JHEP01(2021)118 , or Springer July 7, 2020 : October 31, 2020 January 20, 2021 December 3, 2020 : : : Received Revised Accepted Published Published for SISSA by https://doi.org/10.1007/JHEP01(2021)118 [email protected] , . 3 2006.11317 The Authors. c 2D Gravity, Black Holes, Conformal Field Theory, Nonperturbative Effects

We propose that a class of new topologies, for which there is no classical , [email protected] Department of Physics, UniversitySanta of Barbara, California, CA 93106, U.S.A. E-mail: Open Access Article funded by SCOAP ArXiv ePrint: extremal limit. To make thisof argument, Jackiw-Teitelboim gravity we with dynamical study defects. ato We closely show a related that this matrix two-dimensional theory theory integral. is equivalent Keywords: Abstract: solution, should be includedthat their in inclusion the solves patha pathological integral nonperturbative negativities of shift in of three-dimensional thecalculation the pure spectrum, using BTZ gravity, replacing extremality a and them bound. dimensionally with reduced We argue theory that captures a the two leading dimensional effects in the near Henry Maxfield and Gustavo J. Turiaci JT gravity with defects as a matrix integral The path integral of 3D gravity near extremality; JHEP01(2021)118 38 22 9 39 40 5 39 17 – 1 – 2 28 26 = 0 amplitudes ) black holes. 7 Z , , g 45 44 = 1 26 13 48 6 n 7 43 17 15 40 34 29 7 16 31 2 37 3.3.1 Two3.3.2 boundary amplitude All genus-zero amplitudes 5.1 An ensemble5.2 dual for three-dimensional The pure full5.3 gravity? three-dimensional path integral Consistency with5.4 modular invariance Including matter 5.5 The instanton gas for JT with defects 3.4 Density of3.5 states Higher genus 3.1 The JT3.2 path integral with The defects disc3.3 with defects: The matrix integral dual 2.2 The classical2.3 solutions: SL(2 Kaluza-Klein instantons 2.4 Near-extremal limit 2.5 Instanton gas 1.3 JT gravity1.4 with defects Outline of paper 2.1 Dimensional reduction 1.1 The spectrum1.2 of 3D pure 3D gravity gravity in the near-extremal limit D The instanton gas theory A Weil-Petersson volumes B Lagrange reversion theorem C Explicit higher genus checks 5 Discussion 4 Back to 3D gravity 3 JT gravity with defects 2 The path integral of 3D gravity Contents 1 Introduction JHEP01(2021)118 ], and we 8 ]. Our first major result ]. We also argue that this 7 8 , and perturbative graviton excitations are 3 – 2 – ] constructed a candidate partition function from ], and we by now have a detailed understanding of 10 1 ]. Perhaps the simplest question to ask is whether a consistent quantum 6 – gravity exists. 4 , ] and Keller and Maloney [ 2 9 pure We now review the context of the paper and summarize our main results. We will begin with three dimensional gravity, for which an apparent obstruction to While this question is simple to ask, it is much harder to make meaningful progress. A and Witten [ a pure 3D gravity path integral andas found the the ‘MWK’ corresponding partition spectrum; we functionpath will and refer integral, spectrum. to these by They classifying tookcomputing all a their very classical classical natural action solutions approach and with to all perturbative the appropriate corrections boundary (which happen conditions, to be exact 1.1 The spectrum of 3DGravity pure in gravity three dimensionssolutions has are several locally simplifying isometricdetermined features. to by the empty Virasoro In AdS symmetry particular, of all the theory. classical Using these simplifications, Maloney integral dual to this gravitationaltheory theory is exists equivalent and to generalizes another [ 2Dappearing dilaton from gravity theory integrating with out a the modified gas dilaton of potential, defects. 2D models, which weintegral study of JT in gravity was their recently own solveddraw exactly right heavily by from independent Saad, Shenker their of and work.mechanical Stanford their [ description Their 3D as main origin. a result double-scaled wascontinues The matrix that to path integral. JT hold We gravity when find has we that a include such dual a a quantum description gas of defects in JT gravity. We show that a matrix the existence of ais theory to of find new pureobstacle. contributions gravity to was Our the pointed three-dimensional pathof out considerations integral two-dimensional will in of models lead [ pure which usof 3D generalize naturally additional gravity Jackiw-Teitelboim to ‘defect’ which (JT) consider degrees overcome gravity this of a by freedom. class inclusion Our second major result relates to this class of natural way to attempt toclassical construct a theory, quantum for theory example ofto using pure a gravity make is path sense by integral quantizinglow-dimensional of over the models, metrics, this but in in it whichcomplete is most gravity understanding. very simplifies cases. challenging enough that We we can can ameliorate hope for these a difficulties more by studying through the AdS/CFT correspondence [ the conditions that a CFTlow must energies satisfy [ to betheory well-described by of a local theory of gravity at In the quest tois understand to quantum distinguish mechanical theories lowwithout. of energy For gravity, theories theories one with withquestion central asymptotically becomes problem consistent anti particularly de ultraviolet sharp, since Sitter completions a (AdS) from UV boundary complete those conditions, theory can this be rigorously defined 1 Introduction JHEP01(2021)118 | 2 ), m J | h (1.1) − ¯ h E > = and energies J ]. Besides the | J , 13 | | and J , the density grows 1 1 2 ) to become negative − | − 1 12 c ≥ E E ( ) − m p and perturbative graviton α ¯ h ) J 3 ( 2 0 + , where for a scalar of mass MWK S ( J 2 h / e ρ 1 J The solutions in question are the − = ) ]. 1) | 1 − 11 J E ( ) black holes are negligible, since they 1 Z − | a , ] proposed that the dynamics of the theory + E | ) black holes, is exponentially suppressed by 14 p J ] that the density of states is also negative in ( Z 7 ) , J – 3 – − | are constants. The first term, coming from the ( ) as measured from infinity is lower due to screening from 0 2 1 S E ] and [ α m 7 , a − p α 0 ) e a J (1 ( 0 N , the density of states is nonzero for energies ], generalizations of the BTZ black hole [ 1 S G J 8 e and larger, but it would nonetheless be preferable to have 12 scaling with energy, but grow much slower with spin. 0 a ∼ 2 N / 1 For sufficiently light particles, with G E 1 ∼ 8 − but enhanced very close to the edge. It becomes important when ) 3 ) . | ) ∼ E J N J ( ( 2 ) 0 m S − | mG − e 4 E MWK − ( J p 1 is the Bekenstein-Hawking entropy given by the area of the extremal BTZ ρ  = 1 ) m J ( α is exponentially small, and for odd spins it causes 0 ) Euclidean black holes [ | S Z J , We propose an alternative resolution which does not require a modification of the In order to resolve this negativity, [ To explain this feature, we write the most important contributions to the MWK density A noncompact theory has a continuous spectrum, but with density of states proportionalHere, to the a mass formally means the local mass, for example appearing as the coefficient of worldline length in a The sum is not convergent and must be regularised. The ambiguity arising from the choice of regulator 2 3 1 − | does not appear to be relevant for theinfinite features factor discussed here of [ compatible the with volume a of quantum mechanical target Hamiltonian, space. even forparticle such Here Lagrangian. a the noncompact The theory. energy spectrumgravitational is backreaction. continuous but finite, which is not a consistent theory with only metric degrees of freedom. theory, but rather the inclusion of additionalThese contributions to contributions the path require integral over spacetime metrics. topologies for which no classical solution exists, loop running round thethe horizon density of of the states BTZ proportionalwe black to have hole gives an additionalfast contribution enough to to curewith the Planckian negativity. mass The matter can be very heavy, consisting of particles E in that regime. Termshave from the other same classes ( of SL(2 must be modified by introducing particle degrees of freedom. The correction from a matter where event horizon in Planckordinary units, BTZ and black hole, is positivewhich and comes dominates from in the almost simplest all class regimes.a of relative The SL(2 factor second of term, corresponding to the extremalityvery close bound to of this BTZ bound, black the density holes. of states For behaves large like More seriously, it was recentlya observed particular by [ ‘near-extremal’ regime. of states forquantized) energies angular close momentum toin the appropriate edge units of (in the terms spectrum. of CFT For a quantities, given value of (integer SL(2 vacuum and its Virasoroexcitations, descendants the describing resulting emptywhich spectrum AdS BTZ is black holes supported exist.features. on However, the First, precisely MWK the corresponding density the spectrum of energies is states continuous, has and which several we spins problematic will comment for on later. at one loop) and finally summing their contributions. JHEP01(2021)118 2 / 1 − (1.3) (1.4) (1.5) (1.2) ) ], our | ) is an J 14 1.1 , − | 7 E . ]. We will draw 8 ··· + , ) 2 / J 5 ( 0 − ) has received a nonper- ) | J ( J 0 E − | . The negativity in ( | , ) J E | ], explaining how our proposal is J ( ( q ) 0 16 ) J ], where it was shown to be generic S | ( 2 J , E > E 0 ( 2 − ) J 15 0 S e S J ) black hole in question, and we have − ( e  − | Z e 0 J , J J . E q E p 1) ) 1) J π − ( p − 2 2 − 0 ( ( S  E 1 3 – 4 – − a a p became negative, the later terms in the expansion e ) cos J ) + + J coefficients. Once we sum over them, the density of | ( 2 1) ∼ 0 / J S # 3 ) − e a ( − J 0 MWK − | ) ( ( | a 0 ) black holes, which we have so far disregarded, are them- ρ J E Z ∼ , δE | ∝ − p ) − | ) J J E ( ( E 0 ( J − | S 2 ρ e ) a 0 J ( a + 0 ∼ E ) E ( J ], but here we see a nonperturbative version from pure gravity alone. This ρ ) black hole now appears as the first term in an infinite series of nonperturbative 15 Z -independent prefactors; we have so far been discussing the leading order shift , is a rational number labelling the SL(2 J q p = 2. q An extremely similar phenomenon arises from the perturbative one-loop contribution The other families of SL(2 Before explaining this, let us describe how the additional topologies in the path integral of matter [ was mostly studied from aphenomenon dual for CFT chaotic perspective CFTs inrevisit [ from pure crossing gravity symmetry from this and bootstrap modular perspective invariance. in We [ will where ignored with artifact of truncating theits binomial radius series of for convergence. the square root andselves then the evaluating outside first termsThese in shifts a take multinomial the form series contributing further nonperturbative shifts. The interpretation is that theturbative, spin-dependent black correction hole to extremality its bound classical value corrections. In the regime where are no longer suppressed so allto terms compute must all be terms considered including together.states the to Fortunately, we leading are order able in the expansion near the edge takes the simple form The SL(2 result in a positivenew density contributions of grow more states.at slowly the As with edge. spin, opposed but to Includingthe are some density the more behaving of matter singular like the proposal than most of ( important [ corrections, we find an expansion for an ‘off-shell’ path integral over anpoint appropriate space approximation. of configurations, Integrals rather over thananalysis a off-shell of saddle- the configurations two-dimensional were theory prominent ofheavily Jackiw-Teitelboim on in (JT) this the gravity work, [ recent whichthree-dimensional will problem turn we out are to studying. have an extremely close relationship with the which explains their absence from the MWK partition function. We must therefore perform JHEP01(2021)118 , J ], a in the 19 , π q 2 , and the 17 1 2 = 6 states at the ]. The precise q p − 31 , 29 ]. Allowing additional – 9 ]). They are somewhat ) black holes. The new Z 33 27 , , . 25 q p – 23 , ]). The KK instantons correspond 20 , 35 instantons labeled by 8 k rotation in the circle fiber. The corresponding – 5 – q p ], which are smooth solutions to a five-dimensional π . We can usefully describe the physics using a two- 34 2 ) come from 1.2 in ( ] are evaded. k 7 a ], and was rigorously shown to hold for a generic CFT with large 32 . An instanton creates a conical defect with opening angle where the circle fibration degenerates, and are labeled by a rational Q 2 ∩ 1) , ). The effect we are studying is important only for black holes very close (0 ) comes from including the instantons labeled by 1.2 ∈ 1.5 q p We compute the relevant path integrals in the two-dimensional theory, and argue The KK instantons tend to attract, so the only classical solutions in this class are The new contributions to the 3D gravity path integral we find in this paper come A different negativity of the MWK density of states, taking the form of contributions to the path integralthe we terms are studying with have coefficient multipleshift instantons. in ( In particular, that this gives a good approximation to the three-dimensional pure gravity result in the number two-dimensional geometry, but gives a smootharound three-dimensional the metric defect since is a accompanied closedthree-dimensional loop by topologies a are 2 known as Seifert manifolds. those with a single instanton, and these correspond to the SL(2 analogous to Kaluza-Klein monopoles [ gravity theory which become singular(or Dirac monopoles 6-branes when in viewed typea from four smooth IIA dimensions solution supergravity, which toto locations arise 11-dimensional in supergravity as AdS [ KK monopoles from reduction of from extending the studyeffects. of this The two-dimensional crucial theorywhich new to we call ingredient include ‘Kaluza-Klein new is instantons’. nonperturbative singular that These from the correspond the to theory two-dimensional configurations containsthe geometry, which 2D nonperturbative with appear spacetime, a objects, but conical are defect smooth localized in at three a dimensions (see point also in [ theory which has receivedmap much between perturbative recent physics attention inholes [ JT was gravity described and in in [ central near-extremal charge rotating using BTZ bootstrap black methods. terms in ( to extremality. Ingeometry this of a regime, circle thedimensional fibered black over theory, AdS obtained hole by developsthe reducing a resulting on near-horizon theory the is region circle. perturbatively with equivalent In the to an Jackiw-Teitelboim ensemble gravity of [ fixed spin the near-extremal limit wenew use insight to to study control thesedefinitive them in untamed is this fluctuations no regime. of longer topology a is good required1.2 approximation. to say Some anything 3D gravityWe in now the describe near-extremal the new limit contributions to the path integral that give rise to the additional and how the arguments of [ massless, spinless BTZ threshold, was observedtopologies, in the we original work seem [ tothe topologies lose we semiclassical study control (and entirely likely many for more) these seem to small contribute black at holes: the same all order, and consistent with modular invariance without introducing states far below the BTZ threshold, JHEP01(2021)118 , ). 3 = 2. 3.50 q . Their solution re- k for each defect (proven for ]. Including defects requires us πiα 38 1) which specifies the conical angle , = 2 (0 ], but here we sum over contributions b ]. For one boundary, this gives a closed -parameter generalization of JT gravity. ), which shifts the edge of the spectrum. ∈ 36 F 44 ], and they are obtained simply from the α 1.2 N 42 ], we prove that this extends to all orders in – and number of defects 45 – 6 – g 40 , computed in [ for a defect to appear in the path integral, defined b λ ]) in a genus expansion, and we directly generalize their flavors we have a 2 37 . We may have several ‘flavors’ of defect, each with their own F N 3 2 include energy- and spin-independent prefactors that do not seem q > ]. Using a theorem of [ 8 , so for i ] (see also [ λ 8 ]. 43 and i α ) it sources, and the amplitude α , but conjectured more generally). This fact was recently used in a slightly different 1 2 − We also argue this theory is also equivalent to a 2D dilaton gravity with a modified We then explicitly compute the sum over any number of defects at genus zero, using We first compute the defect contributions to the spectrum at low energy, and show The nonperturbative path integral of JT gravity was solved recently by Saad, Shenker Such a defect is labeled by two parameters, We now turn to a summary of the two-dimensional calculations. ), inconsistent with a unitary theory. This problem in 2D is resolved in the manner (1 ≤ E ( π For multiple boundaries, the result matchesdual precisely generalizing what [ is requiredthe for a genus matrix expansion integral (as well as providing several explicitdilaton checks). potential, obtained by summing the defects as an instanton gas. ρ described above when we sum over an arbitrary numberan of explicit defects. formula for WPform volumes expression derived for in the [ density of states at leading order in the genus expansion ( context by [ that they give rise toThis an expansion gives of a the 2D formonly toy ( model classical of solutions the andthe Maloney-Witten loop disc partition corrections; function, with for whereby the JT we insertion gravity, include this of means a only single the defect. disc This and can produce negative contributions to geodesic boundaries of specifiedto lengths generalize this tovant a volumes have moduli been spaceusual studied of WP already surfaces volumes [ with byα cone inserting points. a Fortunately, boundary the with rele- with any number of defects and any topology. and Stanford [ approach to an expansionquires in the both Weil-Petersson (WP) genus volumes of the moduli space of hyperbolic surfaces with 2 more precisely in section value of A single such defect on the disc was studied by [ 1.3 JT gravity withOur defects three-dimensional considerations have thusof led JT us gravity, naturally by to including consider dynamicalwe a instantons study which modification this source conical theory defects. in In its section own right. Contributions from to be captured fully byis the subleading in two-dimensional the theory near alone. extremalwhich limit. Nevertheless, we their A leave contribution full for understanding future requires work. a full 3D calculation, near-extremal regime. We concentrate on the leading order KK instantons with JHEP01(2021)118 3 κ (2.1) (2.2) we study a . 3  ,  is dimensionless. ) 3 1 θ ` , β , due to the absence on the boundary to + E − 3 t γ 3 κ β, ϕ  γ + , and extrinsic curvature √ 3 E x t ` 2 ( d ∼ ∂ Z ) π ]. + 2 + 2 46  and Euclidean time 2 3 2 ` we study the path integral of 3D gravity , ϕ , AdS length ϕ E 2 3 + t has units of length, while ( ] to which we refer for more details. 3 R 1 – 7 – R 32 ∼ −  )  3 g , ϕ E √ t x ( 3 d , curvature 3 , so that Z g E  N dϕ, we apply what we learnt back to 3D gravity, and we argue that ϕ, t E 1 4 πG dt 2 ). 16 with discussion and open problems. −  4 − 5 = = A related analysis of JT gravity with defects and matrix models was carried γ EH I is a holographic renormalisation parameter which is taken to zero. We choose  We start from the Einstein-Hilbert action in three dimensions: Nonetheless, we will find it profitable to simplify the theory yet further, by studying be a flat torus parameterized by spatial angle where dimensionless coordinate The subscripts on thein metric the boundary termlater. distinguish them We from impose the boundary two-dimensional quantities conditions we that encounter fix the induced metric 2.1 Dimensional reduction We begin by describing thetion two-dimensional of action pure that 3D results gravity, from closely a following dimensional [ reduc- more closely the ‘S-wave’ pathsional integral reduction over on metrics with thewe a symmetry can U(1) solve. direction, symmetry. we Fortunately,near After the will dimen- extremal issues find limit, with a fordiscussed the two-dimensional which in path theory this section integral that reduction of in 3D a gravity good appear approximation in (up the to a subtlety that we might understand itIn completely, particular, if all it classical indeed solutions existsof are as locally local a isometric propagating consistent to quantum degrees empty theory. enjoys of AdS an freedom, infinite-dimensional Virasoro so symmetry can group, whichexcitations be determines around classified the perturbative the in vacuum. some cases. In addition, it 2 The path integralOur of main 3D target gravity negative of cosmological study constant. will This be is a the very spectrum simple of theory in three-dimensional several pure ways, giving gravity hope with integral dual. In section the two-dimensional theory isconclude a in good section approximation in theNote most added. interesting regime.out We independently by E. Witten and appeared in [ The paper is organizednear as extremality follows. from the In perspectivesuggest section of a a new dimensionally class reduced2D of theory. theory topologies This of to leads JT include ussolve in gravity to the with the theory path a in integral. gas an expansion of In in defects, section genus motivated and by number the of 3D defects, considerations. and construct We a matrix 1.4 Outline of paper JHEP01(2021)118 : J (2.8) (2.5) (2.6) (2.7) (2.3) (2.4) , which . To see A . 2 g   3 1 ` , ] − will be imaginary.  i 2 . 2 47 ) 2 κ J (this being the only g c J  24 θ µ √ − Φ 1 2 x 0 and modular parameter = 2 ¯ L ds is the field strength of the ( + d 0 A ¯ ∂ τ 3 is a U(1) gauge field, with ¯ L Z F − πi R , dA 2 A Φ J 0 − i N ) = L + 2  c G 24  ) in terms of the two-dimensional F , Z 2 3 2 2 − ` ) 0 2.1 = = 8 L A + . ( µ ab , so that ϕ + ab . The three-dimensional diffeomorphisms πiτ F of the two-dimensional coordinates. The 2 dλ F ϕ µ. e ab dϕ λ h ab ( J F + 2 i 3 F where 2 A Φ = – 8 – 2 Φ + Φ g 1 4 7→ F , dilaton Φ and Kaluza-Klein gauge field 2 ) = Tr at the cutoff and the proper length of the boundary √ 2 g − x 1 g A 2 iθJ 2 periodicity of − = the angular momentum. We have also written this in d −  R 3 π  J g Z a proper length parameter on the boundary. Expressing βH Φ − s 2 N e for any function g 1 G √ ϕ 32 x , giving an Einstein-Maxwell-dilaton theory [ 2 λ∂ ϕ ] and references therein). Furthermore, by changing boundary d = ) = Tr( 32 Z , we get back to the original action, in the meantime finding that it direction on the boundary. This means that we can write the three-  J β, θ . ( ϕ N iβ Z Maxwell π π run over the two remaining coordinates, 2 I − 2 πG θ . Fortunately, the simplicity of two-dimensional gauge fields allows us to 16 A = a, b generated by − τ is the Hamiltonian and ϕ , ¯ = . Finally, we must fix the holonomy of the gauge field iβ H π β 2 + EH 1 θ This will not be the most convenient form in which to study the theory, due to the Given this ansatz, we can now write the action ( The simplification of the S-wave path integral is to consider not completely general I −  = If we integrate out is related to the field strength via In particular, note that for real Euclidean three-dimensional geometries conditions the resulting effectivethis, action we will rewrite in the fact Maxwell be piece of local the in action terms by of introducing an Φ auxiliary and scalar field is gauge invariant information containedthe in boundary). the one-dimensional gauge field pulled backpresence to of integrate it out (see [ The indices Kaluza-Klein gauge field, and the three-dimensional boundary conditions inwe terms must of have constant two-dimensional dilaton fields, Φ we = find that compact gauge group due to the 2 fields and integrate over in terms of a two-dimensionalcan be metric taken to be independent ofthat the leave coordinate the metricshifts in in this form consistlatter of acts two-dimensional as diffeomorphisms, gauge as transformations well as metrics, but onlytranslation those in which the respectdimensional a metric as U(1) symmetry, which we will take to act as function of the theory, where terms of the left-τ and right-moving conformal generators With these boundary conditions, the resulting path integral is interpreted as the partition JHEP01(2021)118 = J 2 || (2.9) 3 (2.12) (2.10) (2.11) δg we pick a || .  J  (a Neumann 3 1 ` J − = 2 κ J  Φ ds ∂ Z , ) + 2 ). ), leaving behind the local effec- . But we can in fact identify  µ. F β, θ J Φ ( 2.12 2 3 2 Z ` iθJ Z 2 . Similarly when we integrate-in − 2 + for spin. This means that we would like J iJθ U(1) symmetry of the boundary torus. to be not only constant but an integer, 3 1 2 = θ δA − 3 × J + ) black holes’, for reasons that will become Φ dθ e A Φ 2 Z θ π R ) J , 2 i J J 0 = black holes. – 9 – 2 i ∂ N Z 2 Z ) G || = π 1 Z (8 − 2 , δA || − in the Maxwell action eventually lead to spin dependent one- = 2 3 ∂ ) = I SL(2 R β Maxwell 4 ( Φ I , which suggests that it is nicer to work in an ensemble of J  J Z 2 g ]. They fall into a discrete set of topological classes, with one √ 9 is a symmetric tensor quadratic in the inverse 3D metric) implies a natural . This precisely imposes the constraint that x 2 θ G d partition function Z for the Maxwell field.  J µ (where N R π -periodic variable. cd 2 2 πG π J δg 1 2 16 ab δg − is a 2 = abcd θ However, this two-dimensional action does not quite capture the whole theory. We We thus find a theory of metric and dilaton only, governed by the two-dimensional This still leaves us to sum over possible values of Now, we may instead integrate out the gauge field, which imposes the constraint that It might seem that the factors of Φ gG EH is constant, and the Maxwell action becomes 4 I √ loop contributions.R This does notmeasure happen, for since the U(1) ameasure field natural that to does be choice not defined of introducefor by spurious 3D the one-loop path measure factors. integral These defined over reasonable the by choices gauge imply field the contributes final only answer as in ( The finite action classical solutions ofa three-dimensional complete gravity with classification torus [ boundarysolution admit in each classThese respecting solutions are the usually fullclear. called U(1) the The ‘SL(2 other solutions add perturbative excitations of boundary gravitons, obtained will find that wedimensional must theory. include additional To nonperturbativedimensional see theory. dynamical why, objects we in next the two- examine2.2 the classical solutions The of classical solutions: the three- action before integrating over boundary condition fixing the asymptotictive field action strength which requires us to introduce the boundary term since with the angular momentum fixed spin, rather than withto a study chemical the potential spin- J In fact, we see that gauge-invariance requires JHEP01(2021)118 . r ϕ E = 1. by a ϕ, t which (2.16) (2.13) (2.17) (2.15) (2.14) ˜ z θ with ˜ c . d 3 ) such that ˜ − β + = a, b , E ˜ Q t 2 − r ˜ θ, , − c d − + r . . itself, corresponding ϕ 2 + b − d ( ˜ 3 r iβ π ) ) elements, but by the + + ∼ 2 + − c πi Z r ) , 2 θ we can choose the second cτ aτ E i + ˜ t 5 + = , = + d c d π, r ˜ τ ( ), write the real and imaginary π (and by choice of orientation we + c 2 E +2 ˜ t − − i cτ ϕ c, d ( ˜ = = τ, τ + τ, ∼ ), and write the metric of AdS ¯ τ ϕ + ˜ ) β + ˜ ( ˜ i z E π ˜ z 1 ˜ t 2 + ∼ ∼ ˜ θ ϕ, = ( ( ˜ to pick out any simple closed curve in this way, π ) black hole, we simply define new coordinates 1 ). z + 1 2 – 10 – + 1 Z Z w z , , ˜ z = ) transformation, not to be confused with the central charge. , ∼ Z ) in terms of the original untilded coordinates 2 τ ∼ , . The solutions are thus in one-to-one correspondence , ˜ ϕ b c ˜ d d z , where infinity accounts for AdS SL(2 d + + (which is ambiguous up to the addition of a multiple 2 through ˜ 2.15 for coprime integers − ∈ Q , θ Q r b c a z d )
