Matrix Quantum Mechanics on S^{1}/{\mathbb Z}_{2}
MMatrix Quantum Mechanics on S / Z P. Betzios, U. G¨ursoy and O. Papadoulaki
Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,Utrecht University,Princetonplein 5, 3584 CC Utrecht, The Netherlands.
E-mail:
[email protected] , [email protected] , [email protected]
Abstract:
We study Matrix Quantum Mechanics on the Euclidean time orbifold S / Z .Upon Wick rotation to Lorentzian time and taking the double-scaling limit this theoryprovides a toy model for a big-bang/big crunch universe in two dimensional non-criticalstring theory where the orbifold fixed points become cosmological singularities. We derivethe MQM partition function both in the canonical and grand canonical ensemble in twodifferent formulations and demonstrate agreement between them. We pinpoint the contri-bution of twisted states in both of these formulations either in terms of bi-local operatorsacting at the end-points of time or branch-cuts on the complex plane. We calculate, in thematrix model, the contribution of the twisted states to the torus level partition functionexplicitly and show that it precisely matches the world-sheet result, providing a non-trivialtest of the proposed duality. Finally we discuss some interesting features of the partitionfunction and the possibility of realising it as a τ -function of an integrable hierarchy. Keywords:
Matrix Quantum Mechanics, Matrix Models, String Theory, String Cosmology
ArXiv ePrint: a r X i v : . [ h e p - t h ] J un ontents c = 1 Liouville Theory on S / Z n = 0 case 153.1.5 The n = N/ n = N/ n n = 0 305.3 n = N/ – i – Oscillator wavefunctions 37
B.1 Normal Harmonic Oscilator 38B.2 Inverted Harmonic Oscillator 38B.3 Mehler for parabolic cylinder 39
C Representation in terms of angles (Wilson-lines) 40
C.1 Measure 42C.2 Pfaffian in regular representation 42
D Grand Canonical for n = 0
43E Hilbert transform properties 44F The Kernel 45
F.1 Kernel in Energy basis 45F.2 Kernel in elliptic functions 46F.3 Trace of the kernel 46F.4 1-particle density of states 48
G Approximate methods for large β G.1 Generic n in angles 49G.2 Generic n in eigenvalues of M n = N/ n = N/ Little is known in Quantum Gravity about how the space-like singularities in general, andcosmological singularities in particular, can be resolved—if they can be resolved at all.Some of the questions in this context are: is string theory able to provide consistent, non-singular dynamics around such singularities? What is the set of possible initial conditionsfor the cosmological evolution starting from a big-bang singularity? What are the pos-sible initial wave-functions of the universe at the big-bang? How is the evolution of theuniverse determined following the big-bang, et cetera. Quantum gravity is, notoriously, asubject where problems vastly outnumber results especially for the physics near spacetimesingularities. At short distances strong fluctuations of the metric are expected to cause abreakdown of classical geometry and the notion of space and time might lose their meaning– 1 –nd become emergent concepts of a more fundamental theory. Nevertheless, various effortsto understand the initial conditions of the Universe based on the semi-classical approxima-tion to the path integral were made in the 80’s, most notably the no-boundary proposal ofHartle and Hawking [1] and the tunnelling boundary condition of Linde and Vilenkin [2, 3]but it is fair to say we do not have a unique sensible answer to the aforementioned problems.It is natural to ponder what string theory has to offer in this context, and whetherit can resolve these problems or at least provide a new perspective. In string/M-theorythese fundamental questions have been addressed in the various approximations and usingvarious models in the past. Some notable work includes the study of time dependentorbifolds and the null-brane construction [4–9], the Bang-Crunch scenarios [10–17], tachyoncondensation [25–27], constructions attempting to address cosmological singularities viastring theory [18–24], or via AdS/CFT [29–34] and pre-Big Bang scenarios [35] amongmany others. For related work on string cosmology with a view towards inflation see [36–38]. Motivated by these difficult questions, we ask a more modest question in this paper:What can string theory teach us about the cosmological singularities in the context of a toymodel: the two dimensional non-critical string theory or c = 1 Liouville theory [48, 54]?The idea here is the following . Start with the Euclidean
2D non-critical string theorywith Euclidean time direction, τ , compactified on a circle with radius R . This theoryhas a well-known dual formulation in terms of Matrix Quantum Mechanics (MQM) of aHermitean N × N dimensional matrix M at finite temperature T = 1 / πR in a doublescaling limit [57–59]. Now, consider a Z orbifold of the non-critical string theory (NCST)in the Euclidean time direction where one identifies τ ∼ − τ . The following identifications τ ∼ τ + 2 πR, and τ ∼ − τ , (1.1)restrict the domain of the Euclidean time to the line segment 0 ≤ τ ≤ πR . Upon Wickrotating to Lorentzian time, the fixed points of the orbifold at the points τ = 0 and τ = πR correspond respectively to the big-bang and big-crunch singularities of a toy, cosmologicalbig-bang/big crunch universe in two dimensions. The questions posed above are expectedto have a much simpler formulation in this toy universe, since the only non-trivial physicaldegrees of freedom in the bulk are a massless closed string “tachyon field” in case ofthe bosonic NCST with an additional RR scalar C in case of supersymmetric type 0BNCST [49, 51, 52, 69, 70]. This is to be contrasted with the infinitely many physicalexcitations of the critical bosonic string in 26 dimensions and supersymmetric string in 10dimensions. The 2D toy model also enjoys the following great advantage: Resolution ofcosmological singularities in string theory is expected to involve not only the full set ofcorrections in the string length scale α (cid:48) but also the perturbative corrections in the stringcoupling constant g s [43] . This seems an insurmountable task for critical string theories More precisely c = 1 Liouville theory is an exact CFT equivalent to 2D string theory in a linear dilatonand exponential tachyon background in the Liouville direction φ . The non-critical is an adjective referringto the number of dimensions. The basic idea and some of the calculations presented in this paper are due to discussions that one ofthe authors (U.G.) had together with Hong Liu in 2005 [28]. and possibly corrections non-perturbative in g s . – 2 –unless one attempts to use the BFSS [64] or related matrix model formulations, as in someof the references above). In the case of 2D NCST however, the dual formulation in terms ofHermitean MQM comes to the rescue. The partition function evaluated via MQM involvesat least the full set of perturbative g s corrections in the dual string theory and in additiona lot is understood for the non-perturbative corrections as well [73–76].The duality between 2D NCST and the Hermitean MQM was discovered in late 80s [55,56, 77, 78]. Starting from a Lagrangian of the form L = Tr (cid:32) (cid:18) ∂M∂t (cid:19) + 12 α (cid:48) M − κ M (cid:33) , (1.2)where M is a Hermitean N by N matrix, one constructs the web of Feynman diagramsthat arise from the cubic interaction vertex. This web of Feynman diagrams then providesthe dual lattice of the one obtained from triangulations of a string world-sheet a la ’tHooft [124]. As one increases the bare coupling κ one discovers that the average numberof triangles on a given world-sheet begins to diverge at a critical value κ c . Then, takingthe double scaling limit N → ∞ , κ → κ c with N ( κ c − κ ) kept constant, one obtains acontinuum formulation of the 2D string theory in terms of matrix quantum mechanics.The crucial point here is that a universality arises in this double scaling limit, that focuseson the tip of the potential provided by the mass term in (1.2). Therefore, the theorydual to the continuum limit of the 2D string theory is just described by Hermitean matrixquantum mechanics with the inverse harmonic oscillator potential . In this duality, thetime direction in MQM provides the time direction for the 2D space-time where the stringcan propagate. In addition, the eigenvalues λ i of the matrix M provide the extra space-likeLiouville direction φ in the 2D string theory picture.In some sense this duality is the oldest example of the open/close dualities in stringtheory, much before the famous AdS/CFT correspondence in the critical IIB string theory[125]. The lessons learned from AdS/CFT, in particular the role of D-branes in this cor-respondence, ignited a revival of interest in the old matrix quantum mechanics in the 00s.A gauge/gravity type of interpretation focusing on the target space physics arising fromthe matrix model has been proposed in [67, 68]. According to this picture, MQM describesthe field theory living on N D B fermionic NCST also admitsa non-perturbative formulation where the cubic potential in case of the bosonic NCST issimply replaced by a quartic potential. Therefore, unlike the bosonic theory, 0 B fermionicNCST is believed to be non-perturbatively stable [69, 70]. One important insight thatarises from the D-brane interpretation in the string/matrix duality is the need to introducea non-dynamical bulk gauge field A ( τ ) in the matrix path integral. Integration over thisgauge field then projects to the singlet sector of the MQM. The gauged matrix model thencaptures the physics of the so-called linear dilaton background of the 2D string theory.In this paper, we consider a toy cosmological universe with a big bang/big crunch sin-gularity in the context of bosonic and 0 B NCST. As explained above, a natural model that This gauge field is necessarily non-dynamical in two dimensions. – 3 –s suitable for this purpose is a space-time where the (Euclidean) time direction is com-pactified and orbifolded as S / Z and coupled to the Liouville direction. If the Euclideantime direction in this model admits an analytic continuation into Lorentzian signature, onecan interpret the orbifold singularities as cosmological singularities. One also hopes thatinformation about the initial and final wavefunctions is encoded in the twisted sector ofthe orbifold that describes states localized at the orbifold fixed points. One can go furtherand also ask if one can compute the transition amplitude of the universe in this model.We take the first step toward this aim in this paper and focus on the calculation of theorbifold matrix model partition function in Euclidean time using the machinery of matrixquantum mechanics. In particular, we show that • the orbifold operation is represented in the matrix model by the operation diag( − , − , · · · − , , , · · · , (cid:63) where (cid:63) acts on time as (cid:63)t = − t(cid:63) and with n eigenvalues withthe value − n = 0 , · · · N/ N − n fractional instantons that are stuck at the orbifold fixed points and n D-particles free to move along the t-directon. We argue that the correct choice cor-responds to n = N/
2, where there are no fractional instantons. • Using the matrix model techniques we calculate the torus partition function in thelarge R limit for the n = 0 and n = N/ n = N/ • The calculation of the full orbifold partition function in the canonical ensemble inthe large-N limit proves hard. However, we manage to represent the grand-canonicalpartition function in terms of an integral kernel whose spectrum gives the single-particle density of states. We obtain this density by two independent methods thatagree with each other. • We further discuss certain aspects of this matrix model in connection with the corre-sponding 2D string theory. Finally we make various comments on how to implementthe Wick rotation of the Euclidean time orbifold partition function to Lorentzian sig-nature. We leave the full Lorentzian space-time interpretation of the possible initialand final boundary conditions at the cosmological singularities and a more thoroughstudy of the semi-classical geometry that the matrix model describes, to future work.The organization of the paper is as follows. In the next section we first outline the necessarymaterial on the orbifold c = 1 Liouville theory. In particular we present the torus partitionfunction of the bosonic, super-affine 0 B and 0 A NCSTs including the contribution from thetwisted sectors. This is achieved by considering the possible Z orbifolds of these theoriesand using self consistency CFT techniques that relate the orbifold with the circle CFT at– 4 –ifferent multiples of the self-dual radius. In this section, we also introduce some more de-tails of Matrix Quantum Mechanics in 2.3 and set up our conventions. Finally in section 2.4we make use of the D M and the gauge field A consistent with the orbifold projection. Interestingly, we finddifferent representations of the projection classified by an integer ≤ n ≤ N/
