JHEP07(2017)123 Springer July 19, 2017 July 26, 2017 : : + 3 there are March 24, 2017 : p ≤ d = 8 supergravity in Accepted Published Received N c [email protected] , Published for SISSA by https://doi.org/10.1007/JHEP07(2017)123 and Irene Valenzuela b [email protected] , . 3 Angel M. Uranga a 1702.06147 + 1)-form gauge fields, and for spacetime dimension The Authors. p c Black Holes, Global Symmetries, Black Holes in String Theory, Superstring

In this paper we study the consistency of generalized global symmetries in , [email protected] Fohringer Ring 6, 80805 Munich, Germany E-mail: Institute for Theoretical Physics andUtrecht University, Center Princetonplein for 5, Extreme 3584 MatterInstituto CC and IFT-UAM/CSIC, de Te´orica Utrecht, Emergent F´ısica The Phenomena, Netherlands C/ Nicol´asCabrera 13-15, Campus deMax-Planck-Institut Cantoblanco, 28049 fur Madrid, Physik, Spain b c a Open Access Article funded by SCOAP Keywords: Vacua ArXiv ePrint: sor) gauge fields is in theterms. Swampland This unless its conjecture effective implies actioncharged that includes particles such many Chern-Simons familiar required theories, byfour like dimensions, the QED are Weak (even inconsistent including GravityChern-Simons in the Conjecture) terms. quantum or gravity unless they are completed by these pairs, in which thesemetry, problems via are (suitable evaded versions bythe of) either absence Stuckelberg gauging of or or such 4-formglobal breaking couplings, couplings. symmetries the the by global generic We cubic argue solution sym- fields. Chern-Simons that in We terms string even conjecture involving theory in that different is any antisymmetric the theory tensor breaking with of (standard the or higher-degree antisymmetric ten- theories with ( obstructions to their breakingwith even a by 2-form quantum gauge effects fieldholes of (or charged with with objects. an quantum axion In hair,describe scalar), 4d precise these a theories fields mechanisms, global endow and Schwarzschild examples charge black from leading string to compactifications and usual holographic trouble with remnants. We Abstract: theories of quantum gravity, in particular string theory. Such global symmetries arise in A Chern-Simons pandemic Miguel Montero, JHEP07(2017)123 7 22 21 10 26 5 18 31 24 41 12 41 29 9 – 1 – 24 7 14 36 17 17 40 17 34 28 2 13 38 28 4.2.3 Final remarks 4.2.1 The4.2.2 AdS case The dS case 6.1 KK photons 6.2 Type IIA/B6.3 flux compactifications Digression:6.4 an M theory Other BGHHS examples black hole 5.1 Rationale for5.2 the conjecture Implications for Einstein gravity and supergravity 4.3 Symmetry breaking4.4 in a holographic Field example theory4.5 symmetry breaking by Field topological theory mass symmetry breaking by Chern-Simons term 4.1 Gauging 4.2 Presence of charged states 3.1 Global hair3.2 for black holes Production of3.3 BGHHS black holes Quantum effects and observability of quantum hair C 2d instantons and Chern-Simons terms 8 Conclusions A Confinement of strings B Self-dual fluxes in Taub-NUT space 7 3-form global symmetries 5 A conjecture about Chern-Simons terms 6 Examples 4 Breaking generalized global symmetries 3 Black holes with generalized global symmetry charge Contents 1 Introduction 2 A brief review of generalized global symmetries JHEP07(2017)123 ] +1 4 , p + 3 3 p + Λ = +1 d p ] (see [ C 2 , 1 → dimensions is in +1 D p in spacetime dimen- C +1 p C ]. In fact, the absence of exact 7 ]). ]). 35 33 + 1)-form gauge field in – p ]. A prototypical example arises in theories , for which the global symmetry is associated 12 φ 36 – 2 – , with charged operators given by (exponentials +2 p + 1)-form gauge potential F p = 0; they lead to generalized global symmetries with , which in the absence of charges have fields strengths ∗ ] for additional ideas, in particular the Weak Gravity ] (in string theory, there are stronger arguments at the ]. +1 = +2 p 6 p 0 , 10 C 34 5 j , F , 9 , 4 ∗ d +2 p most notably String Theory [ F 1 . In four spacetime dimensions, which is our main focus, this occurs = = 0, ], or in holographic setups [ j 7 dφ ], i.e. effective field theories which cannot be embedded in a theory of +2 8 p = j ] and related recent activity [ dF 11 + 3. This can be used to study Quantum Gravity constraints on theories with a closed form. Illustrative examples of the two mechanisms are: p + 1)-form gauge fields +1 p = p obeying d There are several ways in which a theory may escape this fate within field theory, The conclusions generalize to a ( In this work, we show that arguments along those lines can also be applied to gener- Most discussions about global symmetries in Quantum Gravity focus on symmetries For alternative viewpoints, see [ 1 +2 p which roughly amount either tobreak promote it. the To global introduce symmetry them,with note to Λ that a the gauge symmetry symmetry, acts or by to shifting (arbitrary-rank antisymmetric tensor) gauge fields.to Such higher constraints dimensions, can be ifIn extended we particular, even assume in the principle any theorydanger theory should of with make falling a sense into ( and upon in lead compactification. the to Swampland, problematic since global it symmetries. can be compactified down to the corresponding 2-form globalthe symmetry, incompatibility leading of to generalized troublesfooting global with to remnants. symmetries usual with This global Quantum symmetries. puts Gravity on a similar sion explicitly even in the vacuumfine e.g. charged objects via loop-effects, and prevent butsymmetry this in of kind lower a of dimensions 2d breaking. IR periodicto Concretely, effects axion-like this the scalar con- applies current to thefor shift the dynamics of aendow 4d 2-form Schwarzschild gauge black field; holes we actually with show quantum the hair, presence of which this manifests fields can as a charge under with ( F conserved currents of) generalized Wilson linescycles. of In high the enough electric dimension, the and existence magnetic of charged gauge objects potentials breaks these on symmetries non-trivial some plausible assumption abouta the number consistent of gravity possible theoryperturbative remnants [ level not [ getting to large in alized global symmetries, in the sense of [ Quantum Gravity (see e.g.Conjecture [ [ under which charged objects areglobal particles. symmetries involves The evaporation main of black argument holes supporting by the emission violation of of charged particles, with Global symmetries are a powerful toolbeen in Quantum argued Field to Theory, yet be they havefor incompatible convincingly with recent theories discussions), including Quantumglobal Gravity symmetries [ isthe the Swampland prototype [ of a set of ideas distilling out the Landscape from 1 Introduction JHEP07(2017)123 - d → 3 ]. +2 p 44 C via a couplings 2 4 ] in a recent − Stuckelberg p 42 − φF ]. . These string d 5 with lagrangian BF C S . An example is 39 3 , c × +1 38 ]. p 5 ]; also example involving stringy among NSNS and RR 43 ] and in [ 5 p transforming as 46 S − , . A prototypical example 41 7 × , by introducing a standard 2 +2 5 45 F − p φ 40 p p ]. On the other hand, there C − cannot have simultaneously a C d 3 2 47 , b H dC 37 4-form coupling, since together they = 4 1 as follows. Even in the absence of the − . Another example is the gauging of the 4 p φF . 2 for some non-abelian group, both can be present − | 2)-form gauge potential 2 d 2 A B ] of axion monodromy [ F by coupling it to a 3-form − F − 2 has a 37 p b Another example is given by – 3 – φ = tr dφ − | 4 2 d , with F 1 − ; the global symmetry thus simply becomes the global p +2)-form gauge potential − ]; for d p +1 p F couplings play a fundamental role in making certain U(1) 36 q + , there is an explicit solution in the AdS p BF C 4.3 with lagrangian -form. For instance, a 4d 2-form A p , for general Λ ; as in the 4-form description introduced in [ 4 +1 , as in the dual description [ p 3 φF | Λ 3 d c + − 2 +2 p db application to axioncouplings monodromy. in 4d, insymmetry which shifting coupling periods to of a the standard 2-form U(1) gauge field breaks the global | Breaking it by couplingdimensional coupling it to ais the ( breaking ofstrength the shift symmetry of a 4d axion by coupling it to a 4-form field C part of a gaugethe removal symmetry, of obtained the by globalU(1) restricting shift gauge symmetry to field for closed asymmetry Λ 4d associated axion to a 2-form field Gauging it by introducing a ( Stuckelberg coupling if its dual axion The solution to this conundrum is that string theory provides a slightly more subtle The above topological couplings are often present in string vacua. For instance, in It is interesting, and clear from the above examples, that considering the symmetries present. In fact this is the case even for many supersymmetric AdS vacua with well- There is a mixed ’t Hooft anomaly (from a Green-Schwarz diagram) which prevents us from gauging the As we will discuss in section See also a related realization involving D-brane instantons in string theory [ 2 • • 4 2 3 F 2 two dual potentials at theif there same are time additional [ chargedanother matter scalar, fermions which contributing mixes to withand the the one anomaly; axion with the in 4-form fermion two coupling. phase orthogonal contributes combinations, one witheffects. Stuckelberg However, coupling even in this example, the mechanism described in the main text is present. (yet related) mechanism to breakabove quadratic the couplings, symmetries, string modelsarising contain upon cubic reduction Chern-Simons of couplings, the e.g.forms 10d those supergravity in couplings type II, I theories, and generalizations (or heterotic dual versions) thereof. In not defined holographic duals, includingcompactifications the and celebrated type constructions IIBsymmetry would on problem, seemingly AdS yet suffer they correspond from to the consistent string generalized theory global configurations. 4d string compactifications, gauge fields massive (inunderlie particular, stabilization anomalous of ones), axion seeare components e.g. many in [ examples moduli in [ which such couplings (or their analogues in other dimensions) are breaking of its dual, andpresent vice-versa. for It the is same alsob interesting that both mechanismscombine cannot into be a Green-Schwarz-likeIn mechanism string rendering theory the realizations, theory this not follows from gauge microscopic invariant. consistency conditions [ associated to a gauge field and its magnetic dual, the gauging of one is equivalent to JHEP07(2017)123 ]. 2 5 48 [ -field 248 B 1)-form presents − U(1) . Section 6 discusses a d 6 × 3 8 -field hair can E B and 5 and . 8 496 We provide examples of seem- 5 = 8 SUGRA. extends the conjecture to ( explores the different ways to solve this provides a brief introduction to the as- N 7 4 on, particularly sections 2 4 – 4 – ], which focuses on the black holes with 49 = 1 supergravities with groups U(1) N -field quantum hair, showing that an ungauged/unbroken couplings, bringing us to the solutions described above. On B 4 φF . Those who care mostly about the conjecture, implications, and or 4 BF and 3 4 dimensions, or four-dimensional ≥ dimensions. Finally, we present our conclusions in section d d We have tried to structure the paper in such a way that the different sections are as The paper is organized as follows: section We have learned about the work [ From this latter viewpoint, we can promote our observations to a conjecture, in the Topological Chern-Simons terms have played an important role (albeit seemingly unrelated to ours) in 5 “experimental” support, can read from section is a review which includes no new material. outcasting to the Swampland certain 10d examples in support offields the in conjecture. Section self-contained as possible. Readersfocus interested on mostly sections in black holes with pects of generalized globalSchwarzschild symmetries black which hole will with be usefulsymmetry later leads on. to a Section problem, remnant either problem. by Section gauging orpresents breaking our the conjecture symmetry and via discusses different implications mechanisms. and Section rationale for it. Section quantum hair which were ouroration original to motivation, and obtain studies newthe the cutoffs generic dynamics on mechanism of the by their which evap- effectiveresolved the in field problems string theory. with theory. this The and present similar paper situations focuses seem on to be breaking the corresponding generalized globalingly symmetries. reasonable theoriesafter which the do inclusion of not thegravity survive charged in particles it, required for by the instance Weak Gravity U(1) Conjecture, gauge pure theory even for the presence of these terms in the string effectivelandscape/swampland action. spirit. Any fieldfields theory with is (possibly not antisymmetric compatible tensor) with gauge Quantum Gravity if it does not include Chern-Simons terms can be used to nucleate bubblesof inside this which kind the nicely flux illustrate is theto turned breaking arbitrary on; of numbers effects the of of symmetry. virtualof dimensions. The bubbles argument string Therefore, easily compactifications generalizes the with generalizedthe (possibly global presence antisymmetric-tensor) symmetry of gauge problem cubic fieldssiderations Chern-Simons is of terms solved gauge-gravity (we by holography). also provide Conversely, additional this support provides a from con- partial explanation displays quadratic the other hand, inthe the absence 4d of effective internal theoryproblem. flux, seems there However, to the is presence describe no ofing actual a the of quadratic massless cubic the coupling, gauge Chern-Simons symmetry, and term even field in also with produces the a the fluxless global break- vacuum: symmetry the theory contains domain walls, that the presence of internal fluxes in compactification to e.g. four dimensions, the 4d theory JHEP07(2017)123 + 1- (2.2) (2.3) (2.1) p -form Hodge -form global d p must be closed, p λ , so that p . p A must actually be a singlet O p A O (we will only be concerned with -cycles p λ . λ. p extends all the way to infinity. In x = 0. O Z +1 ! +1 p p p +1 A ) = J λ p x p ∗ C defined on ( , and also demands that correlators including A p d ), the operator λ p Z +1 – 5 – A p dλ