2 − , z ) black holes are simply solid tori; there are many such + ˜ r , − + Z 2 ∪ {∞} E r c d d a b + , r − 2 r and it Q ˜ cτ i − π r Q 7→ + z 2 + 2 by d a + r + 1 + ˜ r ¯ cτ . It is convenient to write the solution in terms of new parameters τ ϕ π ( c Q ∈ 2 ) mapping class group of the torus. To see this explicitly, we first 2 c + τ, = Z = 2 E dϕ , ˜ , where in order that these form a basis we must choose ˜ t = = z z or b d . Equivalently, they are classified by the rational number τ ) β Z 2 + / r ) . The solutions are thus classified not by PSL(2 β, θ here parametrizes the SL(2 lattice identification as ˜ π aτ or Z c 0): z , respectively) does not affect this metric, since it acts only by shifting d c > = (1+ ˜ = 1. We thus have 2 bc and ds = 0. Now, to understand this solution in terms of the dimensionally reduced theory we Now, to construct a Euclidean SL(2 Topologically, the SL(2 The factor 2 − c c 5 , related to − 3 `  would like to express theTo do metric this, ( we simplyas make the the relevant coefficient substitutions of andr define a new radial coordinate multiple of 2 coset PSL(2 gets mapped to infinity by with rational numbers to Note that theof choice of generator as ad as real and imaginaryparts parts of of the ˜ ˜ chosen as the spatial circle: may set Having picked the first generator of the lattice to be on a lattice, These coordinates pick outBut the we can spatial choose circle arunning as different from coordinate a the line origin to running from the origin to sufficiently rapidly to be regarded as part of thesolutions, gauge since symmetries. we haverelated a by choice the of SL(2 whichuse boundary complex cycle coordinates to to fill write the in. torus Different as the choices complex are plane identified by translations by acting with an asymptotic symmetry generator, a diffeomorphism that does not fall off JHEP01(2021)118 . ). is θ J 2.8 , and (2.22) (2.23) (2.18) (2.20) (2.21) (2.24) (2.19) − , where , we see in ( and the r + π . 2 r F ) ρ θ 2 − = ) with real r ]. Indeed, for . + + r − r − 2 13 2 + r r 2.15 + β, ϕ r ∼ + N = 0 looks like a cone 3 + E ` G t r ρ 4 ( − ∼ = r ) J π . ), so we may immediately write 2 and the field strength ⇒ , + 2  , = J 2.4  E 2 . 2 E , ϕ  E µ dt E E dt − π dt t ) β defined as r ( 2 dt r  + ( ρ  − r ∼ f − 2 r 2 2 r ) r N ρ + 3 r + + , the metric close to r is an angular variable with period 2 2 ` G 1 r , ϕ c i ) c 4 2 i r E E i ( t t – 11 – − + dr ( − f π β = 2 2  θ r, β dϕ E 3 3 , so we can write dρ , we find ), when we go around the spatial circle we must also  ` `  E θ dt  2 , ∼ 6= 1, since we have slightly different identifications. r r = = ) dt ∧ + 2 c 2 − 2 when we Wick rotate the time coordinate. However, it is + g Φ = ∧ A r g 2 dr 2 E , so that − − 3 − − β, ϕ dr r ` β is imaginary for the Euclidean solutions of ( 3 dt r 2 = 1. For larger ) r + r + − r 2 c r ( r )( E r i t f 2 + ( in terms of r . + = 2 β ∼ π c − ) 2 2 ) . With this understood, we can write our solution as follows: r 2 ( can be reconstructed (at least locally; we will see later how the holonomy has period r dA E dr , ϕ ( f t ) for A E θ β E = t t = F = ) = 2 2.16 r λ ( ds is positive, and f 2 = 0 this is the BTZ black hole, in Euclidean signature for imaginary which does not have such a shift when we go around the Euclidean time circle. − 3 is smooth only if + ` E r 2 t , d g θ β All seems well so far. But if we look more closely, we will see that this solution is This metric is manifestly in the form of our ansatz ( where has its largest zero. Using a new coordinate − = 1 Recalling that that with opening angle imaginary for the Euclidean solution. apparently singular! To seef this, we examineexpression the ( metric near the origin This is the usualalso expression for be the read angular off momentum from of the the BTZ asymptotics black of hole, the which three-dimensional can metric. Once again, The gauge field arises) from the relation between theThe angular momentum volume form is given by From the two-dimensional perspective,parameter this acts as a gauge transformation with gauge it down in terms ofthat two-dimensional in fields. the current Before form we weidentification do have ( this, a nontrivial we gauge first bundle addressperform at the infinity: a subtlety due gauge to transformation. theϕ nontrivial We can remove this by using a new angular coordinate This result takes thec familiar form of theLorentzian signature metric for of real thenot BTZ quite black the hole BTZ [ black hole for We then have where JHEP01(2021)118 . c d 7→ 7→ nπ π θ τ 2 (2.27) (2.26) (2.25) + 2 − θ = = 0. This 7→ A ρ θ by multiples R d , ) ]. β, ϑ = 0 corresponding to the 33 ( z . Q , we can shift N 3 n iϑJ ` G − ) with integer spin, it is sufficient dϑe |  J , Z 1) (with 2.10 E , π − | β 1 dt 2 for integer c d E iJθ < E π e 2 dt 0 – 12 – J β πn ) black hole solutions, we record their contribution X , the classes correspond to elements of the double 2 ∼ − Z , nπ + , A ) = N 3 A ` + 2 nπ G has a delta-function source of this strength at in the interval [0 θ 7→ F c d + 2 7→ A direction by the holonomy. A closed curve only arises after |  − θ J | ϕ β, θ = ( z Q n X , where the groups of integers on the left and right both act by ) black holes (though not the later generalizations) have real three- Z Z ) to the path integral, we can rewrite a sum over its images , / , have complex Kaluza-Klein gauge field. These are real Euclidean solu- ) J Z ). , β, θ times. A nice description of this is also given in [ ( z c 2.18 PSL(2 ; they are not gauge equivalent because the identification with the boundary metric \ Z nπ To complete our review of the SL(2 We note that the solutions we have been led naturally to consider, using the ensemble We thus see that in the fixed spin ensemble ( Before giving a complete two-dimensional description of these defects, we note one To see how this is possible, given that we started with a smooth three-dimensional . This moves us between isometric but inequivalent gravitational solutions, with c + 1. This is also the classification of solutions of a given angular momentum and either +2 to the density ofextremal regime, states. we will Since give the we answer will only in be that studying limit, the taking theory exclusively in the near- tions for the two-dimensional metricEuclidean and metric. gauge field, The butnature. off-shell have configurations complex The we three-dimensional SL(2 generalizedimensional to Lorentzian later sections, will whichseen be are from of nothing ( the but the same usual BTZ metric as can be identification by such shifts coset τ mass or temperature, which is more pertinent for us since we workof an ensemble fixed of integer fixed spin. and we see that the Poisson dual variable isto precisely restrict the angular to momentum. rational usual BTZ black hole). Under the classification which identifies geometries only up to related in the same way), which cancontribution be accounted for simply by quantizingusing spin. the Given Poisson some summation formula, important feature of the two-dimensional solution.that wind By making around large the gaugeof transformations circle, θ is different. We can simply sum over these solutions (or, more generally, any configurations means that traversing anot small give loop rise around to the atranslation origin closed in in curve the the in angular two thegoing dimensional three-dimensional round metric space; rather, does there is a nontrivial solution, we must also look at the gauge field near the conical defect. We have so that a small closedEquivalently, the curve field strength surrounding this point has nontrivial holonomy JHEP01(2021)118 . Z q , but ∈ (2.31) (2.28) (2.30) c d 1 , −  − = ) ]. | q ) J Q 1 14 ) , − | . 7 − | p J (  p E ) ( | (BTZ) (2.29) − | J N 3 ) E ` G | ( 8 . Since they are smooth − | J N . 3 π q ` q G 2 E  8 − | as ( π q ) 2 | N E J 3 q  J ` ( G 8 N 3 − | ` , which is the Brown-Hennaux relation G q 8 cosh N 3 π E . ` ) G ( 3  | J 2 q ( N J 0 3 | ` G S 8 q 1 N 1 = coprime). They provide a delta-function 3 cosh e ` G ) q − | q 4 , that is the solution to J N c q π J ( | 0 2 q G 3 ; in the previous section we had S 4 ` π and  1 2 Q e p – 13 – s | ∩ N J  sinh | ) = 2 1) modulo G 3 q ) , ) J 4 ` 1 : we choose p ( J q c , with ( 0 (0 − s 0 p 6 S ( S J e πi 1) 2 | Q ∈ p < q e N − ) black holes, coming in families labeled by the rational pa- J ( | G , and create a conical defect of angle Z − ≤ 3 , contribute as , 4 N q ` p 1 3 ) as the Bekenstein-Hawking entropy given by the area of the ]. We discuss here their incorporation into the two-dimensional Q ` (1 π G we have J  s 8 ( q p ∩ 34 J π 1 2 0 q p 2 S ⊃ 1) = = πi , ) ⊃ 2 Q ) black holes, we have seen that smooth configurations in the three- E ) (0 − Q ( e Z E J ∈ , q ( ρ J denotes the inverse of q p N 3 of strength 2 ρ ` has a renormalisation ambiguity from gravitational quantum corrections, we define it more q G ) = 8 F N 1 − G Q ⊃ p ) E We will take these boundary conditions at the defect as a definition of a KK instanton, We have a countably infinite set of species of instantons labeled by rational numbers First, we define ( Since 6 J ρ but it can also be helpful to understand themprecisely from via the an dual action CFTwith formulation. central a charge natural To one-loop impose shift. the strictly between zero and one, here we will write source for in the three-dimensional geometry,the they action should once not these boundary provide conditions any are singular obeyed. contribution to dimensional theory give risetheory, which to are particular localized in singularanalogy spacetime. objects to We the call in Kaluza-Klein these monopoles the objectsspacetime which dimensionally Kaluza-Klein arise dimensions instantons, from reduced [ in compactification fromtheory. five to four 2.3 Kaluza-Klein instantons From the SL(2 where ( In particular, for The other families oframeter SL(2 Now, we must treatcontributes the to usual the BTZ density black of Virasoro hole primary solution states as of a spin special case in this limit. It units. We will apply this limit to the exactclassical result, extremal which BTZ can black be hole: found in [ This means that the temperature is low and the horizon is large when measured in AdS JHEP01(2021)118 is a as a (2.32) (2.33) J q p πiJ 2 for a scalar − ). Due to this by the integer δf J R 2.32 . After integrating J ) is modified to 2.8 q p J πi inserted to change to the fixed spin 2 δ, ]. − q p iθJ π 35 N Φ − π G 2 + 2 = 4 µ ) after rewriting the Maxwell theory in terms ∂  , the instantons are nonperturbative, solitonic I q 1 , which can be computed by the Gauss-Bonnet – 14 – J N 2 i 1 ) term in the dilaton action, this sources a conical 2.5 G − R 2 = 1 R  2.11 F ). Here, the dilaton Φ is evaluated at the location of background with the dilaton approximately constant, so = ) then remains unfixed, and this can be determined by 2 J ). This amplitude or measure may depend nontrivially on 2.9 at the location of the instanton). We still have that p/q I f a two-form delta-function defined so that 2.5 ) is of order δ using ( accounts for the Maxwell piece of the action and induces the correct 2.32 J q p ) defines the KK instantons classically, but a full quantum mechanical J πi 2.32 2 − ) precisely cancels the dilaton term in the instanton action ( gives the value of f 2.12 The action ( The advantage of formulating this two-dimensional theory with KK instantons is that Since the action ( The term chosen by boundary conditions. We can think of the resulting action background fields, in particular the dilaton.dimensional Properly analysis. determining this However, requires ingeometry a approaches the full a three- near-extremal rigid AdS limitany we natural describe measure in appears to aconstant reduce moment, (which to the can a depend simple preferred on choice. Only a normalization integral over off-shell configurations. Theas resulting Seifert three-dimensional topologies manifolds. are known treatment requires more input.for the In insertion particular, of each atwe instanton, one may or insert loop a it measure we for (see must integrating section specify over the an locations amplitude at which smooth solution of 11-dimensional supergravity [ we can now considerdo the not effect of obtain includingto new multiple minimize classical instantons the in solutions action. the in path Nonetheless, this integral. we way, We can since in the several instantons circumstances tend perform to the attract path contribution to the boundaryensemble, term coming ( from the additional holonomy sourced by the‘D-instanton’-like instanton. objects. Anwith solitonic analogous D6 phenomenon branes arising occurs as in the Kaluza-Klein type monopole IIA upon compactification supergravity, of a including a sourcefunction (with constant, which our boundary conditionsout set the equal to gauge the field desiredJ spin we can therefore simply replace the auxiliary field source for the field strength. With this addition, the relation ( the instanton. In concertdefect with of the the desired Φ strengthfunction at contribution the to instanton’s location. thetheorem curvature This applied conical to defect gives a aaction small delta- ( disc containing thecancellation, defect. the overall The contribution resulting to the singular action piece from of this dilaton the piece is zero as desired. to the Einstein-Maxwell-dilaton actionof ( the auxiliary field boundary conditions for a KK instanton, we can add an action JHEP01(2021)118 , , ) 0 0 S 2.12 (2.36) (2.34) (2.37) (2.35)  ∂ φ  , which shifts . βJ , so we may solve  0 2 1 ` of the boundary to S ∂ and expand to linear − L 2 φ κ N satisfying 1  G ∂ . φ dsφ N + 4 is the extremal entropy of BTZ, ∂ 3 G 0 ` Z 0 S 16 − =  , 2 2 γ 2 J. ` N 3 J ` . Second, it induces boundary conditions 3 G + ` J N 4 2 G = R r space is the near-horizon geometry of a near- 4 π  where 2 p M – 15 – φ . Its coefficient 2 = 2 = solutions of the theory. These must have constant g χ 0 0 ] and references therein for details. 2 √ S Φ x β, 32 2 2 γ d ` . This AdS Z = geometry, we write Φ = Φ 3 ` 1 2 2 ∂ ∂ 1 2 , setting the ratio of the proper length φ L − ∂ = χ 2 0 ) black holes. A more careful understanding of this is an important ` S Z boundary conditions, this classical solution from the region far from , − (with both taken very large): J ∂ = φ radius JT 2 I ], which is the analogue of near horizon extreme Kerr). To incorporate fluc- 48 . This leads us to the action of JT gravity, φ With our fixed As an example of a more generic phenomenon, ourWe can Einstein-dilaton begin theory by looking ( for AdS the energy by the valueat of our extremal artificial BTZ, boundary the dilaton value is sufficiently small that the linearizedthis approximation leading region, to quantum JT corrections gravity are is suppressed valid.the by Outside theory the classically. large parameter the black hole has two important effects. First, it contributes an action which we take tolocation be where very the large. valuedemarking of the edge The the of the final dilaton ‘near-horizon’ is boundary region. set term Inside this at is near-horizon a region, introduced the constant dilaton at an artificial The first term hereproportional to is the the Euler characteristic two-dimensional Einstein-Hilbert action, which is topological, giving the AdS extremal rotating BTZ black holegeometry (completed [ to thetuations three-dimensional ‘self-dual away from orbifold’ thisorder AdS in dilaton at a zero of the dilaton potential, which for us sets The two-dimensional AdS radius is then set by the gradient of the potential at Φ = Φ as well as beingphysical tractable, effects. this is in fact thereduces regime in where the theyreview near-extremal this, give limit referring rise the to to reader Jackiw-Teitelboim to important [ gravity. We first very briefly problem for the future. 2.4 Near-extremal limit One regime in whichnear-extremal limit we of are rapidly able rotating to black study holes at the low-temperature. Kaluza-Klein We instantons will in see detail that, is the matching to the SL(2 JHEP01(2021)118 (2.40) (2.41) (2.39) (2.38) . .  ) 3 x ( p/q I − in terms of the JT e ) x . appears as the coefficient 2.32 q p , and we integrate over all γ . x ) Ω( ) ], x x πiJ ( ( , 2 2 20  g ) p/q − I !. The sum over all sectors then x p k − ( φ below which quantum fluctuations x e 2  ) 1 p/q q d 1 x I − γ − − Z   1 ) Ω(  x ( π 2 ) exp g x = exp – 16 – + 2 p ) l 0 Ω( x x 2 S 2 ( d g  p/q q 1 √ I Z x − 2 − − e d ) for an instanton at the point ) 1 l I  x Z → = 2.32 -instanton contributions by ) is simply linear in the dilaton, so the instanton gas adds an I ) Ω( k l p/q x I ( 2.32 2 g p l sets the energy or temperature x 2 γ d ) is the action ( Z x ( k =1 l Y ! p/q 1 I k Now, we can include many instantons by simply inserting many such factors into the Note first that we can introduce a single instanton by an insertion into the path Now we simply need to incorporate the Kaluza-Klein instantons into this near-extremal =0 ∞ k X integral. In other words, it is equivalent to adding a new local term to theOur action, instanton action ( exponential to the dilaton potential. exponentiates: But now this amounts to an insertion into the path integral of the exponential of a local we will be contentdiscussion. to leave Ω as an unknown functionpath in integral. this section’s Totons, more avoid we qualitative overcounting must equivalent divide configurations the from permuting instan- possible insertion points. Wefields have in included the a theory. measure Thisbe determined is Ω, if either which we taken use is as aour possibly definition part theory independent a of obtained of function the from the dimensional definition of actionin reduction. formulation, of as the the The is instanton, measure the classical is or case limit, of for must subleading and importance Ω should approach a constant in the near-extremal limit, so integral of where possible alternative approach,incorporates summing the the effect of KK theis instantons instantons usually into as introduced a an as shiftthey of an are instanton well-separated the approximation, gas. and dilaton taking do potential. notfundamental a pointlike interact. This This objects sum method Here, in the over however, two-dimensional if theory, instantonsrequired we there is in think in no of principle. a such our approximation limit instantons as that We will study the theory of JT gravity2.5 with such defects Instanton in gas detailBefore in section turning to the analysis of JT gravity with defects outlined above, we mention a become strong. In the Schwarzian descriptionof of the the Schwarzian theory action. [ theory. We can dovariables: this by simply rewriting the instanton action The parameter JHEP01(2021)118 (3.1) and a (2.42) (2.43) 2 g ], making 8 ) as . 2.12 1) − 2 , κ , (  N Φ dsφ π (Φ)) G ∂ Φ. The instanton gas from the 2 4 U 2 3 Z 2 ) ` 1 − − − − 2 q 3 R − − + 2) (Φ Φ (1 2 2 2 of instanton insertions and summing over ) g R − J k (  √ N φ x 2 2 G g exp – 17 – d (8 q √ p 1 2 Z x 2 , leads us naturally to a two-dimensional theory of , is a theory of a two-dimensional metric πiJ d N 2 e π Z 2.4 2.4 2 πG -instanton path integrals. Conversely, when we have a ≈ (Φ) = 1 2 k 16 U − − (Φ) χ 0 = p/q S I U − = by addition of a term proportional to JT . I U , with action φ 5.5 shifts . This occurs close to the threshold for the lightest black hole at small spins. N p/q but very low temperature. This means that Φ is still large so the new exponential I G J We now have two potential ways to compute the effects of instantons, either by com- The second important regime, which we concentrate on in this work, occurs at large First, these terms will be of leading importance if we have a regime where Φ is of More explicitly, we can write the bulk part of our dilaton action ( , or by using the instanton gas potential from summing their effects. These approaches the necessary extensions toextremal incorporate limit defects. of BTZscalar JT dilaton in gravity, section as encountered in the near Jackiw-Teitelboim gravity with thestudy inclusion this of family fundamental of defects. theories in In their this3.1 own right. section, we The JTWe path here integral with review defects the genus expansion of amplitudes in JT gravity following [ this in section 3 JT gravity withOur defects analysis of dimensionallythe reduced near-extremal limit three-dimensional of gravity section above, in particular in have their utility in complementaryton regimes. number is For circumstances small,able where the to the first typical explicitly approach instan- calculatehigh is density, the useful; the effects indeed, maypotential, in be arising the describable as next the in section collective semiclassical behavior terms we of from will a the be condensate instanton of gas instantons. We will discuss terms from instantons are small,of but the there ground are state physical energy. effects nonetheless, such asputing a the shift path integralk with a fixed number There, higher topologies, includingKK (but instantons, not are limited no to) longerphysics the suppressed, semiclassically. manifolds and we represented appear by to the lose all hopespin of describing the where we should remember that the prefactor is subject to determinationorder of the measure Ω. where the dilaton potentialaction is JHEP01(2021)118 n ) (3.5) (3.3) (3.4) . We k 5.5 as a large in total: , . . . , α 0 i 1 k S i α to infinity. We ; n ; we explore this P ∂ of the connected λ , with parameters L = F g k , , . . . , β i 1 held fixed. (3.2) to unity. The coefficient ) gives us a defect q p β 2 ( ,...,N ` using the Brown-Henneaux , with γ β 2.38 i πiJ = 1 c as the renormalized length of g,n,k 24 = corresponds to temperature of i Z + 2 β ∂ ∂ γ = . ], we can include a defect by an 0 ! ) φ L S i x ! N ( 36 i i k 3  φ G k ` λ ) q 1 α 16 . This amplitude can be expanded as i − , − C Y = (1

