 as in ( ) are the phase shifts. The trace ( c S k k k C ( B ( λ + = ]. λ ρ δ subsector. Using a standard result ( 63 E – Asymptotically, the metric on λ Index theory on non-compact manifold and its relation to scattering theory has been considered earlier 61 8 to compute threshold boundinvariant. states The [ relationin between [ region. By substituting the asymptotic wave-functions ( with that depend on theabout details of the the manifold functional dependence on exactly. Remarkably, the the asymptotic data as a consequence of supersymmetry relation ( the in section shifts In general the individual phase shifts and density of states are nontrivial functions of where where The asymptotic form of the scattering wave functions is then fermionic scattering states shifts. separation of variables to firstDirac diagonalize operator the on operator manifold operator first term can bethus identified need with to the show index that the contribution from the continuum equals the We show this by relating the supertrace to the difference in the density of bosonic and Since the continuum of states is separated from the zero energy states, it is clear that the JHEP11(2019)080 . c M of the (3.25) (3.26) (3.27) (3.24) λ we show -invariant . Now the η 4.3 c M ) 2 λ ). We can diago- + -invariant obtained B 2 with APS boundary η k γ ( one can use its path- β + ¯ − η e ×N 3.1 1 2 u i + ∂ ) π ( R − k u ( + ! α γ λ − 2 β ) ρ -invariant of the boundary oper- 2 = M λ λ η λ r − Z | / e ) + − D λ + λ | k 2 ( ik ik
such that metric is of the product form k λ + ( ρ (0)) = π h ). Effectively, for each eigenvalue M ) for the ln − c W ) erfc i dk λ ) to get 2.31 − − 3.15 ∞ ) = ) – 18 – 0 k ), ( Z sgn( ( ∞ . ) = C.13 ( λ − λ k λ λ ( ρ c W extend the manifold to a non-compact manifold X X 2.30 λ − − δ ) in ( 2 = = ) k i − ( b H ) λ + (0) + ( 3.24 β k ρ ( − trivially c W e λ + we obtain δ F = 2 λ 1) I are the same as the scattering states of − , we have a supersymmetric quantum mechanics on a half line. The ( c B h M s Tr SP 2 invariant. First, we will explain the space-time picture and then we will map , given a manifold with boundary -invariant gets contribution only from the scattering states of η -invariant and path integrals η η 3.2 , the The Dirac operator on the half line is given by Can one find a way to formulate a path integral that directly computes the Now we use the formula ( c M it to a world-line computation. nalize the boundary operatorboundary as operator in ( APS boundary condition is essentially Dirichlet boundary condition for one chirality and in section near the boundary weIn can scattering states of condition at the origin.to We use compute this physical picture to find a path integral representation its index to the Witten indexevaluate of the the infinite Witten cigar. indexfrom In using this operator simple localization methods. example, and one can compare explicitly with the without the bulk Atiyah-Singer piece? This can be achieved as follows. As we have observed 4 The Given the definition ofintegral the representation noncompact and Witten use index localizationhow methods in this to section works compute for the it. two-dimensional In surface section of a finite cigar with a boundary by relating ator. We have thus proven which is the Atiyah-Patodi-Singer index theorem. This is precisely the regulated expression ( After summing over all in each eigensubspace with eigenvalue and therefore, JHEP11(2019)080 ] u 64 , × N (4.2) (4.3) (4.1) 10 + R invariant 1 to 1 as P transform as − u = 0. One can we can map this u . B R keeps only parity-even u N × -invariant does not change projected onto = η ψ ) without any boundary. We f M ) on the half-line, the extension 4 × N B → − γ R ) (4.4) +¯ B := u γ )¯ ∂ ( u f M ( u . (see figure ε γ i P + = u , ψ u / e × N D ∂ with APS boundary conditions can be obtained – 19 – 1 + ( h R → − B → −B u 1 2 γ u := × N = + f / e M D R : ) is a step function with a discontinuity at u . The doubled noncompact cylinder P ( ε to obtain a smooth Dirac operator. One example of such function ∞ f M Figure 4 to + ) where has parity symmetry ) to be a smooth smearing function which interpolates between 2.25 −∞ u ( ε ). This does not change the conclusions because the is the parity operator. Invariance under the reflection of × N u P ) we are left with the choice R Now we return to the world-line picture. For each eigenvalue of Since we are interested in the operator 4.1 := f varies from is tanh( under deformations that do not change the asymptotics. problem to a world-line path integral problem. The effect of the eigenvalue of the boundary the doubled cylinder thus takes the form instead of ( also take of this operator should transformin as ( an eigen-operator under parity. Given This ensures that Dirac operator as a whole is invariant. The extended Dirac operator on where wave functions in the tracedition on for the one half chirality, effectively line.boundary imposing condition Supersymmetry as Dirichlet ensures required boundary that by con- the thefor APS other a boundary chirality more conditions. satisfies detailed See, the discussion. for Robin example, [ integral on the original manifold by considering the path integralstates. on This the is manifold effected by the insertion of the following operator M that is consistent with supersymmetry and leaves the supercharge invariant. The path Robin boundary condition for thewith other these chirality. boundary To obtain conditions, ato path it obtain integral is a representation more noncompactextend cylinder convenient it to in ‘double’ a manner the that manifold is consistent with the APS boundary conditions. The manifold JHEP11(2019)080 is a (4.7) (4.8) (4.9) (4.6) (4.10) M . i , we compute − = 4.2 N . )] ) by setting ) on the non-compact + real fermions we have U β ( ( 2.6 ψ n e η n − = 0 of the Euclidean and the 2 0 βS = 1 and ψ . . In section of ( ∂ . It reduces the problem to ) − . With this construction, we ∂τ n n 9 c h u . λ )) (4.5) 2 0 M ( . 