= 2.3 A O ]). Since local operators are inserted at a point, ) Z = ) , unless x 50 p ( p x ( λ φ O φ λ δ δ . These symmetries no longer act in a natural way on p λ ) is still a symmetry. In the 0-form example this amounts to -form, 2.2 p , which transforms by + 1) conserved currents, . The original generalized global symmetry still survives, as the subgroup . p p +1 λ ] may safely skip this section. p = 0. There is a similar relationship between generalized ∗ A C 36 j = ∗ d +1 p closed. Because of the coupling ( A p ∂ λ vanish unless they also contain a factor to vary from point to point, even if the symmetry is global (for instance, this is how Gauging the symmetry amounts to dropping the requirement that Noether’s theorem relates ordinary global symmetries with conserved currents, 1-forms Generalized global symmetries are constructed by taking the symmetry parameter to Ordinary symmetries are described by a parameter p λ A where with under transformations withcompact closed spaces this cannotunder the happen, original and generalized thus global symmetry; every this correlator is is Gauss’ law. forced to be invariant insertion point. The storyform is potential similar in theO generalized case; one introduces a local while demanding that ( the familiar condition that symmetrycase, parameter invariance is demands now the allowed to introductionlocal be of operators nonconstant. an must In additional be this 1-form dressed potential, by and a charged gauge field Wilson line along a path ending at the Thus, correlators are invariant under infinitesimal shifts of the satisfying symmetries and ( be an arbitrary closed local operators, but rather on operators Using Hodge duality, we might asdual well describe parameter the global symmetry with the mations on fields, suchof that expectation values areWard identities invariant. are It derived, isnamely see very on e.g. a natural 0-cycle, [ to thenatural think infinitesimal pairing action between of cycles the and symmetry forms, can be recast in terms of the We will start by reviewingus a organize few and aspects homogenize offamiliar the generalized with discussion global [ symmetries in which the will rest help of theabelian paper. rank 1 symmetries). The This reader parameter already generates a one-parameter family of transfor- 2 A brief review of generalized global symmetries JHEP07(2017)123 . . p A dλ have (2.6) (2.5) is the + 0 , where +2 d +1 p p +1 F p C ∗ , breaking its → + Λ φ +1 , and the conser- +1 3, where p p F C C − 0 the electric current. → d , e has vanishing divergence j  = which does not vanish at +1 p A p p +1 (plus two other symmetries C p λ C J 1 Z i , with S e  j . From this perspective, it is clear = 4 ) are manifestly not invariant under exp A F  . 2.6 ∗ R 1 λ d +1 . p = C C 6= 0 (2.4) Z φ 4.2 i J  – 6 – ∗ d D-brane , they enjoy a gauge invariance exp Z +1 → p C  + 1)-form global symmetries A p C Z i  + 3, generalized global symmetries can be spontaneously broken = 0. This symmetry is broken explicitly by the introduction of become instantons for the three-dimensional axion F exp 1 ∗ S d > p d -form, not necessarily exact. These symmetries are explicitly broken p -form gauge potentials from RR and NSNS fields (or heterotic counterparts). p . As an illustration, consider QED in four dimensions. There is a putative 1-form +1 p Furthermore, if Perhaps the most familiar examples of generalized global symmetries in string theory Generalized gauge symmetries lead to lower-degree generalized symmetries under com- We can alternatively break the symmetry explicitly. In the 0-form case, this means is any closed J ∗ p number of noncompact dimensions,subject requires of further this paper, consideration, as and explained will in be section in the the main vacuum. Foris instance, in spontaneously the broken, above and 4d the QED photon example, is the global precisely 1-form the symmetry Goldstone mode. This in turn This symmetry is gauged,invariant and under so the ungauged, is ( exact.Λ On top ofby this, the the D-branes supergravity themselves, action sincethe is their also symmetry. couplings ( Anothernonvanishing divergence, way as of explained saying above. this The is case that when the associated currents infinity generates a symmetrytheir of couplings the theory, in which the D-branes get a phase given by shift symmetry. is given by Focusing on e.g. the RR potentials The charged operators are associated to D-branes. A closed coming from the magneticsymmetry, restricted potentials). to the The three-dimensionalsists 3d gauge of 1-form field, shifts symmetry and of isthat the the charged the 0-form three-dimensional objects same symmetry axion in con- 4d thecles 1-form higher-dimensional running theory on break the the symmetry: electric parti- that is, they arevation multiplied equation by a is phase.electrically The charged associated particles, current since is we just now have pactification. For instance, the0-form 1-form and symmetry one of 1-form QED symmetry when we just compactifying discussed on leads to one global symmetry, which actsThe by charged shifting operators are the Wilson periods loops, of since the electromagnetic potential that the divergence of the current is now nonvanishing: A similar statement holds for higherd symmetries: the current JHEP07(2017)123 6 (3.1) (3.2) ] , if we had such a symmetry, 34 , 4 2 can arise directly from dimensional 2 dB , B 2 = 3 + Λ 2 B ,H – 7 – 2 | → 3 2 H | +3, in which the Coleman-Mermin-Wagner prevents B contains noncontractible 2-cycles Σ, we can compute p 1 2 , to which the interested reader may jump safely. 4 ≤ 4 Z X d -form potentials, or as the four-dimensional dual potential to an p ] for alternative viewpoints. 6 , 5 : if the 4d spacetime is any closed 2-form. This is an example of 2-form global symmetry, as discussed 2 2 . The latter arises in particular in any axion inflation model (although the standard φ This system enjoys a symmetry given by Although the argument is not to be taken too seriously, it does tell us that we should be See however [ . Such systems are ubiquitous in string theory: 6 2 where Λ in section where by “roughly” we meansuch as that the the dilaton kinetic or term other coefficient moduli can of depend the on compactification. other fields, reduction from the higher axion inflationary description in terms of theof axion the does action not is make roughly it manifest). of the The relevant form part wary of any theory inbackreact which on one the can metric. endow blackon In holes a the with simple “global following four-dimensional hair”, weB theory which consider which does one includes not such gravity, realization. plus a We dynamical will 2-form focus field folklore theorem that theoriesThere of are quantum several loose gravity arguments do forwe not would this. be have For able instance exact to [ build global waywhich too would symmetries. lead many (actually to infinitely a many) pathology charged atdue black arbitrarily hole to low states, energies. the Unruh For instance, effect the for energy density a uniformly accelerated observer in flat space would diverge. for other generalized globalrecapped symmetries. and extended in The section main discussion for3.1 the latter are Global hair anyway We for will now black address holes the problem that originally motivated this work. It is a widely accepted role in the physics ofanalysis Schwarzschild black explains holes. the The physical purpose blackof is hole generalized two-fold: arguments global underlying symmetries on with the one quantum compatibility hand,the gravity, placing problems this problems then of on similar usual footing globalsystem with symmetries. can be On regarded the as other a hand, prototype the of a Euclidean 2d version compactification, of providing the a good template be concerned with theories with spontaneous breaking of the symmetry. 3 Black holes withIn generalized this global section symmetry we charge consider generalized 2-form global symmetries in 4d, and explore their forces the low-energy lagrangian to be the standard Maxwell term. However, we will also JHEP07(2017)123 2 ]. S 34 (3.6) (3.3) (3.4) (3.5) volume . 2 ) is just a B S 2 S 3.2 R Ω corresponds ) was proposed d = 0 and there is no 3.5 = Q over the horizon 2 ≡ 3 Q H ) is however not a local 3.5 ]. Classically, the no-hair the- Using the unit sphere 53 7 , -field period i 52 B Q | of length 1 (since Λ . ) ) 1 . Ω πinQ -field value, since Z R 2 2 S 2 , , B 4 4 B B Q d ) ). In this case, the symmetry ( X X Σ dQe Z ( ( – 8 – -field background of the form = Z , 2 2 Ω shift the 4 B Z H H B X λd ( = 2 i vanishes everywhere in spacetime (at least outside of = H n 2 | H ∈ field does not a priori affect the mass of the black hole or ] 2 B field yield a one-parameter family of Schwarzschild solutions, , it follows that quantum superpositions takes values on an B Q ) with Λ Q 3.2 , and so under the global symmetry. This is a global U(1) charge, which can be λ , they render the theory pathological in the same way as remnants do [ ], hence we dub it the BGHHS black hole. The flat B-field ( n Q 51 Ω, we can introduce a non-trivial d If we take the BGHHS black holes as legitimate states, with no further restrictions on This solution was first discussed by Bowick, Giddings, Harvey, Horowitz, and Stro- Different values of the In this section, we will be interested in a particular solution with nontrivial 2-cycles: The presence of a non-trivial 2-cycle is actually more manifest in the Euclidean theory, see later. 7 have charge added to a black hole with no backreaction on the metric. Notice that this is not the same any other property, like e.g. theclassically, one dynamics can of Hawking build evaporation. black holes It follows with that, arbitrary at values least the of value mass of and Since the symmetry shifts orem prevents a four-dimensionalfrom black its hole mass, from charge,observable; and carrying on any angular any contractible momentum. other regiondiate information it The contradiction can apart B-field arises. be ( simply Since the gauged horizon), away. Therefore, turning no on imme- this all with the samebackreaction. metric independent of the minger in [ as an early example of quantum hair on black holes [ The global symmetry ( by an amount to a large gauge transformation). Schwarzschild spacetime. The nontrivial 2-cycleconsidering is the the singularity homology as class excised of fromform the the spacetime. event horizon, once we account fornontrivial identifications cohomology provided class by [Λ largecontinuous shift gauge symmetry transformations for (which the have periods of which are higher-dimensional versionscounterpart, these of periods Wilson take lines. values in In analogy to their their 0-form the periods JHEP07(2017)123 m Q q times -field, to the n B n e q m πq . N should be quantized. Q which, when winding e q field are strings; magnetically charged B . Notice that the same argument cannot be Q – 9 – , as the cases we have been considering, then the field corresponds precisely to turning a charge of H ] for a general upper bound for reason why the vortex charge B 54 via B a priori ) at infinity. In the vortices we are considering, however, there is π -valued; see [ N Z is an example of a “vortex charge”, and not an electric or magnetic ]. It seems one can tune the value of the black hole magnetic charge Q = 0 everywhere, and so without changing the asymptotics. Furthermore if 61 – H 58 ] for a discussion when Chern-Simons terms are present). Vortices for 1-form 55 ]. A spacetime containing two black holes has a nontrivial 3-chain which stretches in the two black holes. Q 57 , − Since the amplitude for producing these black holes is nonvanishing, the states should We have discussed how BGHHS black holes hide a global charge, at least semiclassically. The crucial difference between a vortex and the other objects (charged under gauge The charge 56 -field, with to any real number.to the However, path when integral, computing onebackground. has the These to actual will integrate include contribution over chargedaround quadratic of particles fluctuations the of the around charge Dirac instanton the string instanton connectingamplitude. the This two will black only holes, be will well-defined yield if a Dirac phase quantization 2 is satisfied. indeed be part of the theory,made for for arbitrary charges real affected by Diracthere quantization. is For an instance, instanton if producing weblack consider holes a a [ par U(1) of theory, oppositely charged magnetic Reissner-Nordstrom in [ from one horizon toB the other. Onthe the lagrangian boundary only of depends thisaction on 3-chain will remain it unchanged. isand This possible to turn on 3.2 Production of BGHHSPerhaps black the most holes straightforward way toconsider argue an for euclidean the instanton presence producing of BGHHS a black pair holes of is black to holes such as the ones considered single-valued (modulo 2 no Higgs field whatsoever: the geometry itself supports a flat connectionBefore for attempting the to gauge fix field. this,in we the will theory, argue and that discuss these how objects quantum can effects generically impact be the produced picture. potentials) is that there is no Quantization of electric andnot magnetic apply charges to stems vortices.the from Certainly, abelian field Dirac’s Higgs theory argument, model, vortices which vortex are does charge often is quantized; quantized for so instance, that in the phase of the Higgs field is flat connection of theof 2-form a gauge vortex field. in 2+1 Thisline dimensions; is for in the completely fact, BTZ analogous in black(see to 2+1 hole, e.g. the dimensions yielding [ description one a can lower-dimensionalgauge analog turn fields of a are the standard particles BGHHS Wilson in blackfield three hole are dimensions particles or in strings four in dimensions. four; vortices for a 2-form gauge instanton effects; this will bethe clear above later charge on is in the Euclidean picture. For a massive charge. Electrically charged objects coupledobjects to the are instantons. The BGHHS black hole is a defect around which there is a nontrivial as the U(1) charge associated to the shift symmetry of the dual axion, which is broken by JHEP07(2017)123 . , 2 ), 2 ) A the B R , to (3.8) (3.7) 2 r -hair. R S πnf R Q -fields). (2 . As this B ∝ πif T πnf = 2 dt, axis. The energy ∧ z -charge. dφ -field near it. If the scattered all the way Q 1 dz , far away from S B R H ∧ R profile corresponds to an dr H R 2 2 r − ≈ even far away from the string, and in dt field, dt ∧ ] that the dimensionally reduced system B dφ ∧ 64 dz dz ∧ ) r dr ). It is energetically favorable, for large – 10 –  R -field radiation, some of which will be absorbed by = ln( R B 1 log( B + r R -field is now coupled to an emergent three-form, as we instead. Actually, once quantum effects are taken into B + R R 1 − r ], which extends the discussion to the case of massive . , so that the integral of the axion profile  63 θ ≈ 4.1 ∼ H φ R ] and [ can be measured in principle by the holonomy operator exp(2  lim 62 r Q ] addressed the BGHHS black hole, questioning the observability of field sourced by the string adds up instead of cancelling out. As a result, 52 H is the radial distance to the string, which extends along the r While these considerations show that the dynamics of lassoing a black hole with a This problem can be cured by considering closed strings, or configurations with strings We could also try to build a BGHHS black hole directly: consider an ordinary instantons. A slightly moreEffectively, detailed this discussion means of that thiswill the effect discuss can in be section found in appendix string are far more complicated that one could have expected, they do not show that such the configuration will scaleaccount, as the situation can becomei.e. much the worse: strings are it confined. isof possible Indeed, a for it is the U(1) known energy gauge [ tothe field grow behavior as in of (2+1) thepotential dimensions Wilson between can loop. two show charges confinement, The is as naive corrected shown logarithmic explicitly to dependence by a of linear the one electrostatic by quantum effects involving which is indeed compatiblethe with flat large asymptotics. However,the near energy the center of of thenucleate the string a string, scales concentric as string with opposite orientation; as a consequence, the energy of and anti-strings together.fieldstrengths of If both the halves string of the forms string a add loop up with of opposite radius signs, so that for a single string.axion of This the can form beis shown a topological explicitly: quantity, this thefact forces above using nonvanishing the stress-energy tensorat for infinity, which the contradicts scalar flat field asymptotics. one finds nonvanishing infinite string sources a logarithmically divergent where density of this field curves space so strongly that it is not possible to have flat asymptotics Reference [ The value of which can be measuredhole (see via also a [ Aharonov-BohmHowever, effect this in picture which ignores a the string backreaction circles of the the black fields sourced by the string. An Schwarzschild black hole andstring a starts to string wiggle, coupled itthe will electrically black produce to hole. the Afterto the infinity, while string the settles part down, that there falls will to3.3 be the some black hole Quantum endows effects it with and observability of quantum hair JHEP07(2017)123 . 