of order 1

π ) . γ  2 . We should therefore regard of each species n 0 1 2 β 0 − . As in [ S β → ∞ i − S e ( λ k 2) h ≤ − ∂ 2 Z − e g – 18 – n , φ √ + exp ∂ ··· < α g x we had ) L 2 ∝ (2 1 on the boundary, which we also take to be large, but d − β ( ∂ e 2.4 p/q Z φ Z =0

parameterizes the suppression of higher topologies, which λ 1, though to be conservative, for reasons that will become is fixed once and for all, and acts only to rescale the units 1 of the boundary, and ultimately take ,... χ γ 2 ∞  X ,k 1 , λ and a weighting ∂ ]. This means that q 1 L < α < g,k α 49 = = α C

. ) n ∂ ∂ KK instanton in our 3D analysis, the action ( φ L β ( Z Boundary condition: p/q ] we denote this by is measured. In section 8 ··· β ) 1 β ( . The path integral is defined to sum over insertions of any number of each species i The general amplitude we are interested in is computed by the path integral with For the Our theory in addition contains dynamical ‘defects’ or ‘instantons’, parameterized in We impose boundary conditions on circular boundaries ‘at infinity’, which means the Z of the Euler characteristic

, α . Without loss of generality, it is sufficient to compute the connected amplitude, and 0 i a sum over sectorsspacetime, give as by well as different the topologies, number labeled of defects by the genus but in this section we take arbitrary parameters. boundaries, each with boundaryβ condition parameterized by itsfollowing own [ renormalized length clear later, we can restrict further to 0 develop below as our precisepath definition integral for definition the defects a and littlemay the more also parameter in have the several species contextλ of of the defect instanton labeled gas by inof an section defect. index We will take 0 terms of a defectinsertion angle in the path integral proportional to For our purposes, we will take this more as a heuristic guide, and use the expansion we in which value of central charge [ order one in Planckthe units, boundary, not or in an AdS inverse units. temperature. We refer to take the dilaton to befixing a the constant ratio The dimensionless parameter S will organize into anparameter. asymptotic series in we fix the proper length We have here chosen units to set the two-dimensional AdS length JHEP01(2021)118 ,

1), ,...... (3.6) 1 − ) with , can be g . geodesic geodesic 3.5 = 1 case = 0 1 n n k + 6( g,n,k , n n Z with = 1 , as in ( , and it becomes