2 00 h . . . dψ ∞ 2 exp [ ]. ( 1 0 ih du d . . . ψ + , f W 1 15 dψ 0 ) = 1 , ) n ) + u ψ dψ − ) = 2 0 ( u n 14 au u ) ( ) − 0 ( 11 i β 00 2 ( dψ . . . ψ . So, the limit − f W 1 0 iF h tanh( iF Z ψ λ = ( + n , g – 20 – ) → , h 2 dF n i 2 2 F ) = 2( F ) = = 0 ∞ du dh 1 2 − β u −∞ ( ( ( 0 Z e η . . . γ ∂ h ∂σ Z 1 ) = ) := du γ + N = u ) = n ( i ) in the presence of a super-potential with a target space ∞ 0 β = , ψ ( −∞ ∂ h β determined by the eigenvalue can be understood as follows. For 2 ∂τ = b η − ( 2 Z i i γ λ ¯ γ ) . We discuss this example first before proceeding to localization. f W − − . We have two real fermions, hence u Tr ¯ ( ∞ n ε ) u, F, ψ i which is the dimension of the spinor representation ) = ( − β n S ( ) = u < u < = ( W ( = 2 h 2 N γ −∞ which is much simpler than the path integral on f M A particularly interesting special case is Note that in the Euclidean continuation 9 Lorentzian actions gives different actions for the supersymmetric integral. Moreover, Tr ¯ which implies The normalization factor The action can be obtained from the euclidean continuation The path integral is now just an ordinary superintegral with flat measure where Some of the essential‘worldpoint’ path-integral points where about the athe base real space noncompact line Σ path is integralThe a supersymmetric can point worldpoint and be action the illustrated is target given by space by a It is straightforward tocylinder write a path integralit representation using for localization and relate it to4.1 Callias index [ Supersymmetric worldpoint integral computing Witten index being half-line appropriately extended tousing the path full line. integral. Thesuperpotential The Witten index corresponding can supersymmetric beconclude quantum computed mechanics will have a operator can be incorporated by adding a super-potential JHEP11(2019)080 λ (4.11) (4.12) (4.13) (4.14) yields different ). This is not a λ and is independent 5.6 ) can be expressed in . u ∞ ( 0 5 2 gives h )) ) at F u ( u -invariant and its connection 0 ( η 0 . h dx ( ) h ! ) varies, and makes the integral i | 2 u ) 2 β u y ( λ ), when ( u | 0 − 00 − 2 − h β h ( 2 β θ 4.10 r dy e ] and also in the definition of the com- λ − r
λ 2 β ) exp 2 ) in ( β = u u 50 q , q ( ( ) is single-valued, and the integral reduces to ) erf − 1 00 θ y λ Z h ( . Inclusion of fermions effectively limits the λ | → ∞ u π – 21 – 1 u dx h , dy a | √ sgn( ) ) = u ∞ − − ( u −∞ 0 ( Z 0 h = h 2 π β ) = -invariant when an eigenvalue of the boundary operator β 0 and is ill-defined. Regularizing with 2 η β ( r r ) which appears naturally in this integral makes its appear- ) is monotonically increasing or decreasing depending on if = W − ∞ × u ( y y 4.13 , ) = β ∞ ( 0, the action reduces to that of a free superparticle. In this case, the W to → ) of a mock modular form and, in particular, in ( λ −∞ 5.13 ) is bounded above for large . Integrating out the fermions and the auxiliary field u crosses a zero in a spectral flow as explained before figure λ ( 0 pletion ( coincidence. The two turnprecisely out to to the be ordinary related superintegralple through considered is a above. particularly path important For this integral forto reason, which our mock this localizes discussions modularity. exam- of the terms of the Heaviside step function: ance in the proof of the APS theorem [ B of any deformations thatobtain do the not same change result in the the asymptotics. limit In particular, one would finite. integral is of theanswers form depending on whether werelated approach to 0 the from jump positive in or the negative side. This is h integrand to the region close to the origin where goes from The answer depends only the asymptotic behaviour of In the limit Without the fermionic integrations, the integral has a volume divergence because The error function ( The worldpoint integral illustrates several important points. u 3. 2. 1. 4. As is positive or negative; the inverse function One can change variables for real JHEP11(2019)080 0 M ≥ s (4.15) (4.16) (4.18) (4.19) ). The QV u ( + = 0. This h ψ ] V − X [ 2 ]. After local- ψ ) Q ] but now for the u 15 ξQV ( , P } 00 − ] λ ) with target space 14 { ih X Tr [ [ S 1 2 2.2 we can use separation ), one can apply local- + − e + F ] 3.2 ) 2.21 u . is chosen such that F X ( [ is 0 F . 1) V V ] ih 1) -invariant we need to have to U − η X ( [ − Q + ( ] 2 P Tr ξQV F dX with eigenvalues 1 2 − 2 1 [ ] = Z + N X,β [ . + of maps from Σ to the supermanifold ) = S ˙ on ψ ] F − } + e ) X , the problem reduces to a supersymmetric = 0 B [ 1) ] ψ σ λ -invariant ( − (4.17) 2 1 – 22 – η ∂ ( ) ∂σ ) ξQV dX . As discussed in section X [ β -exact term: P + { − ( ] f Q 2 4.2 M Z − ˙ ψ f W X,β [ (1 + − 1 2 S ) = 6= 0). To compute the invariant by computing path integral for supersymmetric F ψ − 1 2 e η 1) λ ] ) + β, ξ − ( + β ( X ( [ 2 1 f W ˙ u f W Tr 1 2 QV 1 2 1 2 (where ] u = dτ to the action does not blow up the path integral) and dX is a fermionic vector field on this supermanifold so one can use Stokes . For each eigenvalue [ β ) = 0 β Q f Z Z ( M QV tanh λ f W ) = ] = ) = β, ξ U, β -invariant supersymmetric measure. The functional ( u [ the worldline version corresponding to the path integral for a supersymmetric ( Q 0 S f W ]. The action for SQM can be obtained as a specialization of ( h by setting Given a path integral representation for the Witten index ( d 4.2 dξ 65 R implies The supercharge theorem to show that the above integral vanishes. Usually one use the compactness of the The integral is over thewith a supermanifold (so that adding The path integral for the firstfor term bosons is and the fermions. same asfermions In before have the with anti-periodic path periodic integral