2 to S 1 (3.9) S . The ) in a (3.10) (3.11) 2 B B R 2 S . The effect R over the πi φ for the axion = B c φ ). The topology . + 2 φ M B is the usual periodic 2 charge problem which → compactified on S ) plane, an the 2-form τ = 2 R φ Q A r r, τ = 2 2 Ω φ d 2 r in the lower-dimensional theory, of smallest volume, at the origin + φ 2 S r M 2 2 to a Aharonov-Bohm experiment and ) plane, where ], this symmetry is actually broken by dr − of the Euclidean Schwarzschild solution. 67 Q 1 r, τ – A 2 1 S + iqφ S 65 2 + , Z S dτ 12 − = – 11 – is the ( e  φ 2 r M R has a contribution 2 corresponds to the angular variables, and its volume 1 2 − S S 1 . The  wrapped on the 2 S = yields an effective 2d theory on the ( B 2 × 2 2 ds S R wrapping the compactification to two dimensions, in which the 2-form symmetry of q 2 S . The Euclidean BGHHS black hole has a nontrivial period of . This reduces to an ordinary global shift symmetry r A One could think that same effect would break the 2-form global symmetry acting on This is superficially analogous to an ordinary 4d gauge field The Euclidean perspective suggests a possible solution to the We could also drop any pretense of relating the periods of a gaugeIndeed, field reducing on the global symmetry becomes an ordinarydesired global shift instantons symmetry are for Euclidean the strings axion wrapping the to the path integral, which manifestlyof breaks the these continuous particles shift symmetry can of betower of understood KK as modes, a and provides 1-loopsuch a effect that periodic in the potential for the continuous symmetry effective theory is involving spontaneously the broken. periods of field. However, as isthe well-known (see coupling e.g. of [ electricparticle particles of charge in the higher-dimensional theory to the gauge field. A 3d, leading to an axion The higher dimensional theory in principle has a 1-form global symmetry, which shifts the is not easily discussed ina the priori Minkowskian perspective: break the thesystems effects 2-form as of an global euclidean strings symmetry.interest can becomes This an ordinary is shift easier symmetry to for the describe 2d by axion regarding the does not have anynow horizons, noncontractible and (the the 2-sphere representativeof parametrized of the by minimum solution the area is angular isEuclidean thus coordinates time at is coordinate.depends The on black hole background. To doThe this, discussion it is is very particularly convenient clear to because switch to the the Euclidean Euclidean Schwarzschild picture. solution terribly unstable in practice,some but (incredibly it suppressed) isphase. amplitude a Thus for legitimate the it state proposed of to “experiment” might the return still theory to possible andsimply in its there principle. try initial will to state, be understand the times spectrum some of the gauge-invariant operator exp(2 a procedure is impossible in principle. The state of a single macroscopic string might be JHEP07(2017)123 ) = 3.6 B 2 S R states ( i n | over every sector, . n Z 4.2 n P + 2)-form gauge field. However, p . Before that, let us make a few 4 charge actually vanishes in the quan- Q ], which guarantees no spontaneous symmetry – 12 – 69 , 68 charge leads to the same kind of trouble as any other Q Thus, the expectation value of any charged operator, such 8 . Thus, we seem to have a continuous shift symmetry which Q = . The total partition function is the sum φ n ] corresponds to the partition function to the black hole with hair φ [ , which will be the subject of section Z 2)-form gauge potential. We arrive at the conclusion that the most generic φ − ) plane. However, the action of this configuration diverges once we take into p ), vanishes. Thus the naive semiclassical r, τ − iφ d Gauging Breaking We start reviewing the first option in which the global symmetry becomes the global To sum up, the Coleman-Mermin-Wagner theorem makes the global charge completely Classically, the BGHHS charge corresponds, in the 2-dimensional euclidean perspec- In this way, the symmetry remains unbroken even after accounting for effects of global -independence means that also the partition function for the different Strictly speaking, a Schwarzschild black hole in flat space is not a thermal state, because it evaporates. • • φ 8 , to a ( mechanism which works for any dimension and isHowever, present the in proof allsolution, of and known the so (at Coleman-Mermin-Wagner it theorem carries least over only for to depends the us) Euclidean on Schwarzschild solution. the IR of the Euclidean part of a gaugethere symmetry are associated examples to in an whichbe additional this broken. is ( not the Forsymmetry, case the including and charged rest states, the exotic of global stringy symmetry the effects should and section therefore explicit we breaking by analyze coupling possible mechanisms of breaking the troubles in quantum gravity.gravity, Therefore, there are when essentially faced only with two a options: global symmetry in quantum global charge in quantum gravity, so it should be dealt4 with accordingly. Breaking generalized globalWe symmetries have seen in the previous section how exact generalized global symmetries can lead to as exp( tum theory. unobservable from outside of theof black hole black — hole but at microstates: the expense the of an infinite degeneracy is spontaneously broken. However,Coleman-Mermin-Wagner this theorem cannot [ happenbreaking at either the in quantum the level,black vacuum due hole or to morally in the representsapplies a a in thermal thermal this state. ensemble case at Since as Hawking the well. temperature, Euclidean the Schwarzschild theorem and it diverges asConsistency it with would do quantum for gravitysymmetry black should of holes exploit with other anfurther mechanisms ordinary comments to exactly on remove how conserved quantum the charge. effects shift impact the observabilitytive, of quantum to hair. the vev of account backreaction, as will be discussed in more detailstrings. in Since section φ is independent of of the ( JHEP07(2017)123 p B . In (4.3) (4.4) (4.5) (4.1) (4.2) +1 p ) around C 2 = 0, which B p . R B +1 in p can be achieved C p B , then the operator is the asymptotic peri- . We may turn on a flat +1 β p R , V p × ∂ , and modifying the kinetic 2 Λ d S = +1 p p + × C 1 W +1 S +1 -form potential p , p 1) branes cannot exist on their own; p 2 C . | C − Ω 1)-brane would couple electrically to +1 → d +1 p p , p − p V ) to go to a gauge in which ∧ +1 p Z ) as it winds (here = 2 in four dimensions, the case relevant to B p mC 4.2 p p m µdt − W – 13 – βµn − p Z ). However, if = ,C − p p 3 dB 4.2 , gauging of the symmetry has dramatic effects for B . For | C p p -brane which couples electrically to 1 2 2 + 1) form potential + Λ p B W p p Z B . Arbitrary shifts of the axion are now symmetries because 2 | 1 → p A -dimensional worldvolume. However, it is clear that this is not B + p ]. dφ | 42 , 40 massive. This is a higher-dimensional version of the Stuckelberg mechanism; it s the brane +1 is the gauge coupling. This system enjoys a gauge invariance p p C W m We now describe sketchily how gauging helps with the remnant trouble presented We can always use the gauge freedom ( As mentioned briefly in section connection at infinity, 3 This implies that as wethe parallel-transport time a circle, charged it operator acquiresodicity such a of as factor the exp( exp( Euclidean time coordinate). The path integral representation of a chemical they can always be compensated by transformations of thein gauge the field. previous section.Schwarzschild metric asymptotes This to is theC best flat metric done on in the Euclidean perspective. The Euclidean they are always the boundary of a makes manifest that by gaugingthe the 2d symmetry case, we the are above1-form mechanism actually gauge corresponds field in via to a a Higgsed Stuckelberg phase coupling for of the axion to a invariant under the gauge symmetry ( is indeed gauge invariant. In other words, the ( via a worldvolume coupling where which includes shifts ofthe periods BGHHS of black hole ofdual the formulation previous [ section, this is the 4-form coupling lagrangianthe in charged its objects coupled to it: a charged ( term to where breaking comes from the ubiquitous presence of4.1 Chern-Simons terms. Gauging Gauging of the shiftgenerically symmetry by of introduction the of periods a of ( a examples of string theory is an elaborated version of the latter, in which the explicit JHEP07(2017)123 2)- + 3. − p d ≤ + 1)-form d are not all . If they do, p n A Z , the partition does not have n 1 Z − d dimensions (we will C d ), with a noncanonical is unbroken, even after is coupled to instantons, in ) gives 2 µ 4.1 φ ( +1 B p Z C are not mass-degenerate, avoiding the , as discussed in appendix n φ dependence means that the µ – 14 – in the 4-form fashion via eq.( . Nontrivial B -field confine; equivalently, there is an emergent dynamical n ), the partition function with a chemical potential for the B µ ( Z -field of the previous section. The dual axion ]. B -form gauge symmetry with gauge potential 43 , p , the global symmetry acting on the periods of , which couples to 40 2 3 B C , we actually get µ Consider a From this perspective, it is clear that this is a generic feature of continuous shift As a result, in this example the symmetry is gauged by an emergent three-form. In We should also remark that the particular case of interest here, gauging of a ( By computing the Euclidean action on the Euclidean Schwarzschild solution with global symmetry, and discuss thecharged IR objects. divergence that We prevents will theclearly existence also interesting. of consider the AdS required and dS asymptotics for thework metric, in which the are Euclidean picture). We will assume that the theory is lagrangian, and that the symmetries in two dimensions, andtheory of we gravity will or have compactification theapplies of same to string trouble global theory in to symmetries any twosense in dimensions. two-dimensional high upon Therefore, dimension compactification. it if also we We impose will that now the establish theory this should in make general, for any ( the symmetry via loopFor effects. instance, However, the we action havegauge of seen field these objects in diverge theintroducing for quantum previous effects from section charged strings. that,compactification, In the in terms problem of a the canwould 4d 2d break be theory theory the obtained traced upon axion with back shift a to symmetry 2-form actually the turn fact out that to the have formally instantons infinite which action. 4.2 Presence of chargedThe states first mechanism one mightof think electrically charged of states. to break For high the enough global dimension, symmetry these is states the will introduction explicitly break kinetic term [ any case, there are plentygauged of — examples this in will string happenaxion theory whenever — the in so instantons which we do the must not shift also induce symmetry explore a is the potential not second for option, the breaking dual the symmetry. emergent field which is not part ofconsider the again tree-level the description of the theory.and As these an may illustration, or may notthen generate the a strings potential coupled for to the three-form function in the sector ofequal; charge in other words, statestrouble with with different remnants. charge form global symmetry, canpropagating be degrees subtle, of in freedom. that Because the of gauging this, potential it can often show up “for free”, as an undergoing parallel transport alongflat the connection describes time a circle. chemical potential In for other the 2-form words,nonzero symmetry. this asymptotically global symmetry turned on. The inverse Laplace transform of potential is obtained by specifying that charged operators should pick up a phase after JHEP07(2017)123 + 3, (4.8) (4.9) (4.6) (4.7) d > p . In Q ), the electric fields 0. The generalized , + 1)-form symmetry. -brane which couples 4.6 p represents a nontrivial p + 3, which also contains = 1 +1 p 0 p .A d C ≤ 2 2 dimensions. In what follows , the above theory has electric d -brane, which explicitly breaks ≤ 2 D 1 − p +1)-cycle. In any given configuration . p − + 3) electrically charged objects fail to , +1 d . p p Q, 2 +1 | C p = ≤ = 0 C +2 +1 + 1)-form global symmetry can be generically d d p p p C dφ +1 F p | – 15 – ∗ Z 2 1 W ≡ Z Z a canonical kinetic term, provided that the volume φ φ . Reducing Gauss’ law of the higher-dimensional gauge + 3, the ( 3)-form generalized global symmetries, which act on periods 4.2.3 ) gives − d > p 4.6 p − = 2 case; the results are similar for d 0 d +1)-dimensional worldvolume of the -brane charge wrapped on the ( induces a term in the action p p +1 p + 1)-cycle. Since we will be discussing an IR effect, let us work on the C p + 1)-form symmetry, which survives as an exact global symmetry. is the ( is homologically trivial, the wrapped branes have finite action and therefore the p +1 is the net p +1 and of the magnetic potential. is finite. We will assume this from now on, and will comment briefly about alter- p Q W C +1 +1 More precisely, consider the theory on a manifold with If The reason for this is that in low dimension and for the action ( For the remainder of this section we will focus on the electric ( p p C C + 1)-form symmetry is very generically broken. However, if + 1)- and magnetic ( + 1-cycle, the branes that break the symmetry carry nontrivial conserved charges. We p p theory yields where in the path integral sum, symmetry breaking effects are proportional to Dimensional reduction of ( of native situations in subsection dimensionally reduced theory, which lives in we will focus inglobal symmetry the now becomesdimensional an axion ordinary 0-form continuous shift symmetry of the two- sourced by the branes decayishes. so slowly This that is the is action theorem. diverges same unless kind the of total behavior charge that van- leads to the Coleman-Mermin-Wagnera the- nontrivial ( ( p will now show thatbreak for the low ( dimensionality ( where the symmetry. of For high enough dimension broken by electrically charged objects,electrically as to discussed in section We will also allow fordepend interactions only with on other the fields, fieldstrengths.