g g ) α β ( = 1 in figure Z

surface with defects in n g k (without defects) can be com- ‘trumpets’ with one geodesic = 1 contributions for . = 0, with some number of defects n 2 g,n β b , g Z 2 γ , g − = 1 e = 1 n are included to account for overcounting n γ πβ are of the same or different species. We 2 can be obeyed on the trumpet geometry, α r g,n,k β Z ) = – 19 – β, b ( , where the dotted line corresponds to the geodesic of asymptotic boundaries has a unique geodesic homotopic (the ‘convex core’), and 1 n trumpet n ] was that the amplitudes Z 8 = 0 amplitude with one boundary and disc topology, every ), so acts as a Lagrange multiplier constraining the curvature g . The second row shows the k , . . . , b ! in this definition of λ 1 3.1 i 0 b k S e = 1, n . We then have cut the spacetime into a genus n . k , of order λ 0 S , . . . , b . The first few topologies contributing to the expansion of ,... 1 − 2. The resulting metrics are locally unique, so the path integral reduces to an . On the asymptotic boundaries, we must integrate over all ways in which the b 2 e b , − 1 , The remarkable result of [ = = 1 : = 0 : = 1. The top row shows the topologies for the disc 2 = 0 g g Next, we must integrateboundaries over of all the constant specified curvatureand lengths. it surfaces turns of This out is genus that a the compact correct space measure of on this dimension space 2 is provided by the WP symplectic boundary and one asymptoticin boundary. the bottom This diagramlength split of is figure shown forboundary the conditions of renormalizedwith the length result exceptions of the constant curvature surface with to each boundary. Welengths can cut theboundaries surface of along lengths these geodesics, which we take to have puted from the Weil-Petersson (WP)surfaces volumes with of boundary. the Tolinearly moduli understand this spaces in result, of the we constant-curvature actionR note ( first that theintegral dilaton over appears locations of thefinite-dimensional boundary moduli (governed space by of the surface Schwarzian theory), with a and given over the topology. More explicitly, with the by permutations of identicalregarded defects. as With distinguishable, thisirrelevant labeled definition, whether only the two by defectssketch their with the the defect first same parameters few terms in this expansion for a single boundary n k of order The symmetry factors Figure 1 JHEP01(2021)118 b . ]. ) n ] in 38 is a (3.7) (3.8) (3.9) 43 ). inst α δ , . . . , b . ). Finally, 1 − b n ) ( 2 (1 . Fortunately, β 2 π g,n β V + g,n 1 ) , . . . , b V 1 β ), where 1 n β b √ ( ( , b , we impose constant inst π n δ φ 2 g,n β ) ( V α informally understood as ) = − , with degree equal to the b 1 n , b . (1 2 trumpet γ β β π 2 ( Z 4 π n ) is exponential in the dilaton, it 2 e − , . . . , b db 1 3.3 n b 3 b 3 trumpet ∞ + 2 γ πβ 0 Z 2 2 ) Z R , b s ( ··· 1 φ ) β 1 ( R ) = , b 1 2 – 20 – β 1 . Putting these pieces together, we have ( ], giving the expansion of the amplitudes without ), with the factor of − β 8 2 ( of the geodesics separating the trumpets from the 1). b =0 bdb trumpet ,k b ; the first is the disc which we have already addressed, − − =1 1 g 1 b ,n trumpet ( bdb Z δ Z =0 1 + 6( 1 1 g ∞ b 0 n Z = 0 double trumpet, which follows from the above if we set db Z 1 b , g ∞ ) = 0 2 2 ) = ) is only as useful as our knowledge of the volumes b 2 Z = 2 3.7 , β − 1 n ) = 1 2 β n b ( ( δ =0 2 1 ,k , . . . , β =2 1 ) = ,n β 2 = 0 disc, for which we have ( =0 , b g =0 1 , g Z b Now, just as before, we are left with an integral over surfaces of constant curvature, This concludes our brief review of [ The expression ( There are two exceptional cases for which this formula does not apply. The first is the ( 2 , = 1 g,n,k 0 Z allow homotopies to pass through anybut defect. the This first split two is diagrams madeand in along figure the the second dotted line will forexcept be all that discussed we in must a usehas moment. volumes a We of natural thus moduli have generalization spaces the to with same this conical formula setting. points. as before, The The fact WP that measure this is the correct measure for curvature everywhere exceptsources at of the positive curvature. locations These of are conical defects, defects, where withnow defect we with angle have 2 conical delta-function points.surface has a And unique just geodesic as homotopic before, to excepting each asymptotic some boundary, special where cases, we every do not such volumes of moduli spacesa of somewhat hyperbolic different surfaces context. withintegrate conical out First, defects, the let also dilaton. us usedcan understand Since by be the the [ thought defect effect of insertion ofinclusion, as ( the another the defects term bulk in when JTdelta-function the we at action action the becomes which location is of linear the in defect. the When dilaton. we integrate With out this they can beIn efficiently computed particular, due the todimension volumes of a are moduli recursion even space relation polynomials 2 found in by Mirzakhanidefects. [ It is now a rather simple matter to include the defects, using a result on the The second is the V coming from the residualof gauge the symmetry separating of geodesic: rotating both boundaries along the length n we must integrate overconvex the core lengths with the correct measure form. The result of the integral is thus the WP volume, denoted JHEP01(2021)118 , = 1, ) ) for = 1) k (3.12) (3.13) (3.10) (3.11) n χ 3.11 . ) k , . . . , α 1 = 1, which we α πiα ; k of the boundary n k = 2 k ) does not obviously + ) to 7 ); this area is negative n , . . . , b α 1 3.10 . 3.7 b ) ), by rather formal use of ( − . 2 (1 , . . . , b πiα 3.10 1 α g,n,k γ β P V 2 = 2 ) π π πiα n 2 1. To see that this occurs, we can = 0 and a single boundary e , b g + 2 ≤ β, b = 2 n ( ]), with the restriction essentially due ) π β geodesic boundaries with the specified γ πβ 2 ( α ) which we applied earlier to the double +1 2 n 2 42 − . n , − b 1 2 r ) for the amplitudes, along with ( , b = 41 − trumpet n (1 ≤ 1 Z trumpet A b ) = 3.10 P , with α Z ( α g , δ – 21 – n ; , . . . b π 1 ) = 1 including cone points. Remarkably, this does not 1 = 1. The relevant single-defect path integral was b β db α b ( n boundaries, simply be evaluating ; ( k b k : β =1 k ( ) = + g,n,k ∞ ] (see also [ ,k 2 0 + V =1 πiα g,n Z 40 =1 , b n [ ,k V 1 ]. We thus can simply generalize ( ,n b 1 2 ) = , is correct even when the argument that led there does not ( ··· = 2 =1 k =0 43 2 ) , ,n b g ) = ≤ 1 0 k Z V =0 g,n,k , b α g 1 V Z β , . . . , α ( 1 , . . . , α α 1 ; n α ; ], with the result trumpet n Z cone points with the specified defect angles. We take these amplitudes as a 36 1 ]. 1. are the WP volumes of genus k , . . . , β db 43 , . . . b 1 1 1 β b α < b ( g,n,k ( ∞ without defects but with V 0 k Z g,n,k We now address the special case We now turn to the volumes Before discussing these new volumes with cone points, we address the special cases + g,n,k This definition of the theory explicitly excludes the possibility of a defect hitting the Schwarzian bound- Z 7 V g,n We take this as evidencethe that volume the polynomials formula ( ary curve. As weas will long see as later this process is suppressed anyways for the Schwarzian boundary conditions, In fact, rather remarkably, thisthe result two-boundary can volume be obtained fromtrumpet: ( gravity in [ computed in [ lengths at imaginary values This has been proven for to the special cases discussed above. This formula was recently applied to de Sitter JT treat as a specialour case attention below. to We sufficiently can strong avoid defects, meeting any other specialrequire any cases new by ingredient! restricting TheseV volumes can be computed from the volume polynomials apply. This occurs onlywhen for the disc total topology, defect with angleuse genus is the less Gauss-Bonnet than theorem 2 with to geodesic compute boundary and the defects,if area finding the of total defect a angle negatively is curved too disc small. ( This always occurs for a single defect lengths and constructive definition of what we mean by JT gravitywith with defects. boundaries whichinto are trumpets not and surfaces homotopic with to geodesic boundary, geodesics, and formula so ( we cannot split the surface where our integrals was derived in [ JHEP01(2021)118 i ) n ) is β β ( ( (3.15) (3.14) Z Z , ··· β is the flat ] is closely ) . 1 38 1 2 β such that the ( dH are the integral ≤ Z V h , φ α ) as moments of this H i n ) β n − β e ( expansion agrees with the Z ), we recall the interpreta- Tr( L ··· ··· 3.10 ) of a dual quantum mechanical 1 ) β H H ( in JT gravity without defects. As 1 β Z i . We will mostly restrict to a single ) h − ) k e H n ( Hermitian matrix (and β V ( and tune the potential L Z Tr ) Tr( L × amplitudes H ··· − for any ( e L ) 1 → ∞ ,k – 22 – ). This equation is equivalent to the statement ], Mirzakhani’s recursion relation [ ∝ β = 0 ( ) =1 L ], implying that the level sets of dH µ Z 3.1 50 ,n , g h H , 51 ( [ Z =0 µ 45 g 2 = 1 g = to be an Z i n from the JT path integral. From the agreement between ) H n µ β ) is the corresponding partition function. However, since we ( Z βH − e ··· this Hamiltonian is not unique, but rather selected from a random ) , taking 1 H β V ). The structure of such matrix integrals is extremely rigid, since the ( 1), though strictly speaking they are proven only for , Z 3.8 h ) = Tr( (0 . β ( ∈ -boundary path integral Z is a Killing vector for g, n n α φ b ∂ ab  for all First, we note that most contributions to the path integral discussed above do not come In the following we will find that such an interpretation exists for our modified theory of Before beginning our analysis of the amplitudes from ( g,n from classical solutions. Onevariation way to of see the this metric isthat in to use the the action equationcurves ( of of motion coming the from Killing vector, coinciding with orbits of the symmetry it generates. For a scaling limit. 3.2 The discWe with begin defects: our analysis ofdefect JT gravity insertions, with the defects amplitudes by lookingspecies, at commenting the later disc on with the any extension number of to multiple types of defect. Z JT gravity with defects.are We consistent will with show analytically thewill that moments allow the of us amplitudes some to ensemble find of an Hamiltonians explicit at solution any genus. for the This matrix integral measure in the double measure on independentlimit components). of such More integrals,resulting precisely, average where density we of we must states take atdisc take leading amplitude order a ( in ‘double-scaled’ the large leading order density of states (or equivalently the disc amplitude) completely determines genus expansion, the moments agreea with matrix a integral with particularly measure special type of ensemble, namely for some potential and learn about theMirzakhani’s measure recursion relations and topological recursion, we learn that at any order in the are integrating is totheory, be and interpreted as theare Hamiltonian integrating over distribution determined by the measurea in random the variable. matrix We then integral,distribution, interpret so the that amplitudes for each for general tion of the observed by Eynard andrelated Orantin to the [ topological recursion obeyed by matrix integrals. The matrix over which we apply. We would therefore be extremely surprised if the results we will obtain fail to apply JHEP01(2021)118 ) . | γ λ 2 | 3.18 (3.18) (3.19) (3.17) at one . Now, 1 − E < E −∞ . 0 ) (3.16) β ( 2 , E > α =1 γ β # 2 ,k
π 2 =1 , ]. The defect in question in e γE ,n 7 2  =0 √ γ πβ g γE 2 γE λ 2 2 πα ) takes precisely the form of ( λZ r 2 √ 0 √ λ S , two defects on a constant curvature 0 e + 1 2 S 2.31 e cosh ] in three dimensions. Indeed, the sum of , but enhanced by a factor of ) + + γE λ β α < 2 12 γ β , a classical solution to JT therefore exists if ( πλ 2 2 p π g =0 2  + 2 ,k e – 23 – ) density ( γ π  3 =1 Z 0 3 , S ,n γE . γ e πβ 2 2 2 =0 / g ∼ ) 0, our naive density of states violates this for p SL(2 J Z s ) black hole [ ) ( π 0 0 0 1 2 Z E 2 S S S , ( e e  λ < ). We can then extract the corresponding density of states − = e = = J Q naive 3.12 sinh 1) ρ " − naive KK instanton described in the previous section, whose one-defect 2 ( i . In the 3D gravity context, this can be achieved by introducing a ) π γ | 1 2 ) and ∝ 2 β λ , for which we can bring the defects as close as desired, but they recede ( 0 ) and ( λ = 1 2 S Z 2.29 e ≥ | h Q 3.8 = 0 λ α ) = E ( 1, the first term is almost always dominant, except at very small energies, where naive |  ρ λ If we include only classical solutions and fluctuations around them, we must conclude In the context of three-dimensional gravity, this is precisely the problematic feature of As a result, if we were to take the approach (analogous to Maloney and Witten in the | for an appropriate that this theory does notforced have a to unitary alter dual. the dynamics To ofof solve the the defect theory, negativity for with problem example we by would introducing be an additional species positive density of states. For the Maloney-Witten-Keller partition function pointedthat out case by [ is the solution corresponds to an SL(2 the BTZ density ( we have so the second term isany suppressed consistent unitary by a dual factor (even of one involving an average over Hamiltonians) must have For where we used ( by taking an inverse Laplace transform, finding case of three-dimensional pure gravity)perturbative of expansion classifying to all classical allcontributions, solutions, orders, computing and the summing the results, we would only have two disc with all defects localizedmetric at are the origin. always separated For cusp, by so a we finite can minimal nevermarginal distance case have at more than which one theto defect surface infinite in develops distance, a a limiting single to location a (and cusp likewise geometry). for the and only if the metricsuch has symmetries a globally are defined thedoes continuous disc symmetry. not The and have only double the surfacesend). trumpet desired with (for boundary Once we which conditions, include the sincelie defects, at corresponding the the fixed dilaton solution dilaton points must must of go be the to symmetry. stationary at The only their additional locations, classical so solution they is therefore the given surface with constant curvature metric JHEP01(2021)118 . ) is β 1. ( , 5.4 , but ,k g,n (3.22) (3.24) (3.20) (3.23) (3.21) λ V = 0 =1 ,n k is breaking =0 (0)). Roughly g k ψ Z − ) , E ), we have ) ( 6  ψ − ( 2 k . 3.11 / ··· 2 3 . n b − ( + )) 3+ 1 O 2 − / − g 3 + dE E 3 β ) ( 2 ]. 2 ,  ∞ 0 ). − O λ 2 R 2 γE k b 14 β πγ = 0 using ( ,  2  3.20 7 ], finding g 2 ) = 2(2 2 6, we are picking out the monomial b (1 + )! E s ( − 3 2 n 52  − , ψ , defined through its pairing with a smooth test − ) g 2 2 k ) = 38 / / 2)! γE 3 3 + 3 1  λ 2 − 1 + 6 − − γ ) is not an integrable function, so strictly β − E α, α E √ n k E ;  g P P ( ( β + π (3 ( – 24 – R 2 ! 1 g =2 ) = √ γE by g k ,k 2 1 ∞ =1 p ) = + (24) ,n  β with all other lengths or defect angles fixed. Since = 1. This volume is clear from the observation that there ( → =0 ∼ γ , . . . , α π )): 3 The new term is once again suppressed by a factor of g ,k 2, but the result also is valid for the special cases ), as discussed in the introduction and later in section , 1 0 ) E 0 S Z α 8 N using results from [ =1 3.5 V ≥ e ; gives us the Laplace transform ( G b ,n → ∞ k ( A b, . . . b βE =0 ,k ( ) = (16 − g e / E =1 Z g,n 1 ( , so we cannot trust the perturbative series in the regime where the ) using ,n V (supported at positive λ ) = ≤ =0 disc E 2 g ( ρ / 3.11 m V 3 ψ − E of order ) and ( which decays rapidly as E , and the result is independent of all other lengths or defect angles. We compute ψ g 3.10 − 0 in an appropriate way. g In fact we can do better, since we can compute the leading order term in However, it turns out that the negativity is automatically resolved by including the Specifically, it is the principal value distribution +6 8 → n 2 The derivation applies for function speaking, we have subtractedintegral. Choosing an ‘infinite delta function’ at zero energy to cancel the divergence of the and integrating this against the trumpet gives simply the coefficient in appendix Specializing to the case of multiple defects on the disc at low temperature explicitlyThe limit and when sum our asymptotic thesponding boundary geodesic resulting becomes length very series long ofan corresponds even taking leading polynomial the order corre- in corrections. b the lengths of order 2 speaking we must regard theE last term as amore distribution singular which at regulates small the energy.down divergence This at at suggests that thenegativity expansion occurs. in powers of factor of 2 in the definition ( The density is a unique hyperbolicso surface the with moduli three space boundaries in of question fixed is length a (i.e. single a point. pair We of find pants), the result which we can include in our expansion of the density of states (recalling the symmetry This is the two-dimensional version of the proposal ofpath [ integral over off-shellhow configurations this including might multiple occur,from defects ( we on look the disc. first at To the see two-defect result, which is simple to obtain particle of mass JHEP01(2021)118 , ··· , 1 (3.29) (3.30) (3.26) (3.27) (3.28) that is , k = 0 and growing p , ) 2 . / 4 1 λ for can then be read − ( ··· p β 0 O + E λ. 1 2 1 β )) + ) + − 2 2 4 . γ simply add, α α )) 0 2 1 E − ∼ − − β + 15 0 ( (1 2 2 O α λ 18 . ) + (1 + − i 2 1 ) (3.25) λ λβ α β (5 . The next order in 1 ( 2 4 0 − ,k , E > E π γ E X 1 − ) 3 , e 0 . The higher order corrections are no longer 3 1 0 λ 3 2 λ (1 E − Z 1 = 0 term describes a change in the functional γ !  k − λ − – 25 – k λ γ ) p β ) including the required symmetry factors, simply )( 2 E 2  ( α ∼ − =0 ∞ λ 2 3.5 k γ X π 0 is not simply a uniform translation of the Schwarzian 2 2 0 = 6 defects using the list of volume polynomials in 1 contains terms proportional to + E − S 1 √ p e k ,k disc λ (1 ) arises from taking a binomial expansion and truncating 1 γ π , 2 defects include new terms decaying as ( ρ 2 = as in ( 0 0 2 π S ) = π ≥ Z 2 k e β 3.19 , which themselves have the potential of causing a negative λ ( k − β ∼ 2 − ) disc λ λ E Z ( − − 1 λ disc ) = ρ ]), we find that all higher corrections fit into the same pattern, shifting − λ . Explicitly, the first few orders for a single defect are given by 2 there is a unique coefficient determined by lower orders in ( 42 α 0 = 1 term, and finally the ) = ≥ p λ γE ( p 0 γE 2; for In the next section, we will perform these calculations exactly and explain more directly The contributions from At this leading order in the low-temperature limit, it is straightforward to generalize Now we simply sum over − off from the form of the density ofdensity states, of so states. the origin of the structure that emerges from the perturbative calculation done above. We emphasize that this structure placespolynomials. extremely stringent constraints The relating the amplitude volume k compatible with the interpretation as a shift of and for two species of defect we find density of states orcalculations other (we pathological compute low-energyappendix up behavior. B to of Experimentally, [ bythe explicit edge of the spectrumindependent at of higher orders in to include several species of defect, and their contributions to as odd positive powers of The naive density of statesafter ( the first twoseries terms. fails to The converge. negativea It density rather was of elegant this states manner an appears by artifact including in of the the a off-shell truncation, regime configurations and where of has multiple that been defects. resolved in In term of theenergy, density of states, this corresponds simply to a shift in the ground state finding that the defects give an exponential correction to leading order at low temperature: JHEP01(2021)118 ) is x ( (3.34) (3.35) (3.31) (3.32) (3.33) JT . , ) u =0 x πiα

2 i at genus zero, ) we give a more x , C ( ) i 2 ) 0 JT β 3.5 2 u + β 1 ( boundaries. We used πiα, . . . ,  β , ) ( Z 2 k 0 ) ,k , x 2 1 2 E ( , β 0 − , b ( i JT e 1 ) u 2 Z b 2 2 h ( β β β p k ( πx , 1 ] for the genus zero WP volumes: n + β Z b 2+ , 1 ) 44  0 √ = 2 1 β 0 V β  I π ( = 0 is included in the sum with the 1 ) ! 2 ) Z x k i h ( ··· ! = k . In a moment we will briefly review the , b  k i JT λ x ) β u x ( ( MM =0 , ). ∞ p k X 2 JT π b =0 – 26 – 2 u = −  , C,g p trumpet 0 1 1