because boundary for of conditions the the second insertion term of bothization the methods bosons by and deforming the it by a The Euclidean action for the components of the superfield with evaluate the projected Witten index ( path integral for this problemin can [ be readily written downas and has been considered earlier In this section, we computequantum the mechanics with target space of variables to first diagonalizeentire the manifold operator quantum mechanics with a one-dimensional target space and a superpotential a Hilbert space. Totion see the connectionquantum with mechanics, the and relate canonical it formalism, toization, Callias-Seeley-Bott we the index consider theorem worldline [ path in integral sec- will reduce to the4.2 worldpoint integral considered above. Callias index theorem and the The worldpoint integral does not have an operator interpretation in terms of a trace over JHEP11(2019)080 ) ) is and 4.23 (4.25) (4.22) (4.24) (4.20) (4.21) (4.23) u β, ξ ( W [2] ) ) for any value +0 V ψ + 0 β, ξ Q ( − . The integral ( ( ψ ξ ) W D 0 − . In this case, the path- u ] is ( F 00 , β 1) V h 0 ( η − ) X ( ξ [ µ ¯ iβ η 1 P ! ˙ S √ ξ u 1 − 1 2 β − + √ 2 + r iψ )) SDet( | -invariant supersymmetric configura- 0 ] . +0 λ i u | ,β Q ψ ( ) ψ 0 ) = ] exp
µ + = 0 + † ( h ). Hence we obtain ˙ = + ) 0 ψ ( i dη + ψ [ + X + 2 β = 0. The critical manifold is parametrized [ + ) erf ψ S 4.11 +0 † Qψ − ] for a recent review. λ Q ) − ( ( + e – 23 – 66 dψ i ] + ] 2 Qψ X ¯ u ψ u ψ sgn( − ˜ u ψ dµ exp ξ ]. A canonical choice of [ − ξ = 1 = ˙ = dψ 1 √ X X +0 √ ][ [ Z V ). See [ ˜ V V u ) = . In this limit, the functional integral localizes onto the + d µ dψ + β ( ][ QV 0 . Small fluctuations around the saddle point is given by 0 ( 0 ) = Q 1 u . The path integral localizes to an integral on the critical − + = 0 + β = 0 and ( X } ( dF ψ f W [ → ∞ = u µ dψ 0 { f W ξ u Qψ 0 du πβ 2 du = 0 = Z √ i u in modes and after evaluating the non-zero mode integrals we obtain − comes from the determinants computed in section Z . This implies that one can perform the integral i − πβ ξ ψ 1 − 2 satisfying ] = ) is the superdeterminant of the quadratic fluctuation operator coming from satisfying periodic boundary conditions. So we have, β √ to argue that there are no boundary terms. In the noncompact case, there } [ µ ) 1 η ) = is a real supercharge. The path integral localizes to constant modes of µ f β ( W M ( 0 + 1 and X Q f W { u and in particular for action expanded around For the problem at hand, we choose . Fluctuations around the constant modes are given by ξ + It remains to compute theintegral localizes other to piece due to insertion of The factor of is identical to the worldpoint superintegral ( Expanding the ψ with ˜ where manifold where SDet( QV The critical points oftions this function are simplythe the collective coordinates can be a contribution fromarguments the outlined asymptotics in in our theindependent discussion field of space of but the it noncompactof can Witten still index. vanish by Hence the critical points of the functional manifold JHEP11(2019)080 . k √ cigar (4.34) (4.33) (4.27) (4.28) ) for each finite ! 4.27 ) (4.31) 2 β with radius = 1 (4.26) r 2 θ r | [2] λ | )
V (tanh r ) by setting + . ) . We get ( Q dθ r ( 2.5 k ∂ λ )erfc ) ) (4.32) ξ 2 1 2 r r λ 1 2 − . = + i ] is a two dimensional ) (4.30) tanh sgn( ψ with the Killing vector ! 2 , β tanh( tanh( θrθ 0 2 θ β = λ , k . The manifold has a boundary at k k M c X X [ r + √ √ r dθ r = Γ | S 2 = ψ λ = (0 ) (4.29) = = ≤ | − β j θ 2 θθr 2 r (
!# ) obtained from scattering theory. Hence b η K 2 β β ≤ ij ( + tanh g ) erf is the circle parametrized by b η ] exp r = 0 2 ) = | λ ¯ – 24 – η = β λ i − d N | dr ( i ( ][ ψ e η
and 0 K ) for sgn( − k )Γ 1 2 invariant to be π r erf = dψ ) η − 2 3.26 where ][ 2 − ¯ u I ds d 1 k e kdr e 4.17 = 0 1) ) = ][ , " i √ √ β (tanh ) ( F r N × λ dF = = [ f W = (0 r 1 -periodic boundary condition. 1 k ∂ i Z e e 1 2 sgn( i K ) we get the − − anti λ X = 4.5 ] = β rθθ [ ) = 2 Γ β satisfy 1) supersymmetric action can be obtained from ( ( , -invariant of a finite cigar f W e η η η is a periodic with period 2 = (0 and ¯ θ with a product form N u c r = The The orthonormal forms and the nonzero vielbeins are The cigar has a Killing isometry under translations of r The non-zero Christoffel symbols are It is instructive toputation apply for the a general simple considerationswith and in metric illustrative earlier example sections where to an explicit com- where In conclusion, thedependent Witten as index a consequence for of the the noncompactness worldline of4.3 quantum the target mechanics manifold. The is temperature- which reproduces the expression ( We have performed theeigenvalue. computation From ( for a single eigenvalue So the full answer is given by ( Here ¯ JHEP11(2019)080 = i and θ θ X (4.44) (4.39) (4.40) (4.41) (4.42) (4.35) (4.36) (4.37) (4.38) (4.43) ˙ θψ ψ ) r 2 [2] and θ QV j (tanh β ψ r ) and integrating 0 ∂ ij Z j r πimτ/β ξ 2 K ψ i 4.35 e σ kψ − θ m − ) τ − ψ θ iwψ r D j ˙ i . ψ m − ψ ) X r i j iψ r , θ, ψ r ˙ η r ψ 0 + i η K kψ r + τ j θ ξ 0 j , ψ 1 . We have: ) in action ( + D ψ kη 0 √ i r w∂ X r r ˙ ψ ψ ( r πwK σ + − 2 + r r ∂ ij 4.37 2 ) = i r 0 ψ ˙ = τ χ G ) direction to convert the action on the ψ kψ ( dτL X i and k r θ ) + 2 tanh ( σ + β 2 ψ σ = r = σ ψ ∂ 0 k j σ ( 2 dτ r r i Z ∂ we set − K β w i X − j 0 tanh δψ [2] r Z + K k X – 25 – 2 ( ψ 2 τ r ˙ = and ) = θ w ∂ rr QV ∂ χ ψ i r π ij θ [2] G ξ 2 ] exp i πimτ/β X 1 G θ 2 τ √ = + 2 e ) we get: ∂ iwψ + ] along the sigma direction using the Killing vector gives m dψ σ wK + tanh V j ( θ ][ − i 0 ij k dτ QV 67 = X r 4.38 θ ˙ dθ β X τ i m [ + ψ σ g X 0 ∂ is deduced from the transformation of the superfield = r 0 θ 2 i 2 X Z ˙ d i r r + σ ξ ) has unit Jacobian. We can now mode expand X dψ rψ k ∂ ψ τ 0 Z 2 -invariant for a finite