2. Differentrepresentations are found to be related via the action of a certain kind of “loop-operator”at the end-points in 3.1.3. In section 3, we also compute the canonical (finite N) partitionfunction by representing it as a path integral over the eigenvalues of M . In addition we findthat this partition function admits a natural continuation into Lorentzian signature, henceprovides a possible connection to the cosmological toy universe. In particular it has a nicestructure from which the initial and final wavefunctions and the transition amplitude of thetoy cosmological space-time can be read off. These wavefunctions are expressed in termsof determinants of eigenvalues of M at t = 0 and t = T . We further argue that the regular n = N/ A .Section 4 is devoted to the computation of the MQM grand partition function for the “reg-ular” n = N/ n = 0 representations. The grand canonical partition function is helpfulin taking the double scaling limit [63], hence connecting the MQM partition function to thegenus expansion of the dual string theory. This section contains one of the main findingsin our paper: here we show that the calculation of the grand canonical partition reducesto the computation of the spectrum of an integral kernel which we express in various use-ful forms. The equations that determine the spectrum of this kernel can be expressed asintegral equations. By deforming the contour of integration in these integral equations,we identify contributions to the untwisted and twisted sectors in the free energy of theorbifolded 2D NCST.It proves hard to evaluate and express these contributions in terms of the dual string theoryquantities in the double scaling limit. In section 4.2.3 we perform a partial matching ofthe various expressions for the kernel by computing its trace, from which we can read-offthe one-particle density of states that we express as a sum of the usual harmonic oscillatordensity of states including a twisted state contribution.Finally in section 5 and in appendix G we attempt to the twisted states at the orbifoldend-points by performing a large radius expansion of the canonical partition function. Wemanage to do this precisely for the n = 0 representation and the “regular” n = N/ sine-kernel which expresses the probability that all the energy eigenvalues takenfrom a random Hermitian Hamiltonian lie outside the interval [ − µ,
0] and thus form thefermi sea. This is also called the level spacing distribution E (0 , µ ) in the random matrix This possibility was observed earlier in the unpublished work [28]. – 5 –arlance [103]. The initial and final wavefunctions take the form of “square-roots” of thisdistribution.Throughout the text, we discuss similarities and differences with established results in theliterature such as the circle and the 2D black hole [57, 66]. We also discuss the possibilityof realising the grand canonical partition function as a τ function of an integrable hierarchywith a Pfaffian structure. Finally, in section 6 we discuss our results and provide a lookahead. Several appendices contain the details of our calculations. c = 1 Liouville Theory on S / Z One computes the orbifold partition function at the torus level in string theory as folllows.Let us call the bosonic matter field X restricted to the line segment − πR ≤ X < πR andobeying the following identifications under translation and reflection X ≈ X + 2 πR and X ≈ − X . (2.1)The modular partition function of the theory is (see for example [39]) Z orb ( R, z ) = 12 (cid:26) Z circle ( R, z ) + | θ ( z ) θ ( z ) || η ( z ) | + | θ ( z ) θ ( z ) || η ( z ) | + | θ ( z ) θ ( z ) || η ( z ) | (cid:27) , (2.2)where Z circle ( R, z ) is the modular partition function for the circle, η ( z ) is the Dedekind η function, θ ’s are the elliptic functions, R is the radius of the circle and z is the modulusof the torus. The fist term in (2.2) gives the contribution from the untwisted states andequals half the partition function of the circle. The contribution from the twisted states isgiven by the R independent part. To obtain the full torus partition function on the orbifoldone should couple the ghost and the Liouville modes to (2.2) and integrate over the moduli z Z orb ( R ) = − V φ (cid:90) F d z (cid:18) | η ( z ) | z (cid:19) (cid:0) π √ z | η ( z ) | (cid:1) − Z orb ( R, z ) , (2.3)where the integral is over the fundamental domain F , the first term in the integrand isthe contribution from the ghost sector and the second the contribution from the Liouvillemodes. V φ is the contribution from the Liouville zero mode, shown to be proportional tothe renormalised volume in the Liouville direction log µ with µ the renormalised stringcoupling [50]. Upon performing the integral over z one finds the following answer Z orb ( R ) = 12 Z circle ( R ) + c, (2.4)where c is independent of R and Z circle ( R ) is the partition function of the circle coupledto the Liouville mode computed by the worldsheet methods in [50] Z circle ( R ) = − (cid:18) R + 1 R (cid:19) ln ( µ ) . (2.5) This field corresponds to the Euclidean time τ in the previous section. – 6 –o determine the constant c for the orbifold partition function one may use the relationbetween the circle and the orbifold at the self-dual radius [45]: Z orb ( R = 1 , z ) = Z circle ( R = 2 , z ) . (2.6)Then substituting in (2.6) to (2.3) and combining them with (2.5), one finds the final result: Z orb ( R ) = − (cid:18) R + 1 R (cid:19) ln ( µ ) −
116 ln ( µ ) . (2.7) The classification of ˆ c = 1 CFTs has been performed in [40][41]. According to this classifi-cation, the continuous lines of theories include two lines of “circular” theories and variousorbifolds of these theories. The “circular” theories consist of the circle CFT and a super-affine CFT. The coupling of these “circular” theories to super-Liouville is discussed in [69].We summarize their results: • Circle CFTs : The usual fermionic circle theory (compact X + Ising) gives rise totwo theories when coupled to super-Liouville: 0 A and 0 B depending on the GSOprojection. Their partition functions are: Z cirA ( R ) = − √ µ (cid:18) R + 1 R (cid:19) , Z cirB ( R ) = − √ µ (cid:18) R + 2 R (cid:19) . (2.8)These theories are interchanged under the T-duality: R → /R . At the specialradius, R = 1 there is enhanced SU (2) × SU (2) symmetry. • Super-Affine CFTs : The usual super-affine theory is obtained by modding out theusual fermionic circle theory by the following Z :( − F s e ipδ , (2.9)where ( − F s is defined as +1 on the antiperiodic fermions and 1 on the periodicones. e ipδ is a shift operator that shifts by a unit vector on the self-dual lattice.When Coupled to super-Liouville one again obtains two theories: Super-Affine A andSuper-Affine B theories with the following partition functions: Z saA ( R ) = −
112 ln µ (cid:32) R √ √ R (cid:33) , Z saB ( R ) = −
124 ln µ (cid:32) R √ √ R (cid:33) . (2.10)These theories are both self-dual under R → /R . At the self-dual radius R = √ SO (3) symmetry.Apart from the type 0 theories, there are other “circular” ˆ c = 1 theories with type I GSOprojections. These have been classified in [42]. In addition to the “circular” ˆ c = 1 theories,there are three families of orbifold CFTs [40][41].– 7 – Orbifold I : The first class of orbifolds is obtained by modding out circular theoriesby: R : X → − X, Ψ → − Ψ . (2.11)Both the left and right handed fermions on the world-sheet are transformed in orderto preserve world-sheet supersymmetry. R as defined above is a symmetry of onlythe 0 B theory since in the 0 A theory states in the Ramond sector have odd fermionnumber. Therefore one obtains only one orbifold CFT by twisting the 0 B theory by R . The partition function is obtained by noting the following two relations [40] whichcontinue to hold after coupling to super-Liouville: Z orbI ( R ) = 12 Z cirB ( R ) + const, (2.12)and Z orbB (1) = Z cirB (2) , . (2.13)The result is: Z orbI ( R ) = 12 Z cirB ( R ) − √ µ . (2.14)We also find two other continuous families of orbifold theories, discuss them andpresent their torus level partition functions in appendix A. We now provide a very short review of Matrix Quantum Mechanics (MQM). For moredetails the reader can consult existing reviews in the literature, for example [58, 59,62]. Gauged MQM is a 0 + 1 dimensional quantum mechanical theory of N × N Hermitian matrices denoted by M ( t ) and a non dynamical gauge field A ( t ). Thegauge field acts as a Lagrange multiplier and projects onto the singlet representationof the SU ( N ) gauge group. The path integral is defined as (we work in units where α (cid:48) = 1): (cid:104) out | in (cid:105) = (cid:90) D M ( t ) D A ( t ) exp (cid:20) iN (cid:90) t f t in dt Tr (cid:18)
12 ( D t M ) + 12 M − κ √ N M (cid:19)(cid:21) , (2.15)where D t = ∂ t + [ A, M ]. This model has an SU ( N ) gauge symmetry. One candiagonalise M by a unitary transformation M ( t ) = U ( t )Λ( t ) U † ( t ) where Λ( t ) isdiagonal and U ( t ) unitary. One then picks up a Jacobian from the path integralmeasure for every t D M = D U Haar N (cid:89) i =1 dλ i ∆ (Λ) , ∆(Λ) = (cid:89) i 12 ( D τ M ) + ω M (cid:19) , (2.18)where τ is the Euclidean time variable, β = πR , and the covariant derivative withrespect to the gauge group is D τ M = ∂ τ M − i [ A, M ] . Here, anticipating the largeN limit in (2.15) we have dropped the interaction term in (2.15) and we have allowedfor a more general mass term ω . For real values of ω this action corresponds to thenormal harmonic oscillator potential—the inverted one can be obtained upon theanalytic continuation ω → iω .The action (2.18) is invariant under the SU ( N ) gauge transformations: M ( τ ) → U ( τ ) M ( τ ) U † ( τ ) , A ( τ ) → U ( τ ) A ( τ ) U † ( τ ) + iU ( τ ) ∂ τ U † ( τ ) . (2.19) This cubic potential is always non-perturbatively unstable, but the supersymmetric version of the model(0 B ) has a quartic stable potential and is thus non-perturbatively well defined. This procedure has been worked out in [28]. – 9 –he theory also has a Z symmetry corresponding to τ → − τ, M ( τ ) → M ( − τ ) , A ( τ ) → − A ( − τ ) . (2.20)One can gauge this symmetry by projecting to the invariant states. We will use amore general gauging, by combining the reflection symmetry with a Z subgroup ofthe SU ( N ) gauge group (see also [44]). We define (up to a change of basis- note alsothat Ω is defined up to a minus sign)Ω = (cid:32) − n × n 00 1 ( N − n ) × ( N − n ) (cid:33) ∗ , (2.21)with ∗ f ( τ ) = f ( − τ ) ∗ , ∗ ∂ τ = − ∂ τ ∗ , ≤ n ≤ N , and then require:Ω A ( τ )Ω − = − A ( τ ) + 2 n i /βδ ij , Ω M ( τ )Ω − = M ( τ ) . (2.22)with n i ∈ Z which is allowed since the eigenvalues of A are periodic variables withperiod β . This term turns out to be unimportant since it can be gauged-away. Thisprocedure naturally splits the matrices into (even/odd) blocks that need to satisfydifferent boundary conditions. We get M ( τ ) = (cid:32) M ( τ ) Φ( τ )Φ † ( τ ) M ( τ ) (cid:33) , A ( τ ) = (cid:32) A ( τ ) B ( τ ) B † ( τ ) A ( τ ) . (cid:33) (2.23)One immediately sees that the n × n M , A and the ( N − n ) × ( N − n ) M , A matrices should be Hermitian while the n × ( N − n ) Φ , B are complex. In addition,consistency with 2.21, 2.22 requires that M , M , B are even while A , A , Φ are oddfunctions of τ . From the gauge transformations (2.19) the ones that are consistentwith the action of Ω are U ( τ ) = (cid:32) V ( τ ) W ( τ ) W ( τ ) V ( τ ) (cid:33) , (2.24)with V , V even and W , W odd.After orbifolding the fundamental domain is 0 ≤ τ ≤ β . In the bulk of the domain thetheory is as before. The changes come from demanding different boundary conditionsfor the fields in (2.23) imposed at the fixed points 0 , β due to their symmetry. Inparticular we need to demand A (0) = A (0) = Φ(0) = 0 = A ( β ) = A ( β ) = Φ( β ) (2.25)The SU ( N ) gauge group gets broken to SU ( n ) × SU ( N − n ) at the boundaries, andas we will see the initial and final wavefunctions contain two separate sets of n and N − n fermions. This breaking also means that the zero-modes of A come solely fromthe off-diagonal elements B . – 10 –he two most special cases are n = 0 and n = N/ 2. The first is the simplest casewhere there exist no zero modes of the gauge field, while the second describes theso-called “regular” representation of the orbifold which is expected to give the MatrixModel dual to the orbifold Liouville theory. Further reasoning for why n = N/ n correspond to adding fractional D-instantons at thefixed points. This becomes clear in the T-dual picture. In particular, upon T-dualizing the Euclidean circle to a radius 1 /R and then orbifolding, the original D branes become D-instantons whose position on the dual circle is governed by the zeromode of the gauge-field. This means that for n = 0, A ( τ ) = n i πR δ ij , n i ∈ Z and all theinstantons are stuck at the fixed points 0 , π/R . For the generic n representation, onehas n -zero modes with arbitrary angle in the T-dual circle and thus the configurationcontains n -physical instantons at angles θ i together with N − n stuck at the fixedpoints. This makes clear that the regular n = N/ The partition function for a generic n representation of the orbifold is then obtainedby integrating over the non-vanishing components of the matrices M and A in (2.23)at the initial and final points—that are the even components M , M and B at τ = 0and τ = β , and performing the path integral of the full matrices between these points.Thus, for a generic n-representation we have, Z = (cid:90) D B (0) D B ( β ) D M , (0) D M , (0) (cid:90) A ( β ) A (0) D A ( τ ) (cid:90) M ( β ) M (0) D M ( τ ) e − S . (3.1)The next step is to reduce this matrix integral to an integral over eigenvalues. Onecan show that (cid:90) M (cid:48) ( β ) M (0) D A D M e − (cid:82) β dτ Tr ( D τ M ) + ωM = (cid:90) U ( N ) D U (cid:104) U M (cid:48) U † , β | M, (cid:105) . (3.2)In our case the propagator is the (Euclidean) propagator for a matrix harmonicoscillator given by (cid:104) M (cid:48) , β | M, (cid:105) = (cid:16) ω π sinh ωβ (cid:17) N / exp (cid:16) − ω ωβ (cid:2)(cid:0) Tr M + Tr M (cid:48) (cid:1) cosh ωβ − M M (cid:48) (cid:3)(cid:17) (3.3)These two equations can be combined beautifully using the Harish-Chandra-Itzykson-Zuber integral (cid:90) U ( N ) DU exp (cid:16) g Tr M U M (cid:48) U † (cid:17) = N − (cid:89) p =1 ( p !) g − N ( N − det e gλ i λ (cid:48) j ∆( λ )∆( λ (cid:48) ) (3.4) From now on we assume even N . – 11 –hat will allow us to reduce the integral to eigenvalues. This is possible since the onlyterm that couples different matrices in the propagator is precisely of the form thatcan be reduced to eigenvalues via the HCIZ formula.Before moving on, let us note the following two options: we can either first diagonalise M and then integrate over U using the HCIZ formula or first diagonalise U and thenintegrate over M . In the orbifold case one also needs to take care about the orbifoldprojection which is implemented through the block structure of the matrices. In thenext section, we will follow the first procedure and compare the results for the circleand orbifold. In section 3.2 we will follow