1 =0 b ) Z J b E ( i ) H  δ C,g 2 x db 1 i 0 1 β i b ( ) I b 2 − h 2 e JT β , 3 ( u ) = − Y =1 Z 2 n i ) p . In this section we will obtain exact expressions at genus 1  ) Tr( , b λ

1 β H ( b ∂ 1 ∂x ∞ ( β Z 0 2 , h  − Z 0 e V 2 1 in the matrix model, but for now we will simply take the formula as matches precisely with a matrix integral. In section ≡ is the modified Bessel function of the first kind. The function Tr( 0 C ,k JT I i 2

) = , u ) 0 n n i 2. ) β ( 2 ≥ β ( n , . . . , b Z 1 ...Z ) b 1 ) ( 1 β We now make use of the explicit formula derived in [ To compute this cylinder amplitude, we write the answer in the defect expansion ,n ( β 0 ( Z V h Z defined implicitly by and primes denote derivativesrole with played respect by to given and apply it to our problem. valid for The quantity in thethe right fact hand stated side above issimplify that the the the WP expression, volume volume with theunderstanding with defects case that 2 is without + a defects simple analytic continuation. To where the sum is over the number of defects present in the gravitational path integral, and model. If our model defectsthat gives there this is result, a it is dual a matrix ‘smoking integral. gun’ giving us strong evidence position of the edge of the spectrum where we include the annotation MM to emphasize that this result comes from a matrix 3.3.1 Two boundaryWe amplitude will begin byincluding considering defects. the This two correlator boundaryIts is correlator independent universal of for all the double-scaled spectral (GUE) curve matrix which models. defines the theory and depends only on the In the previousthe section weighting we parameter studiedzero JT from gravity resumingh with over an a arbitrary gasformal, number of but less of defect explicitly, defects proof perturbatively that and in is show valid also that for the higher genus result contributions. for 3.3 The matrix integral dual JHEP01(2021)118 . for =0 =0 x x B

(3.38) (3.40) (3.36) (3.41) (3.42) (3.39) (3.37) defects   k k k   , from these )) )) x =0 x ? ) is called the ( ( x x

x . ( JT JT ! u u  u  ) k , ? p , p  x ( =0 )) =0 πα πα ,g x x, x JT

( (2 (2 ) u 0 = 2 0 MM JT β J J

p  u )  +  ), ) 0 ) 1 2 ) H x x β  p πα 2 β 2 ( ( ) 2 )( β β 2 + u u x  β 1 + − πα ) for the Weil-Petersson volumes ( 1 β 0 e u + p ( β 1 (2 ), recovering the JT gravity string γ 0 1 β 2 0 )( ) with a minus sign relative to the x which produces the prefactor and a λJ E 3.34 x πα ( J )( − x ( − x 2 2 (  e + ( e ) Tr( b JT 0  JT u 2 2 2 x 2 u u JT  H 0 β β β ) 1 defects. At this point, we can explicitly γ u β 1 1 2 2 β 1 = γ 1 + β 2 β + and e λ I − k β ? ) 1 + e e 1 → − √ 1 1 x x β √ +  β b β ( )  π – 27 –  )( Tr( 1 π 1 x 1 ) x 2 0 JT ( 2 (

− x ). We can eliminate the variable u u k ( JT  x = with  ) u = = ( ) to simplify the expression. In the old matrix model u 2 1 γ 1 ∂ β − 2 p =0 2 ix ∂x JT , k ) implicitly through =0 ( e π β u  +  n x )) 1 2  C,g ( ) . Putting everything together we obtain 1 J x C,g i β   2 ∂ 2 u 0 sends ( i ) n β ∂x 1 1 ) ? β 2 √ ( β 2 − I 2 −  x β i π β → k ( β ( + ) 2 1 2 (  1 x Z β λ β JT and ( Z ) π β ) = 1 = ) u 1 √ ( u ∂ πγ 2 1 x β ∂x 1 β 4 π ( β β ( p 2 √  n =0 ( is related to the leading KdV parameter. This function characterizes 0 limit of the expression above we get the final answer for the two-loop I Z ≡ − ! k Z h x = = k ) C,g h λ → i x ] this equation is called the (genus zero) string equation, ) ,k ( x 2 2 =1 , ∞ u β k X 0 56 ( i , ) Z 2 ) 53 1 + β 1 ( β

( Z ) × Z 1 h Taking the So far, we have just been plugging the formula ( Using this explicit formula for WP volumes we can find the contribution with β ( Z h amplitude corresponding to JT gravity with a gas of defects heat capacity and the theory completely inzero the density of double states scalingIt (or limit is equivalently evident and the that isequation. spectral taking equivalent curve), to which giving we the will genus compute later. expressions to obtain a more direct implicit definition for where we used that literature [ where we defined the function For later convenience, weconventional have definition chosen used to for define perform the sum over alla defects using statement the of Lagrange the reversionscaled theorem theorem). matrix (see integral, appendix We given obtain by precisely the answer expected from the double- into the expression for the amplitudes with To derive this formulasimple we exponential integrated of over to the double trumpet JHEP01(2021)118 . 0 of λ =0 E F C,g (3.46) (3.43) (3.44) (3.45) N of the i 0 gives ) 0 n . β E → ( x Z =0 x

···  -loop amplitude k ) . 1 n  β )) ( . x = 0 . Taking Z ( = 0). The zero-point x h 1). Using the explicit  ’s, comparing with the = 0 0 JT x λ ( u >  u 0 γE x, 1 p F 2 − = ) γE N p πα 2 γ i
n u (2 β p n . Using the second version of the 0 πα √ β F J 2 i + = (2 πα   2 0 0 we can show the two-loop amplitude ··· 0 πα  ···  I 1 2 E ) 0 B i β n + β λ 0 ,...,N √ 1 I + F β i πλ I ··· =1 N i 2 X λ = 1 + / 1 i π F β 2) + 2 =1 – 28 – N i X )( − 0  x n for + 2 S 0 ( ( + ) i ) n  JT
γ α 0 γE − u u 2 γ (2 1 (2 2 √ 2 γE e e p π 2 n/ ), and we checked that they agree up to seventh order in  π 2 π  2 p should be thought of as a function of 2 (and summing over species if  3 1 π ) from summing over defects thus takes the precise functional u I 3.30 2 1 − k k I = u  π and angles ) but with a string equation given by + 0 1 2 3.41 ), ( √ n i I =0 λ  0 γE 3.38 2 C,g 3.29 ∂ i ∂x γE , which we will treat as a special case in the next subsection). We will ) p 2 ) for all  n n β ! p k ( 3.2 k λ Z =0 ∞ ··· , . . . , β k X ) required for a double-scaled matrix integral. 1 ). This is completely obscure from the expansion in terms of Weil-Petersson ) 1 boundaries involves a sum over defects for a genus zero topology, adding β × ( β ( 3.31 n 3.44 Z ,n,k The gravitational calculation for the connected amplitude We find it remarkable that the defects appear linearly at the level of the string equa- This result can be straightforwardly generalized to allow a number of flavors h =0 g becomes find that these all match the results from thewith matrix integral we metZ above. formula for the WP volumes, for the case of one specie of defects, the 3.3.2 All genus-zeroThe amplitudes procedure in the previousany section number can of be boundaries generalizedtively to (including in the the genus section one-boundary zero amplitude disc with amplitude studied perturba- perturbative results ( tion ( volumes. As weedge will of see, the all spectrum. the nonlinearities can be traced back to the shift These expressions can be verified by Taylor expanding in all the where we leave implicit that the equation that the zero-point energy satisfies form ( defects with weighting Lagrange reversion theorem described intakes appendix the same form ( energy can be written moreof explicitly, the using equation the string equation, as the largest solution The gravitational result ( where the exact edge of the spectrum is given by JHEP01(2021)118 . ). B . 3.44 (3.51) (3.49) (3.50) (3.47) (3.48) =0 ]. The x

) 44 . n β ), this gives n ! + β
··· u + + 3.44 1 √ . . i β ··· )( , JT gravity with a x ) is given by ( πα ( = 0 0 + MM ). Therefore we have x 2 u ( S

1 γ  1 ) u 2 0 1 β I H − 3.47 , ]. Using ( i n e β γE β u not only through the linear ) 2 2 x − 61 πα ( √ , λ − e 2 u p of the string equation. In the n i i . γ 59 1 2  λ u λ ), where Tr( − πα ∂ i ∂x 2 ... X dependence of the zero-point energy  3.47  ) 0 dx e + λ I H 2 1 /
i ∞ , we have the explicit expression ]; for a recent discussion see [ β u 0 2) λ u − n Z − √ e β 58 n i , π ( X ) 2 γ πβ Tr( ··· 57 γ π 2 ). The matrix model multi-loop amplitude at

1 0 – 29 – (2 I β r 2 = + 2 ), see for example [ 0

√ 3.40 n/ S x  = 2 we will just quote the final result: u ( e π 0 u 2 n − = 0 γE S JT+def ) 2 du i =0 n γE ) g ). Once again, this matches with the answer for a matrix p − 2 i n ) π (2 β √ 2 ( β e ( ]. From this perspective the spectral curve plays an important 3.44 0  8 = γE 1 Z 2 h γE I 2 ...Z 0 =0 Z ) 1 0 C,g γE β S π i ( 2 ) 2 ) appearing here is exactly the same one that appeared above for the Z n γ e x p h β ( = 2, defined thorugh ( ( u = Z n =0 ) is defined by ( g ··· i x ) ) ( 1 u E β ( ( ρ The density of states extracted from this formula can be obtained by an inverse Laplace The partition function with a single boundary can be derived using the same method Z h h . This is defined as the largest root of an equation that we repeat here for convenience: 0 right hand side the densityterm of in states the depends integrand, on but theE also coupling through the implicit transform. Changing the integration variable to The integrand is basically the derivative with respect to where integral with heat capacity equalus to an exact expression in the defect weighting parameter will analyze this quantity. as before, combining thetheorem. exact formula We obtain for the the result genus zero WP volumes and the reversion before discussion higher genus amplitudes. 3.4 Density ofWe states show that JT gravity with defectsof has random a matrix Hamiltonians integral [ dual,role interpreted as and an is ensemble related to the averaged density of states of the dual theory. In this section we When multiple defect flavors areAgain, this present, is we easily derived obtain using ( theIn second the version next of section the we reversion theorem will in analyze appendix the density of states one can derive from these formulas, special case genus zero was derivedexpression some derived in time those agoshown references in here is [ precisely that equivalent at togas genus ( of zero, defects or is equivalently equivalent to to a leading matrix order integral in is almost identical to the case with The function We can again evaluate this sum using the Lagrange reversion theorem. Since the derivation JHEP01(2021)118 and 0.0 (3.52) α 1. This 08 -0.2 . = 0 -0.4 α > λ E 2, for -0.6 / . γE 2 = 1 is positive. However, 2 -0.8 √ α . Now that we have an πα disc 0 2 (c) -1.0 ρ λe

0.04 0.03 0.02 0.01 0.00

) ( ρ E 0 and the theory is fine (c) For ) in figure ∼ < E > E ) 0 as expected and the theory is fine 0 3.50 E 0.25 E > ( 0 for which ρ 0 627 is the ratio between the first zero . E 01 E λ . 0 0.20 − ≈ = 0 E c . 0.15 1 and λ 0 E α 1 is not unexpected from the gravitational p 06 we get . 2, 0 0 we see / γE ≈ 0.10 2 – 30 – α < ) . The prefactor is proportional to the integrand < ≈ = 1 α > is positive for all E 1 → 0.05 . ( 2) for α 0 3.2 / u , we can explore this for a range of parameters − disc . There are several cases to consider; we show some disc λ 0.00 γE (b) ρ ρ λ = , the density of states is positive for all energies, for any = 1 2 1 √ λ

α J

0.10 0.05 0.00 0.35 0.30 0.25 0.20 0.15 0, the theory is well behaved for couplings smaller than a

π , there is a range of ) ( ρ E 0 the density of states is positive. An example is shown in ( 2 c α e λ and λ > ∼ c λ < 0.4 ) 1 ). This critical coupling is given implicitly by solving the following . E < α and 0 α ( and ( 2 ρ − c c c / 0.3 1, where the critical value λ = = 1 λ . E 2) the density of states becomes negative in a finite range of energies. λ α 0.2 / 2, 01 smaller than / (a). . < α < (1 1. < α < α c < α < α 2 and the first zero of c = 1 α 0 = 0 0.1 . Density of states of JT gravity with (black) and without (dashed blue) defects, for J > λ α λ For 0 figure For 0 critical value For of choice of 08 . < α < (a) We find that for all such First we can look at large energies. In this regime, it is possible to approximate the Now, we should check whether the model we have arrived at makes sense as a double- 0.0 • • • = 1. (a) For