cigar. To deform the action we choose ∂ η β 1 2 dr π πpτ ij 4.42 1 4 2 G dτ Z tanh i = β k 0 − I under the Killing symmetry. Using ( dτ ) = Z + i τ ) in the action ( ( Z θ ) = ] = 1 2 direction we get the Euclidean action after a Wick rotation: ¯ β θθF 4.30 ( = is the winding number. We have σ + k, w c W i S ; w β ¯ [ θψ Our goal is to evaluate the path integral on the infinite cigar using localization and Scherk-Schwarz reduction [ S + i The transformation ( we have so that the quadratic fluctuations are given by This localizes the integral to constant modes of To compute the quadratic fluctuations then connect it to the Plugging ( where the derivative of x over the where We dimensionally reduce along2-torus the to a worldsheet collection of actions on a circle. The Lorentzian action is given by: JHEP11(2019)080 , we i 0 (4.47) (4.48) (4.50) (4.51) (4.45) (4.46) (4.49) r w 2 2 − w k n + !# 2 2 tanh 0. w w β πp and comparing , we have βk 2 h + θ 1 2 → u 2 ψ ) . Using the inverse β )Θ 0 c = r r ( 11 ) 2 y βn πp β and ) with r = 2 θ 0 2 θ r r tanh k n , ψ )) we obtain, r 0
βk − + sgn( 1 y tanh 1 2 B.7 , ψ − − 0 w ! − e r
" ( w k ) w 2 θ − w k − n sgn exp πp β + θ . dτ S 2 2 ) k ), we obtain n w K ) − n i∂ β
. Identifying ( + X 0 β πp 1 − w 2 πp β Z w k 2 S ! θ ( 2 3.15 kβ iw − − i∂ − )) = + (see equation ( ( 1 0 r βk θ θ 0 1 sgn 1 2 p − 0
∞ γ i∂ 1 2 r iw − ( ( Now we can take the limit
– 26 – e exp ) − − n c r W r on a boundary located at ) + erfc ) 10 X p 2 r ] = − X 1 B β πp θ 0 2 w dy B [ tanh )) w K (0) 1 − + dψ (0) = (tanh βk r − 0 − r 1 2 c W k n wβ r c r W 0 iw ∂ dψ k∂ ( Z ) vanishes. Using ( 0 i∂ 1 2 ( r sgn( 0 ∞ r 6=0 6=0 dθ sgn ( p p X X γ iγ 0 1 2 dr nβ i i (0) = 2( ∞ c β β W dr = = 0 η − − − Z / )Θ( Z D ), the Dirac operator near the boundary takes the form i p from the determinants as before. Substituting = X wβ πi ) = πβ βnw i 2 β 2 1 − πβ 4.32 2 e − − is due to the presence of the non-zero central charge of cigar supersymmetric quantum k, w ; n , the boundary manifold is a circle = +sgn( c X β ) = ) we find the boundary operator ( r βnw − c e W ) = k, w 2.25 ; β ( After integrating out the fluctuations and non-zero modes of β The The last two terms vanish in this limit which is easier to see before Poisson resummation. ( c W 10 11 c W with ( mechanics. For large It is instructive toasymmetry of compare the this boundary operator resultvielbeins with from ( a target space computation of the spectral It is easy to check that where Θ is the Heaviside step function. After Poisson resummation with respect to The factor of obtain JHEP11(2019)080 - η η has B (4.56) (4.57) (4.52) (4.53) (4.54) (4.55) the on the η k direction is -invariant is η θ . The elliptic with k , and hence the 5 defined by n wk into i 2. This is as expected of . − function i wk s 1 2 wk h ) ζ i 2 wk + of h k n − q c wk 1 h − − wk + w b n ) = 1 need not in general be an integer. ) = ( i k c is positive. This matches with the , q } =0 wk sgn ∞ k (0 } h X n wk Z Z , Z ∈ + ). ∈ U(1) WZW model at level X n (0 ∈ − b / s ζ is an integer, the boundary operator has a n ) ) . The regularized version of the | -invariant vanishes. As one varies n i | 4.49 , ζ R k n η ) = wk is not an integer, the boundary operator , ) + wk n s – 27 – i wk − inθ k ) = 1 − q − i e wk 1 − h -invariant wk { + )). This behavior is plotted in figure { η n wk w k n ( h − h of SL(2 crosses, an integer it jumps by ( 1 k 2.38 , =1 12 ∞ =0 sgn( ∞ X n X n (0 Z ζ − ∈ . As long as X n − k ) WZW model, the parameter ) = to be a positive integer and consider a worldsheet path integral R = ) = √ , s both integers, the k s, q η ( -axis. ( invariant is then given by: η is not an integer in order to avoid zero eigenvalue of the boundary (0) = ζ x w η η wk and -invariant of this operator can be computed readily. Since η k on the is the greatest integer less that wk c wk b -invariant can now be expressed in terms of the modified η In the infinite sum, one can absorb the integer part For the noncompact SL(2 invariant is expected to depend only on the fractional part 12 -axis and In this section, we take for the superconformal fieldas theory a for coset an conformal infinite field cigar. theory The SCFT has a representation zero eigenvalue. There is anfrom ambiguity zero in the (see APS discussion theorem belowy about ( sign of the contributions 5 Mock modularity and the to obtain Note that for invariant changes and every time from level-crossing because precisely when The where where in the lastinvariant step computed we from have the used path integral the ( fact that η The radius of the cigarno is zero modes. The operator. The periodic, the eigenfunctions are given by the set with eignevalues Here assume that JHEP11(2019)080 ,k 1 A (5.3) (5.4) (5.6) (5.1) (5.2) 2 | b + N aτ + u | 2 . τ πk ]) e ) k n 2 4 ) (5.5) 40
/ q 2 ) 1 r z − τ, z ] using the path integral − ( y q 2 k z k,l − 68 2 k of the Appel-Lerch sum , ϑ 2 ) z k ,k k + 1 + 39 ] using canonical methods. )(1 ¯ 1 τ πτ ; n ˆ u ¯ τ . A τ, 64 + r − ( yq , | t τ, = 1. The same result was obtained ∗ ` kt u r τ, | 2 g − 44 y q − y N Z ,k 1 2 1 m τ, − ϑ )(1 ˆ kt 2 ( A n 1 X / q 1 ) q Z ϑ ) for Z ) erfc 3