( dilaton As couplings, described etc. in as section long as they leading contribution is the standard canonical kinetic term JHEP07(2017)123 ], φ 52 (4.10) (4.11) ] by a ]), this φ , which [ φ 52 eff S ] and us are ], obtained by 49 φ [ ) also asymptotes eff ]. S c 4.9 : + φ φ , and so the dilaton and [ r only modifies Z 0 ]. x ], which focuses on the black ] = . 52 in this background. It is clear, of the Euclidean Schwarzschild φ 49 [ 2 X . Z ] ]. S c → ∞ X,φ different from + 2 [ S − φ p [ X − , which forces the inclusion of backreac- dr − Z d φ . We must emphasize that the divergences r Xe φ = 0 solutions at all. → D ] Z , which at first sight might seem at odds with Q ] one should not take into account backreaction φ 2 φ Z [ φ – 16 – [ ), so to compute it, we fix a generic background Q Z ) together with the canonical kinetic term for x = eff ( ] ∼ S φ φ 4.9 [ 2 ] that instantons generate a potential in the quantum | eff S 49 dφ | − e 1 2 ] discusses the quantum effective action = 0 is imposed, and the two-dimensional partition function is Z background. 49 Q φ (which are responsible for the backreaction on the instanton) are φ + 3, however, ( ); one does not have to solve the equations of motion for . In particular, an instanton located at p 0 φ 2, these terms contribute to the path integral and effectively break the x ≤ > − 0 d x d ( -field hair we discussed in the previous section as motivation. The conclusion δ ) B x fields on ( φ X R As a result, the conclusion of [ There is actually no contradiction, and both results are correct, but [ At this point, we should compare with the results of [ The conclusion still holds in presence of extra fields or dilatonic couplings: the allowed ) and compute the path integral over every other field x ( effective action is indeed correct.function However, as one we also discussed integrates above, to overtion compute fluctuations when the of partition computing instantonbackreaction contributions. makes the As instantonstill we action has discuss formally the above infinite, problematic shift (see so symmetry that also the [ partition function in this language, that when computing of the term would result in a modified The effective action is aφ functional of solution do indeed generate athe potential discussion for in this section, as well as withdiscussing the different results objects. in [ [ performing the path integral over any other fields to the vacuum, butthings so are even slowly worse that whenthe it one metric, there couples cannot are to avoid no gravity: a asymptotically because divergence. flat the gauge Asholes field mentioned with backreacts in on [ of this reference is that Euclidean strings wrapping the the 2d axion, we mean the partition function symmetry configurations should asymptote to theany other vacuum fields at should large asymptote enough finite to asymptotic their value vacuum expectation for the values. dilaton. We should The require gauge a field demanded by ( As a result, a selection rule independent of the particular vev forwe are the discussing axion, occur only ifover the the full partition fluctuations function of is computed,included. after the path For integral the rest of the paper, when we mention the continuous shift symmetry of symmetry. In imply that in any such configuration the action suffers from an IR divergence, that is, when JHEP07(2017)123 , 2 ]). B 52 . For (4.13) (4.12) with its 2 → ∞ S r is ], and therefore 2 71 is just AdS 2 the above propagator . The two-dimensional r dS r 0 . φ 2 B + 2  S Z r/l r/l of the four-dimensional axion dual to = . Intuitively, the two-dimensional instanton r 2 − = 0 is again imposed. However, in this case, S 1 1 + changes as a function of – 17 – , φ  infinite, so the instanton action is still divergent; Q 2 2 | ln S ]. The latter two explicitly break the global shift → = 0 now becomes Gauss’ law. We thus see that this dφ ∼ | r Q 2 ) 71 leads to a two-dimensional coupling , r 1 r ( 2 70 φ 3.1 , rather than logarithmically (as already noticed in [ 3 R , the zero-mode over ϕ field in section 2 B . This means that the IR divergence present in flat space is absent, so it would /r ). As a result, the selection rule is the AdS radius. Unlike its flat space counterpart, for large l On the other hand Neumann boundary conditions forbid a logarithmic field such as 4.12 We see that the couplingin diverges as fact, itOn now the diverges other as hand, this ingredient is actuallySchwarzschild metric essential the volume for of the someaxion decay discussions; constant will for therefore dependaction instance, on will in remain the divergentinstance, Euclidean as the long as the coupling does not go to zero at standard metric. The selectioncase rule is qualitatively similar to Miknowski. 4.2.3 Final remarks So far we have implicitly considered compactification on manifolds without warping, but that the symmetrytherefore must consistent be with explicitly the absence broken, of much the exact as4.2.2 global in symmetry the in The quantum flat dS gravity. The space case other case. case of This interest is is of course de Sitter space. Euclidean masslessness forces the dual operatorthe to field sit is right at a theto noninteracting unitarity be singleton. bound explicitly [ If broken.theory we In with want different the boundary it context conditions to gives of riseto be AdS/CFT, to be different interacting, it CFT’s. interacting the often no If happens symmetry we matter want that has what the scalar boundary the conditions same we bulk impose, we reach the conclusion fields. In theor present mixed case boundary of conditionssymmetry of a [ the scalar axion, which field,the forestalls electric any we branes discussion can of merely spontaneous generate specify symmetry a breaking; either contribution to Neumann, thein Dirichlet, vacuum ( energy. where goes as 1 seem that charged states can break theTo symmetry. have The a story well-defined is a theory little in more subtle, AdS however. space, we must specify boundary conditions for the There is an obviousmost caveat of to our these examples considerations, thisthe the is AdS reliance the case. case on The of flat field interest, sourced asymptotics. but by later For the on instanton we in will global also Euclidean be interested in 4.2.1 The AdS case JHEP07(2017)123 . 1) 3 ]. 4 − ( (4.14) D , resulting 3 C . This can be 3 C . We know that -field indeed leads us to 0 C to yield a five-dimensional B 5 S ], which means that we should 72 , like in the AdS case), and the . This means that the instanton 1 . Another possibility is a charged 2 − 5 r /r 4 F AdS ∂ – 18 – . Similarly, configurations with nonvanishing near the boundary, which would conflict with the Z 3 0 3. For instance, chiral matter can result in Chern- C dC can be wrapped on the 8 = p < C − 4 F d horizon, around which one can wrap a period of 3 is the dual potential to the RR axion S 3 , which leads to a problematic 3-form generalized global symmetry 3 C C would lead to nonconstant . The IIB RR potential 0 5 . Actually, we can phrase the problem in terms of black hole physics as in section S 5 dC × 5 Our first step is to determine the boundary conditions obeyed by The above considerations do not mean that low-dimensional charged objects can never We have seen that avoiding trouble with four-dimensional To sum up, we have established that charged objects are not enough to ensure the = 4 F in the path∗ integral, where In 5d, black holes havein a a five-dimensional AdS version of the BGHHS blackdone hole. by noticing that instantons are allowed with the usual boundarysum conditions up [ over configurations with different values of graphic example. Notably, the problemin of generalized the global canonical symmetries isAdS example already present of thethree-form potential, AdS/CFT correspondence,in type AdS IIB string theory on topological nature. We willscalar see with an a example potential in allowing section spontaneous symmetry breaking4.3 in three dimensions. Symmetry breakingHaving in stated a the holographic problem, example we will now provide a concrete solution in a controlled holo- break the symmetry; onlydimensions that does the not generic workSimons mechanism for terms via under compactification, instantons which available canSimons in break term the higher 2d is axion induced symmetry.electric via The charge Chern- a and closed therefore loop it of does the not chiral suffer fermions, any which does IR divergence, not as have net expected due to its a similar conclusion by demandingthe absence case of black for hole absence hairdimensions. of in black two We dimensions. hole emphasize However, hair thatour motivation in at comes two no entirely dimensions point from is higher-dimensional we much examples. discuss weaker two-dimensional than black in hole higher hair; breaking of generalized globalpactifying symmetries to in two quantum dimensions, gravity.the they resulting In axion fail scalar, particular, to so when completelyglobal we need com- symmetries. break another the mechanism to global guarantee shift the absence symmetry of of exact require absence of a particular kind of global symmetry in two dimensions. One might get action is now finiteinstantons (the are IR allowed to tailThis contribute of illustrates to the that the field our pathdeep goes results integral, IR. require as as Otherwise, nonsingular they it kinetic is do terms possible in for for four the the dimensions. 2d instantons we axion consider in to the have finite action [ now has a two-dimensional coupling which goes as 1 JHEP07(2017)123 , , 3 2 = 5 C F . 1 CS (4.17) (4.15) (4.16) n S = tr × 3 3 S dω period over the non- 3 0-brane; this is the bulk C D , 3 ω is allowed to fluctuate and it = 4 SYM. Since 3 ∧ N 4 ]. C enjoys in the bulk is mapped to F 73 sources a deformation of the CFT, d 3 5 on the boundary is a source for the 4 , we should include other saddles of ∗ C , 4 F 3 N F ∗ ω creates an unphysical singularity in the 3 CFT 5 ∧ C Z 4 3 regime), the dominant bulk saddle contains S F ]. ∼ d 3 AdS Z ) 5 S ∗ 76 – 19 – F = , ∧ φ 75 ], in which the boundary SYM lives on CFT F Z 74 [ tr ( 9 0 C on Euclidean 3 CFT on the black hole non-contractible 3-sphere seems to inherit a C Z must be a three-form which is well-defined up to a total derivative. 3 3 as the Chern-Simons three form of C ω 3 2, the singularity is replaced by an unstable ω / = 1 in the Euclidean Schwarzschild solution associated to the 5d black hole. To 3 3 C S . 6.4 near the boundary. In the high-temperature phase (large In the gravity dual, we would like to consider the behavior of The would-be continuous shift symmetry that Hence, by the standard AdS/CFT dictionary, The status of the global symmetry in the vacuum can be carried out in the general analysis of section 4 9 2 correspond to sphalerons) and hence nonvanishing energy coming from the Yang-Mills / F However, this is not theinstance, for case in stringdescription theory, where of the the singularity CFT is sphaleron, smoothed see [ out. For see section obtained by wrapping global 0-form shift symmetry.the Euclidean However, path at integral, such finite value as asymptotic euclidean AdS. value of Inbulk, supergravity, setting which up formally some finite has infinite action and should not be included in the path integral. the dual pair in the thermal state an euclidean Schwarzschild black hole. As before, the two-dimensional axion contractible understand the breaking of thedescription 3-form of global the symmetry, sphaleron wewill potential, only settle since need at the to the provide value the minimum of bulk of this potential. The analysis is most intuitive if we consider constant shifts of thethe Chern-Simons symmetry three-form. is explicitly Fromnumber broken: this on any point the field of sphere configuration view, must1 with it have non-integer a is Chern-Simons nonvanishing clear fieldstrengthkinetic that (configurations term. with This is the so-called sphaleron potential [ which indeed shows thatChern-Simons the three-form. normal Finally, derivative this of picture isboundary just condition. a higher p-form version of a Neumann and the dual operator It is natural to take this can be motivated by rewriting the theta term as is subject to Neumann∗ boundary conditions, which fix the value of the normal derivative Dirichlet boundary conditions for this field. As a result, we are forced to conclude that JHEP07(2017)123 . 1 − R , but (4.19) (4.20) (4.18) 3 ∼ C in which the two- 0 is broken in the full α 3 √ C . The AdS and the AdS- ∼ 3 . S 4 . / 3 1 R C πλ 2 2 YM , collapses to a point in the former, g √ 3 N 4 2 S . Computing the potential explicitly λR π , = 3 3 C 2 YM is = R Nλ g πN 4 / 2 R R 3 2 ∼ √ λ π 4 3 2 R YM – 20 – g ] = BH 76 0 = M α 0 tachyon. 2 √ Sph D √ s g M where the dual theory lives, is only valid for small ’t Hooft ) must have vanishing expectation value. We conclude, as = 3 0 brane, 3 0 C D S limit we are interested in, the height of the sphaleron is given D R λ M πi , this is indeed the case. , that there is no associated quantum hair for the black hole; it is 3.3 → ∞ 4 / 3 . For the large Nλ λ is the radius of the R It is also interesting to look at how the symmetry-breaking effects look like from the Naively, one would expect symmetry-breaking effects to become dominant when the Technically, that solves the problem; the shift symmetry of In other words, the AdS saddle is manifestly not invariant under shifts of on the AdS saddle as a “bubble” of sorts — a region of size 3 two-dimensional point of view obtainedSchwarzschild upon saddles reduction only on differ the inbut the not fact on that the theS latter. From a two-dimensionaldimensional perspective, low-energy one theory may fails regard the to collapsed provide an accurate description of the physics. A potential. The mass of a black hole of radius and since lower than the height ofblack the hole sphaleron barrier. has This sodominate is many the due microstates to partition entropic as function effects.symmetry at compared Because temperatures is to the far approximately the below restored AdShole their in contribution, mass typical the they at energy. black the start Since hole the Hawking-Page to phase, threshold we to would be expect greater the than the black height of the sphaleron On the other hand, the Hawking-PageSo phase transition it takes seems place at that a temperature we transition to the symmetry-preserving phase at a temperature much where coupling by the mass of the unstable washed out by the quantum mechanics of the zerotemperature mode drops of below the height ofwhat the happens. sphaleron potential. The Yang-Mills However, this result is [ not exactly just a reflection of theof fact the that it sphaleron is potential. restoredAdS contributes at and temperatures In we much the can higher thinktheorem infinite than of ensures the temperature the height that symmetry limit, as it exact,operators, where remains the such only Coleman-Mermin-Wagner unbroken as Schwarzschild- in exp(2 thewe thermal did state; in as section a result, any charged rather scans the sphaleronin potential the as bulk we requirestachyon shift potential string for field the theory, unstable as it istheory, equivalent as to it computing should. the The fact open-string that it seems to be there in the Schwarzschild-AdS saddle is JHEP07(2017)123 , by 0 must must F is the (4.23) (4.21) (4.22) L 1 B R such that dA -valued, due V = ) comes from N 2 strings coupled Z times and the effective F 4.21 N 10 N are B . This implies that the 1 . This means that when on the compactification terms plays out directly 1 A F Nλ d 4 BF + ∗ V , the coupling ( → 2 B V , . 