0.8 0.6 0.4 0.2 0.0

= 0 ) ( ρ E . For simplicity we will restrict to the case of a single species of defect. γ this can fail forrepresentative examples sufficiently from large numerical integration of ( change of theperspective, ultraviolet behavior since for suchsubstantially defects modify) would the be asymptotic regionon favored of 0 to the geometry. proliferate As and before, destroy we will (or now at focus least require that the densityexpression of for states this density atλ finite density of states by as we found perturbativelyin in the section expression above evaluated at scaled matrix model (at least perturbativelynonperturbative in instabilities the for genus expansion; now we which leave aside are possible also present in JT gravity). For this, we Unless the solution toa this smooth equation density is with degenerate, a this square-root immediately edge implies that we have Figure 2 2 (b) For λ JHEP01(2021)118 , z E ). ) sat- 0. A (3.53) (3.54) (3.55) (3.56) z 03 and ( . < 0 3.10 zy ) 2 . The sec- ). Solving ≈ c E − α λ ( ( ]. To simplify c (0) 8 λ c , . disc resolvents, ) = λ ρ z ( , n ) vanishes. Elimi- ) 1 = 0 = 0 , . 2 n 0 . z   ) correlator which we c c W 3.52 − conn βH = 1 the resolvent of the γE γE  − 2 2 e for which ,..., H , n p p 2 1 2 z − jumps to a smaller value, and 1 πα πα − ]. First define the coordinate n = 0 2 2 0 ( 8 (b). For the KK instantons, the   E E g 0 1 g,n 2 I I Tr ). For example pure JT gravity has R c c n E ( ... < E < E . The spectral curve is defined through πλ (c). 1 E παλ gives the desired relation H . . . z disc 2 E − ρ 1 + 2 c 1 − 0 03, the theory has a positive density of states z , so this is the relevant case. + 2 . E  S n 1 = 0 c 0 2  − S – 31 – E 2 c n ) goes smoothly between values . z γE Tr 1) iπe α 2 λ γE (  as − 2 , where we also write down the recursion explicitly. − c p λ ), the density of states is no longer positive for all π ≡ p C E α 2 π ) ( ) =  c ) = ( 2 n z n 1  ( I 0 y c I contribution to the connected correlator of c λ > λ g ). For the exceptional case γE ,...,E , . . . , z n 2 1 γE 1 2 z E p ( ). Then define the functions and ( p n π c g,n 1. An example is shown in figure R (4 , . . . , β in this section. Since the proof is rather formal, we also present explicit / W 1 is exponentially small in 12. For small enough ) . β 1 2 ( 0 λ πz α < ) determining the edge of the spectrum = ≈ g,n < α < α γ ) Z c 3.53 α ( c we then find arepresentative example finite is range shown in of figure energies coupling For 0 so we cannot interpretto our ( model as a double-scaled matrix integral. The solution degenerate, which means thatnating the the suppressed critical prefactorthis zero-point numerically, in energy we ( find that λ for any The first equation givesond the equation zero-point determines energy the at point the where the critical solution coupling to the first equation becomes two equations simultaneously ) = sin(2 z We begin by introducing some notation following [ • ( JT The expectation value is takena within the Laplace matrix transform modeldenote to ensemble. by This the is connected relateddouble-scaled through partition integral function is not Tr( convergent, but we can instead define proportional to the genus checks at higher genus in appendix in terms of thethe matrix disc eigenvalue density ofy states as gravity with defects byfor explicit the computation. volumes of moduliapproach. We do space In this at not section, higher we have giveisfy genus, a a the so formal proof topological closed we that recursion must form the of amplitudes takeWe expression a with thus a defects matrix have slightly ( integral, a more using duality indirect awe to theorem will of all set Eynard orders and in Orantin. the genus expansion, generalizing [ 3.5 Higher genus We have now defined a matrix integral which reproduces all genus zero amplitudes of JT JHEP01(2021)118 (3.60) (3.62) (3.59) (3.61) (3.57) (3.58) (which λ . ) k are given by ˜ z . ) g,n,k ,..., λ 1 ; W ˜ z , ) we have a similar n ) , ). ) (which for simplicity n . λ . n ) z ; ) ( 3.57 1 n λ f ), the ; , . . . , z 2 . 1 z, z ) , . . . , z ( , z 1 z 2 , . . . , β , 3.59 1 ( z , . . . , z 1 z, z 0 ( 1 ( ,k β k z 1 ( W , z, z ( + ) +1 0 ( z 3 g,n , ( g,n W g,n 0 Z ). The undeformed case corresponds f ! g,n,k k W λ ), and from ( W k W λ ) ; ) W πi ) k dz n z ] and for a summary of other interesting ! 2 z k z ( ( =1 k ∞ ) will be equivalent to the expansion coeffi- (˜ λ 3.58 γ k X f 62 ) and f n f I n z k 1 =0 πi ∞ 1 πi 2 ˜ dz z πi k dz X 2 z 1 2 , . . . , β 2 d – 32 – 2 ]). This starts with a given solution of the topo- 1 γ − γ γ β I , . . . , z ) − I ( ) = 63 I , . . . , z 1 z 1 λ z ( z ( ; g,n y ( ··· ) = ) = ) = n Z ) λ λ λ 1 ; g,n ; ; g,n,k z ). We require that the spectral curve and the cylinder ) = n 2 1 (˜ W z λ λ W ), up to an integral transform to convert between resolvents f ( , z ; , . . . , z 1 1 y ), z ) and 1 ˜ z z πi ( z The statement of Eynard and Orantin’s theorem requires the 3.5 z ( λ ( d 2 ∂ y ( : in the complex plane and a function 2 ; , . . . , z ∂λ y , 1 γ λ n 0 From the above, it is simple to construct a solution to topological γ z g,n I ( W ) in ( W . n g,n ∂ ) = λ ∂λ ) in a simple way. Since the resolvent correlators are invariant under ] (see also section 4.3.2 of [ , . . . , z n W z 1 ( z 45 ∂ ( f ∂λ , . . . , β 1 g,n β , . . . , z ( 1 W z ( ), g,n,k 0, returning us to λ ; Z g,n,k z → ( The main result we will use is the deformation theorem derived by Eynard and Orantin, W y λ and function The same equationexpansion applies for for the spectral the curve, cylinder ( cients and partition functions.multiple By integrals, repeated as use of the relation ( For our model, the amplitudes γ permutations of its arguments theside choice is of irrelevant. the variable being integrated in theSolution right at hand finite recursion to all orders in Then, applying topological recursion withsatisfies these deformed amplitudes as input, the solution The deformed amplitudes are thus completely determined at all genus by the closed curve to Deformation theorem. choice of a closedwe curve take to beamplitude behave independent as of follows under the deformation: Theorem 5.1 of [ properties of the topologicallogical recursion [ recursion and considersfor a us one-parameter will deformation, beby parameterized the by defect amplitude). We denote various quantities after the deformation JHEP01(2021)118 ), g,n,k 3.11 (3.64) (3.65) (3.66) (3.67) (3.63) W , this is . )) to be b . Taking ) . variables . Now we λ ) } m ; n z 2 2 b πiα +1) 2 , z . m 1 ). ) to implement . 2( z β 2 ) z k b 4 ( πβ {z ( − , . . . , β n 2 , 1 − z f √ 0 3.10 e β 2 so that the integrals ( W in a neighborhood of γ πiα, . . . , + 1)!˜ 2 | , . . . , b λ ) = 1 , g,n,k b n m Z ( and ) as a monomial β, b defects ( ) and 2 n f ( z g,n b λ k n V ; β z n , . . . , b and function z + ( 1 n y b ... to implement the inverse Laplace b γ ( trumpet ) and setting lengths in the Weil- + − 2 1 e , z g,n ··· n 1 ) V β ˜ ) z 3.63 ( n db z z − (˜ n ,Z n πα e b ) b f πα n ), with the trumpet factor replaced by the − 2 ˜ z ∞ πi e d 2 0 β, b n 3.7 ( Z sin(2 H 1 db ) to the deformation of the topological recur- – 33 – . . . dβ z n 1 1 b b ) = ), using the WP volumes with cone points ( ∞ − z 3.64 0 dβ e (˜ trumpet Z 1 f Z ∞ 3.10 1 2 0 z db 1 1 Z b βz b n − − . Because the volumes are all even polynomials in We will now apply the general result above to our specific e ∞ 1 0 Z dβe . . . z πiα db 1 1 ∞ b z ), it is roughly clear what we need to do: the extra ˜ 0 ∞ n ) = = 2 Z 0 n b ], Z z 3.60 64 ) = 2 = 2 ) = n n , . . . , z bz 1 z − to be any curve encircling the origin, as we will check in detail now. ( e γ , . . . , z g,n 1 , . . . , z ), we should choose the integral 1 z W ( z ) are precisely those given by the path integral with ( 3.61 g,n,k Each term in the WP volume will depend on the lengths Comparing the gravity answer ( From equation ( 3.61 g,n,k W W equivalent to choosing and taking the Laplace transform, this contributes to the JT resolvent as (2 sion ( transform, stripping off somePetersson volumes of to the integrals in ( with the identity This can be derivedcorrelators by combining the relation between resolvents and partition function This is equivalent toLaplace the JT transform path so integral thatdefect ( we expansion get of the theis resolvent resolvent then rather ( than the partition function. Our act to replace thosePetersson boundaries volumes with are simply defects. relatedLaplace Let to transform us the [ expansion make of this the more resolvent explicit. in JT The gravity Weil- by a problem. We will findthe an deformation appropriate induced choice by of inclusion contour in of ( defects, before showing that the coefficients give us extra boundaries in JT gravity, and we need to choose the origin, this seriesthe converges topological to recursion the (theanalytic, solution. deformed since spectral In the curve fact,example topological of it recursion the is sum preserves sufficient over analyticity. defects. forJT the This input gravity is with to the defects. case for our If the family of solutions to topological recursion is analytic in JHEP01(2021)118 ) 9 γ ˜ z πα πα (4.1) 2 (3.68) sin(2 +1) m , which we take γ 2( − z ) is given by +1)!˜ 3.2 m . ) 2 = 0 case, which is the disc λ ( , g O + = 1 ) motivated us to study a simplified . n ) ), can be treated similarly. There is | 2 ). λ J we can choose a fixed curve ; παz z z 2 a ( − | 3.64 = 1 disc amplitude with a single defect, y E cos(2 ). We saw that this problem was naturally . When we included only classical solutions, ( , k λ 3 . N at which we are evaluating the spectral curve. 3 1.1 ` − G = 1 – 34 – z 8 ) → ∞ , n ≡ πz a π . The overall result is therefore to evaluate the WP ]. Then for any 4 m 8 JT = 0 2 ) E 2 and all genus. The sin(2 g , as required by ( ≥ πiα ) = n πiα λ ; z ( ) black holes, and we instead have a nonperturbative shift of the y Z , ), where the integral picks out the residue of (2 as in section 2.4 of [ a 3.61 replaced by 2 b = 0, which is precisely (2 Here, we will use a shifted and rescaled energy which is natural from the near-extremal To establish this mechanism conclusively, and for a more detailed quantitative study, it This establishes that the addition of defects is a special case of Eynard and Orantin’s This analysis applies for The theorem still applies, since we can perform the deformation before taking the double-scaling limit, z 9 introducing a cutoff to infinity as we take the double-scaled limit where the effect isreduced relevant, theory we is have a principledand good reasons quantify approximation. to various believe sources To that of that corrections end, the to in dimensionally this this approximation. sectionpoint we of will view, measured also in discuss units of the characteristic scale of quantum fluctuations: arising from the SL(2 extremality bound. is important to solve thegral complete on Seifert three-dimensional manifolds. problem Nonetheless, of in computing the the regime path of large inte- spin and very low temperature JT gravity with defects, asthis studied in model section hadMaloney-Witten-Keller a partition potential function problem ( resolved with by including a the negative pathin density integral the with of edge multiple of defects, states, thein and similarly spectrum instead the of we to full energies. found the three-dimensional a Our problem: shift proposal is a that sum the over same Seifert mechanism applies manifolds cures the negativity 4 Back to 3DOur gravity analysis of 3D gravitymodel near extremality for in the section path integral on Seifert manifolds. This simplified model is an example of must be large enough to enclose the point deformation theorem, and hencecursion. the The resulting argument genus generalizesspecies expansion straightforwardly of to obeys defect multiple the as defects, topological a by separate re- treating deformation. each where the corresponding Weil-Petersson volume isorder not in defined. the deformation This contributes of to the leading spectral curve, which from section ( This can be checked separately to follow from the same deformation, though the curve at ˜ volume with with defects giving theone deformed spectral exception, curve coming from the insert this into ( JHEP01(2021)118 (4.5) (4.6) (4.7) (4.2) (4.3) (4.4) . Go- 2 and ), then / JT J E = 1 . . ) ) α J J ( ( 0 0 0 (odd S S . 1 2 1 2 < − − 1 2 e e J J λ fixed 1) 1) ) − − J ( ( . ( ) obtained earlier. 0 N N  . S ) 1 2 G , given near extremality by G ] at very low energies above  J 3 2 3 2 2.38 e JT 2 ( 1 2 7 ` / ` 0 π 1 π JT E ) is shifted by a non-perturbative S JT λ = 1 2 E 32 J √ 32 E ( − 1 2 Q 1 2 0 p and include the path integral over e + ∼ π J E − ≡ 2 | JT 1) 1 2  3.2 1 2 = JT E − λ | E ( (0) JT p 2 ) sinh  ) 1 π ) , λ J J ( ) black holes is relevant at leading order, but all (2
– 35 – ( 0 0 with ,E Z S JT S , e e | − (0) JT E 2 ∼ J JT E ) black hole with π | √ ) 4 , E ). From the 2D JT perspective this is the contribution Z π − 0 , J √ JT ∼ ( ) = , the extremal value 0 E JT ) → J ( E S ( , this agrees with the action ( E 0 cosh 0 JT J ρ JT 1 2 q E E ) 1) ( λ J family of SL(2 ) ρ ( − J 0 1 2 ( ,E S 0 e S = e 2 → Q π → ∞ ) 4 J the density of states can become negative. JT ∼ | E ) 1 2 ( λ ρ JT | defined above. For the exponential classical dependence on parameters, including E ( < 2 1 / ρ 1 JT λ Now, there is one important contribution to the three-dimensional path integral which But now, we can apply the lesson of section The potential pathologies in the spectrum occur at exponentially low temperatures, E = correction anticipated in the introduction is not captured by the dimensional reduction in this limit, namely the one-loop effect of Here, we wrote the shift ining the back to extremal the energy original is energy written in terms of the variable multiple defects. Here, thismanifolds defects with U(1) are symmetry. KK In instantons, thethe which near-extremal full limit lift path we to are integral studying, smooth bythe we Seifert the approximate previous three- dimensional section. reduction, The so resultingwe we density have can of a directly states shift apply is of now the the free results extremal of of negativities, energy: and instead In our double-scaling limit, thesefor two terms are of the same order. If In this regime the others are negligible. Then the density of states from classical solutions is approximately λ the important factor of ( so we willextremality: focus on the following ‘double scaling’ limit [ contribution comes from the SL(2 From the JT perspective this is precisely the one-defect contribution with BTZ black hole where we ignore a (spin dependent) multiplicativeas constant a that can logarithmic in correction any case to to be absorbed the partition function from the disk with no defects. The leading correction to this The leading contribution to the density of states near the extremal limit is given by the JHEP01(2021)118 . , ], JT 32 −Q E (4.8) 1 and √ ) black / Z 0  , S and 1 ) 2. At small 0 q 1 ) black hole S Q − ) Z q 1 (1 , q > − − ) BF theory, which 1 2 e R ( , A second source of ]. Their partition func- − ∼ e 66 ) all these contributions ]. From this, the BTZ E , these phases give rise 0 ( q S q 10 1 2 , ρ = 2. − q 9 e , 7 ) ∼ with the coefficient in the exact J ( 0 λ JT S ) E 1 − q ). The corrections can be suppressed − (1 4.4 e J q p coprime to a given πi p 2 – 36 – − e ∝ ) contribution proportional to J q p 10 ( 0 = δE Q ). The imaginary part cancels between terms labelled by First, we look at contributions from KK instantons with 2.29 ]), and they should be included. Intuitively, this loop determinant accounts ) black holes become important. These should also give rise to shifts of the 65 2. Since we are focusing on energies of order Z , ) contribution from three dimensions, since we have the correct energy dependence. Z q > , Finally, we analyze the order of magnitude of corrections to this result and see that If we were to go to lower temperatures still, the contributions from other families 2 do not grow any faster at low energies than those with We can explicitly check this intuition in a similar situation using the results of [ 10 Corrections to dilaton potential fromcorrections dimensional is reduction. the fact that in the throat,tion the for primary 2D states dilatonsince reduces gravity to the the which WP JT measure appearsin gravity used from answer turn in for is the the the a JT double reduction gravity trumpet. of calculation Chern-Simons. This can We is be expect not deduced the unreasonable from same SL(2 to be true for the geometries considered here. with are exponentially suppressedtherefore in can the be large neglected.q spin It > is limit crucial by for this a conclusion factor to hold of that contributions from they are negligible ineither the due to double the scaling large limit spin limit ( Other or saddles. due to theenergies small their energies contribution above to extremality. the density of states is of order that the vacuum hascontribution in a ( null descendant);When this we can add be theto seen contributions Kloosterman from from sums, the which all extremality appear SL(2 bound in appears the to MWK depend density on [ spin in a complicated number-theoretic way. extremality bound, with the at leading order. Thephase proportionality originating constant includes with some the additional one-loop spin-independent effects discussed above (in particular, from the fact for multiple KK instantons,gravitons at propagate least only at at leadingfull the order. three-dimensional boundary, This calculation. but is demands not more unreasonable, careful since justification the fromof a SL(2 states carry a very small amountby of matching their energy the in JT descendants.energy density This dependence intuition of is of supported states, theholes. as low-energy spectrum well For of a as primaries single theSL(2 KK from density instanton, BTZ with we and can a SL(2 simplyOur single match underlying defect, assumption to is the that the graviton KK modes do not introduce new effects is very large sosee these modes also are [ notfor heavy boundary (though gravitons their on interactions topin are of the suppressed our dual. [ solutions, whichstates, We correspond not therefore to of expect Virasoro all descendants our states; results these to are not capture greatly the different, density since of at Virasoro large primary central charge, most Kaluza-Klein modes of the graviton. In the large spin limit, the transverse circle direction JHEP01(2021)118 ε ∼ (4.9) JT ]. We ] have 7 ) black E ], which Z , where 10 , 2 , 67 ], replacing 9 7 εφ ]. + . 8 0 φ ). This produces S 2 1 2 . ) − − 0 e 2.12 E . In the limit we are ∼ ) = from the SL(2 − 2 JT φ ( 0 E εE ( U E δρ/ρ 0 3 S ∼ 2 e 0 . We discovered that this family − 3 = δρ/ρ , ξ = 3 KK instantons become relevant. At i . The shifts in which is also negligible: from our very low 2 this is of order 0 q ) 0 0 ξ S 1 2 /S – 37 – 0 Ai( − S ξ e E < E , computed in detail by Kitaev and Suh [ − 2 e − ∼ − 2 ) ∼ JT εβ ξ 0 ( 0 E which can be read off from equation ( ∼ 0 Ai Z h δρ/ρ 0 /S 3 S 2 log e δ Another source of corrections are higher topologies appearing in the ) = . This grows at low energies, but not sufficiently quickly to overcome E ( 3 JT ρ . The relative shift in the density of states from adding a handle on the disc /E 2 0 S 2 are therefore not particularly meaningful, since the spectrum is expected to 2 − e q > ∼ , which is the same scale at which the 0 We conclude with some comments on open questions and future directions. In the process we needed to understand the path integral of JT gravity with a sum Interestingly, these higher genus corrections become important at energies S 2 3 δρ/ρ − integral, and we argue thatit they with cure a the nonperturbative negative shift density in of the states energy found ofover in defects, extremal [ which rotating we BTZ studied blackof in holes. theories its own is right dual in to section a matrix integral, generalizing the pure JT result of [ sional theory of JT gravity,the with reduction the from addition three offibration dimensions. new over instanton-like These which objects we objects are inherited correspond reducingis from becomes to singular degenerate, points while so where the the the two-dimensionalthree-geometries full geometry circle three-dimensional — geometry Seifert remains manifolds smooth. — The are corresponding the new topologies we include in the path have proposed a mechanism topath resolve integral this for tension, whichintegral by including there over certain such is topologies topologies noproblem in in classical by the the concentrating solution. on full thetant. theory, We near but In have extremal instead this not limit, reduced regime where computed to the their the a physics effects path is are more approximated most tractable by impor- a dimensionally reduced two dimen- 5 Discussion Previous attempts to defineconsidered the only Euclidean the path quantumresults integral fluctuations in of around tension pure with saddle an 3D point interpretation gravity geometries, as [ and a obtained unitary quantum mechanical theory [ We do not then haveand a has sharp an edge exponentially ofholes small the at tail spectrum, for since itbe is smooth smoothed at out the on relevant this scale. scale, the exponential suppression: for e very low energies, weintegral expect with a the square-root handles density to at genus resum zero, into namely the universal form for a matrix energy perspective it is a UV effect,Other and topologies. we are interestednear-horizon in region. the deep We IR. canhandles to study AdS a classis of these which arise in JT gravity, by adding correction to the dilaton potentialis in a the fixed near coupling extremal of limita order is shift 1 in the freetranslates energy in a relativeinterested shift in of this is the of density order of states reduction of 3D gravity has non-linear corrections to the dilaton potential. The leading JHEP01(2021)118 ], ]. 73 69 ], so , 8 68 ] for discussion of an 75 , ] appeared while this paper 74 66 ] as an ensemble average (rather 77 , 76 ] in an asymptotically AdS context [ 72 ] for first steps in this direction). Such re- – 70 78 , – 38 – 66 ). The simplest of these corresponds to a three-dimensional geometry with the 3.3.1 We also have more dynamical data and constraints coming from boundaries of higher For two-dimensional gravity, a dual description is a one-dimensional quantum me- The ensemble interpretation offers an explanation for the path integral over geome- than a microcanonical average)from for a the gravitational OPE calculation coefficientslations of of with pure genus different 2 state boundary black partitionproperties conditions holes, functions. of arising and More a topologies gravitational putative willsults calcu- ensemble be provide (see helpful us [ to with elucidate the the opportunity of a window into the statistics of chaotic quantum moduli space of theories,appropriate but ensemble rather may a look set likeresemble of in pure isolated this gravity, CFTs; case. which it mimic Perhaps is a there hard continuum is to in a imagine a large what large setgenus. an central of For charge CFTs example, limit. that we can interpret the results of [ strongly suggest that theby spectrum a of Virasoro matrix primaries integralcontains at near no a sign extremality. given of spatial spin However, locality.correlations this is between An well-described cannot sectors ensemble be of of consistent different the localand spins seems whole CFTs (for likely certainly story, example to requires since to also impose requiretor it deviations modular of from invariance), fixed random spin. matrix statistics In even particular, within without a marginal sec- operators we do not expect a continuous chanics, for which matrixwe integrals instead provide require a a naturalensemble two-dimensional class to CFT of select dual, ensembles. from, andensemble For at of there 3D free least is gravity, theories for no and irrational such the obvious theories possibility candidate of (see an [ exotic gravitational dual). Our findings and put the ‘babyAt universe’ least discussions naively, of these [ wormholesso give the rise partition to function ‘connectedpartition on correlators’ functions two on between disjoint each boundaries, tori torus. isbe This different interpreted is from as incompatible a the for result single product local of of dual statistical the theory, correlations separate but in can an ensemble of theories. section ( topology of a torusinterval; times work an computing interval, the with pathwas integral two on torus in this boundaries preparation. topology living [ aries Similar at have ‘spacetime the also ends wormhole’ made a of geometries recent the connecting appearance disjoint in discussion bound- of the information paradox [ interpretation would explain the secondthat pathology it of is the continuous: MWKfrom this an density is of average over not states, a consistent namely family for of a theories. single theory,tries but that is connect the several expected disconnected result Euclidean boundaries, such as those discussed in Our cure for thebe pathological a negative consistent density theory ofThis of states relied pure revives on 3D the quantum two-dimensionalwe hope gravity, calculations are with that closely led only there naturally related metric may tothat to degrees the is, those of idea in freedom. we that JT do a gravity dual not [ CFT have description a should be single of dual the CFT, same but form: rather an ensemble of duals. Such an 5.1 An ensemble dual for three-dimensional pure gravity? JHEP01(2021)118 ]. ]). 7 84 , 83 ] argued that a positive 7 ]. We will state a precise and ], so we may be able to draw 86 89 , 81 , region where quantum fluctuations are 15 [ 2 | J − | E – 39 – ], which we briefly preview. ], as well as a generalization giving (for instance) ]. At least in perturbation theory, 3D pure gravity 16 ]; while not directly related to gravity, this model is 16 66 82 ] and [ 85 and fixed twist | → ∞ ]). J | 88 , ]. Another possible source of inspiration is provided by the two-dimensional ] in a companion paper [ 87 7 81 – 78 Modular invariance gives universal results for the density of states in an asymptotic Our results suggest that there is one regime where topological fluctuations become Due to the emergence of the near-horizon AdS The geometries we studied are Seifert three-manifolds, which may be a tractable class argument of [ limit of large spin rigorous version of thisthe statement density in of [ even spin states minus the density of odd spin states, following ideas of [ After observing the negativitydensity of of the states MWK requiredhole threshold, spectral the from density, existence modular [ of invariance.indicates On that sufficiently the there light other is states, hand,We a our well will cure gravitational below explain to calculation the the how negativity this black without is introducing consistent any with such modular light states. invariance and the loophole in the over positive values of thebreaks dilaton), but down at entirely, least so formallywill the we be note infinite of that set equal the of importance genus expansion topologies at leading we order. consider5.3 (and likely may Consistency more) with modular invariance manifolds seems beyond reach, weto would the like most to relevant find topologies. some organizing principle to guidecompletely us uncontrolled, namely forapproximation small is non-rotating not useful black there holes. (for example, Our it two-dimensional becomes important that we integrate only inspiration from previousexamples studies are of [ Chern-Simons theories on Seifertimportant, it manifolds is (some plausible thatthose the we most study, important respecting topologies the inwe the symmetry are near-extremal of limit not the are large guaranteed transverse that circle this direction. will However, be the case. While a systematic study of all three- (such as correlations between sectors of different spin) inof a manifolds putative on which ensemble to dual. studygeneralizing the the gravitational path methods integral of more [ completely,is for closely example related to a noncompact Chern-Simons theory [ ory to a two-dimensional sector.tion We have to argued the that physics this of truncationthis interest is to near a extremality, good a but approxima- it fullmodel is is three-dimensional nonetheless indeed calculation. important a to goodto upgrade approximation, Besides extend such the verifying a calculation range that would of the be validity useful two-dimensional away in from its extremality own and right, study more detailed statistics analog of the SYK modela studied tractable in example [ of an ensemble of irrational CFTs5.2 (other relevant models are [ The full three-dimensionalTo make path our integral off-shell path integrals tractable, we truncated the full three-dimensional the- systems [ JHEP01(2021)118 directions, on which we 2 1 , the resulting . S N m 1 G ], for light matter 8 15 ] to cure the negative 14 m < , 7 . , where we instead perform ) J ( ). To cancel the contribution 0 2.5 S N 3.19 mG 4 − e ∼ , a defect insertion in JT gravity is equiv- . For a particle of mass λ 3 3.1 and – 40 – and m N G 2.5 4 − = 1 α ] and sections 36 growing sufficiently rapidly with spin, which means λ As discussed in [ Of course, we can now include multiple matter defects, and as a result find a shift in the This gives a two-dimensional perspective on the proposal of [ However, this is not quite the limit in which the negative density of states appears. to pursue is provided by the instantonthe gas described sum in before section doingequivalent the to path a integral. theory of This dilaton leads gravity. us toalent propose to that inserting this an theory operator is also exponential in the dilaton. Including an integral over the we used in this paper. 5.5 The instanton gasWe for gave JT a with solution defects definite to number of JT defects, and gravity then with summing their defects results. by A constructing possible alternative the approach path integral with a analysis of one-loop corrections(excluding backreaction). to Such an the effect extremal wasusing also BTZ conformal demonstrated for bootstrap geometry a generic methods, in irrationalapplied which CFT to [ a gave gravitational a dual. result Theand consistency that between the includes this full two-dimensional backreaction three-dimensional reduced when analysis theory gives us extra confidence in the approximations density of states with matter, asof discussed a after equation KK ( instantondefect defect with without considering off-shell geometries, one canblack add hole a extremality matter bound. Precisely this effect was obtained from a three-dimensional reduce. Taking the worldlinesthey of such appear particles as toexamples pointlike be of objects independent the of in defects thedefect the AdS we is two-dimensional studied parameterized theory. by in section Indeed, they give new 5.4 Including matter If we consider a three-dimensionalour theory near-extremal of two-dimensional description. gravity with We obtain matter,the a we useful matter can perspective as by ask including how ‘first-quantized’description. it particles, affects An using interesting effect a arises point-particle from particles action which rather run round than the a field important at very lowit twist, it implies does a not smallrequired require correction correction the to to existence the the ofgiving density light degeneracy an of operators. of exponentially states small black Instead, relative correction hole to to states the the grows entropy. Cardy polynomially formula. with The spin, the operators of lowest twist. Instead, to probe thislow regime twist, we which require is a morenot combined involve subtle limit new to of terms study. large which spin Since grow and rapidly our exponentially with proposed spin, cure but for instead the terms negativity that does become more The most important terms in this large spin expansion arise as the modular transforms of JHEP01(2021)118 . ). λ 5.1 (5.1) (5.2) (5.3) , and φ . First, ), giving 3 3.1 is not too large, λ has two zeros. The ). This is a model of U to arrive at an integral . Here, we will highlight D φ D , , , and now a theory of dilaton φ , the potential is positive for ) c )) α 0, since the potential increases φ ]), which approaches JT gravity 3.3 − ( (1 ) 92 U π x – ( 2 λ > λ > λ + , the potential φ − ) c 90 2 α λ e − ) φR (1 ( α π 2 2 − g − < λ < λ e √ (1 2 x – 41 – g π ] (see also [ 2 d √ 19 ] for a discussion on the relationship between the x + 4 Z 2 92 φ d 2 1 − Z 1, since otherwise defects will proliferate at large , we now can insert any number of defects in the path = ) = 2 ). At this point, we could carefully relate this to our earlier φ ( IG 2.5 α < I 3.3 ; this leads to classical solutions with energy unbounded below. U is monotone with a single zero, which means that the model is U → −∞ φ 0 the potential , and as a consequence we do not have classical solutions below a critical temperature: We discuss the semiclassical physics of this model in appendix This argument leads us to conjecture that we have an equivalence between three theo- As described in section λ < φ the model is notthere stable is even perturbatively some at hopeperhaps low that energies. we the are As solutions long saved thatto from as the lead this spacetime fate to quantum pinching thepotentially mechanically. off large instability For near quantum of example, the corrections. the minimum theory of are the dilaton themselves potential, unstable where we have classically stable. However, thiswithout is bound as no longerFor true coupling for smaller than aconsequence critical is that value, we 0 have sensibleat black least hole solutions make for sense everyall temperature, of and the we model can perturbatively. For dilaton gravity theory and the sum over defects. one aspect, determining thefeatures classical of stability the of dilaton thefor potential, model. for which This we depends have on three qualitative cases, shown in figure destroy or at least modify the asymptotic region). ries: JT gravity with defects,gravity. the We matrix make integral of some section veryfuture. preliminary comments, See and the leave more independent detailed work study [ to the with the normalisation chosendilaton to gravity match in parameters the (see classat appendix studied large by dilaton [ (as long as This can be incorporated as an additional local term added to the JT action ( with potential over moduli space, checking that wesuch land on a the calculation Weil-Petersson volume would form.We determine In will the particular, not normalization follow of thismethods. a route defect here, insertion instead fixing in the terms normalization of by comparingintegral the and two sum over them including symmetry factors, giving the exponential of ( in the path integral, ascalculations in by gauge-fixing ( diffeomorphisms and integrating out location of the defect, we can include defects by insertions of an operator proportional to JHEP01(2021)118 . U φ ] and with no 96 < λ ]). From the c is sufficiently λ λ ) < λ 95 , φ c ( λ 43 U (c) close to one. Another α . The model is well-behaved (see also [ 2 0 and 3.4 the largest and (c) φ 0 φ λ > . This can be intuitively understood as 0 with only a single zero at a positive value c 2 in section λ λ < ) φ ( < λ < λ U – 42 – (b) 0 ]. For a dilaton potential with a certain shape it is possible φ 94 , 93 with two zeros at negative values and , but does not make sense for low temperatures when c 0 λ ) φ ( λ < U (a) < λ < λ . Shape of the dilaton potential for (a) (b) 0 We can construct more general dilaton gravity models by including multiple species At a very qualitative level, this stability analysis chimes with our findings for the φ We thank N. Benjamin, Luseful Iliesiu, discussions. A. HM Maloney, is T. fundedNSF Mertens, by grant a D. PHY1801805, Len Stanford and DeBenedictis receives and Postdoctoral additionalGJT E. Fellowship and support is Witten under from supported for the by University of a California. Fundamental Physics Fellowship. using the exact density of statesWe as leave a such seed explorations for for the future solution work. of the Wheeler-de Witt equation. Acknowledgments a condensation ofexample defects is that given modifies byto the [ produce geometry flows deep wheresemiclassical in the analysis geometry the we inside bulk. see thepossible this bulk application may An is is be important to dS possible understand the for path integral over finite spaces following [ of defect, each ofIf which our should simply conjecture addmatrix is another integral correct, exponential duals. we termmodels thus to Through have the nonperturbatively. this a potential dual, largebetween An we geometries class interesting have that of the application are dilaton opportunity not is gravity necessarily to to theories AdS understand studying with such flows in the bulk density of states in thefor matrix any model negative at finite large and positive. We potentially havethe a theory semiclassical interpretation is of well-defined, the which regime deserves in which better understanding. Figure 3 of zeros. JHEP01(2021)118 ] 52 = 0. (A.4) (A.5) (A.6) (A.1) (A.2) (A.3) 1 i> . Finally α n and n , . . This is precisely 1 1 1 − b − 3 + ..., g,n g,n − ) . 1 g + 0) , − )) n . This calculation is easier n 1 n n α is independent of − 1 = 3 2 n . b 6+2 1 ! ,..., ( − g 0 g,n α g O g , 6 1 . . . b 0) 1 1 , . . . , α b 1 1 α )! − (24) 2 1 are the coefficients multiplying each (1 + n − b ) where one of them has a very large n i n ,..., n )! = ! 0 g,n n i g,n 3+ ) 3 + ) 3 + α n n, (not to be confused with the defect angles − g,n n g − i − 3 3 + 0) =1 , . . . , b , . . . , α α g 1 n i 3+ g 1 1 2) − b (3 / Q α − =1 g 2 1 ( n | n i , . . . , α – 43 – g , . . . , α b α ,..., 1 | 1 = ( ( (3 = (3 6=0 3+ 0 P 2 α i ! α − X b ( α g g n, = 1. This was computed by Itzykson and Zuber [ = g,n g 3 | 1 = 2 α n 0) α ! X | 3 + g (24) g,n g − 1 0) ) = ,..., g , b 0 ) = 1 (24) ( (3 b − n, ( n g,n ) = V g,n b 3 + V ( . Then the string equation is − n , . . . , α g,n g 1 V α (3 ( ) we wanted to prove. The dilaton and string equation prove more useful in 3 + . The leading term is this limit is given by − A.1 g → ∞ are a set of positive integers, ( ]. For our purposes since we are interested in taking only one boundary length to = 3 geodesic boundaries of length 11 i 1 | b 38 α n α | We thank A. Maloney for discussions on this. and 11 where the dots indicate termsthe relation in ( the polynomial withorder lower to powers derive of this result,relation. and also to derive subleading terms, than Mirzhakani recursion Putting everything together we get the leading power we needed This equation shows that the coefficientwe (3 need to compute this coefficientto for do some for particular a value of singleobtaining boundary so that 7.2 of [ be large, we willThe consider string the equation simpler becomes case very with simply top power have which Mirzhakani showed to be equivalent to her recursion relations in theorems 7.1 and where power of the boundary lengthwhich and will not appearlength in this can appendix). be obtained The with coefficients the of string highest equation. order on This the applies geodesic when, in this notation, we To show this we first write down the general WP volume as In this appendix we willg give a simple derivation of thelength WP volume on a surface with genus A Weil-Petersson volumes JHEP01(2021)118 , β ) x ( , and (B.4) (B.1) (B.2) (B.3) (A.7) (A.8) (A.9) JT u γ 1 , defined 2 v ) indicates e x ,...,K ( 0 → = 0 using the g ) x = 1 ) for genus zero. x 3 derivatives act ( i , g at − A.1 , =0
x n x )