r ∈ ) − t X τ, z τ 5.3 ) + u ( 2 ( 2 (1 1 τ η – 28 – d sgn ( ) = iϑ =1 τ, z ∞ C Y ( Z n . Spectral asymmetry. − k Z 8 ,k τ, z / ] using free field calculation and modular properties 1 N ( 1 +2 ` Z X ,k A 40 ) = ∈ = 1 z X r | A πz q a,b ¯ τ 1 2 ) = Figure 5 z k ) agrees with ( − τ, N ; ( ) ¯ τ b 2 2 χ 5.1 z τ, ) = ( k +4 ¯ τ wk ) = 2 sin k ,k τ, 1 − ( 2 b | τ, z ∗ A ` ] that ( z ( | g dependence explicitly to emphasize the nonholomorphicity. The cigar 1 c b ( -orbifold of this theory was computed in [ 39 ϑ τ 2 π N τ 2 Z e ) is the odd Jacobi theta function ) = τ, z ) is the Dedekind eta function. The completion ] using localization of the generalized linear sigma model that flows to the cigar ( z τ | 1 ( ¯ τ ϑ η 43 , τ, Our aim is to re-re derive the elliptic genus of an infinite cigar using localization of ( 42 b χ It was shown in [ in [ geometry. The cigar elliptic genus was re-derived in [ the path integral directly in the nonlinear sigma model connecting the computation to the with is given by and was shown to be given by (in a notation slightly different from [ where We display the ¯ elliptic genus was a computed in [ where genus for the for the coset theory JHEP11(2019)080 13 (5.7) (5.8) (5.10) ). The 1 2 − 1 as . Using the u v < | − − y | w < (5.11) | ) l q z completion of a mixed | + | . τ ks ( 2 ) (5.9) . y z | πizJ sl m,` τ 2 5.2 ( + ϑ of weight ( ) of weight 2 e . 2 ) 0 τ k,l ) ¯ ( ˜ τ ks L l ϑ τ ¯ τ q = 0 is counted with multiplicity g ( . The theta expansion can once τ, ) ` πi ( ` ¯ α τ 2 g ∗ ∗ ` 0 − g τ, e ( ], better suited to the problem at hand. Z l< πi l 0 X 2 2 m h b 0 2 − Z / s< X ) transforms like a vector-valued modular X πiτL Z m 2 ¯ τ = ∈ 2 e ` + / ) J X ) τ, Z and index ∗ as in the compact case. The theta coefficients ¯ ( τ ∈ + 0 ∗ ˜ ` l – 29 – J u ≥ ¯ g τ, τ J τ, z ) l X ( ( 1) ∂ 0 ∗ ` f − ≥ ) + τ, z ( s X ∂g ( τ ( f v ) + ` − ) SP Tr h 2 τ, z ) = ] the Appell-Lerch sum in the range ( πτ z ) = 2 | φ z (4 ) = ¯ ) := τ | ¯ ¯ τ τ τ, -invariant for the boundary of the finite cigar. For this purpose, ) = ( τ, z τ, τ, η z ( b ( ( φ | ` b ,k χ ¯ τ h . The holomorphic modular form b 1 ) can be written as 1 2 τ, A ( 5.9 which is nonholomorphic but transforms like a Jacobi form. ) is a solution of the following differential equation. b − φ ¯ τ . The elliptic genus for the cigar turns out to be a ,( u ` τ τ, ), we expect the elliptic genus to transform as a Jacobi form of weight 0 ˆ h guarantees that it will be elliptic. Consequently, by the arguments outlined ( − we defined the elliptic genus of a compact SCFT. One can generalize the as in the compact case. However, unlike in the compact case, it need not be ∗ ` ) is a jacobi form of weight g J 2.3 w m 2.3 completion of a vector valued mock modular form τ, z = ( f v We recall a few definitions about (mixed) mock Jacobi forms and their completions. Our definitions are a slight variant of the definitions in [ 13 completion of nonholomorphic sum form where where where again be seen byare bosonizing the the current independent of ¯ mock Jacobi form A completion of a mixed mock Jacobi form admits the following theta expansion: integral of the SCFTinvariant. with Hence a it covariant isthe regulator expected current is to be manifestly modular. diffeomorphismin and section Moreover, the Weyl ( spectraland flow index symmetry for as for the noncompact Witten index One can then writestarting a point path for integral using representation localization of methods. the elliptic Even genus in which the can noncompact be case, a the path 1/2. This is the form in which5.1 we will encounter it Mock in Jacobi section In forms section definition to a noncompact SCFT by generalizing the trace to include the scattering states it is convenient to re-express [ where the asterisk in the sum indicates that the term earlier discussion of the JHEP11(2019)080 n by − q ∗ ` (5.16) g (5.17) (5.18) (5.13) (5.14) (5.15) = 0, then we 0 , b is related to its = 1. n . ) commutes with ) v h − 3 ) ) q z 5.8 | τ τ, z ) ( ( τ 2 ) if 1 ( η 2 πizJ θ . The full elliptic genus 2 1 e πτ m,` 1 or that πnτ ) dz . − , we fix the choice of 0 ϑ 4 ) ) (5.12) 4.3 ˜ n L ) z ) q | ¯ v, z − πz τ v > 0 τ log(4 ( − ( l − L `,n ( ( πizJ 0 is the contribution coming from g b 1 2 g b ¯ k,l Z e ϑ 0 − v Γ(1 πiτ Notice that for the cigar case, the m ) r ≥ 2 2 − oscill τ n / e mw ) = (2 sin ( ) X 1 Z `,n Z l τ 14 0 b ¯ ∈ P , h ˜ L ` ) 1 πiτ 2 + Z ) + 2 1 − 0 z e log cosh v − m πτ ) = ( L 2 y ( n − τ τ, z / ¯ τ 2 ∞ n ( ( X Z 0 0 − q 2 = 2 g d πτ ∈ oscill ) Z l 2 – 30 – X n> n β − 1 ) − q πi f ·Z e − 2 ˜ + v ) implies that the completion of ) = Φ ) J − )(1 r τ, z 2 − ( ( y 1) d +1 n (1 π f i πτ ) denotes the incomplete gamma function defined as v = − Φ 5.10 q 2 1 ) converges despite the exponentially large factor ( − ) (2 ) ) . If we assume either that − z are KK momenta and winding respectively along the cigar − 2 v, x ) = | H x c Tr W ¯ v τ 5.13 − (1 w ¯ πτ τ − e ) = , τ, z = τ, ∂ v ¯ (4 ( τ ( n ) = − b φ φ 0 appearing in the definition of elliptic genus ( =1 ∞ z τ, x b Y ¯ | n ∂ ( ( ¯ τ J ∗ ` v O -invariant for a finite cigar can be used to compute the full elliptic = g ) τ, 2) SCFT on an infinite cigar. η ) = 2 ( , ¯ τ b χ ) = πτ ) and Γ(1 oscill τ, τ ( (4 Z ) is holomorphic, ( ∗ ) has the Fourier expansion ` U(1) coset has is a non-trivial background ‘spacetime’ dilaton = (2 v, x ) is the Witten index with / g τ alternatively by the integral ) 2 ( ` − τ, z R N ∗ ` ( g , πτ g φ = Im( (2 2 τ ), and where the first term must be replaced by c W Note that the series in ( B.5 The SL(2 direction. The contribution coming from the oscillators is given by 14 -symmetry generator .) Since which ensures that the theorycouples is to conformal the even worldsheet though curvature, the it target plays space no is role not if Ricci the flat. worldsheet Since is the a dilaton torus as in our case. where left-moving oscillators and θ computing the non-compact Witten indexis computed given in by section 5.2 Elliptic genusThe of computation an of infinite cigar genus for a R the right moving supercharge. Hence for the right movers the computation reduces to shadow by We will apply these definitions in the next section to the special case of our interest. can define (The integral is independentz of the path chosen because the integrand is holomorphic in where in ( because Γ(1 Given that setting where JHEP11(2019)080 . ) = m J 4.47 cutoff (5.23) (5.20) (5.21) (5.22) (5.19) -symmetry. is an integer. R
by noting that k L k n 2. 0 J √ s y − 1 and the bosons k wk k = 4 + − w / 0 n ) using the following