the second and in section 4.2.3 we willperform a matching between the two methods. To set up our notation, we define K E ( λ i , λ (cid:48) j ; β ) = (cid:104) λ (cid:48) j , β | λ i , (cid:105) the Euclidean oscillatorpropagator as follows K E ( λ i , λ (cid:48) j ; β ) = (cid:16) ω π sinh ωβ (cid:17) exp (cid:16) − ω ωβ (cid:104)(cid:16) λ i + λ (cid:48) j (cid:17) cosh ωβ − λ i λ (cid:48) j (cid:105)(cid:17) = (cid:80) ∞ n =0 ψ n ( λ i ) ψ n ( λ (cid:48) j ) q n + , (3.5)where the second spectral representation is also known as Mehler’s formula. In thisrepresentation q = e − ωβ and ψ n ( λ i ) are the Hermite functions. Note that upon ana-lytic continuation ω → iω the Hermite functions turn into parabolic cylinder functions D ν ( z ) defined for complex ν, z see Appendix B. One can also resolve the inverted os-cillator propagator in terms of parabolic cylinder functions from the start [60], therelevant formula is presented in Appendix B.3. As we discuss below the possibility ofanalytically continuing the propagator in the parameters ω, β is the reason we expectto obtain the Lorentzian transition amplitude in this 2D toy universe directly fromthe Euclidean description. We first review the case of circle [57, 63]. For this partition function on S we justhave to demand periodic boundary conditions ( M (cid:48) ( β c ) = M (0), β c = 2 β ) Z N = (cid:90) D M (0) D U (cid:104) U M (0) U † | M (0) (cid:105) = 1 N ! (cid:90) N (cid:89) i =1 dλ i det K E ( λ i , λ j ) , (3.6)where we diagonalised M (0) = U Λ U † and integrated over the matrix U = U U † . The∆ (Λ) in the numerator from the measure of M, canceled the similar term producedby the HCIZ formula. The term (cid:81) N − p =0 p ! in the HCIZ formula got canceled by thesecond integration over the gauge group which is (cid:82) D U Haar = π N ( N − / / (cid:81) Np =0 p !. Inthe end the 1 /N ! term is due to the left-over permutation (Weyl) symmetry betweenthe eigenvalues. For more details on factors of N for this and more general cases– 12 –ee [90].The result is the partition function of N free fermions in the harmonic oscillatorpotential: Z N = q N (cid:81) Nk =1 (1 − q k ) , (3.7)with q = e − ωβ c . One can also expand this result for large β c and recover the zerotemperature free energy F = β c ω N β c E + O ( e − ωβ c ) , (3.8)where E = (cid:80) N − k =0 ω ( k + ) the vacuum energy of the system of N fermions. In con-strast, in the orbifold case at least for the torus contribution we expect a subleading β independent term due to the presence of twisted states since these are localised atthe end-points. The orbifold partition function for generic n after we integrate over the propagationbecomes Z n,N − n = (cid:90) D M D M (cid:48) D U (cid:104) U M (cid:48) U † , β | M, (cid:105) , (3.9)with M = (cid:32) M ( n × n )1 M ( N − n ) × ( N − n )2 (cid:33) , D M = D M D M (3.10)and similarly for M (cid:48) . We now use the HCIZ formula to evaluate the integral overthe unitary matrix U . If we define the eigenvalues of M , as x i , y i respectively, (cid:81) ni =1 dx i /n ! ≡ d n x and similarly for y , the result is found to be Z n,N − n = C N,n (cid:90) d n xd N − n yd n x (cid:48) d N − n y (cid:48) ∆ n ( x )∆ N − n ( y )∆ n,N − n ( x, y ) det K (¯ x i ; ¯ x (cid:48) j ) ∆ n ( x (cid:48) )∆ N − n ( y (cid:48) )∆ n,N − n ( x (cid:48) , y (cid:48) ) , (3.11)with ∆ n,N − n ( x, y ) = n (cid:89) i =1 N − n (cid:89) j =1 ( x i − y j ) . (3.12)First of all we make the following crucial observation: the form of this Euclideanpartition functions in (3.1) and (3.11) are appropriate for analytic continuation intothe Lorentzian time. The analytic continuation is obtained simply by changing β = iT in the propagator. Therefore, after the analytic continuation we can simply interpretthese Lorentzian partition functions as transition amplitudes from an initial state ofthe universe at t = 0 to a final state at t = T (cid:104) ψ f , T | ψ i (cid:105) = (cid:90) D ¯ xD ¯ x (cid:48) ψ ∗ f (¯ x (cid:48) ) det K L (¯ x, ¯ x (cid:48) ; T ) ψ i (¯ x ) , (3.13)– 13 –here we introduced the compact notation ¯ x = ( x, y ). Here the initial and finalwave-functions in this toy universe are of the form ψ i (¯ x ) = ψ f (¯ x ) = ∆ n ( x )∆ N − n ( y )∆ n,N − n ( x, y ) . (3.14)One can rewrite these wavefunctions in the form (cid:81) i,j ( λ i − λ j ) q i q j in terms of fermionshaving positive ( q i = +1 , ≤ i ≤ n ) and negative ( q i = − , n + 1 ≤ i ≤ N ) charge(or spin), with same charge fermions “feeling” repulsion and opposite ones attrac-tion. They represent a Coulomb-gas in one dimension. From this point of view, therepresentation n = N/ . Then theyreappeared in connection to the description of effective IR superpotentials of N = 1gauge theories and in studies of supermatrix models (see [85, 95, 96] and referenceswithin). If one replaces rational with hyperbolic functions, a similar ratio can alsobe found in studies of superconformal Chern-Simons theories of Affine ˆ D -type, at thequiver end-nodes [101]. Finally there is recent interest in these wavefunctions [138]in the context of non-Unitary holography. One can connect different n representations by inserting operators in the end-pointsof the path integral of the form (cid:81) j ( x − y j ) or (cid:81) j / ( x − y j ) to lower/raise the valueof n. This form of operators is known as loop operators. We first define the loopoperator that creates macroscopic holes/boundaries on the worldsheet (this meansthe string gets attached to a D-brane, the so called FZZT brane) in matrix modellanguage [62, 71, 94, 98, 99]: W ( x ) = 1 N Tr log( x − M ) . (3.15)The function that creates a coherent state of them is: e NW ( x ) = det( x − M ) = N (cid:89) j ( x − λ j ) . (3.16)In these equations x can be thought of as a chemical potential µ B (or a boundarycosmological constant). For c < n to the n = 0 representation n (cid:89) i N − n (cid:89) j ( x i − y j ) = det ( M ⊗ N − n × N − n − n × n ⊗ M ) , (3.17) In some of these studies the divergence coming from the denominator is avoided, since it has the form x i + y j with the variables restricted to be positive. We will regulate this divergence taking the principalvalue in section 4.2. – 14 –here the determinant is in the tensor product space. Similarly one can transformthe generic n representation to the n = N/ M act aschemical potentials for the eigenvalues of M and vice versa and have to be integratedover (they do not represent external parameters as in the familiar computationsthat involve FZZT branes). The open strings are the ones stretched between thetwo sets of n and N − n D SU ( N ) → SU ( n ) × SU ( N − n ). Using a Miwa-stylerepresentation [86], the authors of [72] found that in the case of the Normal matrixmodel, these determinants/inverse determinants decrease/increase the closed stringtachyon coupling thus deforming the closed string background. Therefore there existtwo complementary ways to understand these operators (open/closed duality). Inour case let us note that similarly we can write det ( M ⊗ N − n × N − n − n × n ⊗ M ) = e [ ( N − n )Tr log M + n Tr log M − (cid:80) ∞ k =1 t k M k − (cid:80) ∞ k =1 ¯ t k M k ] (3.18)where we chose to expand each determinant factor in a different way and t k =Tr( M − k ) /k , ¯ t k = Tr( M − k ) /k . These are the closed string tachyon couplings inMiwa variables. The logarithmic terms in the exponent appear in versions of thePenner model, for more details one can consult [93]. This description makes clearthat there is a backreaction effect where M deforms the closed string background of M and vice versa.One might furthermore try to use grassmannian/fermionic variables to exponentiatethese factors [98, 99]. In particular we getdet ( M ⊗ N − n × N − n − n × n ⊗ M ) = (cid:90) dχ † dχe χ † ( M ⊗ N − n × N − n − n × n ⊗ M ) χ , (3.19)with χ αj , χ † αj fermions transforming in the bifundamental representation of SU ( n ) × SU ( N − n ) that exist only at the orbifold endpoints. One could also endow thesefermions with a kinetic term (dynamic-loops on the worldsheet) as in [61], that wouldcorrespond to the T-dual picture (Neumann conditions in Euclidean time for openstrings). This construction also indicates that determinants correspond to fermionicopen strings streched between the branes, while inverse determinants to bosonic openstrings [72, 98, 99]. It would be very interesting to study further our model from thispoint of view and connect it with various ideas related to FZZT branes in the existingliterature and possibly understand non-perturbative effects as well. n = 0 case This is the simplest case, where the zero modes of the gauge field vanish. Theline segment partition function for n = 0 has a structure similar to the two-matrix Integrating-in fundamental fermions had been already used in the context of c = 1 open string theoryin [61]. – 15 –odel [79, 80]. Z ,N = (cid:90) D M D M (cid:48) (cid:104) M (cid:48) , β | M, (cid:105) = C N (cid:90) N (cid:89) i =1 dλ i dλ (cid:48) i ∆( λ (cid:48) ) det i,j K E ( λ i ; λ (cid:48) j )∆( λ ) , (3.20)with C N a constant. One can also compute the canonical partition in this case usingthe methods in the appendix of [79] or by direct Gaussian integration to find Z ,N = (cid:18) πω sinh ωβ (cid:19) N / . (3.21)Defining the partition function of a single harmonic oscillator with open boundaryconditions [126] Z op = (cid:90) ∞−∞ dxdx (cid:48) (cid:104) x | x (cid:48) (cid:105) = (cid:18) πω sinh ωβ (cid:19) , (3.22)one finds that the n = 0 partition function is just N copies of the single particle one Z ,N = ( Z op ) N . (3.23)For large β = β c / F ,N = 12 β c E + N C + O ( e − ωβ c ) , (3.24)with the β independent term depending on the normalization of the partition func-tion. We evaluate this β -independent term from the canonical partition function inan unambiguous manner in section 5 and try to directly perform the double scalinglimit. n = N/ Case As we discussed, the regular n = N/ (cid:81) i 2, one can pass to thegrand canonical ensemble using the pfaffian formula 4.4. We also present the n = 0case with an alternate method in Appendix D. Combining equation 3.29 with 4.4,the result for the regular representation can be written in a nice operator form as Z G = (cid:113) det( I + e βµ (cid:98) ρ ) , (4.8)with (cid:98) ρ = (cid:32) ˆ O e − β ˆ H ˆ O e − β ˆ H − ˆ O e − β ˆ H ˆ O e − β ˆ H ˆ O− e − β ˆ H ˆ O e − β ˆ H e − β ˆ H ˆ O e − β ˆ H ˆ O (cid:33) , (4.9)where we defined the bi-local operator (cid:104) x | ˆ O| y (cid:105) = π ( x − y ) that acts at the orbifoldend-points and ˆ H the usual harmonic oscillator hamiltonian. The evolution is for β = β c / 2. If we furthermore use the Mehler formula, equation (3.5), we find thatthis operator acts on the harmonic oscillator wavefunctions (Hermite functions) atthe segment endpoints as (cid:104) x | ˆ O| ψ n (cid:105) = 1 π (cid:90) γ dy ψ n ( y ) x − y . (4.10) One should remember to set ω = i in case of the inverted harmonic oscillator potential. – 23 –ow it is important to properly discuss the contour of integration γ since the inte-grand is singular when x = y . This is related also to the problem of the singularnature of the integrals we’ve encountered so far when two eigenvalues of M , coalesce.To avoid the singularity one can adopt an i(cid:15) prescription to go around the singularityeither on the positive or negative imaginary plane, and using the Sokhotski-Plemeljtheorem 1 x − y ± i(cid:15) = ∓ iπδ ( x − y ) + P x − y , (4.11)one learns that these two independent possibilities are either to encircle the singularityand pick a delta function or to adopt the principal value prescription. It is easy to seethat the first prescription of the delta function trivialises the action of ˆ O and one justfinds eigenfunctions of the matrix kernel as the vectors v T = ( e − β ˆ H ψ n , ψ n ) and theeigenvalues as λ n = q n + , q = e − ωβ c . The free energy in this case would then just beone for the circle divided by two (due to the pfaffian/square-root of the determinant).This makes clear that the prescription that contains the non-trivial twisted statecontribution should be the other one, namely the principal value prescription. Inaddition, this prescription is consistent with the fact that the original integral is for y ∈ ( −∞ , ∞ ) and the principal value is the natural regulating prescription for thesingular kernel 1 / ( x − y ) in this range. One can therefore understand the operator ˆ O acting as a Hilbert transform to the Harmonic oscillator wavefunctions (see appendixE for the properties of Hilbert transform.) (cid:104) x | ˆ O| ψ n (cid:105) = (cid:104) x | ψ H n (cid:105) = 1 π P (cid:90) ∞−∞ dy ψ n ( y ) x − y . (4.12)One can also notice that the kernel ˆ ρ can be written as the square of a more elementarykernel ˆ ρ = ˆ¯ ρ with (cid:104) x | ˆ¯ ρ | y (cid:105) = 1 √ ∞ (cid:88) n =0 q ( n + ) (cid:32) − ψ H n ( x ) ψ n ( y ) ψ H n ( x ) ψ H n ( y ) ψ n ( x ) ψ n ( y ) − ψ n ( x ) ψ H n ( y ) (cid:33) . (4.13)One can easily extend these definitions, using from the start parabolic cylinder func-tions which are the eigenfunctions of the inverse harmonic oscillator and the appro-priate Mehler resolution of the propagator, see appendix B. It is also possible then towick rotate β = iT to discuss the real-time propagator as well. It is also importantto note that one can also write the orbifold kernel ˆ ρ in the energy basis in terms ofhypergeometric functions, see appendix F.1.Finally, as an interesting result coming from eqn. 4.9, one can compute the traceof the kernel, if one resolves the operator ˆ O in momentum basis as (cid:104) p | ˆ O| p (cid:105) = Note the similarity with kernels arising in the study of Riemann-Hilbert problems [88–90]. – 24 – i sgn p δ ( p − p ) (see appendix E). Expressing the oscillator propagator in momen-tum basis one computes ( ˜ β = ωβ c ).Trˆ ρ = 12 π sinh( ˜ β/ 2) tan − β/ . (4.14)This is an interesting expression from which we will manage to extract the one-particledensity of states -see section 4.2.3 - and match it with the analogous expression arisingfrom the representation