1 2 2 over the 2d spacetime, or of its 2d dual | and integer and where B -valued. 2 , dλ Σ 2 F R N N coupling, rendering the 2-form and the U(1) on a non-trivial 2-cycle, + ; namely, monopoles with worldline NB Z iN L 2 B φ X − e is B BF 1 – 21 – because of Dirac quantization, Z V → B dV , with k L | Σ = R 2 R B 1 2 ∂ i e NF = 2 X ]. More in detail, introducing the dual photon 46 , 4 can nucleate a hole bounded by a monopole of ) by arguing for the quantization of , albeit localized at the bottom of the AdS gravitational potential well. 2 B gets a contribution to its potential energy from this small bubble, which is only 4.21 6.4 3 , has quantized periods over a compact spacetime. C strings with worldsheet on Σ. Conversely, the worldsheet of , the Lagrangian can be completed to 2 4.5 F X N ∗ We can also see how the quantization coming from the A natural possibility is Let us focus on the global shift symmetry of a 2d scalar axion. Due to the CMW the- coupling ( = In a more direct 2d language, without resorting to a 4d lift, one can obtain the same result from the 2 10 in two dimensions.manifold Upon is constrained reduction to to be an 2d, integer the integral of φF using Dirac quantization for the corresponding domain walls, see later. where Σ is aexpel surfaces with boundary electrically to using the string tocontribute measure a the trivial period phase; as of a result, This theory enjoys a gaugemonopole invariance operators must be dressed into gauge invariant combinations of the form dimensional reduction of a 4dgauge Stuckelberg boson massive. Into the four-dimensional Dirac theory, quantization the periods [ dV of where field strength of anpicture ordinary in two-dimensional which gauge the field. axion arises Lifting from to a a 4d four-dimensional 2-form orem, the breaking cannot bewhich spontaneous, preserves but the due axion to discrete an periodicity. explicit This term restricts in us the to Lagrangian, terms of the form The AdS/CFT example shows thatto in the rescue specific to circumstances break exoticgeneral the stringy nor problematic physics global the comes symmetry. simplestwithin However, this the possibility. is realm neither We the ofsuitable now most avatars) field this show theory. is that the the In general solution breaking fact, seemingly can in present occur in subsequent generic sections purely string we vacua. will argue that (in well described by stringy physics.it The doesn’t field backreact outside or of sourceinterpret this the stringy this region two-dimensional stringy axion sees bubble no field,sections as potential; as a an particular instanton avatar would. of We the bubbles4.4 in the general Field setup theory of symmetry breaking by topological mass nontrivial JHEP07(2017)123 . 2 = φ 0 G (4.25) (4.24) N φF minima N . b via a φ are quantized, and (in . Electrically charged 0 N that this is actually the G 5 = 0, it would seem that the 0 G . We recognize the 2d version of the 4- 2 . | 2 φ, . The values of F 2 2 0 Nφ F G G ]) the theory now contains membranes under − . However, if ∧ 4 k | = 0 – 22 – j 4.4 is G . The theory thus contains different phases, each ∗ φ N . Therefore, the only presence of the Chern Simons Z d 1 6= 0, then the above argument applies, and the shift now has a nonvanishing divergence, -form gauge fields encountered in many string vacua. -form gauge fields are neither gauged nor broken (and p 0 p dφ G = j ) does not have any effect, and that the symmetry is unbroken. couplings (or their generalizations), and there is no evidence for act as domain walls separating one phase from another. It is clear 0 4.24 G ]. There is a multibranched two-dimensional potential, with BF coupling with different values of the coefficient 42 coupling breaks the continuous shift symmetry, it differs from the holo- 2 2 . The potential explicitly breaks the continuous shift symmetry of bF shifts by an integer amount coupling, implemented by triple Chern-Simons term instead, is a new ingredient, a (nondynamical) 0-form field strength. This is best regarded N φ F 0 0 2 k/N G G bF = If we are in a phase with Let us consider the 2d setup with a 0-form global shift symmetry for an axion scalar A more general problem is that in many (in fact, most) string compactifications, the 6= 0. Within it, the U(1) gauge field is in the Higgs branch, eating b Chern-Simons term ( However, even in this case,of we motion can which construct manifestly a breaksN finite-action the configuration symmetry. of Consider thecoupling. a equations finite This size is region illustrated in with figure that the associated current symmetry is broken, exactly as in section as the 2d dual ofaccordance another with gauge the field completeness strength principlewhich [ having a particles coupled to The key observation isvia that a there is a slight modification of the breaking shift symmetry where breaks generalized global symmetries.Chern-Simons couplings The in mechanism the is effectiveconfigurations based actions, where on breaking the vacuum the the value symmetry of existenceevant, explicitly; the of the fields even breaking cubic seems in is to manifest render the the Chern-Simons nucleation irrel- of bubbles in which the symmetry is broken. miraculous exotic stringy physicsmore flexible coming version down of tois the symmetry their compatible breaking rescue. with mechanismWe in the Fortunately, explain the massless there previous it is section, ingeneral which a the mechanism following, in and which moreover string claim theory in (and arguably section any theory of quantum gravity) graphic example above in onespacetime, while important in aspect: the the holographicin breaking example a is the “bubble” explicit symmetry-breaking of in effects stringy every were size. point localized of global symmetries associated to made massive) by form lagrangian [ at 4.5 Field theoryWhile symmetry breaking the by Chern-Simons term potential for the two-dimensional axion JHEP07(2017)123 . The q leaves the c + φ ∼ φ 6= 0, is enough to 0 G Chern-Simons term in is no longer a symmetry. c 2 ]. F + φ 2 = 0 [ φ B 0 Z 0 G → G φ . After computing the action for . In this case, the gauge symmetry 2 | 0 r < R pA , one has to integrate over all fields in the − Z 6= 0 dφ | 0 – 23 – G , it is also possible to have bubbles with an effective by a constant everywhere, is indeed broken, as above. 6 φ was freely fluctuating outside, it falls to a definite value . We now have a triple 3 , and should also sum over the charge of the bubble, B 6= 0. Outside of the bubble, in the green area, R 0 -field is quantized, and related to the electric charge of the black G ]. When computing B φ [ Z , and the lagrangian is ]; even if 0 outside of the bubble — and is an exact symmetry of the theory. But profile that solves the equations of motion is very simple — it is constant ] = A 52 c [ φ φ 0 + G φ [ The will enforce the exact discrete shift symmetry of Z , and falls of exponentially to zero for q 11 . Triple Chern-Simons terms mean that we must include into the path integral configura- . term inside the bubble, even in vacua when the axion symmetry has been gauged R only shifts 2 0 Finally, as we will see in section We can take a step back and see how the triple Chern-Simons terms affect e.g. the Inclusion of these bubbles in the path integral leads directly to a loss of the shift r > R The precise details of the computation depend on the two-dimensional metric and gauge couplings, and A 11 , we should integrate over by a second U(1) of the global symmetry, which shifts are different in each case. configurations which however do notis lead similar to to a the IR stringyis divergent bubble a action, in simple as the field instantons holographic theory do. example, description This with both the within advantage and thatNφF outside there of the bubble. the action, and canbubbles, nucleate the bubbles value within of the whichhole the gauge modulo field iswithin Higgsed. the Within bubble. the Thistwo-dimensional is axion a feels reflection within of the the bubbles. 4-form We coupling thus break quadratic the potential symmetry that via the localized for R sum over BGHHS black hole of section ensure breaking of the global symmetry in the fullsymmetry theory. path integral — andradius this now includes bubbles —. Consider a two-dimensional bubble of Figure 1 tions with a bubble inaction which invariant, but in the blue area it does not. Asterm, a result, which allows for the possibility of nucleating bubbles with JHEP07(2017)123 ) 4.24 ], which 8 , so that no global : we understand how 2- -form potentials and weakly 3.1 p ]. . 31 6 ] or [ 27 , , suggesting this is the general solution realized 22 6 – 24 – . 6.4 admits field-theoretic bubbles related to Chern-Simons 4.3 every consistent theory with higher In the following we will display the rationale of the conjecture and the arguments why It is worth emphasizing that we are requiring a consistent quantum theory of gravity to This conjecture, if correct, is another example of a Swampland criterion [ As detailed in the next section, we have found that the triple Chern-Simons terms ( Many classes of string compactifications providing explicit examples of the Chern- in string theory supporting the conjecture for section 5.1 Rationale forThe the conjecture examples in thetups, next where section the Chern-Simons provide terms evidence break for the the generalized conjecture global in symmetries very a gener- variety of se- terms, as we will discuss in section the Chern-Simons provide in oursymmetries. view the Then most we generic will mechanismour discuss conjecture for as breaking some well the examples as global possible of implications effective for theories the which SM, leaving do the not exposure fulfill of examples allows us to further discriminate betweenUV effective completion. field theories If with correct, and itstring without theory a strongly are sensible suggests a that generic the featureconjecture, Chern-Simons of the a terms real consistent ubiquitous support theory in for of itbeen quantum comes gravity. able from Like examples: any to other in comeexample all up the discussed stringy with, examples in the we have section conjecture seems to hold. Even the original holographic requirement for any consistent theory. be well-behaved under compactification.proposed Such Swampland a conjectures, requirement such is as present [ in several recently coupled gravity there shouldsymmetries have actually the survive in appropriate two Chern-Simons dimensionsfor no terms matter any how we global compactify.which symmetry Equivalently, the in symmetry string iswalls theory, either taking it gauged us from or seems one explicitly that to broken, we the and other. can there always Our are conjecture find always elevates domain phases this in generic property to a which give rise to field-theoretic symmetrywe breaking have bubbles are considered. present ina For all the rule, the examples and remainder will ofSimons propose this pandemic, that one, consistent i.e. we theories will of upgrade quantum our gravity suffer observations from to a Chern- We have now formallyform solved global the symmetries problem insolved stated four by in dimensions symmetry-breaking section can effectsaction lead instantons from to in a charged the remnant objectsbreak two problem, the (namely, dimensional which symmetries there version), explicitly. is are but not we no have finite- learnt the mechanism to Simons mechanism are presented in section in string theory, andsome arguably general in lessons in any the theory next of subsections. quantum gravity.5 Accordingly, we draw A conjecture about Chern-Simons terms JHEP07(2017)123 , 4.3 , the -form -form p p 4.3 one requires the shift Take e.g. Euclidean if 12 ]. Bulk Chern-Simons 77 ] that different boundary break the symmetry in a 70 4 . = 0 case, where Chern-Simons , the bulk Chern-Simons terms p - form gauge field be dual to a p p j p instead, whether or not the KK C ; a Taub-Nut background seems A 2 T × 2 AdS – 25 – it is possible to have a consistent theory in which is contractible. However, the Taub-Nut is a KK if 1 S , plus a gauge field (but no charged particles). Consider 2 , so that the original motivation for requiring the shift symmetry of the T 3 × 2 is clear: the global 3-form symmetry needs a nontrivial bulk 3-cycle R . It is part of the standard dictionary [ ] to be broken is lost. We are merely pointing out that p 4.3 φ [ C Z is not conserved. This is familiar from the . If we were compactifying to p j . -form gauge field is dual to a boundary current 4.2 C p It is not true in general that one can pick whatever boundary conditions and end up The Chern-Simons terms may be understood from a different point of view in the We see no reason however to think that this mechanism will work generically for any Of course, a generic two-dimensional compactification cannot be related straightforwardly to black hole 12 -form boundary field (the boundary Chern-Simons form), rather than a conserved remnants, as we didpartition in function section symmetry to always be broken,the the fact CS that terms are theseto a always generic be seem way broken to of in be achieving generic this. present compactifications, suggests Turning although around that we the there do logic, may not be have it. a reason for the shift symmetry ensure that terms provide theterms bulk are description the of onlyterms an without way also anomalous to spoiling break current the invariance [ local under gauge invariance large of gauge transformations via bulk boundary current. with a consistent theory. Nevertheless, the bulk context of thegauge AdS/CFT potential correspondence.conditions Consider for a the bulk bulkappropriate fields AdS boundary result theory in conditions different with demandp CFT’s. that a In the the bulk example of section contrast, the triple Chern-Simonsvery term generic discussed way. incases, In section instanton-anti fact, instanton in pairscase a to some certain break resemblance sense, the toappendix the symmetry, its Chern-Simons giving higher-dimensional the terms siblings. two-dimensional allow, We in elaborate certain on this point in to break the symmetrymonopole, because and because the one of2 the in transverse the directions is non-compact compact, space,in it so section has its real action codimension hasmonopole the is same kind allowed of would IR depend divergence as on discussed the boundary conditions for the KK U(1). By 3-form symmetry at all. low-dimensional compactification, especially in thequantum Minkowski gravity case. on one of the global 1-form symmetry for the periods of two-dimensional symmetry breaking can alsosimple be 2d accomplished field by theory stringyhappens effects description. in without From section a a four-dimensionalto point exist of and, view, because however,we gravity what must forces include us configurations to where sum the over boundary topologies 3-cycle with is the contractible same and asymptotics, there is no ically. Still, this can only be part of the answer, because as illustrated in section JHEP07(2017)123 , 2 14 dC can ]. (5.1) ∧ 2 55 4.3 NB R 4 dimensions -form field in p ≥ d ]: since gauge strings are 36 solution. We merely want the bulk theory [ 3 16 A AdS . N Z For the case of a ), we can make a more compelling 13 corresponds to its Wilson line in the πn. example, we find such a term , the real part of the torus complex ] should not work, at least without inclusion ], this means that there should be a Chern- 4.3 4 which is compactified on a circle with 4 by compactifying to four dimensions 80 78 KK , KK π φ φ ) + 2 79 x /CF T ( d > 5 + 2 ϕ ϕ AdS – 26 – ∼ ) = π ϕ This is easily seen by compactifying four-dimensional + 2 15 x ( ϕ -field U(1) 1-form symmetry to admits a boundary condition As a first simple example, pure gravity in B π field discussed in section 3 4. + 2 C , compactification to three dimensions should give a gauge field x 3d compactification, and ≥ p j ∼ d . Indeed, in the standard . This allows to consider compactifications with a Scherk-Schwarz → 2 ϕ x B : we get a unstabilized 2d axion 2 -form symmetries in the dual field theory. T p in the 4d compactification. The 0-form global shift symmetry for this scalar requires some ], as follows: a general axion 1 = 3 pure gravity is also incompatible with a strengthening of our conjecture which we discuss in S (this includes the KK , dual to a boundary current. Conformal invariance demands this current to be 81 . 