2 α and ) y 2 s wk nw 4.47 + q -current is given by kπτ n ] uses = 0 by putting an ( R p 0 q 40 r k wR 4 k . / wk< 2 − X ) − = 0 as we have assumed. We can erfc while [ n n R wk r 0 ) near k θ − 2 w n √ ψ X ( w< ) mod 2 r = − πτ − by the sum over 0 in the end. With this regularization, − q iψ R k (2 n wk 0 w ≥ − → − c W i wk n X of a mock Jacobi form of weight 0 and index ∂θ w sgn − ) vanishes and we get (somewhat surprisingly) and 1 k , p n 1 2 – 31 – − n 0 of the cigar is r ) = ( ≥ k n 4.47 i X w R n X wk = wR w ) h J ) + + w ) from the zero modes of left moving fermions since they -current is given by τ X n completion ( τ, z n R ) over by R ) ( 3 )Θ πz ) 1 η ], the left-moving fermions have charge τ = ( ϑ ( τ, z βn i sin ` = 5.19 40 ( 3 [ i 1 η ) and then taking L ϑ . Now we can substitute the contribution. Using equation ( p 15 i sgn( ) = R z − − | 4.45 ¯ τ τ, ) = ( h z . In terms of the left moving momenta, the | b χ ¯ τ /k τ, = 1 so that asymptotic radius ( 0 b χ α integral in ( . The left and right moving momentas are given by: 0 L r The holomorphic piece is given by: Note that on the cigar, the We use /k p 1 15 We can replace the sum in ( The expression for the ellipticMore genus precisely, is it non-holomorphic transforms as but a it is modular if have chargep 1 and as a consequence not onlyWith the this fermions but normalization bosons are also charged under the correct answer byholomorphic this piece slightly heuristic and procedure. notthe the holomorphic In anomaly holomorphic any is case, anomalyasymptotic determined region this which is by affects is a the good only our quantumanomaly scattering the number main is for states, our not interest. winding-number purposes. affected in by As Since a the this result regularization. the holomorphic leading to anumber divergence is for strictly not this adeal term. conserved with quantum number it This at byin is regularizing the the a Witten consequence index the of contribution the from the fact last that term in winding ( To obtain the above expression wereasoning. have dropped At the the last tip term in of ( the cigar an infinite number of winding modes become massless are charged under U(1) and the contribution fromgiven left by movers, we conclude that the elliptic genus for the cigar is We get a contribution of (2 JHEP11(2019)080 0 16 ks k +2 ` (5.28) (5.29) (5.30) (5.24) (5.25) (5.26) (5.27) y ) which k z 4 . | / k 2 ¯ τ ) 4 z k 0 / τ, 2 ks ( τ, r − +2 nh z k q ` which matches b ,k ( χ 1 q 1 k τ, k 2 A because of the rescaling 4 z k + k ) / k 1 k 2 πτ ) 1 2 4 ) k,` τ, / τ ϑ 2 ( . τ, z ks r = r ) ( | 0 3 1 τ r η ) contribution, the elliptic +2 | k,` ( ` ϑ ks r q m ( ϑ i ) k (1) k,` − m 4 − has index q 5.26 ϑ + 2 / 2 2 1 z k Z ` = r z k k ) erfc 2 2 − ) (5.31) τ, = = r X / τ, k q (mod 2 τ kw Z X πτ k v ( ` ∈ +2 ,k ,k ≡ ` ` wk 1 2 1 r r (1) k,` sgn( | b ) y b k A ϑ 1 2 A + ) πτ . = ) ks w` τ 1 2 Z ) ( τ, z πk k + ) with r but ( 3 ]. 1 τ | k =0 8 1 ( τ, z η 2 z + 2 r k +2 | ( 3
ϑ ` 40 √ w X ` ) 1 is the central charge of the coset. We can now η | 5.15 q = k – 32 – ϑ − r π 2. The total index is c 2 i / τ, z 0 ( − ks , n r ) = `< X ) = ) erfc = 1 k,` − τ 0 ) erfc ¯ r τ ( + 2 ϑ ) = v ` ) and non-holomorphic ( = ` τ, ks X g w< d ( dz 6 where ) ∗ ` τ, z sgn ( = − z ( | c/ + 2 5.25 2 1 πi 0 1 b ¯ τ χ 2 ` ≥ ¯ τ Z ) we conclude that our elliptic genus is a mixed mock Jacobi ) we find that shadow vector is = ` wk X k τ, ∂ ) transforms with index 0 ( − m b ≥ +2 χ ) = ` sgn ( 5.11 X τ, z 5.29 X w n ∂ τ ( = 1 2 , g ( 1 1 and hence r 2 0 ) ϑ ) / − ) Z (1) k,` ) 1 ) is a Jacobi form with index k s,s τ X ) ϑ τ 2 = ( τ, z 2 ( τ, z / Z ( 3 ) and ( ) and ( τ, z ( 3 k Z u ( 1 X η 1 η 2 πτ ∈ ϑ ,k / ϑ ` 1 i Z (4 i X b 5.26 5.15 ) ∈ A − = 0 3 ` is the unary theta function which is defined as ) ) w ) = τ, z τ ( ( (1) k,` ) = 3 1 τ, z η ϑ η ( τ, z ϑ 1 ( i τ, z To find the shadow, let us focus only on the non-holomorphic piece h ϑ ( Note that . The theta function − i b χ f z 16 = − Comparing ( of where with the expected index compute the holomorphic anomaly using ( form with and with genus of cigar is given by Comparing ( Combining the holomorphic ( which matches with the result obtained in [ equals Hence the holomorphic piece can be written as Or equivalently JHEP11(2019)080 , l } 1 2 Q ) 1 = ) to τ = ( b ` (5.35) (5.36) (5.32) (5.33) v 5.6 f { and ], 3 k ) in ( ¯ τ = . In this limit, τ, k 0). Consider the a ( 2 ? ` < g 2 and are naturally re- τ l < ( . 1 2 ) − k ≤ 4 Q ) (5.34) H / πiτ 1 ). In this case with 2 ]. 2 ) . ∈ τ ( − k ) has some property of conti- 70 1 kn γ e 4 ) transforms as 1 5.30 / 1 h and 0 τ 2 +2 τ ( r ` ( := γ ( k 1 = 1 f h q (˜ πiτ . In the terminology of [ πiτ 2 b d 2 R − − e + ] for a discussion of quantum modular + e ∈ , τ 0) but it can equally well be obtained as ) 1 1 2 ) 38 r ) ) 1 1 τ a > cτ aτ b 2 τ kn -invariant appears here too but now in a sum 2 ( η + ` τ ) as sgn ( n ( f f 1 ( + 2 and ) for positive Z τ – 33 – 1 2 a w + k ( ` n ) q ]. See [ γ ` ) can be expressed in terms of the complementary ) of the non-holomophic part − H d ) ˜ f +2 + ` 38 n X + 0 ] the holomorphic anomaly was related to a one-point = limit of the false theta function for sgn ( 5.13 r 1 ) = Z 69 ¯ τ − → sgn( cτ plane ∈ 0 ( ). X Z ) = sgn( n τ, ) = 2 τ ( ∈ 1 τ − ∗ X ` n τ → kn B.6 ) ( g ) and the ‘period integral’ ` 1 ) = is an unary theta series as in ( 2 0 . More precisely, the vector τ 1 f τ Z ] defined by ( + 2 τ ) = } → R , ( ) 2 q f lim ` ` τ (˜ τ ) that appears in the 72 f , ( r ` a,b g SL(2 ) using ( 71 F { x ) have appeared in the computation of topological invariants of Seifert ∈ 1 , one can rewrite the theta function defined on the lower half plane τ ! ( . However, the difference between the function and its modular transform ` kn f R c d a b -invariant and quantum modular forms false + 2 η