of the kernel in terms of angles that we now turn to. One can find an alternative representation of the kernel in terms of angles usingequation (3.41). To pass to the grand canonical ensemble in this case we used refer-ence [111] that treats the same structure as we have in terms of angles. The kernelin this description acts to functions X ( θ ) asˆ ρ (cid:34)(cid:32) X X (cid:33)(cid:35) ( θ ) = (cid:90) π dµ ( θ (cid:48) ) ρ ( θ, θ (cid:48) ) (cid:32) X ( θ (cid:48) ) X ( θ (cid:48) ) (cid:33) , (4.15)with the matrix ρ ( θ, θ (cid:48) ) = (cid:32) ρ ( θ, θ (cid:48) ) ρ ( θ, θ (cid:48) ) ρ ( θ, θ (cid:48) ) ρ ( θ, θ (cid:48) ) (cid:33) = (cid:32) ρ ( θ, θ (cid:48) ) ρ ( θ, − θ (cid:48) ) − ρ ( − θ, θ (cid:48) ) − ρ ( − θ, − θ (cid:48) ) (cid:33) ρ ( θ, θ (cid:48) ) = 1 q − / e iθ − q / e iθ (cid:48) + 1 q / e − iθ − q − / e − iθ (cid:48) , (4.16)and the measure dµ ( θ (cid:48) ) = dθ (cid:48) πi q (cid:112) (1 − qe iθ (cid:48) )(1 − qe − iθ (cid:48) ) , (4.17)that contains two branch-cuts in the complex z (cid:48) = e iθ (cid:48) plane, emanating from fourpoints z (cid:48) = ± q , ± q − . For more details see figs. 2, 3. The relevant Riemann surfacecan be understood by gluing two spheres along two branch-cuts, the resulting surfacebeing a torus. In the next section, we see that this kernel simplifies greatly usingJacobi’s elliptic functions. We find that the simplest representation of the kernel follows by going to the dou-ble cover and using the doubly periodic elliptic functions. Similar transformationsand kernels can be found in studies of Ising, Ashkin-Teller and other models of sta-tistical mechanics [127–129]. For more details on elliptic functions the reader canconsult [116]. In particular we define z = e iθ = q sn( u, q ) with sn u Jacobi’s ellipticsine. Note that q ≡ k = e − ωβ plays the role of the so-called modulus. With thissubstitution we find (cid:90) π dθµ ( θ ) → − q (cid:90) − K + iK (cid:48) / K + iK (cid:48) / du πi , (4.18)– 25 – HP LHP UHP LHP LHP UHP -1 LHP -1 UHP -1 Figure 2 . The geometry in the complex z plane is of a two-sheeted Riemann surface. The ellipticsubstitution makes clear that this surface is a torus. K K+iK'/2K+iK'-K IIIIIIIV IIV IIIII ΑΒΓ Δ Ε ΖΑΒΓΔ Ε Ζ k-k -k -1/2 k -1/2 Η ΘΗ Θ z= snu k Figure 3 . The mapping of the rectangle to the upper-half plane via z = k sn( u, k ), with a matchingof corresponding points. The branch cuts are between H ∆ and Z Θ. Both pictures correspond tothe U HP quadrant of 2. which is a great simplification for the measure. To find the new range of integrationone can follow picture 3. The eigenvalue equation for the spectrum of the kernel cannow be written as follows λ (cid:18) X ( u ) X ( u ) (cid:19) = − q (cid:90) − K + iK (cid:48) / K + iK (cid:48) / dv πi (cid:18) ρ ( u, v ) ρ ( u, v + iK (cid:48) ) − ρ ( u + iK (cid:48) , v ) − ρ ( u + iK (cid:48) , v + iK (cid:48) ) (cid:19) (cid:18) X ( v ) X ( v ) . (cid:19) (4.19) One notices a consistency condition X ( u ) + X ( u − iK (cid:48) ) = 0 arising from the matrixequation. We conclude that one need not study a full matrix problem, since the– 26 –igenvalue equation reduces to λX ( u ) = − q (cid:90) − K + iK (cid:48) / K + iK (cid:48) / dv πi ρ ( u, v ) X ( v ) − q (cid:90) K − iK (cid:48) / − K − iK (cid:48) / dv πi ρ ( u, v ) X ( v )= − q (cid:90) C + C dv πi ρ ( u, v ) X ( v ) , (4.20)with ρ ( u, v ) = 1 − q sn u sn v sn u − q sn v . (4.21)Let us also note that the Jacobi’s sine and thus the kernel, are doubly periodic withperiods 4 K , 2 iK (cid:48) i.e. sn( u + 4 K + 2 iK (cid:48) , k ) = sn( u, k ).It is interesting to note that had we instead used the closed contour C + C + C + C (see fig. 4), we could have then solved the integral equation by picking the poles ofthe kernel at sn u = q sn v ∗ , finding λX ( u ) = q − cn u cn v ∗ X ( v ∗ ) , (4.22)which is solved by X ( u ) = cn u sn m u, m ≥ q − − m (if we demandeigenfunctions that are analytic in the interior of the strip of integration/ interior ofunit circle), or by X ( u ) = cn u sn − m u, m ≥ q − + m (if we demandeigenfunctions that are analytic in the exterior of the unit circle/strip of integration).This is analogous to the discussion in section 4.2, where we find an alternative contourthat also gives half the free energy on the circle.Comparing the integral equation with the contour C + C comprising of two hori-zontal pieces with the one defined via the closed contour C + C + C + C , we findthat we need some extra monodromy data around the torus to relate them. This isalso to be expected since, the orbifold we consider is more than half of the circle be-cause of the contributions from the twisted states localized at the fixed points of theorbifold. What we have shown above then means that the information about thesetwisted states should be contained in the contours C + C . This contribution can bedetermined either from the contour integrals around the branch cuts or equivalentlyfrom monodromy data around the fundamental cycles of the corresponding torus.We were not able to solve the integral equation including the contribution fromthe branch-cuts. Therefore we do not have the full-spectrum of the theory in the n = N/ UHPLHPC CC CCC C UHP zu LHP Figure 4 . The original contour C + C is drawn by red lines in the u and z plane. In addition wedraw also the two extra segments C + C with which the contour can close. In order to relate theclosed with the open contour, one needs to know either the contribution around the torus, or thedifference of the integral above and below the branch cut. A consistency check that can be performed in all the different descriptions we havefor the kernel is to compute its trace. From 4.16 and 4.17 we can compute ( ˜ β = ωβ c )Tr ˆ ρ = 1 √ β/ (cid:90) π dθ π sin θ (cid:113) cosh ˜ β − cos(2 θ ) = 12 π sinh( ˜ β/ 2) tan − β/ . (4.23)This equation matches perfectly with eqn. 4.14, derived from the alternative represen-tation of the kernel and thus provides a good consistency check of the two approaches.This equation is to be contrasted with the one-particle oscillator partition functionon the circle Z pH.O. = 12 sinh ˜ β/ . (4.24)By rotating ω → − iω , one then finds the inverse oscillator result for the orbifoldTr ˆ ρ inv = − π sin( ωβ c / 2) tanh − (cid:18) ωβ c / (cid:19) = (cid:90) π dθ π cos( θ/ ωβ c ) − cos( θ ) (4.25)for more details see appendix F.3. This expression has poles as the usual circle par-tition function for one particle at ωβ c = 2 nπ , n ∈ Z and also branch cuts emanatingfrom ωβ c = 2 π ( m + 1), m ∈ Z due to the two logarithms from the inverse hyperbolictangent. One could try to derive the density of states of this partition function usingthe definition via the Laplace transform ρ d ( (cid:15) ) = (cid:90) c + i ∞ c − i ∞ dβ πi Z ( β ) e β(cid:15) , (4.26)– 28 –ut the branch-cuts pose some difficulty. In particular adding the piece at infinity,the contour will enclose all the poles for n ≤ 0, which gives the same density of statesas in the case of the inverted H.O. but in addition one picks contributions from all thebranch cuts for m ≤ − 1. To simplify things, we rewrote this expression as an integralwith the integrand having simple poles. Exchanging the integrals, one can formallyderive a single particle density of states, ρ po ( (cid:15) ) = ρ H.O. ( (cid:15) ) + ρ twisted ( (cid:15) ) + ρ Im ( (cid:15) ), withthe “twisted” piece ρ twisted ( (cid:15) ) = 14 π sinh( (cid:15)ω π ) (cid:20) Im Ψ (cid:18) i (cid:15) ω + 14 ) (cid:19) − Im Ψ (cid:18) i (cid:15) ω + 34 ) (cid:19)(cid:21) = 14 π sinh( (cid:15)ω π ) Im (cid:90) ∞ dt e − i (cid:15)ω t cosh( t ) , (4.27)with Ψ( z ) the digamma function. For more details of this derivation one can seeappendix F.3. Finally, the density of states contains an extra imaginary piece ρ Im ( (cid:15) )(see F.3), that might have some interesting interpretation in terms of decaying states,since the decay/tunneling rate of a metastable physical system is related with theimaginary part of the free energy Γ ∼ Im F [141]. The contribution to the partition function from the twisted states can be isolatedby considering the limit β → ∞ . This limit reduces the free energy to the groundstate contribution as F = β c E ground / β -independent constant piece Θin this expression is the twisted state contribution to the ground state energy of theorbifold. n One can obtain a closed form expression for this constant piece in the generic n representation of the orbifold, in the formulation in terms of the eigenvalues of M asfollows:Θ = 2 log (cid:90) d N x det ≤ i ≤ N ≤ k ≤ n ≤ p ≤ N − n (cid:20)(cid:90) dy ψ i − ( y ) x k − y (cid:90) dyy N − n − p ψ i − ( y ) ψ i − ( x k ) (cid:21) (5.1)In this expression the determinant is of an N × N matrix with rows labelled bythe index i and the columns separated into three pieces whose size is governed bythe range of k and p . Derivation of this expression can be found in appendix G.2.Another expression in the second formulation in terms of eigenvalues of A is presentedin Appendix G.1. We are unable to obtain the analogous expressions after takingthe double scaling (large N) limit however. The latter is necessary to make directconnection to the Liouville theory. In principle, one should be able to express Θ in the– 29 –rand canonical ensemble. For example it may be possible to obtain it directly usingthe twisted contribution to the one-particle density of states in equation (4.27). In theprevious section we also identified the origin of the twisted state contribution throughthe contour around the branch cuts in figure 4. However, none of these alternativeformulations has practically helped obtaining the final expression in terms of Liouvilletheory quantities. Instead, we perform the calculation for specific values of n below. n = 0Starting from the canonical ensemble, we have managed to treat the n = 0 repre-sentation in terms of even/odd parabolic cylinder functions and write the twistedstate contribution in terms of the chemical potential µ . The relevant calculations arepresented with detail in appendix G.3. This result is found to beΘ = 12 (cid:90) µ ρ H.O. ( (cid:15) ) (cid:90) µ ρ H.O. ( (cid:15) (cid:48) ) log | (cid:15) − (cid:15) (cid:48) | d(cid:15)d(cid:15) (cid:48) . (5.2)One can then use the asymptotic form ρ H.O. ( (cid:15) ) = π (cid:0) − log (cid:15) + (cid:80) ∞ m =1 C m (cid:15) − m (cid:1) of thedensity of states to derive the asymptotic genus expansion, which we now describe.In particular one defines the cosmological constant ∆ = π ( κ c − κ ) that is related tothe renormalised string coupling µ as ∆ = − µ log µ in the limit κ → κ c . One thenfills up states up to the chemical potential µ . The relevant equations are N = 1 (cid:126) (cid:90) µ d(cid:15)ρ H.O. ( (cid:15) ) , ∂ ∆ ∂µ = πρ ( µ ) . (5.3)One can invert the second equation above, to find µ ( µ ) in an asymptotic expansionwhose first term is µ = µ , see [58]. After that we can use an asymptotic expansion ofthe twisted states Θ( µ ) and turn it into an asymptotic expansion in the renormalisedstring coupling µ .Θ = µ (cid:18) − π 24 + (cid:18) π − (cid:19) log µ + 74 log µ − 12 log µ (cid:19) − (cid:18) π (cid:19) log µ + 1 µ (cid:18) (cid:18) π − (cid:19) log µ (cid:19) O ( µ − ) , (5.4)with µ the renormalised string coupling. One notices that the torus contribution isnot the same as in equation 2.7. n = N/ n = N/ O ij , O ij = 2 (cid:90) ∞−∞ dxdy ψ + ( (cid:15) i , x ) ψ − ( (cid:15) j , y ) x − y . (5.5)– 30 –e have calculated this expression using both the Hermite functions and the delta-function normalised even and odd parabolic cylinder functions which are the eigen-functions of the inverted oscillator, see appendix B. Details of the calculation of(5.5) are presented in appendices G.4 and G.5 respectively. For the normal oscillatorone finds that the result can be expressed in terms of a determinant of sine-kernel ,see G.34. For the inverted oscillator the result is in terms of continuous labels O ( (cid:15) , (cid:15) ) = 1 π | Γ(1 / i(cid:15) / / i(cid:15) / | (cid:15) π (3 (cid:15) + (cid:15) ) / sinh (cid:0) π ( (cid:15) − (cid:15) ) (cid:1) (cid:15) − (cid:15) . (5.6)Substituting this in (5.5), we see that the determinant becomes the product of di-agonal pieces times the determinant of the sinh( (cid:15) − (cid:15) ) / ( (cid:15) − (cid:15) ). It is easy to seethat the diagonal pieces do not contribute to the 1 /µ expansion in the double scalinglimit, only giving contributions to non-perturbative terms in µ . Therefore the twistedstate contribution to the perturbative expansion in g s = 1 /µ is determined by thekernel of the operator sinh( (cid:15) − (cid:15) ) / ( (cid:15) − (cid:15) ) in the double-scaling limit. We shouldalso remember to solve µ ( µ ) to derive the correct asymptotic expansion.It is, as far as we know, not possible to calculate the spectra of this kernel withthe currently available methods. However the determinant of sine kernel where onereplaces sinh( (cid:15) − (cid:15) ) with sin( (cid:15) − (cid:15) ) is possible to be calculated in an asymptoticfashion as was done in the 70s [104–106]. Luckily, we can make the replacement (cid:15) → i(cid:15) in (5.6), hence transform the sinh(∆ (cid:15) ) / ∆ (cid:15) kernel into sin(∆ (cid:15) ) / ∆ (cid:15) kernel byconsidering the following, alternative calculation. The canonical partition function(3.11) with the propagator (3.5) is invariant under ω → iω , β → − iβ .This is because one can Wick rotate the integrals over the matrix eigenvalues as x i → e − iπ/ x i , y i → e − iπ/ y i in the partition function. To see this consider theintegral along the contour C = ( −∞ , → ∞ ) ∪ ( ∞ , ∞ e − iπ/ ) ∪ ( ∞ e − iπ/ , −∞ e − iπ/ ) ∪ ( −∞ e − iπ/ , −∞ ) where the second and the last pieces are on the indicated arcs atinfinity. One can see that there are no poles inside this contour C as follows. Theonly possible poles could arise from the denominator in the initial and final wavefunctions in (3.14). However these poles can easily be avoided by rotating x s and x (cid:48) s (and similarly y s and y (cid:48) s) in pairs. Also, there