3 d A 7 2d +2 p A simple modification of the theory to render it consistent with our criterion is to e.g. To sum up, it is unclear that the gravitational effects that work in section Thus, in this picture, bulk Chern-Simons terms merely represent the absence of contin- This would imply that the asymptotic safety program [ For This aligns nicely with the fact that non-abelian gauge theories doThese not comments are admit very sketchy continuous — elec- the current can in principle mix with other currents, or compact- AdS → 15 16 13 14 ification may change theto sign emphasize of that the theassumptions vacuum Chern-Simons be energy terms related so we to that use familiar CFT there to phenomena. is break two-dimensional no of symmetries matter fields. can under some section tric/magnetic 1-form global symmetries, sinceusually the center dual of to the fundamental group isSimons strings discrete term in [ involving the bulkwhich description breaks [ the global part of the include a 4d axion ansatz [ periodic coordinate 3d Chern-Simons term for its breaking,We which is can not reach present the in same theon theory, conclusion a so in manifold it is without the inconsistent. isometries case and then proceeding as before. Einstein gravity in is incompatible with our conjecture. gravity on a structure. Performing theboson compactification in two steps, there is a 3d KK U(1) gauge 5.2 Implications forTo Einstein illustrate the gravity power and of our supergravity ing Chern-Simons Swampland theories criterion, which we do discuss not somewith comply interest- quantum with gravity. it, and which we therefore claim are not compatible anomalous, which translates to a Chern-Simons term involving break the symmetry inthe generic “wrong” situations, boundary both conditions. ingeneric On Minkowski breaking the and of other in thesome hand, AdS symmetry, cases. the examples and Chern-Simons with are terms required induce by a the AdS/CFT correspondence in uous higher AdS argument. If itis is a consistent boundary to current choosein boundary conditions such that the dual operator JHEP07(2017)123 1 2 ]: A KK 11 φ KK φ . . How- 6 q A, A P already in (where the 1 A gM , the 2d axion A , coming from η φ ≤ KK , but not for should be regarded m 2 φ . The trouble with 6= 0 which suffice to n KK n KK A φ with and . This provides a mass for the 1 2 transforms into a Chern-Simons A A KK term, where φ φ KK . Upon dimensional reduction, the nA KK 1 , one obtains an axion − nA 1 KK S − n η F dϕ A A final notable example of theory which dφ the theory lacks one of the required triple 2 and – 27 – T A ]. In our case, consistency of the theory could be 82 may be identified with the real part of the complex under the KK photon, and thus, the gauge-invariant n KK . 2 φ discussed in the previous paragraph. We also have three Similarly, the conjecture puts four-dimensional Einstein- 2 KK T A = 8 supergravity. and the dual field to the axion, as we will explain in section coming from holonomies of the gauge field, and has charge , and N 2 1 2 can obtain a potential if we take a Scherk-Schwarz ansatz. In this way, B iϕ A . The axion , there is triple Chern-Simons 1 e 1 KK φ , φ A 1 φ KK A A A, A φ . In this case, the theory can be made compliant with the conjecture above by = 0 there are effects from bubbles nucleating regions with = 8 Supergravity. Simply put, the theory contains a 2-form gauge fields leading to n can also be solved by including Chern-Simons terms for the gauge boson KK The axion This solution, more than a particular case, turns out to be generic in string com- This is equivalent to the Chern-Simons criterion, as follows: in terms of N φ . Upon T-duality, the Stuckelberg coupling 2 2 A a 2-form generalized globalit. symmetry without Actually, the this Chern-Simonsarguments theory terms has required related to already to break beentoroidally the claimed compactified impossibility to string theory be ofachieved [ in reaching by the including is the Swampland, necessary as by Chern-Simons different a terms, in suitable particular decoupling those involving limit the of φ four dimensions. Four dimensional does not comply with4d our criterion, and we therefore claim belongs to the Swampland, is we can obtain aor Chern-Simons term forintroducing (the 4d charged dual chiral of) matter,terms which via gives rise the to parity3d three-dimensional anomaly Chern-Simons Chern-Simons both term for gives the desired 2d terms for dimensions: reduction of structure parameter of the gauge fields, ever, we claim thatSwampland, even since this upon versionChern-Simons of reduction terms: Einstein-Maxwell on upon with reduction a chargedsubscript on reminds matter the us first is that in this(the axion the latter comes being from a the KK gauge photon), field), and two the gauge metric. fields Reducing again, we get three axions in two coupling involving Einstein-Maxwell theory. Maxwell theory in theEinstein-Maxwell must Swampland. be coupled Of to course, light we superextremal knew states as with much from the WGC [ as a background geometric flux.with According to ourbreak general the discussion, global even symmetry. in the vacuum pactifications, where gravity isB always accompanied by the antisymmetric two-form field quantity in the dimensionally reducedKK photon, theory implying is that, after furtherobtained compactification from to its two dimensions, Wilson the line axion is also massive anddual the to global the axion shift symmetry is broken. This means that JHEP07(2017)123 = 8 (6.1) (6.2) N . Com- 1 . Further translates S 5 in the four- × M , leading to a 5 1 πN A and the 2-form on S M 5 A ) + 2 C y ( φ ) = πR + 2 y ( φ 2 − p − d A F ∧ coming from the period of ∧ ∗ p ]) trying to figure out whether or not φ F – 28 – 85 – ∧ Ndφ 83 B coupling between the KK photon Z BF using a Scherck-Schwartz ansatz 1 S first, we get an axion field . In the T-dual frame, such a coupling arises from dimensional reduction 5 φ M 4d compactification, other examples being similar. In order to see the Chern-Simons As a concrete example, consider the compactification of type IIA on However, in string theory these KK photons always turn out to have the required set of There is a line of work (see e.g. [ 4d dual to → field. The 10-dimensional action has the standard Chern-Simons terms involving the field, 2 to a Stuckelberg lagrangian in four dimensions. The coupling can be integrating byC parts,into form of a Stuckelberg Lagrangian) arisesRR from field. the Scherk-Schwarz compactification of some pactifying on compactifying on B which upon dimensional reductiondimensional provide effective Chern-Simons field terms theory. involving In the original frame, the Chern-Simons term (in the U(1) KK gauge boson; for5d concreteness we focus oncouplings, type let II us theories perform and aspace, a KK in T-duality photon along which in the the a orbitB KK of photon the turns U(1) into isometry the in dimensional the reduction compact of the T-dual NSNS gauge bosons lead tological 1-form mass generalized or global Chern-Simons symmetries, couplingsfall and ensuring in in their the breaking. general Swampland, lack as These the theories earlier in topo- discussed general examples. Chern-Simons couplings. This can easily shown in KK compactification on conjecture. 6.1 KK photons One prominent example ofis U(1)’s that in of effective KK field photons, theories arising from arising continuous from isometries compactification of the internal manifold. These global symmetries are broken, soit the as corresponding a black complete hole theory remnant of problems quantum excludes gravity. 6 Examples We now discuss several classes of examples which comprise the main evidence for our models as required by our conjecture. SUGRA is perturbatively finite.result, but Our renders statement itthe does less non-perturbative not domain relevant. directly walls exclude allowing Even to this if interpolate potential the to theory vacua where is the perturbatively generalized finite, it misses Romans mass parameter (and other in its U-duality orbit), which are present in string JHEP07(2017)123 , the 17 2 in five 3 dB d F 4 ∗ ∧ 2 = couplings, and as explained in B 0 dφ 4 terms breaking it. G BF F ) 2 . When the couplings BF B 3 d 4 ∗ ( , namely 0 0 2 with a 2-form global symme- G B 2 , leading to 5 B ] for a review. F 86 gauges the 2-form global symmetry ∧ breaks the 2-form global symmetry 4 3 2 F F φF 2 0 0 ∧ B G 2 0 B G – 29 – , while couplings inducing its gauging occurs arise -flux corresponds precisely under T-duality to the 5 F 4.5 . This 5 F 5 M arising from the Scherk-Schwartz reduction. . R . = dφ 5.2 6.4 1 0 S G R couplings. The above picture is happily self-consistent, since bubbles with = as explained in section or its dual field. In terms of the axion dual to 2 N 2 F BF . Notice that both couplings can arise from higher dimensional CS terms involv- B 2 B 0 4.1 G One may worry about the fate of the gauge symmetry in phases with 4-form couplings The general lesson is that, given a 4d 2-form gauge field, the CS couplings are such It is an important point that both kinds of 4d terms, Stuckelberg and 4-form coupling, For concreteness, let us consider again a 2-form field This class of examples underlie the solution to the problems with pure Einstein theory couplings necessarily have the 4-form coupling turned off. Therefore the gauge symme- It can happen that the CS coupling does not appear in the 10d supergravity action since involves the 17 broken by BF presence of locally non-geometric fluxes.as we But explain they in will section appear as geometric CS terms in the T-dual theory, On the other hand, thethe theory flux consistently inducing incorporates domain thecoupling wall 4-form that bubbles coupling phases inside is with which a absent, Stuckelberg but coupling it can is begauging precisely turned the in on. global symmetry, the once absence we of nucleate a this bubble in whose interior the symmetry is that there exist domain wallsdomain interpolating walls to to phases phases with withplings Stuckelberg 4-form present. couplings, This yet there4d implies 4-form cannot that coupling be in which phases compactifications gauges with the where both global the cou- symmetry, internal there are fluxes no induce a cannot be present at the samewould time lead in the to effective a theory, since Green-Schwarzarise their anomaly, both simultaneous as presence from mentioned bulk in couplingsprinciple footnote involving be just avoided fluxes, by their thein simultaneous consistency the presence conditions embedding should on tensor in description fluxes, of like fluxes, the see quadratic e.g. constraint [ ing either above couplings are alsobut responsible for yielding gauging the or oppositewhile breaking result. gauges the the 0-form 0-form The globalwhile global term symmetry, breaks symmetry, and the 0-form global symmetry. try in four dimensions.form The couplings inducingfrom the higher breaking dimensional of version thesection of symmetry are a of Stuckelberg the coupling 6.2 Type IIA/B fluxProbably compactifications the richer class ofplings examples are supporting the the four-dimensional conjecturetype about effective IIA/B Chern-Simons theories string cou- obtained theory.global from symmetries flux In coming compactifications short, from of higher the RR CS or couplings NS in p-form fields 10d are ensure broken that or all gauged. generalized dimensions, with axionic flux mentioned in section on the circle of the Chern-Simons coupling JHEP07(2017)123 . , 4 0 2 p F b 3 = (6.3) (6.4) (6.5) C = 0 3 5 , q H 0 C p is enough A 2 − R , F 0 2 = φ b q 0 involution. The = G 2 which leads upon 3 Z 2 C F B tr . Now we cannot play R 5 4 F p, C 0 0 6 = φ D pφ φ, 2 R ], which were not possible for 3 = 0 asymptotically, we recover F = = ) A H 0 0 2 3 3 = 0. However, the same does not b A G 0 C C R 0 p B ] = [ A G 0 − Z Z 2 [Π to be odd under the , pb , − 2 0 2 A b b ). Therefore, the two terms in the above + ( 2 couplings, which break the symmetry. = = b 4 0 5 , the presence of the CS term 5 F p ) C – 30 – C BF φ + 0 B A 4.5 p 0 2 Z Z − pb 0 , 0 p, p pφ ), satisfying [Π] ( 0 = . By dimensionally reducing the 10d CS couplings ) is ( = 0 2 3 φ 3 b 0 p H H − L ∼ A B − = Z Z 0 0 pφ dφ ∗ . We define and B 2 b we get in four dimensions 2 = F = 0, it is now possible to add D6-branes wrapping a 3-cycle Π (and their ori- 5 dφ p = 0, in which the symmetry is not gauged. However, if we turn off the flux by C ∗ 3 p H One may fear that the above situation is non-generic, and that the presence of ori- To recap, as we explained in section In the following we present a simple example which suffices to illustrate this general and its dual 6= 0. From the brane worldvolume we get a new coupling the same game aswith before to interchange the couplings,setting so one may worryentifold about images the D6’ phase wrappingp Π entifold projections may removethat some the details of of the theconsider required discussion the couplings. change, addition but of O6-plane, the Weorientifold and general will projection consider conclusions now only remain allows illustrate valid.The for Let only the us combinations 4d coupling surviving the orientifold projection is in which the coupling responsiblea for term gauging explicitly the breaking symmetrywhich disappears the the and symmetry. global is symmetry The replaced isin by presence the broken of full is theory. enough some Therefore, to regionsare in guarantee indeed in the that previous broken space-time the example, in in symmetry all the the is full global broken theory. continuous symmetries becomes possible to turn on phases with to break the global symmetry,apply even for in the the term phase gauging the with an symmetry. exact In that global case, symmetry. if We have seen, though, that we can always nucleate a bubble global symmetry is gaugedthat while we for can the always nucleate secondin a one such domain a is wall way that explicitly thatis changes broken. the both now flux Notice scalars broken, values however interchange to the roles. and coefficient viceversa. The of global the symmetry In 4-form that other was coupling gauged words, vanishes and for the a global fixed symmetry combination is of not gauged fields, it when and The 2-form dualLagrangian to actually ( correspond to two orthogonal 2-form fields. For the first one the 2-form where which acts both on the 4-form and BF phasespattern is of broken 4-form in vs theCalabi-Yau Stuckelberg presence three-fold of couplings. (without the orientifolds), Let which bubble. A for us simplicity consider we Type take with IIA one compactified 3-cycle on a try only acts on the exterior 4-form phase and is unbroken; the global symmetry however, JHEP07(2017)123 . 