` = = γ r We obtained the quantum modular form as a limit of the completion of a mock modular In an interesting recent paper [ which can be viewedusing as the a fact thatthe sgn functions ( manifolds with three singularforms fibers and the [ relation to mock and false theta functions in the context of Chern-Simons With form defined on the uppera limit half of a false theta function [ where nuity and analyticity fortransform every element as (vector-valued) ‘quantumgarded modular as forms’ the of theta-coefficients weight of a ‘quantum Jacobi form’ [ and not over has ‘nice’ properties over The characteristic sgn( weighted with a phase. Thisfinite infinite answer sum at is rational not points convergent on but the can real be line. regularized to Thus, get the a function is defined only over We consider the radialobtain limit ( where the incomplete gamma function inerror ( function erfc( function on the torus, which by our analysis should5.3 be determined by the The scattering states. The shadow vector JHEP11(2019)080 (5.41) (5.42) (5.39) (5.37) (5.38) wk k + n y . nw . 1 k . 4 πiτ J / z k 2 2 y , e r ) 1 0 − . τ ¯ L q i -symmetry acts not only − 0 w R ,k L 1 ( →∞ ik 1 − 2 πτ A 2 ) transforms as a Jacobi form. τ
z k n πiτ | ) + r 2 ), one can consider ¯ | τ z e | r | ¯ τ τ, z k ( ) sgn , 3.15 τ, b χ 1 ( 1 2 ∞ τ b ( χ ) erfc − c W ) + -genus’ takes the form k,` r 0 n η ϑ − ) → 2 1 τ τ sgn (
sgn( (0) ( -invariant of the elliptic genus -invariant ( ) – 34 – ]. Further discussion of quantum modular forms ` η − η z f c W | mk ¯ τ 74 Z , +2 k 2 ` X 2 τ, n 0 to obtain / ( = X 73 Z r b w , X χ X ∈ h → 2 1 w ` 37 X τ – − ) ) oscill 3 35 ) ) = 2 3 , z Z 1 ) , z 1 z | 1 τ 1 ) = τ ( 1 τ τ ( ¯ τ ( τ ( 1 η ( 1 η ) = θ τ, -invariant from the perspective of a tower of Dirac operators is not ), the ‘character-valued ]. It would be interesting the explore further these connections with η i ϑ z ( η (5.40) | i ∗ ` 1 − 2 g 74 τ , 5.37 ( 0 − η → 22 ) = ) = 0 lim ¯ τ limit gives zero. It is not clear to us how to interpret this expression. z z | | ) = ¯ ¯ z τ τ | ∞ 1 i τ, τ, τ ( ( ( b b χ χ η 0 → − ) as a particular ‘character-valued’ → z τ 2 →∞ lim If we think of the elliptic genus as a right-moving Witten index, another perhaps more Motivated by our expression for the | lim τ 2 1 τ τ ( Acknowledgments We thank Gurbir SinghPutrov, T Arora, Ramadas, Andr´eBenevides, Jan Troost, Nima and especially Doroud, Francesca Jeffrey Ferrari for Harvey, useful Pavel discussions. The ¯ the fermions but alsois on different the from bosons whatinterpretation happens by of for shifting this the the compactcompletely theta clear elliptic coordinate to genus. of us. Consequently, the the cigar. precise This natural limit is to consider the ¯ The second term mustcoefficients also up transform to ‘nicely’ a with sign,One consistent the with peculiarity same the of period fact that the integrals for cigar its conformal theta field theory is that the Substituting into ( Evaluating the limits, we find Going back to theη definition of the elliptic genus in terms of the trace, we can interpret can be found in [ the nonholomorphic elliptic genus. theory and WRT invariants [ JHEP11(2019)080 n 2 ≤ (A.8) (A.3) (A.4) (A.5) (A.6) (A.7) (A.1) (A.2) +). , − a, b, . . . ≤ . to write down the = 1 = diag( } A ¯ θθ αβ ¯ η , θ θ α σ dθd { and hence you get the above 0 AB Z and charge conjugation matrix ρ 0 ! ρ i ρ 0 = α i 0 ∗ = α AB ∂ AB ψ
) ∂ AB A ) C C α A ) α ) = = ∂ and θ ∂ θ α α 1 α ψ to label the components of a worldsheet α ρ ρ ρ as the coordinate indices 1 ( to label the components of a worldsheet ρ ( i ( i ( } i i 2 n 1 ! B A , + 2 − δ −} − 0 , ρ , = 2 A { ≤ 1 0 – 35 – ! = A ¯ = θ 0 1 ∂ + } ¯ θ − + ∂ − ∂ } ! { B B ∂ i ψ ψ
B ¯ θ 0 − = . In this case,
D ,Q A = = C , ¯ i θ 0 ∂ A i, j, . . . A † = 1 A A ∂
Q ψ ρ Q ≤ D ψ 0 { . The worldsheet chirality ¯ {D ρ = = 1) superspace 0 − , αβ ¯ ρ ψ , η with values in = 2 B A 1) supersymmetry algebra δ Σ with real superspace coordinates ¯ , ρ } − s with values in = = } B = (1 } θ β A N α, β, . . . , ρ ∂ { α ∂θ ρ { A, B, . . . { A convenient basis for the two dimensional Dirac matrices is are It satisfies the following anticommutations To write actions invariant under theSupercovariant supersymmetry, derivative one is needs a invariant supercovariant under derivative. supersymmetry and it is defined by In superspace, the supercharge is given by which satisfy the We use the superspace supersymmertic lagrangian. Weand use integrals the following convention for superspace derivatives of Majorana spinor iscondition. that C A two-dimensional Majorana spinor is a two-component real spinor. The usual definition which satisfy We use indices vector and spinor. On field space,as we the use tangent 1 space indices. We use the worldsheet metric to be A Conventions and (1 JHEP11(2019)080 βnw − (B.7) (B.3) (B.5) (B.6) (B.1) (A.9) . The e (A.10) (A.11) i ]. i Zx w 48 = − i k n K . 2 w is defined by y h − 5.1 )Θ Z dy e A ) βn ∞ . z ρ θ 0 Z j (¯ i π AB ≥ (B.4) ¯ ρψ 2 ) ) i − + sgn( | √ ρ . z 2 ( K A | , i ) j w ) . ) (B.2) D x | 2 D z ) βk ) := z i √ + 2 | z = p z ¯ + ) erf( · ψ i z A erf( dt , x AB b πin erf( t D βnw ) + erfc( 2 − ) erf( erfc( − α − − j − z π e sgn( e ∂ ]. For an off-shell formulation see [ 1 K α √ 2 i − ! ) − ρ 47 ) = – 36 –
2 s ( [ ) = 1 K , z t β πp i k n z 2 ) = ij (2 Z, − y ∞ g − M A x − = 2 , x -invariant. ) ) = sgn( ) = 1 w Z + η βk z z 1 2 erf(
erfc( } 1 2 I ρ θ B dy e (¯ Γ( − 2 i b ) = erf( βk Q z erfc( , 0 + −→ Z r exp A s, x A b I