are no possible divergences atthe arcs at infinity, | x | = ∞ , in the n = N/ . Finally, oneshows that possible divergence that could arise from the det K (¯ x, ¯ x (cid:48) ) in (3.11) on theinfinite arcs in contour C are also absent because one can expanddet K = (cid:88) r ¯Ψ r (¯ x )Ψ r (¯ x (cid:48) ) e iβE r (5.7)where the N-fermion wave functions Ψ r are constructed out of products of the theparabolic cylinder wave-functions, and the latter are convergent on the particularinfinite arcs ( ∞ , ∞ e − iπ/ ) and ( −∞ e − iπ/ , −∞ ), as can be seen from appendix B. Note that this part of the argument would fail for the partition functions with n < N/ – 31 –e conclude that the integral on the contour C vanishes, thus one can Wick-rotate x i → e − iπ/ x i , y i → e − iπ/ y i in the partition function giving rise to the symmetryunder ω → iω , β → − iβ .Thus, one could calculate the twisted state contribution in the Lorentzian path inte-gral instead of the Euclidean partition function. The only difference that this makesfor the twisted state contribution coming from (5.6) is to replace the energies (cid:15) → i(cid:15) ,thus transforming the sinh into the sine kernel. This is also the result for the normaloscillator.Therefore the result, remarkably, boils down to the computation of a Fredholm-determinant of sine-kernel which is a well known object in random matrix theory [103]that corresponds to the probability that all the energy eigenvalues are outside theenergy range ( − µ, 0) and thus form the fermi sea. This object has been computedwith various approaches such as inverse scattering, toeplitz determinants and theRiemann-Hilbert method. Some basic references are [104–106]. This calculation isreviewed in appendix G.5.3 and results inΘ = 14 log E (0; (0 , µ )) = − µ − 116 log µ + 148 log 2+ 34 ζ (cid:48) ( − O (cid:18) µ m (cid:19) . (5.8)We observe that the twisted state contribution to the torus level partition function − log µ matches precisely the world-sheet result (2.7). This provides a non-trivialcheck of the duality we propose between the n = N/ S / Z . In this paper we considered the quantum mechanics of an N × N dimensional Her-mitean matrix M compactified on Euclidean time τ and orbifolded by a Z actionthat contains the reflection τ → − τ , which we also embedded into the gauge group.We provided evidence that this MQM on the S / Z orbifold in the large-N limitconstitutes a good toy model for a Bang-Crunch universe in the context of 2D stringtheory. This is because the orbifold MQM admits a natural analytic continuationinto Lorentzian time as shown in equation (3.13) and in the double scaling limit thetheory becomes dual to 2D string theory with space-like singularities at Lorentziantime t = 0 and t = T . The space-like dimension of this 2D string theory is given bythe Liouville direction that is made out of the eigenvalues of M in the dual MQMdescription. One may be ask how come the twisted state contributions in the Euclidean and Lorentzian pathintegrals give rise to different expressions. After all twisted states that are localized on the fixed points arenot supposed to see the signature of time. This should be true at the non-perturbative level. Asymptoticexpansions can differ, which is the well known Stokes phenomenon. – 32 – artition function - The information that one can practically extract from theLiouville description of this theory is rather limited at the moment. In particularwe managed to compute the torus contribution to the partition function includingthe contribution of the twisted states by indirect consistency methods as shown insection 2.1. On the other hand, we believe that the description of the theory in termsof MQM provides an alternative, richer point of view.As a first step, we focused on calculating the partition function of the orbifoldedMQM. We found that the orbifolding operation in the MQM description can begiven in terms of different representations labeled by a parameter 0 ≤ n ≤ N/ Z into the SU ( N ) gauge group. We argued why the “regular” representation with n = N/ n , N − n D λ i of the matrix M . The final expression for an arbitrary representation n is given inequation (3.11). This representation is useful since as we show in equation (3.13) theintegrand can be naturally decomposed into a piece localized at τ = 0, a transitionamplitude from τ = 0 to τ = β and a piece localized at τ = β . This form of the parti-tion function therefore admits a natural rotation into Lorenzian time where the firstand the last pieces are naturally identified with the initial and final wave-functionsof the toy cosmological universe, and the middle piece with the transition amplitudefrom the big-bang to the big-crunch. These wave functions depend on the orbifoldindex n , hence in some sense provide us with a classification of possible bang/crunchuniverses in this toy model and hence it is crucial to understand the role of n fromthe string theory side as well. We also note a similarity of our wavefunctions withthe ones arising in the work of Dijkgraaf/Vafa on “negative branes” and supermatrixmodels, see [138]. We do not develop these observations further in this paper. Oneshould be really careful about whether the Wick rotation into Lorentzian time ap-plies smoothly near the singularities/end points in time. Finally, there is always thepossibility of inserting excited states at the initial and final states of the universe.Nevertheless, this description suggests an intriguing general qualitative prescriptionfor how to make sense of quantum gravity in a bang/crunch cosmology: express thetheory in terms of a dual open-string description, evaluate the orbifold partition func-tion in Euclidean time to obtain a decomposition into pieces that contain the initialstate, transition and the final state, and finally Wick rotate into Lorentzian time.The second formulation of the partition function involves first integrating over thematrix M and expressing the result in terms of the eigenvalues of the gauge field A . This method gives an alternative form for the partition function in terms of– 33 –ilson lines, the zero modes of the gauge field. The final expression for an arbitraryrepresentation n is given in equation (3.35). This formulation clarifies the meaning ofthe index n : as shown in (3.40), in the T-dual picture, n corresponds to the numberof free D-instantons -free to move along the time direction. There also exists N − n fractional D-instantons stuck at the fixed points of the orbifold. Thus there are nofractional D-instantons in the regular representation with n = N/ n = 0 representation.The n = N/ (cid:80) k t k (cid:0) Tr U k + Tr U − k (cid:1) with t k = q k / k . Similar deformations are en-countered also in versions of the GWW model [81–83] which has a third order phasetransition, as well as in the proposed matrix model description of the SL (2 , R ) /U (1)2D black hole [66]. A possible issue with that proposal is that it is based on the FZZcorrespondence with the Sine-Liouville which holds for the radius R = close to theblack hole-string correspondence point [137]. In contrast, in our case these deforma-tions include all windings and are temperature or radius dependent, which is a quiteinteresting novel characteristic. In addition it is expected that the higher-windingswe find are related to higher spin generalisations of the 2D black hole [134], wherediscrete states are liberated as well [135]. These discrete states are remnants of thehigher-spin excitations that exist in higher dimensions [62, 71] and it is not unnaturalto expect their presence due to the orbifolding and breaking of the gauge group thatliberates SU ( N ) non-singlet states near the end of time. The closed string twistedstates should then be thought of as a condensate of both the tachyon and those ex-tra states. The possible presence of these states due to the temperature dependenthigher winding perturbations can thus lead to quite interesting and rich physics oncewe manage to compute the partition function or other observables for finite orbifoldsize R .The two formulations should of course be equivalent. Even though we have notmanaged to find a direct change of variables that would relate the two in the canonicalensemble, the equivalence can be partially demonstrated at the level of the grandcanonical ensemble. Indeed, in both formulations it is possible to go to the grandcanonical ensemble and express it in terms of a square root of a Fredholm determinantof a one-particle kernel ˆ ρ . The spectrum of this Kernel then determines the full non-perturbative answer. We checked the equivalence of the two formulations by explicitlymatching the trace of this Kernel in the two cases, see equations (4.14) and (4.23). Twisted states - A central focus of our paper is the contribution of the twistedstates to the orbifold partition function. Since these states are localized at the fixedpoints of the orbifold that are supposed to become the cosmological singularitiesunder Wick rotation, they are expected to contain crucial information on the stringdynamics around these points. The twisted states are clearly marked in the toruspartition function of the Liouville theory. Their contribution is given by the constant– 34 – R -independent) terms in section 2.1. One can isolate this contribution in the dualMQM partition function in the first formulation (in terms of eigenvalues of M ) bytaking the large β = πR limit. This limit, essentially decouples the propagation fromthe wavefunctions/states at the endpoints in time and focuses on the ground statechannel contribution to the free energy. The radius independent piece has the form ofdeterminant operators and was denoted by Θ in section 5. We were able to explicitlyexpress and compute Θ in terms of 2D string theory parameters in the n = 0 and n = N/ n = N/ n = N/ ρ which can be determined solving an integral equation. In the firstformulation, section 4.2, the presence of extra twisted states was understood throughthe action of Hilbert transform operators at the endpoints. The large β limit, againdecouples the Hamiltonian propagation from these operators and “zooms in” at theendpoints in time.In the second formulation (in terms of eigenvalues of A ), we isolated this contributionin section 4.2.2. Here the integral is defined on a complex plane with two branch cuts,or alternatively on a two-torus. One obtains precisely half the free energy for MQMon S if one ignores the contribution to the contour of integration around thesebranch cuts, or alternatively the monodromy around the fundamental cycles of thecorresponding torus. Hence in this description the twisted states should be containedin these branch-cut or monodromy contributions. Moreover, let us note that from theMatrix model picture it is clear that these extra contributions can generically leadto both radius dependent together with radius independent terms in the free energy.We also note that in both formulations, the partition function looks very similar toa four point correlation function: in the first formulation it can be thought of as acorrelator between two bi-local operators and in the second as containing four twistoperators creating the two branch cuts. Future directions - In this paper we focused on the closed string asymptoticexpansion of the partition function. We have found that the matrix model alsocontains a wealth of non-perturbative information.It will be interesting to understand further the contribution of the fractional instan-tons present in other representations, which we expect to be non-perturbative in g st .Let us also note that the structure of the partition function in terms of Wilson lines, isvery reminiscent of τ functions of BKP/DKP Hierarchies [119, 122, 123] and it maybe very interesting to pursue this connection. For further progress in this direction,one should study free fermions and τ -functions in the presence of twist fields.– 35 –ome other interesting calculations we look forward to perform in the future includethe disk one point function and the annulus correlation function for two macroscopicloops. Such quantities will be very good probes of the singularities at the endpointsof time.Furthermore, we should develop the target space picture of our construction furtherby using the relation between the matrix eigenvalues and the Liouville coordinate φ . A description of the initial state in collective field theory variables might proveuseful here. A natural question in this context is, what is the spatial extend of the2D universe near the singularities? Is our theory describing one of the known metricsin the 2D string theory literature? The previous probes we mentioned could also helpin giving answers to these questions.As a final observation we recall that [133, 136] the horizon and the singularity ofthe 2D black hole is exchanged under T-duality and that there is a relation be-tween the 2D cosmology with the 2D black hole [132]. This can be shown at leastat the classical level, for the Lorentzian 2D black hole [130, 131] described by the SL (2 , R ) /U (1) WZW coset. It is interesting to note that the Hilbert transform op-erators at the endpoints in time commute with the SL (2 , R ) generators of linearfractional transformations and that the description of the kernel on the torus has amanifest SL (2 , Z ) symmetry. In addition, based on the fact that we have a combi-nation of radius dependent vortex perturbations together with radius independenttwisted states, it would be very interesting to investigate whether we can similarlyrelate our setup with a 2D black hole with a possible interpretation of the twistedstates as black hole microstates. To this end, it is encouraging that the contributionof the end-point wavefunctions to the canonical free energy takes the form of an en-tropy S ∼ Tr logρ twisted (or S = N log 2 for the normal oscillator), which is also thelogarithm of the probability of forming the fermi-sea from an ensemble of randomhermitean hamiltonians (taking the double scaling limit of the inverted oscillator).For all these reasons it would