) for x # x ( . (  i = 0. In this case we can i ) ) JT f , x g x u , ( i ( i 0 ) p y g . x JT . 6 ( n u 0 ) − πα ], given by g x n 2 p n 3.3 ( 2 1 i ) 44 b πx. b f 0 , x 6 h  J ) . through ( 2 1 − v 0 ) with respect to f . is another variable, and − n − ( x I ) h x n i → 2 ) through the equation n ) = 2 y ( v f 1 y ) (  i n , =1 − i Y y x JT JT x n (0)) ( ∂ 3)!2 ) theorem yf u u ∂x f  x K =1 0 JT  √ − ( i + X (0) = 2 and we reproduce ( ! u ∂ π n n ∂x ( x – 44 – 0 JT n y + ( 0 JT (2 u . It is useful to introduce the variable  u 1 2 x = 1 " x ! =1 n ∞ J reversion v 3 X n n y = ≈ − ) v n JT =1 of variables and functions b ∞  u ) + X n ( x √ 3 times, giving ∂ ( ,n K ∂x 0 g − ) +  V (and the parameter x n 2 1 ( x ) = g v implicitly as a function of ( g ) = v b ) with respect to ( can be used to resum series which have the following form x ) are arbitrary functions of ,n ( 0 x 12 ) is a given function defined in section g ( V x g this is approximated by the leading term where the ( we showed that, at genus zero, all the multi-loop amplitudes of JT gravity JT ) is the JT gravity specific heat, defined implicitly through u x ( ) and 3.3 → ∞ x JT ( 1 b u f , and λ When dealing with multiple types of defects we need a slight generalization of this This can be seen more directly for the case of genus zero https://en.wikipedia.org/wiki/Lagrange 12 → we define the variable In the main text we applied thisy theorem with theorem. If we allow a number Then the theorem states that the answer for the infinite sum written above is where the derivative of implicitly as a function of In section with a gas ofto defects show match that precisely wereversion with theorem used the a results theorem, from which a we matrix explain integral. in more In detail order here. The Lagrange We can efficiently compute theLagrange derivatives inversion of theorem. This gives B Lagrange reversion theorem When on the first Bessel function use the explicit formula for the WP volumes found in [ where JHEP01(2021)118 . 2 → , − ) 2 in ) 2 / = 0). (B.5) (C.2) (C.3) (C.4) x z ), and #) ( z ) exclud- i x 0 − ( f = 1 1 i i z f n z, I γ i 1 y y = 0 ( − i ( 0 , and the sum , E ) = ( I n 2 ). The leading P n 1+ ] and we will not , z , !# 0 8 1 → i h z n ) will be denoted by ) ( ) , . . . z W x 2 2 ..., x , ). The partition func- ..., ) with the edge of the ) ( βH z ( 0 , . . . , β π i f − 1 ) can be easily computed f W e n β (4 z, I ) + ) + 3.31 ( / , which is also necessary in z K =1 ..., β ) I 2 ( i Y ). and ( 2 − = ( 2