Q π ) 2 { Γ( Q √ β πp erfc = 2 A 1 + ) := w b Q z − iw ( k n erf( ) is an odd function because the integrand is an even function dp z Z sgn is symmetric, it allows for a central extension of the supersymmetry ) modifies to i is a covariant derivative on β 1 2 to make contact with the j − − 2.5 R AB D ρ = ∈ ) = The upper incomplete Gamma function encountered in section z m ( f One of the integralsfollowing (involving error function) which is useful in our computation is the for A special case that we encounter is For the purpose of this paper it is convenient to use the expression They satisfy the following relation Note that erf( B Error function andThe incomplete error Gamma function and function the complementary error functions are defined by symmetry algebra. Theaction central ( charge is related to the killing vector as where In presence of central charge the super-charge and the covariant derivative are modified as The presence of the killing vector in the target space allows a central term in the super- Since ¯ JHEP11(2019)080 are i (C.9) (C.6) (C.7) (C.8) (C.1) (C.2) (C.3) − E ψ | t 0 ) recursively iH ... − C.4 ]) e so that in the far | + t 75 | i V E − . The time evolution φ 0 | V e 2 H t 0 ) (C.4) 0 i iH . Using the M¨olleroperators 0 e . , t ∓ 00 i H t 0 0 t E ( ) = V 0 i H φ t D | E ( iH U − φ i D ) − | (C.5) 00 e V | E ) t ∓ t i ) | i ( i 0 0 E ∓ V − ∓ V H t, t 00 φ ∞ + 0 | ( , i H V e dt − H U E t (0 0 t U − + t E φ − with 0 D | Z 0 – 37 – = U E i iH E + i H i e i = − E ∓ = E = , . The solution is given by 1 ψ 0 i φ ) | 1 ) = | of the free Hamiltonian 0 V 0 U E H H i φ = | ) = t, t E t, t − 0 ) = ( i ( φ | D E E D U t, t t, t U ψ ( ( U | + ) D D t i ( U U E D φ V | -states that resemble the free eigenstates in the far past and = satisfy the Lippman-Schwinger equations ) = in i 0 i E E φ t, t | ψ ( | D U ) is the time evolution operator of the Heisenberg picture. The Dirac evolution U 0 are the d i dt t, t i − E ( -states that resemble the free eigenstates in the far future. Solving ( ψ | U out It follows that This geometric series can be easily summed to obtain where the gives the Dyson series expansion Consider and energy eigenstate one can obtain the eigenstate of the full hamiltonian: with the initial condition We can now define the ‘M¨olleroperators’ where operator satisfies the Schro¨odingerequation where we have added anpast adiabatic and switching factor and for inoperator the the interaction in far the future Dirac one picture obtains is given the by free Hamiltonian We review how the densityConsider of states the can scattering be problem related for to the the Hamiltonian phase (See shifts for in example scattering [ theory. C Scattering theory JHEP11(2019)080 ) −∞ (D.5) (D.1) (D.2) (D.3) (D.4) , (C.10) (C.11) (C.12) (C.13) (C.14) ∞ ( D 2 U 2 β πn 4 on the base circle: 0 A Y n> 0 i ) 0 dx are defined by 2 H 2 Z A . . d dτ − ) n = . dx − − E E ( ( x 1 . dE 1 δ = ## dδ dE i dδ − n = − s − 1 π x 1 x π ∗ n Ax dS x dE ) | n e † give a Gaussian path integral x = − h H S x β } ∞ 1 ) = 2 . ) 0 2 ,A − = τ Z − E ) − ( ( e , x E τ ) dS dE U x ( ( s n † )] { dS 1 δ dE β † y πn + x τ S h ) 1 2 Γ( ( U τ − – 38 – x ( β πi = S πinτ = , ρ D β 2 Tr [ ) s e S = 1 − dτx E n ( Z πi 1 Z " δ ) = 2 β 2 S ∈ 2 0 i X n = =1 E ∞ e Z X n ln ( dE exp Z d ) = n ) = := πiρ E ) = ) = τ i ( dx ( s E y ρ ( x | n ( = 2 ζ x − S h S -matrix is given by dx S ln dE Z d " 0 Y n> 0 ) can be expanded in terms of the modes of the operator τ dx ( x Z -matrix is diagonal, then in each one-dimensional subspace we obtain -matrix in the interaction picture is just the time evolution operator = S S Z The infinite product can be regularized using a zeta function The Gaussian integral can then be written as The field The quadratic fluctuations of a single boson where the inner product over the field space and the operator D Determinants and ultralocality The density of states is then given by the so called ‘Krein-Friedel-Lloyd’ formula: If the and using the above formula it is possible show that which can be expressed in terms of the M¨olleroperators as The derivative of the The JHEP11(2019)080 (0) 0 and ζ (D.6) (D.9) 0 (D.10) (D.11) (D.12) x i x | x h β 1 2 − e ] 0 (D.7) ) dx n [ . 0 ln ( dx πβ n 1 n 2 X Z ln (D.8) √ 2 n ) ) ln A = − ( s πβ 0 1 − ) the second summation gives 1 e 2 0 β ) . n x over nonzero modes. The factor of 2 det ln(2 A X D.6 =1 ( ∞ 1 2 − 0 p X n A = 1 1 i π − = ) into integral over the zero mode 1 β det x 2 | i 0 = x exp p h (0) = 0 0 D.4 n β Ax 1 | ζ 2 0 0 ln 2 πβ dx x − . s h – 39 – 2 = ln e dx R ). This argument makes it apparent that for each ] β − − 1 πβ 2 √ e 1 2 − dx = (0) and using ( 2 √ [ D.9 (0) e ζ =1 Z i ∞ ] , we get ζ 0 X n x β | πn Z A = x 2 h d dx ds [ β π 1 Z ), which permits any use, distribution and reproduction in 0 β 2 2 − ln dx e ) = ] 0 s n ln ( Z X ζ dx [ = d = ds i Z ) we reproduce ( CC-BY 4.0 Z Ax | of the operator x ln This article is distributed under the terms of the Creative Commons h 0 β 1 D.10 x 2 − e ) is the renormalized determinant of ] 0 A ) the first summation gives ( 0 dx [ 0 D.5 for a bosonic zero mode can be deduced more directly by using ‘ultralocality’ as we dx πβ Z 1 2 √ bosonic zero mode we get a factor of Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. counter-term corresponding to the ‘cosmological constant’. We then obtain Substituting in ( It follows from the factthe that base the (and measure not for this its integral derivatives). involves only Hence, the it local can metric be on normalized to unity by an ultralocal We now to set the normalization by setting where det explain below. Separating thenon-zero path modes integral ( to obtain Using ( Hence we get, Another formula that would be useful is the following JHEP11(2019)080 , , ]. ]. ]. , . ]. , A A 5 33 62 SPIRE SPIRE SPIRE ]. SPIRE Bull. IN IN IN IN , [ [ [ [ (1981) 843 (2016) SPIRE Phys. Lett. 49 Mathematical 88 IN , , in (1984) 533 (1978) 213 J. Math. Phys. (1985) 197 (1978) 3690 , Int. J. Mod. Phys. 93 62 , 100 (1982) 661 [ D 18 Am. J. Phys. Commun. Math. Phys. , , 17 Rev. Mod. Phys. , ]. , Boulder U.S.A. (1983), pg. 559. ]. Phys. Rev. , SPIRE IN ]. 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