be extremely interesting to investigate similar S / Z orbifolds in higher dimensions . Acknowledgements This paper is a continuation of the unpublished work one of the authors (U.G.) hasdone in collaboration with Hong Liu in 2005. Therefore the basic idea and some ofthe results have been obtained in collaboration with Hong Liu and we are gratefulto his collaboration at the early stages. We also wish to thank Marcos Crichigno,Vladimir Kazakov, Alexei Morozov and Elli Pomoni for useful conversations. Finallywe happily acknowledge discussions with Johan van de Leur and Alexander Orlovin relation to integrable hierarchies and Alexander R. Its on the Riemann-Hilbertproblems. In the context of the 4D Schwarzschild black hole analogous Lorentzian Z involutions that involvetime reversal have found a recent interest in [140]. – 36 –his work is supported by the Netherlands Organisation for Scientific Research(NWO) under the VIDI grant 680-47-518, and the Delta-Institute for TheoreticalPhysics (D-ITP) that is funded by the Dutch Ministry of Education, Culture andScience (OCW). A Other classes of orbifolds Here we present the rest of the supersymmetric orbifold theories for completeness. • Orbifold II : The second class of orbifolds are obtained by modding out super-affinetheories by the same reflection symmetry as above. One has the following relations[40]: Z orbsaA,B ( R ) = 12 Z saA,B ( R ) + const A,B (A.1)and the following relation at the special radius [40]: Z orbsaA,B ( √ 2) = Z cirA,B ( √ . (A.2)The partition functions are: Z orbsaA ( R ) = 12 Z saA ( R ) − 18 ln µ , Z orbsaB ( R ) = 12 Z saB ( R ) − 116 ln µ (A.3)These theories are seperately self dual under R → /R . • Orbifold III : The third class of orbifolds are obtained by twisting the circular the-ories by (1) F s R . Note that this is only a symmetry in the 0A theory. One obtains, Z orbA ( R ) = 12 Z cirA ( R ) + const (A.4)and Z orbA (1) = Z saA (2) (A.5)The result is: Z orbA ( R ) = 12 Z cirA ( R ) − √ µ (A.6)We observe that orbA and orbB theories are exchanged under T-duality. B Oscillator wavefunctions We provide this section as a collection of the relations between various representations ofnormal/inverted harmonic oscillator wavefunctions.– 37 – .1 Normal Harmonic Oscilator We define the normal harmonic oscillator time independent Schroendinger equation (cid:126) , m =1 12 (cid:0) − ∂ x + ω x (cid:1) ψ n = (cid:15) n ψ n . (B.1)The Kronecker delta normalised eigenfunctions are ψ n ( x ) = 1 √ n n ! (cid:16) ωπ (cid:17) e − ωx H n ( √ ωx ) = 1 √ n ! (cid:16) ωπ (cid:17) D n ( √ ωx ) . (B.2)These wavefunctions satisfy Mehler’s formula for the propagator (in real time) (cid:16) ω πi sin ωT (cid:17) e iω ωT [( λ i + λ (cid:48) j ) cos ωT − λ i λ (cid:48) j ] = (cid:88) n ψ n ( λ i ) ψ n ( λ (cid:48) j ) e − iω ( n + ) T , which we analytically continued to Euclidean time via T = − iβ to obtain equation 3.5. B.2 Inverted Harmonic Oscillator We want to solve the inverse harmonic oscillator time independent Schroendinger equation.Take the normal harmonic oscillator equation and let ω → i , and x → x/ √ 2. One thenneeds to solve: (cid:18) ∂ x + x (cid:19) ψ = (cid:15)ψ . (B.3)This is a particular form of the Weber differential equation (cid:18) ∂ z + ν + 12 − z (cid:19) ψ = 0 . (B.4)The solutions are the Parabolic cylinder functions (equivalently expressed via Whittakerfunctions W) D ν ( z ) = 2 ν + z − W ν + , − ( z , D − ν − ( iz ) = 2 − ν − e iπ/ z − W − ν − , − ( − z . (B.5)where D ν ( z ) , D − ν − ( ± iz ) are linearly independent. We are in the specific case where ν = i(cid:15) − , ix = z , thus D i(cid:15) − ( e i π x ) = 2 i(cid:15) e − iπ/ x W i(cid:15) , − ( ix , D − i(cid:15) − ( e i π x ) = 2 − i(cid:15) e i π x W − i(cid:15) , − ( − i x , (B.6)are the two linearly independent solutions in our case and there is a degeneracy in thecontinuous energy spectrum. It is easy to see that they are also formally obtainable fromthe normal harmonic oscillator upon substituting x → x/ √ ω = ± i and n = ± i(cid:15) − , thenormalization is different though. – 38 –nother useful basis of solutions are the delta function normalised even/odd paraboliccylinder functions [60] which we will denote by ψ ± ( (cid:15), z ) ψ + ( (cid:15), x ) = (cid:32) π (cid:112) (1 + e π(cid:15) ) (cid:33) / | Γ(1 / i(cid:15)/ / i(cid:15)/ | e − ix / F (1 / − i(cid:15)/ , / ix / e − iπ/ π e − (cid:15)π/ | Γ(1 / i(cid:15)/ | (cid:112) | x | M i(cid:15)/ , − / ( ix / ψ − ( (cid:15), x ) = (cid:32) π (cid:112) (1 + e π(cid:15) ) (cid:33) / | Γ(3 / i(cid:15)/ / i(cid:15)/ | xe − ix / F (3 / − i(cid:15)/ , / ix / e − iπ/ π e − (cid:15)π/ | Γ(3 / i(cid:15)/ | x | x | / M i(cid:15)/ , / ( ix / . (B.7)Their normalisation is (cid:90) ∞−∞ dx (cid:88) a = ± ψ a ( (cid:15) , x ) ψ a ( (cid:15) , x ) = δ ( (cid:15) − (cid:15) ) , (B.8)and (cid:90) ∞−∞ d(cid:15) (cid:88) a = ± ψ a ( (cid:15), x ) ψ a ( (cid:15), x ) = δ ( x − x ) . (B.9)The relation with the previous basis can be established using the following equations D i(cid:15) − ( e iπ/ x ) = √ π i(cid:15)/ e − iπ/ Γ(3 / − i(cid:15)/ √ x M i(cid:15)/ , − / ( ix / − √ π i(cid:15)/ e − iπ/ Γ(1 / − i(cid:15)/ √ x M i(cid:15)/ , / ( ix / D − i(cid:15) − ( e i π/ x ) = Γ(1 / − i(cid:15) ) √ π (cid:104) e − (cid:15)π/ e iπ/ D i(cid:15) − ( e iπ/ x ) + e (cid:15)π/ e − iπ/ D i(cid:15) − ( − e iπ/ x ) (cid:105) . (B.10) B.3 Mehler for parabolic cylinder The delta-function normalised odd/even parabolic cylinder functions ψ ∓ ( (cid:15), x ) satisfy thefollowing formula [60]: (cid:104) x | e − iT H | y (cid:105) = (cid:90) ∞−∞ d(cid:15)e i(cid:15)T (cid:88) a = ± ψ a ( (cid:15), x ) ψ a ( (cid:15), y ) = 1 √ πi sinh T exp i (cid:20) x + y tanh T − xy sinh T (cid:21) , (B.11)which is the analogue of Mehler’s formula for the real-time ( T = − iβ ) inverted H.O.propagator with the Hamiltonian B.3. This holds for − π < ImT < ImT = 0 with ReT (cid:54) = 0. To prove it one can use the general expression (7.694) in [120].An equivalent expression can be found also in the basis of D i(cid:15) − ( z ) , D − i(cid:15) − ( iz ) using– 39 –7.77.3) of [120] (cid:104) x | e − iT H | y (cid:105) = 1 √ πi sinh T exp i (cid:20) x + y tanh T − xy sinh T (cid:21) = (cid:90) ∞−∞ d(cid:15)e i(cid:15)T e − (cid:15)π π cosh( (cid:15)π ) (cid:104) D i(cid:15) − ( e i π x ) D − i(cid:15) − ( e i π y ) + D i(cid:15) − ( − e i π x ) D − i(cid:15) − ( e i π y ) (cid:105) , (B.12)with the same restrictions in T. These expressions are most well suited to compute thetransition amplitude in real time. To recover the Euclidean, inverted H.O. expression oneneeds to set T = − iβ in the above, (note that they will hold for R < 1, otherwise oneneeds to change the contour of integration to make them well behaved). C Representation in terms of angles (Wilson-lines) Instead of integrating out U one can first integrate out the M ’s in the expression Z n,N = (cid:90) D M D M (cid:48) D U (cid:104) U M (cid:48) U † , β | M, (cid:105) = (cid:90) D U I ( U ) . (C.1)If we define A = 1 / tanh( ωβ ) , B = 1 / sinh( ωβ ), and remember to use blocks for thematrices after orbifolding we get I ( U ) = ω − ( N − n ) (cid:0) B π (cid:1) N / (cid:82) dM dM dM (cid:48) dM (cid:48) e T , U = (cid:32) U U U U (cid:33) ,K = − A tr( M + M (cid:48) ) + B tr( M U M (cid:48) U † + M U M (cid:48) U † ) + (1 ↔ . (C.2)Now the U (cid:48) s are complex but satisfy certain conditions U U † + U U † = U U † + U U † = 1 , U U † = U U † (C.3) U U † + U U † = U † U + U † U = 0 , (C.4)and can be diagonalised by bi-unitary transformations such that they leave the measureinvariant. We thus use the unitary matrices V , V (cid:48) , V , V (cid:48) to get U = V CV (cid:48)† , U = V (cid:32) C 00 1 (cid:33) V (cid:48)† , U = − V ( D, V (cid:48)† , U = V (cid:32) D (cid:33) V † , (C.5)with C ij = cos θ i δ ij , D ij = sin θ i δ ij , ≤ θ i ≤ π . (C.6)This can be also easily achieved after exponentiation of the zero mode of A that has onlynon-zero the diagonal components of the off-diagonal blocks. Note that any complex matrix can diagonalized by bi-unitary transformations. Also the first line ofequation C.3 implies that U U † and U U † can be simultaneously diagonalized. – 40 –ince the measure of M’s is invariant under a unitary transformation, we can write the fourmatrix coupling term of K astr( M CM (cid:48) C + M DRD + RDM (cid:48) D + RCR (cid:48) C + SS (cid:48)† C + S † CS (cid:48) + T T (cid:48) ) (C.7)where we have written M and M (cid:48) (which are ( N − n ) × ( N − n ) matrices) as M = (cid:32) R SS † T (cid:33) , M (cid:48) = (cid:32) R (cid:48) S (cid:48) S (cid:48)† T (cid:48) (cid:33) (C.8)with R, R (cid:48) n × n matrices. Integration over T, T (cid:48) will yield a constant factor(2 π ) ( N − n ) (C.9)Integrations over S, S (cid:48) yield(2 π ) n ( N − n ) n (cid:89) i (cid:18) B B sin θ i (cid:19) N − n (C.10)and integrations over R, M give(2 π ) n (cid:89) i,j (cid:18) B sin ( θ i + θ j )(1 + B sin ( θ i − θ j ) (cid:19) (C.11)Thus altogether we get I = (cid:18) πBω (cid:19) ( N − n )22 n (cid:89) i (cid:20) B B sin θ i (cid:21) N − n n (cid:89) i,j (cid:20) B (1 + B sin ( θ i + θ j )(1 + B sin ( θ i − θ j ) (cid:21) (C.12)It is also useful to massage this expression into I = (cid:20) πBω (cid:21) ( N − n )22 n (cid:89) i (cid:20) β − cos θ i (cid:21) N − n (cid:89) i,j (cid:34) β − cos( θ i + θ j )(cosh ˜ β − cos( θ i − θ j ) (cid:35) (C.13)where now the angles run 0 ≤ θ k ≤ π and ˜ β = 2 ωβ = ωβ c .One can also express the part of the integrand of the canonical partition function that isnot coming from the measure as the determinant of a differential operator Q , I = (cid:18) πω (cid:19) ( N − n ) (det Q ) − , (C.14)where Q is a differential operator on a circle of length 2 β . Q = − D + ω = − ∂ + 2 iα∂ + α + ω , (C.15)where α is a constant gauge field in the adjoint representation related to θ as θ i = α i β . Q acts on the matrices M as [ Q, M ] = ∂ M + i [ α, M ] (C.16)– 41 –nd [ α, M ] ij = α adjij,kl M kl = α ik M kj − M ik α kj (C.17)( U M U † ) ij = exp[ iβα ] adjij,kl M kl , (C.18)with α adjij,kl = α ik δ jl − α lj δ ik , exp[ iβα ] adjij,kl = U ik U † lj . (C.19)Thus in the momentum representation one can writedet (cid:0) − D + ω (cid:1) = det matrix ∞ (cid:89) n = −∞ (cid:34)(cid:18) πnβ + α (cid:19) + ω (cid:35) = (C.20)= det matrix (cosh( βω ) − cos( βα )) , (C.21)where α is a matrix and the determinant is with respect of this matrix structure. Ifthe gauge field is A N × N = diag ( α , α , ..., α n , − α , − α , ..., − α n , , ..., α adjij,kl =( α i − α j ) δ ik δ jl and C.14 equals C.13. C.1 Measure One needs also to compute the measure for D U . This is achieved by defining the metric onthe tangent space of the group ds = tr( U dU † U dU † ) and then computing its determinantto get (0 ≤ θ i ≤ π ) J n ( θ ) = 12 n n !(2 π ) n n (cid:89) i In the case of n = N/ Z n = (cid:90) π (cid:89) i dθ i J n ( θ ) n (cid:89) i,j (cid:32) β − cos( θ i + θ j )(cosh ˜ β − cos( θ i − θ j ) (cid:33) (C.23)where the angles are in 0 ≤ θ i ≤ π .One then unfolds the denominator using for example1cosh ˜ β − cos( θ i + θ j ) = 2 q (1 − qz i z j )(1 − qz ∗ i z ∗ j ) , q = e − ˜ β , z i = e iθ i (C.24)– 42 –nd similarly the measure J n = 1 i n n n !(2 π ) n n (cid:89) i In this Appendix we collect some of the properties of the Hilbert transform which can befound in [121].The Hilbert transform on the real line x ∈ R of a function f ( x ) is defined as H [ f ]( x ) = 1 π P (cid:90) ∞−∞ f ( y ) dyx − y (E.1)with P denoting the principal value. Some properties of the transform are • The Hilbert transform commutes with complex conjugation ( H [ f ]) ∗ = H [ f ∗ ] • It satisfies linearity H [ af + bf ] = a H [ f ] + b H [ f ] • The linearity of the Hilbert transform also means that if one has a series expansionof a function f = (cid:80) k f k then H [ f ] = (cid:80) k H [ f ] k . • It has the parity property of exchanging even with odd functions • The Hilbert transform relates the real and imaginary part of a function (Kramers-Kronig relations). As an example if f ( z ) = g + ih is analytic in the upper half complexplane then h ( x ) = −H [ g ]( x ) and thus (cid:82) ∞−∞ g H [ g ] dx = 0. Moreover H [ g ]( x ) = h [ x ]. • The combination with fourier transform F gives F ◦ H [ f ]( x ) = − i sgn( x ) F [ f ]( x ) • H = − I and thus the inverse is H − = −H . The eigenvalues of the Hilbert transformare λ = ± i . • The Hilbert transform is skew adjoint H † = −H• If g ( x ) = H [ f ]( x ) then H [ f ]( ax + b ) = sgn( a ) g ( ax + b ). Generically the Hilberttransform commutes with translation and positive dilations but anticommutes withreflection. • The Hilbert transform commutes with the derivative operator– 44 – The Hilbert transform commutes with SL (2 , R ) generators i.e with unitary operators U g on the space L ( R ) acting as U − g f ( x ) = ( cx + d ) − f (cid:18) ax + bcx + d (cid:19) , (cid:40) g = (cid:32) a bc d (cid:33) : a, b, c, d ∈ R , ad − bc = 1 (cid:41) . (E.2)Moreover the following properties hold:For an integer n ≥ g ( x ) = H [ f ]( x ) H [ x n f ( x )] = x n g ( x ) − π n − (cid:88) k =0 x k (cid:90) ∞−∞ t n − − k f ( t ) dt (E.3)Hardy: (cid:90) ∞−∞ dx H [ f ]( x ) g ( x ) = − (cid:90) ∞−∞ dxf ( x ) H [ g ]( x ) (E.4)Hardy-Poincare-Bertrand:1 π P (cid:90) ∞−∞ f ( x ) dxx − t π P (cid:90) ∞−∞ g ( y ) dyy − x = 1 π P (cid:90) ∞−∞ g ( y ) dy π P (cid:90) ∞−∞ f ( x ) dx ( x − t )( y − x ) − f ( t ) g ( t )(E.5)One can define projection operators as follows: P ± = 12 ( I ± i H ) (E.6)Then one can easily see that they satisfy the properties of projection operators (idempotentconditions) P ± = P ± . F The Kernel F.1 Kernel in Energy basis One can write down the form of the kernel in energy eigen-states and try to diagonalisefrom there. One has (after symmetrising appropriately): (cid:104) m | e − β ˆ H ˆ O e − β ˆ H | n (cid:105) = 2 m + n √ m ! n ! e − ωβ ( m + n +1) √ πn − m (cid:34) − m/ − n +12 ) + 1Γ( − n/ − m +12 ) (cid:35) . (F.1)To prove this formula one first has to compute (cid:104) m | ˆ O| n (cid:105) and it is easier to do so in momen-tum basis where the Hilbert transform just becomes a signum function, see appendix E (cid:104) m | ˆ O| n (cid:105) = − i (cid:90) ∞−∞ dp sgn( p ) ψ m ( p ) ψ n ( p ) (F.2)with ψ m ( p ) the Hermite functions. Note that this is non-zero only if m,n are odd/evenor even/odd respectively. One can also form the diagonal components of full kernel by– 45 –omputing the element: (cid:104) n | e − β ˆ H ˆ O e − β ˆ H ˆ O e − β ˆ H | n (cid:105) = π n n √ n ! n ! e − ωβ ( n + n +1) × (cid:88) m m e − ωβ ( m +1 / m !