1+ ) 1 πz ) and 2 ) y h, , . . . , z B.5 g,n, z β z MM g,n, x 1 ) + non-negative integers ( ( W z z 0 Z W → − ( 0 ( 2 g 2 zy 2 K ,I 1 y ,n λ " 2 λ 0 y z 0 ). 1 2 − J X ) and − λ W ) = sin(2 ) + n ) + n h,I,h z = z ( ), giving ( β ) = ( ) = ( 0 =1 ( 2 K i 1 z ) + 1 JT ) + ( z y , z P B.3 1 ( , g, I 1 g,n, , . . . , z  MM 1 0 g,n, z 1 z, J – 45 – ( = z ∂ in ( W 2 ∂x , 0 λZ λ y − λW 0 n h  ) z, i ! = ( W ). After this shift, we already proved analytically that i ( x n i ) + ) + 1 is the total number of defect flavors. This theorem is ( n y denotes a subset of the labels z z ) + − f β ( z C.1 ( ,n K J ( K =1 1 JT i Y ) and ( g,n JT g,n − JT } J = i g y W Z n 0 X { = W I , and " i ∪ ) ) = ) = ) = λ ) + z z z g, I β I ( ( ( x 1 ( ( y → y = g ). The sum runs over a set of 4 g,n i MM g,n x h ) y , before using ( and ( 2 W 0 Z ), and they satisfy the recursion. , g z ) =
g E ) v 1 ). Define the special cases − ( x + 3.47 n = ( g 2 1 set to zero. The topological recursion relation is E 0 z ) = (C.1) 0 JT ( h u E → ,J ( 1 + 0 p z , . . . , β E i ( h 1 → z β ( πα g,n The calculation is tedious and we will give the main final results at each order, to This form of the recursion applies for a density of states that starts at The pure JT gravity has spectral curve 2 Res W MM g,n 0 where to easecontributions, notation which we match wrote with pure JT gravity, were computed in [ exactly from ( show how surprising it isrecursion. that the We gravitational introduce theory some with notation defects first, satisfies the expanding topological the following quantities as Therefore, when we apply it toto our shift disk density of statesJT for JT gravity gravity plus plus defects defects we exactlyorder need for gives the recursion to make sense. Moreover all where excludes the cases ( tion connected correlator from theZ matrix model perspective Tr( The latter is equivalentspectrum to the universal answer presented in ( In this appendix we includeto some a explicit checks matrix that integral JTthis at gravity section. with higher defects genus. is equivalent To simplify the expressions, we will set J a simple generalization ofapply the the previous multinomial one theorem if to we call C Explicit higher genus checks for any function ing the case where they are all zero. In the main text we applied this with then we can use the previous theorem to show that JHEP01(2021)118 ) ]. n g,n 42 ). W 1 (C.9) (C.6) (C.7) (C.8) (C.5) − 3.10 , . . . , β ) 1 , β ( 2 . , and Taylor παz / 0 5 2 / MM g,n β 5 βE Z π e β sin 2 1 π √ α 6 1 2 √ from the topological 6 + − gives 1 , 2 + . 1 / λ πz ) 3 2 2 / 1 W β 3 z 2 β 2) π (sin 2 ) . 2) − 2 2 πz ) α − πz 6 2 α 2 ( − 6 α 2 − πiα ( / 4 1 2 3 2 cos 2 z / π is defined by the ensemble with the edge 3 b, ’s are defined with respect to the shifted 3 z − ( παz 12 π z 2 Now we expand the density of states and 2 − ). Finally, we multiply this by a factor of , ) we obtain W 4 ) 1 The first correction to the density of states MM g,n z ). To compute )(3 + (3 0 − β ). Correcting by the zero-point energy, which ( V 2 z Z παz E ) y β ( p E α + ) p H – 46 – JT √ 4 − ( β, b ’s and ) y + 2 cos 2 π cos 2 ( α β 4 y (1 − (2 α − 2 + e πz . The leading correction in π 2 2 + , E/ 0 α . We get the next shift to the spectral curve 2 ) = trumpet 4 √ λ z α 12 W ) = ( 4 − 1 1 12 πα 1) cos 2 z y − 0, the quantity ( bdbZ (3 1 − 2 , (3 / 1 ∞ ) and 2 → , 7 2 0 z 1 , and switching to z / 0 cosh 2 π ( Z 2 7 λ y W E π λ 2 π ) = 2 = ) = α , and compare with the answer from JT gravity with defects order by ∼ − β ) = 2 λ ( δρ β 0 ( − E 1 gives the correction to the partition function , ) is given by 1 1 , to correct the shift in energy, obtaining the quantity we call , λ 1 MM 1 , ) (2(1 to quadratic order in JT+def 1 Z n 3.10 β Z 0 . + λ E ) = ... z + ( . This is done to be able to directly compare with the gravitational answer ( 1 2 0 This is equal to the gravitational JT gravity plus defect calculation, which at this order The general procedure we apply below is the following. First, we compute the cor- β y ( E 0 E One boundary, one handle,shift two by defects. where we used the explicit form of the WP volumes tabulated by Do in appendix B of [ following ( A Laplace transform of theexpansion full in resolvent to this order, multiplication by The first term incomes the from numerator the comesrecursion from shift we the in only one-defect need the amplitude, energy while of the second One boundary, one handle,from one one defect defect. is to this order is and from it obtain theexpectation resolvent value correlator. of Third, ae an product inverse of Laplace Tr( transform givesabove, the expand in order in ensembles with edge at at rection to theSecond, spectral we curve insert this from result the in the defects, topological recursion and to shift compute the the correction zero-point to energy to zero. repeat them here. While in this section JHEP01(2021)118 ) (C.14) (C.16) (C.17) (C.10) (C.11) (C.12) (C.13) 2 (which 2 β / +1488375) 3 5 2 1 , + , β z 0 1 2 π , the next β / √ 7 W λ 2 6 π )) β / 4 2 √ 7 6 1 z +2204) +1015) + 6 z β ) 2 2 1 +13)( 2 2 1 ) (C.15) 4 z ) to linear order α 2 / + z α 2 5 α z 2 6 π + β ( / +595) 2 2 6) π 5 2172 πiα y 2 / α ) β − 2 − 3 α 4 2 2 +695 4 3 11 π , / ) (C.18) ) 1080 4 1 α α α 2 3 − 2 z 3 π 4 α ) , b (2( π +3895 2 − 1 4 πiα 4 + 7 +504 +(28 α b π 6 2 2 α 2 − 2(1 2 ( z α +338 3 2 b, )) α , + 449 6 1 29 2(1 π )+3 ( 2 1 2 z ) 2 − 2 − 2 2 β ) / 6 2 + V ( 16 , 6 2 β π 3 π α 2 2 / 2 + to linear order. We obtain α β / √ ! / 2 2) − V − 2 11 3 ) 1 β / ) 11 36 (15 πiα − i , β β (145 1 7 6 π 2 2 2 2 2 β π + 2 π / α , b π )(7 + 3 / 7 )) i + β W β, b 7 2 π 2 2 2 − 1 β +( ( (45+9(5 √ β 6 2 / 8 α 1 12 )) β α ( 2 4 1 z 3 z 2 , and Taylor expansion in z πiα, − 6 β 1 β 4 )( 1 In this case knowing α z 0 12 1 2 2 10 With the ingredients computed so far we − 1 1)(1890 (3 z )+ β − z 10 1 β 2 18 +342) − 2 2 z b, +7083) α βE 2 (3 2 z + 3) +558) 72 ( 8 2 2 2 α e α 1 2 trumpet − 3 ( α − π α β , α trumpet 8) 2 ) +10850) 2 (which we already computed) and 8 360 1 2 2 π − α Z − + 4(2 (13 1814400 1 V α α – 47 – 2 i 3767 , 1814400 10 3767 ) 4 α − + +( 1 − (13 − − 2 1 db bdb Z β π 4 4 i 28)+6( β + ). α W √ 13440 (29( α b )) β, b 3 β − 2 ∞ 2 2) − (3+(5 ( 2 2 / 2 / 0 , √ 4 2 2 − 7 28350) , b α 2 α z Z 11 α +641 2 π 4320 2 / +641 − 2 1 6 π Y =1 α 6 2 1 z ) 4320 i α + 11 − z α 4 , b (( +39 π 2 α 43 +9 1 2 ) 4 /

) = 43 +4344 4 2 b 4 (2 − 9 π trumpet α +3) 6 − z β ( +7 α 8 β 2 2 8 α ∞ ) ( 2 18 3 12 α 12 α , α 2 α 0 α (45 ( +7 ( )( 1 − − α 2 2 6 2 18 464 Z 8 ( 2 16 6 V π π α π bdbZ 15 α − 2 ) 1 α − , 180 8 − β 2 ( − − 16 π 1 1 α 6 α , ∞ )(7 − 9135 1)(2 (1 β JT+def ) = 18 2 π 0 − 2 (44 2 2 − − (15 √ 2 α / − )(7 Z Z (1 9(5 4 2 1 2 / 2 z − α / 3 19 , β ( α 1 π π 2! (2 1 2 15 − π π 388) β + + ) = ) = ) = (2 ( − = 2 2 1 210 2 z ) = α − ( , z , β ) = 2 1 1 β 1 ) = , , β ( 475 z 1 2 β ( β , , ( − ( 1 ( JT+def 1 1 1 4 , ) = 1 , 2 , 1 α , Z 2 1 , 2 2 , W 1 , MM , 2 , z 1 1 MM MM (29 1 1 , ( Z 4 JT+def 1 1 Z , Z W is enough. The recursion involves π This is equivalent to the gravitational calculation This function is precisely equal to the gravitational calculation, which is 1 Z , 30 λ 2 + W The partition function is One boundary, two handles,can one continue the defect. topological recursion and compute we can compute exactly in the previous section). We get And the correction to the partition function is using the explicit form of Two boundaries, one handle,at one defect. This is equal to thegiven gravitational by JT gravity plus defect calculation, which at this order is After Laplace transform,correction multiplication to by the partition function is The next correction to the resolvent from the topological recursion is given by JHEP01(2021)118 ]. α is , so U 92 ~ , (D.6) (D.3) (D.4) (D.5) (D.1) (D.2) 0, 90 < and coupling i α . For Λ 3 1, which is expected plays the role of , α α <  . ) − ) φ α ) ˜ φ i , ( α − ˜ has a minimum at U − )) (1 φ (1 . + U ( π π log Λ + 1 φ ) 2 2 ) U as long as α − α . e φ ˜ − + ˜ φ i φR − ) = − 2 (1 λ  ) e π 2 (1 i 2 min g π φR α − φ ( 2 e ( √ 2 − x + 2Λ g 2 ˜ φ (1 d √ and Λ independently once quantum corrections π x + 2Λ – 48 – ,U 4 0, the potential 2 Z α ) φ d ) F > α ) = 2 =1 α N i ˜ X φ Z − ( . − 1 2 + ) = 2 ˜ 1 U (1 log Λ φ − φ (1 ( π , by formally summing defects as insertions in the path π 2 U = 4 → ∞ = − φ ) = 2 IG 5.5 I φ = ( min U . The the potential can be written as φ IG : λ I . Here we will study the case of a single species with parameters φ 2 ) F )) α α − − (1 ,...,N (1 π π (2 = 1 = 2 i ˜ ≡ φ . 1 is a natural classical limit to study. λ Now, we will describe the classical thermodynamics of the model following [ We will here simply analyze the dilaton model at the classical level. The model then To analyze this theory in the semiclassical limit it is convenient to introduce the cou- As explained in section → , with i First, we describe thestability. qualitative There features are of three thedecreasing, different so potential, parameter has which regimes, a determine shown single in the zero. figure classical For Λ The model will dependen nontriviallyare on taken into account. Itα is interesting to note from this that 1 dilaton with The potential approached the JT gravitysince one otherwise for large we get anasymptotic boundary instability driven where by the proliferation of instantons towardsonly the depends nontrivially on Λ, as can be seen by rewriting the action with a rescaled and pling Λ and potential This theory emerges from integratingλ out the species of defects with angle integral, we obtain a dilaton gravity theory with action In this appendix wewe will define as present a more 2Ddefects. details dilaton regarding gravity We the with will aenergy instanton analyze specific gas limit the potential theory, of originating semiclassical which JT fromtwo thermodynamics gravity theories. summing and with over defects, match as it a with basic the check high of the equivalence between the D The instanton gas theory JHEP01(2021)118 c ∼ Λ ) , so is at r 2 > (D.7) (D.8) (D.9) (  (D.11) (D.12) (D.13) (D.14) (D.10) U f t ) h γ r ( 2 U . For Λ  h d φ 2 ρ , ) + r ( 2 , we have one branch c there is a second zero, dρ Λ c dr U Λ . < , h φ , the largest zero of of the dilaton, as h c φ < Λ φ Z Λ ) h , T dS α 0 φ Λ < . < − h = ) = > < (1 φ π ) ( 2 dE − . The classical Euclidean solutions of , and the energy is read off from the min , e φ > φ . For 0 h ) φ . This gives a Euclidean metric as long γ 0 ∀ ( φ h α 2 r, f , Λ) U )) − γ β = − > φ α → −∞ ( our metric goes like h (1 ⇒ 0 Λ = − φ > φ φ φ π W = 2 2 ∂ ρ ∂ – 49 – (1 , φ ) 1 φ L , ) π h 2 − = r dr > h r 4 ( e ) ( (2 , and 0 dr r U r f , φ πφ ( = , so we have chosen the correct normalization of the ) − at infinity. Putting this together, our thermodynamic ∂ c = U h + ∼ 2 h φ φ + 2 h Λ φ 2 πγ φ φ f 1 h ( φ 0 4 coordinate. The proper length of the boundary (at large γ r − 1 dt Z > 2 U S ) t r − Λ βγ ( = = = . r f h 2 S ∼ T E φ − ) , but also a second branch for sufficiently negative γ ∂ 0 φ = ( f 2 > φ h p ds 1 φ − the potential does not have any zeros. For Λ 0, our solutions correspond to ); if we write c βγ h Λ r < . The entropy is given by the value of is the Lambert function or product log. For 0 is the period of the = ) − > h r ∂ πγ β W r ( ) is always positive, so we require 4 L U φ )( As a rudimentary comparison between the dilaton model and the defects, we now For Λ Now recall our boundary conditions at the asymptotic boundary with large positive ( h f r = ( coordinate. The temperature is determined by smoothness at the origin, where ) is and it is manifest that we have the expectedcompare relation the thermodynamics tonote linear that order the in resultthat Λ will the with not exponential the only term effect in apply the of for potential a small is single Λ, a defect. but small correction. also We at sufficiently large energy U β constant term when wequantities expand are given by dilaton and where r t of solutions for we have solutions for all where we have chosen theas gauge where and the potential increases exponentiallythe as theory are uniquely determined by the minimum value so for Λ For Λ larger than a critical value, this minimum is positive, JHEP01(2021)118 ] , 10 02 Int. J. (2010) (D.15) (D.16) (D.17) , 09 JHEP JHEP , , JHEP , arXiv:1407.5597 arXiv:1903.11115 [ , ]. , γE 2 √ Causality Constraints on ) SPIRE α γE IN − 2 (2016) 020 ][ (1 √ π ]. 2 02 γE − 2 ]. e √ Holography from Conformal Field ) α γE πλ λ SPIRE − 2 2 2 Light-cone modular bootstrap and pure JHEP SPIRE IN ]. (1 , √ )) π ∼ IN ][ 2 α ][
−
e − SPIRE 2 γE ) (1 γE IN arXiv:1906.04184 2 α π 2 [ – 50 – ][ JT gravity as a matrix integral ]. ]. √ √ − Λ π (Λ), where the term of order Λ is chosen such that πα 2 (1 2 Emergent Spacetime and Holographic CFTs O π Λ = (2 SPIRE SPIRE 2 ]. + IN IN limit of superconformal field theories and supergravity arXiv:0907.0151 AdS Field Theory from Conformal Field Theory = sinh hep-th/9711200 ][ ][ ), which permits any use, distribution and reproduction in cosh [ to order Λ. From this, we read off the first order variation [ γE N Quantum Gravity Partition Functions in Three Dimensions 2 More Holography from Conformal Field Theory (2019) 066029 δS SPIRE 1 so that we can expect the classical thermodynamics to be E γE √ IN πλ 2 2  ) including a single defect: ][ = arXiv:0712.0155 100 √ [ h = φ γE (2009) 079 3.18 CC-BY 4.0 The Large (1999) 1113 δS 10 This article is distributed under the terms of the Creative Commons 38 arXiv:1101.4163 arXiv:1208.0337 [ [ (2010) 029 ]. ]. JHEP Phys. Rev. D 02 , , arXiv:1006.0976 [ SPIRE SPIRE IN IN gravity [ JHEP (2013) 054 Corrections to the Graviton Three-Point[ Coupling 099 (2012) 106 Theor. Phys. Theory Now, from the defects we find the variation in the entropy by taking the logarithm of To do this, write P. Saad, S.H. Shenker and D. Stanford, A. Maloney and E. Witten, X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, N. Benjamin, H. Ooguri, S.-H. Shao and Y. Wang, I. Heemskerk and J. Sully, S. El-Showk and K. Papadodimas, A. Fitzpatrick and J. Kaplan, J.M. Maldacena, I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, [8] [9] [6] [7] [3] [4] [5] [1] [2] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. where we have taken applicable. 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