2 Z has a B 6-brane 3 C D 6 (2)) = 6 brane and O D ( D 0 π 6 A O 6 plane and reconnects O 6’ couplings of the D = 0 is due to a Freed-Witten p → (2) gauge theory of a BF O B when 6 A D = 0 and with the scalar field enjoying a p . The second configuration in the figure can 6 brane approaches the 2 D 6 A – 31 – O for the dual 2-form). The presence of this coupling 2 6’ F As we cross the four-dimensional domain wall, the brane ] for review. 2 D in four dimensions. The fact that the D6-branes are only b 87 18 2 ) and the configuration without flux (and branes wrapping → F A tr B 0 2 b 6 D 6 A O coincident D6-branes wrapped on the compact manifold. The M-theory lift of . Schematic depiction of the transition that takes place across the domain wall described in N 6’ coupling, we can nucleate a bubble with D Consider compactification of type IIA on e.g. a six-torus to four dimensions, and a Therefore, starting with a vacuum in which the scalar field coming from 4 The existence of these interpolating domain walls introducing wrapped D-branes allows to regard the such couplings as yet another avatar of cubic Chern-Simons couplings. This is particularly explicit in 18 stack of BF models like magnetized D-branes, in whichmagnetic certain fluxes homological charge on classes the are associated D-branes, to see world-volume [ The very explicit realization offollowing the explicit discussion Chern-Simons of mechanism (close in cousins)M-theory terms of configuration of the we BGHHS fluxes consider, black allows the holea the BGHHS in horizonless black M geometry, hole theory. which becomes In will a the resolved allow system us from of to an branes understand ultraviolet in how point the of associated view. problems are in some region ofin the the space-time full is theory. enough to ensure that the6.3 global symmetry is Digression: broken an M theory BGHHS black hole also be understood insitting on terms top of of the the orientifold, field as theory the soliton of withpφF the nontrivial charge under Stuckelberg coupling instead (a anomaly induced on thewall brane interpolating when between the the configuration fluxvanishing with is component flux not along (and vanishing. branesthe wrapping There above 3-cycles is [Π] with indeed 3-cycles). a Thiswrapped domain domain by wall both corresponds sets of toconfiguration branes. a changes 4-chain as connecting illustrated the 3-cycles in figure differently with its orientifoldits image. image cancel; In in the the first third, picture, they the add up. dimensional reduction to allowed to wrap a 3-cycle with component along Figure 2 the main text. As we cross the domain wall, the JHEP07(2017)123 . 3 C dU = 1 (6.9) (6.6) (6.7) (6.8) ∗ is not (6.11) (6.10) N = Q potential . being the 3 Q dM θ C flux, supported 4 ]) G , , with in the case 6 89 3 satisfies . ) for the eigenmodes , 2 R . A valid ω dV 1 2 λ 1 normalizable Poincare ω M ∧ Λ, and find the effective 6.10 which corresponds to the − ∧ + . . Nλ r ) ∧ dθ ω N ω Ω r N + ω ∧ M ) r r ] (see also [ = Ω + Ω, so that we have obtained a ( . d 88 Q along . Finally, dθ ) = QNF d ∧ 3 ( 4 r = 0 − = ( 2 QNd R fibration over ,U G ∧ r dr ) ω 3 = g 1  Ω C 2 M 2 2 S theory circle, and one can decompose flux, and hence, that the charge ) B r ,F ω 0 → − ) + 4 g ), is controlled by the asymptotic radius of ( M 2 + G 1 B M Ω dθ will not be a solution for constant − (  QNF d + 1 Nλ 0 ω – 32 – U ω − ( − g / 2 dθ ∧ − U 1 2 )( of the form is infinite in type IIA string theory because of the Λ = Ur ¨ r g , the center-of-mass self-dual 2-form is given by (see labels points in (  ) + Q ω 3 QNMF d  ∧ N 2 ~x Q F C − d~x = d N dr · 4 = Qω − G Λ = d~x = , parametrizes the = ( IIA 3 3 Λ 4 θ U d C C components, obtaining G = 2 , the asymptotics is 1 vanishing 2-cycles, which translate to = r 2 B ∧ ∗ − . The differential equation obtained from ( ω 4 field at the origin, 1 ds ]. For general Q N 2 G 89 and B F = IIA 3 g . The minimum will correspond to a stable configuration of C ) field at the origin, the uplift to M-theory smoothes out the resulting profile for is Q , the coordinate B r H iωt e -field. The ω It might seem that one can turn on any So let us take an ansatz for Now, consider turning on M-theory 4-form flux g B = in terms of g by the Taub-NUT geometry. Sincethe the M-theory fields Chern-Simons only term dependfrom nontrivially will the on not 4-form four contribute. kinetic dimensions, term. The One only can contribution show then that comes the Taub-NUT, that is, by the IIA coupling. Wequantized. see that To it understand indeedequations what diverges of is in motion; IIA. going the on flux in detail, weaction need for to solve the supergravity We see that atmicroscopic description large of aAlthough (charged, extremal, the singular cost cousinsingular of of a) turning BGHHS on blackthe hole. At large in terms of for this is dual 2-forms, and ancenter-of-mass extra non-normalizable motion self-dual of 2-form can the be system. found in Anappendix [ explicit expression for Here, the Taub-NUTcoordinate space along is the described circle,This as while spacetime has an the D6 brane stack is a multi-center Taub-NUT metric [ JHEP07(2017)123 , Q ) to = 0. (6.12) (6.13) in the 2 c 3 6.10 far away C to diverge Q 4 m/r, 6 system into G must go to zero D g , 6 gauge field arises r = 1 + 2 D ). is to allow ,H 3 is a basis of independent ,
Q charge. Nonzero frequency 2 1 6.13 for small , Q − . /r 1 2 = 1 1 c − i ∼ H , + i U α  N 2 3 ). Hence, it is clear that solutions with λ  r will suffice. r 3 ( 6 = F + T 0 , where 2 6= 0, which corresponds to an asymptotically i 2 – 33 – α 2 Nr c ∧ ,A vanishes everywhere but at the TN center. This is  ω 1 Ω) 4 c d = G 2 = 0 above. The most general solution then depends on r 4 = 0 ω G + g 2 to actually be smooth near the TN centre. This sets dr . Here ) are dispersive; the TN center reflects incoming waves of 4 ( 2 ] 2 G / 1 /F 90 6.11 2 H c 0 for of the form + ]. In order to get a Reissner-Nordstrom black hole with a finite horizon, = 4 . D4 charge is induced simply because there are nontrivial periods of 2 → 90 6 G Q dt r T 2 / 1 as − 2 , we have approximately constant r H r − = It would seem that the only way for the equation of motion coming from ( We now have an uplift of a charged extremal dyonic BGHHS black hole to a horizonless This has a very natural interpretation in M-theory: since the We might wonder if this behavior is an artifact caused by the fact that the four dimen- Therefore, the boundary conditions forbid stationary solutions, and as a result the 2 ; D2-charge is induced by the M-theory Chern-Simons coupling. Finally, D0-charge is 6= 0 do not describe a localized source with finite BGHHS charge at infinity (and actually ds 4 1 hole, and not an artifact due to the absencehave of a a static horizon solution inat with ( the nonzero constant origin. asymptoticsconstant for Then charge the solution with along the compact dimension. geometric background in M-theory, yetthe the same, equations because of noleast motion new if of contribution the we comes three-form ignore fromblack remain gravitational hole the background backreaction. M-theory is a Chern-Simons Thus, genuine term, feature the of at the dispersive M-theory behavior description of of the BGHHS black from dimensional reduction of the M-theorybackgrounds 3-form, for the above gauge fluxes2-cycles lift of to M-theory the G induced because the induced M2-charge sits on a magnetic field, and as a result moves a genuine black hole byD-brane choosing charges background [ gauge fields onwe the need brane to which induce inducefield lower D0-, along D2-, the and three D4-brane independent charge. 2-cycles of Turning a fieldstrength for the gauge Due to the presence of unstabilizeda moduli, naked the singularity solution instead. does bit This present an has horizon, a having simple solution: one can turn the pure D6-brane systemeigenmodes does satisfying not ( reallybut carry is a not nontrivial able to capture any charge. sional dilaton black holeand described dilaton by profile are the [ stack of D6-branes is horizonless. The metric At large c their energy is divergent). Onas the fast other as hand, since from the hole, sotwo we constants, should set A solution describing BGHHS charge would be stationary and have a constant JHEP07(2017)123 ]. 92 ]. (6.15) , while 91 . There 8 8 , or on a C necessary 6 C -type [ 2 q T C goes to 5 . This is consistent 3 C . In four dimensions, to M-theory [ 3 C 0 B F which leads to a 4d coupling ] for review). This is natural, 5 87 F 8 which upon dimensional reduction ) (6.14) qC Z coupling. Direct comparison to the 2 , F 0 } . Upon T-duality, 3 0 5 F 2 2 H C 5 B B 3 − { 0 2 C B 4 F F R has been naively set to zero. A deeper look 8 R ( : the IIA Chern-Simons term F = 0 – 34 – 3 . This is the coupling for the 2-form 5 2 6 G Z H C X ∈ 3 and C 3 H clearly generalize to more general fluxes in the compacti- 3 The discussion of Chern-Simons terms from internal field T R fibration over a three-dimensional basis 6.2 3 = T h is a gauge transformation of the vacuum, so that obtained from dimensional reduction of the ten-dimensional 3 2 C C , and the NSNS flux becomes a locally non-geometric flux of where in order to make the missing CS term manifest, as follows. In the T-dual 5 2 F 3 given by a F T 2 6 X to be an integer. h C upon dimensional reduction on Q 2 F The general lesson is that in string compactifications, whenever there is a BF coupling We now describe an instance of Chern-Simons terms involving nongeometric fluxes, by To sum up, the M-theory uplift allows us to see explicitly that there is a potential for 2 turns into , and that the only smooth stationary configurations have trivial 3 2 missing and the global symmetryphase seems in unbroken, it which is thereveals simply because that tunable we string parameter sit theory insymmetry. provides a the flux-less In requested this Chern-Simonsbubbles case, term with to the a break non-vanishing solution non-geometric the came flux global inside. from the legitimate possibility of nucleating F Therefore, in type IIBh we C indeed have a 10dto coupling break its 2-form generalized global symmetry. there is a 2-form is no apparent Chern-Simons termbecause involving we it are in ignoring the contributions effectivetimes field from along theory, non-geometric but sources. thistype Let is us only IIA T-dualize there three isleads to a 10d Chern-Simons term since they are all onof equal the footing lower-dimensional (due effective to theory. underlying string dualities) from the perspective using T-duality with NSNSmanifold 3-form flux. Consider, for instance, type IIB on Here we collect a few more examples without aNon-geometric detailed discussion: fluxes. strength fluxes in section fication, including geometric or non-geometric fluxes (see [ becomes upon six-torusM-theory compactification picture a is obscured by the somehow cumbersome lift6.4 of Other examples C with the conjecture advocated in section describes a pure gaugeforces configuration in singular gauge. Compactness of the gauge group only consistent if JHEP07(2017)123 . in φ 4.3 term, (6.18) (6.16) (6.17) in this , there 0 1 0 the U(1) F S 2 F F × H 3 2 with standard C when regarded 5 6 CP S X × 5 example of section 5 AdS S × fiber in the ) results in a four-dimensional 5 , we do not actually need this 1 U(1). Denoting by . F S × ∧ AdS 4.3 U(1) F , it would be interesting, but beyond F F . Thus we seem to get an axion 5 Tr( ∧ ∧ S F . 2 ∧ ) 6 as a slightly exotic version of B 4 φF , which breaks the symmetr. This can be B and F  χ F + ) B 2 tr( ∧ | 2 ∧ A – 35 – , which gives a five-dimensional 3 0 2 SU(3) Heterotic string theories have non-abelian gauge − F F χ B B 3 dφ H tr( | 7 6 1 2 . As shown in section ]. In the T-dual type IIA on C X 3 Z ) gives rise to 93 and 4-form like couplings. The low-energy lagrangian H [ 1  2 S 6.16 R BF CP ] for review) = Keeping on with the philosophy of seeking for Chern-Simons terms : they can correspond to gauge backgrounds changing the gauge 3 87 χ H , 7 C 2 CP R revisited. = 5 3 S coupling between the axion dual to C × F 5 Similarly, the dual Green-Schwarz coupling ∧ φF four dimensions with both picture. It isregions quantized, with and different webundle can topology consider (thus four-dimensional movinggeometric domain onto defects a walls interpolating non-standard separating between embedding),(via different or e.g. CY more conifold spaces dramatically transitions). as to we cross the domain wall The prefactor is non-zero becauseso it equals we the get holomorphic a Eulerthought characteristic four-dimensional of of coupling the as CY, a triple CS term by regarding For instance, inembedding, compactifications the of SO(32) gauge the groupfieldstrength, SO(32) is the broken heterotic coupling to ( on SO(26) a CY dimensions, string in four),these holonomies or constitute problematic non-trivial 1-form generalized cyclestheory global should symmetries, in take which string care spacetime. of.Schwarz The coupling The required (see Chern-Simons shift terms [ symmetries indeed come of from the Green- this group. Heterotic CY compactifications. bosons in 10can dimensions. consider In holonomies of either the ten, Cartan or generators four around dimensions black after objects compactification, (7-brane one in ten where term to break theis symmetry also — present, this allowingIncidentally, can us it be to is achieved understand interesting bySince the that this stringy is symmetry it effects. part involves breaking of the Nevertheless,our the in SO(6) it present KK isometry field scope, group U(1) theory of to the associated terms. explore to the the general circle story associated fiber. to the non-abelian structure of involving any kind ofbreaking flux, the we 3-form generalized can globalIn provide symmetry order a to in derive nice the it, description weas again of use a T-duality, Chern-Simons now circle along couplings the fibrationis over a 10d Chern-Simons coupling AdS JHEP07(2017)123 - in p F : it is 3 potential C and 4 D B 0) B-field arises from , 19 . Some solutions, such as 3.1 ] for a recent proposal). . The only remaining interesting 28 Λ which is reminiscent of the massive 4.2 , but the usual trouble-with-remnants 3 C /f P + M -field in section ω 2. These cases can lead to the problematic on such 3-cycle constitute a symmetry of the B 0) theories contain tensor multiplets whose 2- ∧ 3 , − – 36 – becomes the usual coupling between C B d ], support nontrivial 3-cycles. If there is a 3-form 3 = C ]. = 94 4 37 p C = 0 to p ]: one would like to argue that because of the WGC there should Six-dimensional (2 19 , 16 , on a basis of normalizable 2-forms of the ALE space. The 3-form also arises (one way to see this is to reduce to IIA, where the M5’s become D4’s and 4 15 3 , C C 13 , from the decomposition 0) theories. 