( n − m )( m − n ) (cid:32) − n / − m +12 ) 1Γ( − m/ − n +12 ) + perm (cid:33) . (F.3)Now this kernel can be non-zero only if both n , are even or odd and the states that runthrough the sum are then only odd or even respectively. In either case, only one termcontributes in the sum and in particular for n , odd we get ( q = e − ωβ c ): (cid:104) n | ˆ ρ | n (cid:105) = q ( n + n +2) n n Γ( − n )Γ( − n ) √ n ! n ! n F (cid:0) , − n ; 1 − n ; q (cid:1) − n F (cid:0) , − n ; 1 − n ; q (cid:1) n n − n n , (F.4)while for n , even (cid:104) n | ˆ ρ | n (cid:105) = q ( n + n +2) n n Γ( − n +12 )Γ( − n +12 ) √ n ! n ! q n B q (cid:0) − n , − (cid:1) − q n B q (cid:0) − n , − (cid:1) n − n ) . (F.5)which can also be rewritten in terms of F . From this expression we can also match theformulas in G.4 for ˆ O if we set β = 0. F.2 Kernel in elliptic functions One can massage a bit the integral equation 4.20, by adding/subtracting information fromboth sheets. In terms of the torus this means to form (the parentheses in both sides of theequation stand for the even/odd case) λ ( X ( u ) ( ± ) X ( u + 2 K )) = − q (cid:90) C + C dv πi (cid:32) q sn v cn u sn u dn v (cid:33) X ( v )dn v − cn u , (F.6)where the denominator can be also written as sn u − q sn v . One can bring this equationinto the following final form λX ( ± ) ( u ) = − q (cid:90) C + C dv πi (cid:32) q sn v cn u sn u dn v (cid:33) X ( ± ) ( v )dn v − cn u (F.7)with X ( ± ) ( u ) = X ( u ) ( ± ) X ( u + 2 K ). F.3 Trace of the kernel The trace of the kernel can be computed to be (also using equation 4.9)Tr ˆ ρ = 1 √ β/ (cid:90) π dθ π sin θ (cid:113) cosh ˜ β − cos(2 θ ) = 12 π sinh( ˜ β/ 2) tan − β/ 2) (F.8)– 46 –ue to the branch-cut structure of this expression, it is useful to represent this function interms of an integral with the integrand having simple polesTr ˆ ρ = 12 π sinh( ˜ β/ 2) arctan (cid:18) β/ (cid:19) = (cid:90) π dθ π cos( θ/ β ) − cos( θ ) (F.9)(keep in mind that ˜ β = ωβ circle = 2 ωβ orb ). To discuss the inverse oscillator one needs toset ω → − iω . One then findsTr ˆ ρ inv = − π sin( ωβ c / 2) tanh − (cid:18) ωβ c / (cid:19) = (cid:90) π dθ π cos( θ/ ωβ c ) − cos( θ ) (F.10)An analogous formula for the circle is [63] Z invcirc ( β c ) = ∞ (cid:88) k =0 e iωβ c ( k + ) = i ωβ c / 2) = (cid:90) π dθ π ωβ c / − cos( θ ) (F.11)If we define the twisted partition function [63] Z ( θ, β c ) = 1 / ωβ c − cos θ (F.12)we understand both results as a 1-particle partition function derived from averaging overtwist angles with a different weight for the orbifold and circle (after extending due tosymmetry the integrals for θ (cid:48) ∈ [ − π, π ]). Another useful representation is Z ( θ, β c ) = (cid:90) + ∞−∞ d(cid:15) e − β c (cid:15) ρ ( θ, (cid:15) ) = 1sin θ (cid:90) + ∞−∞ d(cid:15) e − β c (cid:15) sinh (cid:15)ω ( π − θ )sinh (cid:15)ω π (F.13)which holds for 0 < θ < π and ρ ( θ, (cid:15) ) is the twisted density of states. From this one findsa closed formula for the twisted dos: ρ ( θ, (cid:15) ) = sinh (cid:15)ω ( π − θ )sinh (cid:15)ω π sin θ (F.14)and also an expression that gives away the spectrum ρ ( θ, (cid:15) ) = ∞ (cid:88) m = −∞ e imθ ρ ( m ) ( (cid:15) ) = 1 π ∞ (cid:88) k =0 ∞ (cid:88) m = −∞ e imθ ( | m | +12 + k )( (cid:15)ω ) + ( k + | m | +12 ) + δ ( θ ) log Λ (F.15)note in particular the logarithmic divergence at θ = 0 that is regulated putting a wallat some cutoff Λ and neglecting any cutoff dependent quantities in the double scalinglimit. In this equation ρ m ( (cid:15) ) = − π Re Ψ( i (cid:15)ω + | m | +12 ) is the Hydrogen atom density ofstates (discrete spectrum) which should be contrasted with the H.O. density of states ρ H.O. ( (cid:15) ) = − π Re Ψ( i (cid:15)ω + ). – 47 – .4 1-particle density of states From the partition function Z ( β ), one computes the density of states using ρ d ( (cid:15) ) = (cid:90) c + i ∞ c − i ∞ dβ πi Z ( β ) e β(cid:15) (F.16)The difficulty in our case is that one needs again to study very well the pole and branchcut structure of the integrand. We will instead try to use the integral representation forthe partition function of the orbifold to write ρ o ( (cid:15) ) = 12 (cid:90) c + i ∞ c − i ∞ dβ c πi (cid:90) π dθ π cos( θ/ ωβ c ) − cos( θ ) e β c (cid:15) (F.17)with c an infinitesimal positive regulator. Interchanging the integrations one picks thepoles at the negative β c axis β c = 2 nπ ± θ and sums over the residues. There is a catchwhen θ → 0, since then two poles merge and the singularity pinches the contour. In anycase, the same singularity appears also in the analogous formula of the circle F.11 and willjust reproduce the irrelevant logarithmic divergence. The result is ρ o ( (cid:15) ) = (cid:90) π dθ π cos( θ/ θ ) sinh (cid:15)ω ( π − θ )sinh (cid:15)ω π = (cid:90) π dθ π cos( θ/ ρ ( θ, (cid:15) ) (F.18)It is thus easy to see that this result is equivalent to the one we would get if we justintegrate over the twisted dos with the appropriate weight. Now this integral can beperformed indefinite to get a result in terms of hypergeometric functions F . Taking thelimit θ → ρ ( (cid:15) ) = 14 π (cid:32) iπ − γ + e − π (cid:15)ω sinh( π (cid:15)ω ) Ψ (cid:18) − i (cid:15)ω + 12 (cid:19) − e π (cid:15)ω sinh( π (cid:15)ω ) Ψ (cid:18) i (cid:15)ω + 12 (cid:19)(cid:33) = i − π γ − π Re Ψ (cid:18) i (cid:15)ω + 12 (cid:19) + i π cosh( π (cid:15)ω )sinh( π (cid:15)ω ) Im Ψ (cid:18) i (cid:15)ω + 12 (cid:19) (F.19)that contains the H.O. dos ρ HO ( (cid:15) ) = − π Re Ψ (cid:0) i (cid:15)ω + (cid:1) , and imaginary pieces. From the π limit we get ρ π ( (cid:15) ) = 14 π sinh( (cid:15)ω π ) (cid:20) Im Ψ (cid:18) i (cid:15) ω + 14 ) (cid:19) − Im Ψ (cid:18) i (cid:15) ω + 34 ) (cid:19)(cid:21) (F.20)One then notices that the 1-particle orbifold density of states is ρ o ( (cid:15) ) = ρ H.O. ( (cid:15) ) + ρ twisted + ρ Im ( (cid:15) ), with the twisted piece ρ twisted ( (cid:15) ) = 14 π sinh( (cid:15)ω π ) (cid:20) Im Ψ (cid:18) i (cid:15) ω + 14 ) (cid:19) − Im Ψ (cid:18) i (cid:15) ω + 34 ) (cid:19)(cid:21) = 14 π sinh( (cid:15)ω π ) Im (cid:90) ∞ dt e − i (cid:15)ω t cosh( t ) (F.21)where we used the integral representation of the digamma function. One nice thing to note is that the twisted part of the dos does not require a cutoff in accordance withthe discussion in [63]. – 48 – Approximate methods for large β Here we give more details on the large β approximation to the canonical partition function. G.1 Generic n in angles One can expand eq. 3.35 for large β and relating q c = q o to find Z n ≈ q ( N − n )24 c q N − nc q n c (1 + O ( q c )) (cid:90) π (cid:89) k dθ k J n ( θ )= q N c (1 + O ( q c )) 1 n ! n − (cid:89) j =0 Γ(1 + j )Γ(2 + j )Γ( N − n + 1 + j )Γ( N − n + j + 1) (G.1)where we used again the Selberg integral to compute the integral. We find that the leadingin β c term will give half the Free-energy of the circle for any n. G.2 Generic n in eigenvalues of M We start by the following generalization of the Cauchy identity [114] (cid:81) ni 2. One can then perform one extra integration to reach the formula eqn. 5.1 of themain text. – 49 – .3 n=0 For the n = 0 representation, we define Dx = (cid:81) Nk dx k /N ! and expand in multi-particlefermionic wavefunctions Z N = (cid:90) DxDy ∆( x )∆( y ) det ij K ( x i , y j ) = (cid:90) DxDy ∆( x )∆( y ) (cid:89) k K ( x k , y k ) == (cid:90) DxDy ∆( x )∆( y ) (cid:88) E n e − βE n Ψ E n ( x k )Ψ E n ( y k ) (G.5)with Ψ E n ( y k ) the multiparticle energy eigenfunctions (cid:104) E n | y , y , ....y N (cid:105) . This in the β → ∞ limit gives Z = e − βE ground (cid:18)(cid:90) Dx ∆( x )Ψ ground ( x k ) (cid:19) = e − βE ground (cid:18)(cid:90) Dx det i,k ( x i − k ) det j,k ( ψ j − ( x k )) (cid:19) = e − βE ground (cid:18) det ij (cid:90) dx ( x i − ψ j − ( x )) (cid:19) (G.6)with ψ i ( x k ) the single-particle wavefunctions and we used Andreief identity [109] to turnthe integral over N variables to an integral over a single one. The Free energy is F = + 12 β c E ground − ≤ i,j ≤ N (cid:90) dx ( x i − ψ j − ( x )) (G.7)where the second term can be interpreted as a radius independent contribution of statesat the endpoints written as a determinant of a matrix F ij . One needs to compute thefollowing integrals F + nm = (cid:90) ∞−∞ dxx n − ψ m − ( x ) → (cid:90) ∞−∞ dxx n − ψ + ( (cid:15) m − , x ) F − nm = (cid:90) ∞−∞ dxx n − ψ m − ( x ) → (cid:90) ∞−∞ dxx n − ψ − ( (cid:15) m − , x ) (G.8)where we have indicated the corresponding expressions for the normal and the inverse H.O.To compute this contribution for the inverse harmonic oscillator we will use the odd/evenparabolic cylinder ψ ± functions of appendix B. We define α = − i (cid:15) and use the followingintegral (c is an infinitesimal regulating parameter) I = (cid:90) ∞ dxx n e − i x F (cid:18) α ; γ ; ix e ic (cid:19) == 2 n e iπ (2 n +1) Γ( n + 12 ) F (cid:18) α ; n + 12 ; γ ; 2 e ic (cid:19) (G.9)This can be proven using Mellin-Barnes representations for hypergeometric functions. Wethen get F + nm = 2 n − e iπ (2 n − C + ( (cid:15) )Γ( n − 12 ) F (cid:18) α, n − 12 ; 12 ; 2 e ic (cid:19) (G.10) F − nm = 2 n +1 e iπ (2 n +1) C − ( (cid:15) )Γ( n + 12 ) F (cid:18) α + 12 , n + 12 ; 32 ; 2 e ic (cid:19) , (G.11)– 50 –sing the following identity for the hypergeometric functions F ( a, b ; c ; z ) = (1 − z ) c − a − b F (cid:18) c − a, c − b ; c ; zz − (cid:19) (G.12)we find F + mn = 2 n − e iπ (2 n − C + ( (cid:15) m − )( − n − e iπα Γ( n − 12 ) F (cid:18) − α, − n ; 12 ; 2 e − ic (cid:19) == 2 n − e iπ (2 n − C + ( (cid:15) m − )(2) n − e iπα √ π × n − (cid:88) k =0 Γ (cid:0) − α + n − − k (cid:1) Γ (cid:0) − α (cid:1) Γ (cid:0) n − (cid:1) Γ (cid:0) n − − k (cid:1) ( n − n − − k )! (cid:0) − (cid:1) k k ! == 2 n − e π (cid:15) m − | Γ ( α m − ) | √ π n − (cid:88) k =0 a k ( n ) (cid:15) n − − km − (G.13)where a k depends only on n . Similarly, F − nm = 2 n +1 e iπ (2 n +1) C − ( (cid:15) m − )( − n − e iπ ( α + ) Γ( n + 12 ) F (cid:18) − α, − n ; 32 ; 2 e − ic (cid:19) == 2 n +1 e iπ (2 n +1) C − ( (cid:15) m − )(2) n − e iπ ( α + ) √ π × n − (cid:88) k =0 Γ (cid:0) − α + n − − k (cid:1) Γ (1 − α ) Γ (cid:0) n + (cid:1) Γ (cid:0) n + − k (cid:1) ( n − n − − k )! (cid:0) − (cid:1) k k ! == 2 n e π (cid:15) m − | Γ (cid:0) α m − + (cid:1) | √ π n − (cid:88) k =0 b k ( n ) (cid:15) n − − km − (G.14)with b = 1. After using determinantal properties, we find thatln (cid:0) det F + mn (cid:1) = (cid:88) i 12 ln (cid:0) e − π(cid:15) (cid:1) = ln(2 π ) − e − π(cid:15) + ... (G.18)contributing only non perturbative terms.Introducing the density of states, one obtains a quite simple result for the twisted statecontribution Θ = 12 (cid:90) µ ρ ( (cid:15) ) (cid:90) µ ρ ( (cid:15) (cid:48) ) log | (cid:15) − (cid:15) (cid:48) | d(cid:15)d(cid:15) (cid:48) (G.19)– 51 –here ρ ( (cid:15) ) is the density of states: ρ ( (cid:15) ) = 1 π (cid:32) − log (cid:15) + ∞ (cid:88) m =1 C m (cid:15) − m (cid:33) (G.20)the coefficients C m are known in terms of Bernoulli numbers. To compute this quantity wetake one derivative wtr to µ to get ∂ Θ ∂µ = ρ ( µ ) (cid:90) −∞ d(cid:15)ρ ( (cid:15) + µ ) log | (cid:15) | (G.21)In this expression one needs to put a cutoff Λ at the lower part of integration and computeit as a series expansion in 1 /µ . After one computes G.19, one has to express it in terms ofthe cosmological constant ∆ in order to be able to compare with the Liouville result (seesection 5). One needs to use ∂ ∆ ∂µ = πρ ( µ ) , (G.22)and the renormalised cosmological constant µ that plays the role of the string coupling,defined via ∆ = − µ log µ . (G.23)In the end Θ can be found in terms of µ as:Θ = µ (cid:18) − π 24 + (cid:18) π − (cid:19) log µ + 74 log µ − 12 log µ (cid:19) − (cid:18) π (cid:19) log µ + 1 µ (cid:18) (cid:18) π − (cid:19) log µ (cid:19) O ( µ − ) . (G.24)One notices that the torus contribution is not the same as in equation 2.7. G.4 n = N/ with Hermite polynomials For the regular case we get (the measures contain appropriate factorials) Z = (cid:90) dxdx (cid:48) dydy (cid:48) det i,j x i − y j det i,j x (cid:48) i − y (cid:48) j det n × n K ( x, y ; x (cid:48) , y (cid:48) ) (G.25)which in the β → ∞ limit gives F = 12 β c E ground − (cid:90) d n xd n y det i,j x i − x (cid:48) j det n × n ψ i − (¯ x j ) (G.26)with ¯ x = ( x, y ). One can use Moriyama’s formula for unequal ranks in the appendix of [102]to get Θ = 2 log (cid:90) d n x det ≤ i ≤ n ≤ k ≤ n (cid:20)(cid:90) dy ψ i − ( x k ) x k − y ψ i − ( x k ) (cid:21) (G.27)– 52 –s we have dicussed, one can also integrate x’s to findΘ = 2 log (cid:2) pf n × n O ij (cid:3) (G.28)with the antisymmetric O ij = (cid:90) dxdy ψ i − ( x ) ψ j − ( y ) − ψ i − ( y ) ψ j − ( x ) x − y = 2 (cid:90) dxdy ψ i − ( x ) ψ j − ( y ) x − y (G.29)Similarly to the main text we will adopt the principal value prescription. This gives O i,j = (cid:90) ∞−∞ dxψ i − ( x ) ψ H j − ( x ) − ψ H i − ( x ) ψ j − ( x ) = 2 (cid:90) ∞−∞ dxψ i − ( x ) ψ H j − ( x ) (G.30)with ψ H the Hilbert transform of ψ and in the second line we used that the Hilbert trans-form is skew-adjoint. For the Pfaffian we have the formula log [pf A pf B ] = Tr log A T B .where we want to apply it for the case A = B = O with O T = − O so that we getΘ = 12 Tr log (cid:0) − O (cid:1) (G.31)One notices that the matrix O is just twice the Hilbert transform operator ˆ O in the energybasis. It is a real antisymmetric matrix with imaginary eigenvalues. Also, since H = − Tr log 4 ˆ I = N log 2. To be more explicit, ifwe perform the integrals we can rewrite O as: O m,n = 2 (cid:104) m | ˆ O| n (cid:105) = ± m + n √ m ! n ! √ πn − m (cid:34) − m/ − n +12 ) + 1Γ( − n/ − m +12 ) (cid:35) (G.32)with 0 ≤ m, n ≤ N − 1. In this expression, only one of the two terms inside the brackets canbe non-zero when m -odd, n -even or vice versa, the odd/odd even/even pieces are zero. Theoverall ± is because the hermite functions are eigenfunctions of the fourier transform witheigenvalues ± , ± i and one finds an overall factor ( − i ) m + n +1 , when going to momentumspace in order to calculate the integral.Using this we can form O as (this now holds for n , n together odd/even!) O n n = 2 n / n / √ π √ n n n − n ) (cid:34) − n )Γ( − n +12 ) − − n )Γ( − n +12 ) (cid:35) (G.33)In this expression, we find that the only non-zero terms are the diagonal. This is alsoconsistent with the appropriate limit of the full energy-basis kernel F.3. Near the diagonalthis expression approaches the sine-kernel O n n ≈ − π ( n − n ) π ( n − n ) (G.34)Taking the limit n → n we findΘ = 12 Tr log (cid:0) − O (cid:1) = 12 N − (cid:88) k =0 log (cid:20) (Ψ( − k/ − Ψ(1 / − k/ πkπ (cid:21) (G.35)– 53 –he expression in brackets has only real part. One also findslim k → N (Ψ( − k/ − Ψ(1 / − k/ πkπ = 4 , ∀ k ∈ N , (G.36)and thus we recover the expected Θ = N log 2 which pinpoints to the fact that we justcount the total entropy of a two state system at the endpoints, due to the spin up/downnature of the wavefunctions. It is tempting to pass to continuous variables via the dos ρ H.O. ( (cid:15) ) = − π (cid:80) k δ ( (cid:15) − (cid:15) k ) which for the inverse H.O. clicks when − i(cid:15) = k + . The resultis Θ = 12 (cid:90) µ d(cid:15)ρ H.O. ( (cid:15) ) log (cid:20)(cid:18) Ψ( 14 + i(cid:15) − Ψ( 34 + i(cid:15) (cid:19) π(cid:15) ) π (cid:21) , (G.37)with the term in the logarithm looking conspicuously similar to the twisted dos equa-tion F.21. One should be very careful though, since the normalization of the Hermitefunctions after rotating is different compared to the one of the parabolic cylinder func-tions and one should really perform the computation from the start using the inverse H.O.eigenfunctions. G.5 n = N/ with parabolic cylinder functions Here we perform the same computation using the delta-function normalised even and oddparabolic cylinder functions of appendix B which are eigenfunctions of the inverted oscilla-tor. Since the spectrum is now continuous, we can imagine obtaining a discrete spectrumby putting a cutoff/wall at Λ which is then send to infinity. We again adopt the principalvalue prescription whenever fourier transforming.We compute (cid:104) (cid:15) | O | (cid:15) (cid:105) = 2 (cid:90) ∞−∞ dxdy ψ + ( (cid:15) , x ) ψ − ( (cid:15) , y ) x − y = 4 (cid:90) ∞ dx (cid:90) ∞ dy ψ + ( (cid:15) , x ) yψ − ( (cid:15) , y ) x − y . (G.38)This expression is non zero and the integrand is even both in x and y . One can thenexponentiate again the denominator using the Fourier transform of the sign function. Thisgives O ( (cid:15) , (cid:15) ) = − i (cid:90) ∞−∞ dt sgn( t ) I + ( t ) I − ( t ) = − (cid:60) (cid:20) i (cid:90) ∞ dtI + ( t ) I − ( t ) (cid:21) (G.39)where, I + ( t ) = (cid:90) ∞ dxψ + ( x ) e − i tx , I − ( t ) = (cid:90) ∞ dyyψ − ( y ) e + i ty . (G.40) Only the energy dependence is important in the overall normalisation of this object. – 54 –he advantage is that now one can compute the resulting integrals using [120] (cid:90) ∞ due − su u b − F ( a, c, ku ) = Γ( b ) s − b F ( a, b, c, ks − ) , (cid:60) s > (cid:60) k, (cid:60) s > , b > | s | > | k | , = Γ( b )( s − k ) − b F ( c − a, b, c, kk − s ) , (cid:60) s > (cid:60) k, (cid:60) s > , b > | s − k | > | k | Or even the following simpler form that can be obtained from the expression above if k = 1, b = c (cid:90) ∞ due − su u c − F ( a, c, u ) = Γ( c ) s − c (1 − s − ) − a , (cid:60) c > , (cid:60) s > . (G.41)Using an infinitesimal regulator e ic we can find for I + ( (cid:15) , t ) with t > I + ( (cid:15) , t ) = N ( (cid:15) ) (cid:90) ∞ du √ u e − i ( + t ) ue ic F (1 / − i(cid:15) / , / iue ic )= N ( (cid:15) ) (cid:114) π it + i − / F (1 / − i(cid:15) / , / , / , e ic + t ) , (G.42)with N ( (cid:15) ) = (cid:18) π √ (1+ e π(cid:15) ) (cid:19) / | Γ(1 / i(cid:15)/ / i(cid:15)/ | .For I − ( (cid:15) , t ), we now have (with t < − ), I − ( (cid:15) , t ) = N ( (cid:15) ) (cid:90) ∞ du √ ue − i ( − t ) ue ic F (3 / − i(cid:15) / , / iue ic )= N ( (cid:15) ) (cid:114) π − it + i − / F (3 / − i(cid:15) / , / , / , e ic − t ) , (G.43)with N ( (cid:15) ) = (cid:18) π √ (1+ e π(cid:15) ) (cid:19) / | Γ(3 / i(cid:15)/ / i(cid:15)/ | . We now encounter a form of non-perturbativeambiguity which has to do with the possible analytic continuations of these hypergeometricfunctions. In particular, the hypergeometric functions F ( a, b, c, z ) have branch points at z = (0 , , ∞ ) and thus the integrals G.42, G.43 have branch points at t = ( ∞ , , − ) and t = ( −∞ , − , ) respectively. We will now assume working in some undetermined branchand naively analytically continue these equations for complex t . In the next subsection weare going to split the t integral into sections and find what are the exact conditions (whichsheet to choose) in order to match the result we find here.We will now introduce the following change of variables z = 1 / ( + t ) , t = (2 − z ) / z to get O ( (cid:15) , (cid:15) ) = − πN ( (cid:15) ) N ( (cid:15) ) ×(cid:60) (cid:34)(cid:90) dzz (cid:20) zz − (cid:21) [ z ] F (3 / − i(cid:15) / , / , / , ze ic z − F (1 / − i(cid:15) / , / , / , ze ic ) (cid:35) (G.44)– 55 –y shifting the corresponding hypergeometric function and performing the integral we get O ( (cid:15) , (cid:15) ) = − πN ( (cid:15) ) N ( (cid:15) ) e ( (cid:15) + (cid:15) ) π (cid:60) (cid:20) i (cid:90) dz ( z − − i ( (cid:15) − (cid:15) ) (cid:21) = − πN ( (cid:15) ) N ( (cid:15) ) e ( (cid:15) + (cid:15) ) π (cid:60) (cid:20)(cid:90) − duu u + i ( (cid:15) − (cid:15) ) (cid:21) = 4 πN ( (cid:15) ) N ( (cid:15) ) e ( (cid:15) + (cid:15) ) π (cid:34) − e ( (cid:15) − (cid:15) ) π (cid:15) − (cid:15) (cid:35) . (G.45)where the last expression holds when (cid:61) (cid:15) > (cid:61) (cid:15) and one can derive a similar one in case (cid:61) (cid:15) < (cid:61) (cid:15) by exchanging (cid:15) ↔ (cid:15) with an overall minus sign . We expect that ouranalytically continued result is valid for some specific branch. A different branch wouldgive a different normalization. This difference in normalization we expect to play a role inthe contribution of non-perturbative states as discussed in the main text. The result canalso be written as O ( (cid:15) , (cid:15) ) = 2 e ( (cid:15) + (cid:15) ) π/ (1 + e π(cid:15) ) / (1 + e π(cid:15) ) / (cid:12)(cid:12)(cid:12)(cid:12) Γ(1 / i(cid:15) / / i(cid:15) / 2) Γ(3 / i(cid:15) / / i(cid:15) / (cid:12)(cid:12)(cid:12)(cid:12) − e ( (cid:15) − (cid:15) ) π/ (cid:15) − (cid:15) = 1 π | Γ(1 / i(cid:15) / / i(cid:15) / | e π (3 (cid:15) + (cid:15) ) / sinh (cid:0) π ( (cid:15) − (cid:15) ) (cid:1) (cid:15) − (cid:15) . (G.46) G.5.1 Calculation of the integrals for segments We will now perform a consistency check and understand better our branch choice. Wesplit the integrals into sections with respect to the branch points. We demand that theparameter t is real and we drop the regulator. Then we indeed find a result that differs fordifferent sections of t . The sections are ( −∞ , − / ∪ ( − / , / ∪ (1 / , ∞ ) . We havecomputed the integrals for each section by taking the limit at the branch points sendinga small parameter to zero (ex. we integrate up to 1 / (cid:15) and then we send (cid:15) → N ( (cid:15) ) , N ( (cid:15) )) • For the section (0 , / I + ( t ) = √ πe π(cid:15) (cid:16) t +1 − (cid:17) + i(cid:15) (1 − t ) / (G.47) I − ( t ) = √ πe π(cid:15) (cid:16) t +1 − (cid:17) − i(cid:15) (1 − t ) / (G.48) • For the section (1 / , ∞ ): I + ( t ) = (1 − i ) √ π (cid:16) t − t +1 (cid:17) i(cid:15) (4 t − / (G.49) These cases probably form different elements of the discrete matrix above and below the diagonal, sincethe poles of the inverted oscillator dos are at (cid:61) (cid:15) = n + . The matrix is then appropriately real andantisymmetric – 56 – − ( t ) = − (1 − i ) √ π (cid:16) t − t +1 (cid:17) − ( i(cid:15) ) (4 t − / (G.50)One can similarly obtain the rest of the sections by t → − t .One can now notice that G.47, G.49 are the same expression if one chooses − e − iπ and G.48, G.50 are the same if we chose − e iπ . This choice corresponds to picking aspecific branch. We already know that the spectrum of the inverted oscillator is twofolddegenerate and our choice just means that the even/odd modes live in a different sheet ofthe complex energy plane. After changing variables z = 1 / + t this choice gives the sameintegral and result as in G.45 G.5.2 The sine/sinh kernel It is now easy to see that since O ( (cid:15) , (cid:15) ) = A ( (cid:15) ) K sinh ( (cid:15) − (cid:15) ) B ( (cid:15) ), the only interestingasymptotic contribution comes from the kernel in the middle. The diagonal normalizationfactors can be shown to contribute non-perturbatively, since they do not admit an 1 /(cid:15) ex-pansion and scale for large (cid:15) as e a(cid:15) with a a parameter depending on the branch we choose.The kernel whose spectrum we want to compute is the analytic continuation of the verywell studied sine kernel K sine ( (cid:15) , (cid:15) ) = sin( π ( (cid:15) − (cid:15) )) (cid:15) − (cid:15) for which various results exist in theliterature in relation to its spectrum and Fredholm determinants [104–106].One way of computing its determinant is to discretise and bring it into a Toeplitz form. Inour case one can put a cutoff Λ and then use the density of states of the inverted oscillator i(cid:15) j = j + which is equidistant, or equivalently analytically continue in ω . Then calculat-ing the determinant of the sine kernel with support on an energy segment one finds that itcan be represented as a Toeplitz determinant in a scaling limitdet K sine | − µ −∞ = det (cid:0) − K sine | − µ (cid:1) , ⇔ lim N →∞ N det C j − k , with C j − k = δ jk − sin( πµ N ( j − k )) π ( j − k ) (G.51)We will now discuss some properties of this fredholm determinant, and provide an asymp-totic evaluation for large µ , with which we can match the torus contribution to the twistedstates. G.5.3 Level spacings The level spacing distribution E β ( n, µ ) of Random matrices is the probability that the inter-val (0 , µ ) contains exactly n eigenvalues [103]. In our case these will be energy eigenvaluesand the random matrix is the Hamiltonian. Thus the Hilbert transform operator effectivelyrandomizes the energy eigenvalues of the system which are to be drawn from an ensemble(GUE/GOE/GSE). The parameter β denotes the ensemble and for us β = 2 (GUE). Wefirst define D ( µ ; λ ) = det (cid:0) − λK sine (cid:1) . We also define K ± = K sine ( x, y ) ± K sine ( x, − y )– 57 –nd similarly D ± ( µ ; λ ) = det (1 − λK ± ). For the other ensembles, β , the kernel is amatrix. Then one has [103] E ( n ; (0 , µ )) = ( − n n ! ∂D ( µ ; λ ) ∂λ n | λ =1 (G.52)and for the other ensembles one can again find formulas involving E ± D ± . We will nowuse the asymptotic formulas in the literature for the level spacings as µ → ∞ much likewhat we want for the asymptotic expansion of string theory µ → ∞ . We provide here themore general result/conjecture for arbitrary β, n [107] that correctly reproduces the provenresult for n = 0 , β = 2 [104–106]log E β ( n ; (0 , µ )) ∼ µ →∞ − β µ 16 + ( βn + β/ − µ (cid:20) n − β/ − βn/ 2) + 14 ( β/ /β − (cid:21) log µ + .... (G.53)We now need to remember that the Pfaffian is the square root of the determinant and thatwe need to divide our result by an extra factor of 2, since we want to match the bosonicstring theory partition function, that has support on the one side of the potential. Aftertaking these into account, one finds the twisted state contributionΘ = 14 log E (0; (0 , µ )) = − µ − 116 log µ + 148 log 2 + 34 ζ (cid:48) ( − 1) + O (cid:18) µ m (cid:19) . (G.54)We see that we correctly capture only closed string contributions with even higher powersof 1 /µ and some of these coefficients can be found in [104]. Moreover this formula predictsthat there is no-logarithmic divergence coming from the genus 0 spherical contribution. Asa bonus, it is interesting to note that one can make the same computation with orthogonalor symplectic matrices in GOE, GSE which can be found to receive open string correctionswith odd powers in µ . These results might be relevant for the unoriented string theoryon the orbifold, where odd powers of µ are known to appear and orthogonal/symplecticsymmetries to be relevant. G.5.4 Properties of the sine kernel The sine kernel has some remarkable properties some of which which we list here • Its eigenfunctions are the prolate spheroidal functions and some asymptotic forms ofthe spectrum exist. • The Christoffel Darboux (CD) kernels approach the sine kernel in a scaling limit thatfocuses on the bulk of the spectrum. • As with all the CD kernels it is a self-reproducing kernel, it obeys K ∗ K = K . • It is the band-limited version of the Dirac delta distribution. To understand thisbetter, let f ∈ L ( R ) a function whose fourier transform has support on the segment– 58 – − πb, πb ] (band limited functions) Then the sine kernel is an orthogonal projectionto this space since (cid:90) ∞−∞ dy sin( πb ( x − y )) π ( x − y ) f ( y ) = 1 √ π (cid:90) πb − πb e ixξ F [ f ]( ξ ) dξ (G.55) • Moreover one can further consider functions f ∈ L ([ − s, s ]). This gives both energyand time band limited functions (in our case s ∼ µ is the energy-band limit while b = 1 / • It is easy to see that it is the natural regulating description of the dirac- δ functionwe were expecting to have (for O ), since at the discrete level we encountered theidentity operator δ nm and we were filling eigenvalues up to the size of the matrix N .It also allows for a rigorous understanding of limiting the energy and defining thefermi surface which corresponds to filling all the negative energy states up to a bandbelow 0 corresponding to the chemical potential − µ . 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