4 0) theory arises from type IIB in an ALE singularity, the (2 , , C This is reminiscent of the situation one encounters when trying to apply the WGC This case doesn’t immediately fit into our framework, because under compactification A 4-form global symmetry is also possible in the Euclidean theory. We would have a flat in our theory, shifts in the period of 19 3 justification for the WGC“decay” does from not one instanton work to in another the (see Euclidean: however [ there iswith no no dynamics natural (andperiods notion no on charged of compact objects, 4-manifolds. due to tadpole cancellation), but which might have nontrivial to be an immediate problemjust with another the seemingly continuous harmless degree zero of mode freedom of associated the to euclideanto solution. axions [ be some instanton with an action scaling as if any at all; weblack do hole not background, know how as tothe we relate did Giddings-Strominger the with wormhole 0 + the [ 1-dimensionalC system to any reasonable theory. This isthere. similar Because to we the are necessarily 1- working in and the 2-form Euclidean examples, theory now, but there doesn’t the seem similarities seem to end possibility in four dimensions is that of a 3-formto global two dimensions symmetry. it leadstion to on a a gauge circle field, leads ratherparticle to than on an an a axion axion. circle. in Additional It 0 compactifica- + is 1 unclear dimensions, what that the is, the problem quantum is mechanics with of global a symmetries in this case, 7 3-form global symmetries So far we haveform only gauge considered fields generalizedtwo-dimensional from global axion symmetries system arising from discussed periods in of section the (2 reduction of from Wilson line examples discussed in [ the symmetry always seems totheory be of a gauged. (stack When of)three-form realizing M5-branes, this the theory symmetry as is thethe gauged worldvolume by coupling the between coupling the tothe tensor the DBI and M-theory action). The same happens for the realization in terms of IIA NS5-branes. When variation of the four-dimensionalof chiral fermions; the usual this Green-Schwarz is anomaly just cancellation. the four-dimensional6D version (2 form gauge fields show anterms associated that 2-form global we symmetry. can There think are no of Chern-Simons for this field, but there is no need for them either, because is not gauge invariant at first sight. However, it is once we take into account the anomalous JHEP07(2017)123 (7.4) (7.1) (7.2) (7.3) in another , leading to 0 φ G 4 , does not vanish 4 G 0 0 G is dual to the field F G , 0 4 G and G 4 F . Thus, if φ in some 4-form coupling. Support 3 0 C G , . = 0, thus killing the 3-form itself. This 2 Z | 4 4 4 dφ ∼ F G ∧ , and modifying the 3-form kinetic term to kD and breaking the symmetry. Therefore, it is 4 3 2. Here the parameter ]. There is however a subtlety not present in ∧ ∗ in four dimensions. The above coupling can − 4 C D 4 − 37 0 4 3 G – 37 – d ˜ F G C ˙ φF | φdt, c d c ≤ 1 2 Z 0 Z . Such a potential can come from higher-dimensional p − = 3 G . From this point of view both 3-forms are on equal C 4 3 Z C G stands for the time derivative of ˙ φ appears playing the role of 3 ]. dimensions, in general), and to explore a few stringy examples. C d 96 , 95 ], which relies on the presence of (many) non-dynamical 3-forms to provide -form symmetries with p 97 1-forms in − , in addition to the one of d 3 This extension of our conjecture may also have implications for the Bousso-Polchinski If the 3-form version of our conjecture applies as well, it implies that all 3-form gauge Symmetry breaking requires to have a potential term in the effective action for the ax- As before, we can either break or gauge the 3-form global symmetry. Gauging would Because of this, the case for both the WGC to instantons and for our conjecture to ˜ C dimensional reduction of Typeappear IIA/B either coupling in an a axion4-form Calabi-Yau with coupling a manifold. 4-form [ coupling There, or all playing the 3-form role fields of mechanism [ The natural question isof whether such a couplingfooting. also breaks In the fact, 3-forma one global non-vanishing can divergence symmetry rewrite of this theenough current term if as the dual 4-form of for coupling this for conjecture can be found on the four-dimensional effective theories obtained upon strength of another 3-formthen field be understood as a field-dependent kinetic mixing between fields should enjoy the appropriate4-form CS coupling term above) (which to in breakconstruction four the of dimensions global axion corresponds monodromy symmetry. to modelsthe This the previous [ 4-form coupling underlies the in 0+1 dimensions. Here identically (as an operator), this term breaks the symmetry. ion in 0+1 dimensions coming from Chern-Simons terms; for instance, a four-dimensional 4-form coupling would reduce to a triple interaction which would have a tadpole essentiallydoesn’t requiring seem a very interesting possibility. 3-forms is weaker than inwe the carry other scenarios. out in In this any(and event, section, it to is explore interesting the exercise, implications which of the conjecturemean for introducing 3-form a fields 4-form gauge potential JHEP07(2017)123 ]. Our . 5 109 = 3, since d -form potentials via charged ] to find a holographic dual to p 108 – 98 correspondence. The breaking is due to the 4 dimensions would be incompatible with our 4 ≥ – 38 – d /CF T 5 AdS 1)-form version of our conjecture holds, it also excludes − = 3 AdS theory of pure gravity. The extended version of our conjecture d )), which turns on 4-form couplings for the Bousso-Polchinski 3-forms. d , we saw that gravity in 7.3 5 in ( 0 G Even though the bubble that breaks the symmetry in this case is of stringy size, we On the face of it, one can take two different viewpoints: perhaps the presence of In section These considerations align nicely with the familiar difficulties with realizing the strict have also exploredtypically bubbles related which to admit tripleand Chern-Simons a off terms field symmetry in breaking the theoryassociated couplings. effective description bubbles field We as theory, to have which found well. be turn these a on Chern-Simons generic These terms feature are and of the string compactifications. They are present Thus, at leastto in be this broken. particularthe example, We BGHHS have there black found is hole thatcontributions a in of this euclidean the very is saddles good indeed withthe the the case two-dimensional same case point for asymptotics in of but theeffective a view, different field symmetry topology; consistent theory these from description embedding saddles breaks of have down, stringy-sized but which “bubbles” nevertheless where break the the symmetry. axion seems to survive. global symmetries is notthe symmetries problematic must on be broken such by anotherthe low-dimensional mechanism. BGHHS systems, We black have or, hole, given a in alternatively, particular which example, an unbroken symmetry would lead to a remnant problem. The motivational observation of this workof is generalized that the global usualparticles symmetries mechanism does of coming not explicit seem breaking from tothose periods work, related at of least to straightforwardly, for two-dimensional low axion enough systems, dimensions, e.g. where a remnant shift symmetry of the would put it in theconjecture Swampland, doesn’t which rule seems out to gravitya align with single minimal nicely axion, matter with which content. recent can results One undergo [ possibility Scherk-Schwarz is compactification adding as in8 section Conclusions conjecture. If the ( the KK photon wouldworth be mentioning that a there 2-dimensional isa a gauge weakly long-standing coupled field quest [ without Chern-Simons term. It is Bousso-Polchinski mechanism in string theory.a To decoupled implement sector Bousso-Polchinski, with weconditions a need that considerable stabilize number thevacuum, of axions which also 3-forms. does fix Otherwise, the notpractice, value the leave of the much minimization the freedom different field towould 3-forms strengths implement necessarily 4-forms always be the in mix the mechanism. the case among if However, themselves, our in conjecture and was with true. other fields, as sented would imply that therechanges should be aIt new would kind be of interesting domainstandard to wall Bousso-Polchinski explore in mechanism. the the precise theory effect (which these new couplings may have on the a landscape finely scanning the cosmological constant value. The conjecture we have pre- JHEP07(2017)123 4 and ≥ CFT d 20 1)-forms puts − d , it naively doesn’t. 2 T × 2 R = 8 SUGRA, as well as pure gravity in – 39 – N , we necessarily generate bulk Chern-Simons terms, as our conjec- 3 AdS current, so that if we start with a higher current in a higher-dimensional 2 4 pure gravity in the Swampland as well, and creates some tension between a strict to be broken or gauged in the UV completion of the Standard Model, but we don’t know at which , CFT In the end, this work has provided a substantial amount of solid support for our It would be interesting to gain further insight into this criterion in other holographic One open question is that we already have a “gravitational” symmetry breaking mech- We have upgraded our observations to a conjecture, claiming that consistent theories L We should remark that the constraints apply to the full theory, and not the low-energy EFT. The − 20 = 3 symmetry-breaking Chern-Simons terms must beB there, but can showscale up the at symmetry any breaking scale, effects much like or we unhiggsing expect takes place. conjecture, coming from diverse examples.have Like only other looked conjectures, at itstrong might special just enough classes be to of that support we stringlandscape/swampland. the vacua. proposal We expect However, of further the thisunderstanding work conclusion criterion in of so as this generalized far a direction global seems generic to symmetries,string feature lead and theory. of to of the progress the in stringy role the of Chern-Simons terms in dence, three-dimensional bulk Chern-Simonsof term a can sometimes beand related compactify to to anomalies ture demands. solve the problems of generalized globalthat symmetries even for in arbitrary this compactifications. holographicto example, Notice break we the can also symmetry. find the appropiate Chern-Simons term setups. A possibly interesting point is that, in the context of the AdS/CFT correspon- this mechanism works in general,we irrespectively consider; of in which some two-dimensional simple compactification By examples, contrast, such the as Chern-Simons a terms singlebreak (or U(1) more the on precisely, symmetry the associated inSimons bubbles) terms any explicitly as background. a general criterion We of therefore consistency take of the the theory, presence which must of be the “ready” to Chern- Bousso-Polchinski mechanism and quantum gravity. anism in the holographicdescription example, of related AdS quantum to gravity,general, the so and one sum if might over so wonder topologies why if we such in need UV the the mechanism Chern-Simons holographic are terms more as well. It is unclear to us whether of quantum gravity sufferture from further a constraints pandemic fieldputs theories of Einstein-Maxwell+WGC which Chern-Simons particles, can terms. bedimensions consistently If in coupled true, the to Swampland. the gravity, d conjec- A slight extension of our conjecture to ( origin, ranging from supergravity Chern-Simons termsChern-Simons to non-geometric terms. fluxes and Another D-brane for way both to breaking put or it gaugingwhich is interpolate in that from different phases one string of to theory the another. always theory, seems and to there are allow domain walls in every string compactification we have checked so far, and have varied 10-dimensional JHEP07(2017)123 (A.5) (A.1) (A.2) (A.4) plane, then xy ) (A.3) V P.D( . to lie in the ∧ ∗ ) . S C . ( dφ ] is the energy of the configuration. χ ] showing confinement in (2 + 1) to )  C . In this limit, the amplitude is a [ z  dz ) + 64 ( ) T 0 dφ E B z δ ∗ ( S is some closed curve in three dimensions, δ φ/f V + Z ) Z i ) is its characteristic function. Then, the C S  i (  S cos( φ f (  χ , where χ  − T ] exp – 40 – C exp ) = [  . It is straightforward to evaluate this around the  E V and ], where V Λ sin − e ] = + Λ(1 − C ,T ] = S [ P.D( S [0 = dφ [ W × φ ∧ W 2 C ∇ dφ 1 2 Z to be this represents the amplitude for a nondynamical string to nucleate S = T S . ) is the Poincar´edual of V ∂V is the region bounded by only to annihilate after a very long time = S S C The term is just the partition function for an axion. If the dilute-gas approximation where equations of motion are where P.D( solution to the equations of motion. If we take the curve where for the instantons is validhave (the an conclusion axion can with be modified modified action significantly when it is not), we then for very large along tunneling effect and is approximately However, one may also write behavior of the Wilson loop If we take the surface A Confinement of strings It is possible to emulatedisplay the the original same computation in phenomenon [ of for gravity strings for in the (3 time + 1). being. For In simplicity, the we euclidean ignore theory, the confinement effects will be shown by the SPLE under contract ERC-2012-ADG-20120216-320421.the IV Max is Planck supported Society. by MMUniversity a thanks of the grant Science Institute from for and Advanced Technologyand Studies for IV at thank hospitality the the while Hong DESY this Kong atmosphere workshop work “New which was Ideas led in completed. to String several MM Phenomenology” fruitful for a discussions. stimulating We thank WilliamQuevedo, Cottrell, Diego Thomas Regalado, Grimm,and Pablo Arthur comments. Soler, Hebecker, MM and Luis issupported Gianluca supported Ib´a˜nez,Fernando by Zoccarato the by grants a for FPA2015-65480-P Postdoctoralcia and helpful fellowship SEV-2012-0249 Severo of from discussions Ochoa” the ITF, Programme “Centro from Utrecht. de the Excelen- AU Spanish is Ministry, and the ERC Advanced Grant Acknowledgments JHEP07(2017)123 , 6 2 B F (B.2) (B.3) (B.4) (B.5) (B.6) (B.1) tr (A.6) 6 B Λ, with an d . ) = is computed by M γ ω + ). . dθ r ( N A.6 U + 1 √ ), where . 2 1 λ Ω V , = . 3  4 e 2 = | Nd z | Vol( ∧ -field, coming from wrapping − γ m Nλ 2 r ) plugging ( B e − = . + e ,U 2 ) θdφ, e 2  unobservable in practice. r A.3 ) dr M NF Ur sin Q = D M + 3 − ∗ F + 4 Ur dθ 2 e N r √ ) arctan dθ )( ∧ ( z r − 1 – 41 – = 1 ( ⇒ e − 3 0 F = U F 2 dU Λ = ) + D NF Ur ) is 3 Λ = d~x ∗ d ) = 4 sign( Urdθ, e = · z 0 ) is ( B.1 √ = = φ F d~x ω = ( B.2 = 1 case) of the form 2 dM U , but focus on a different 4d N = 6 2 ds Udr, e . Plugging back into the path integral, fluctuations around this solution 2 √ . The triple Chern-Simons comes from dimensional reduction of /f = 4 Λ 1 A e p = m First we will find a potential for the Taub-NUT self dual 2-form This area (or rather, volume) behavior of the Wilson loop shows confinement of the on a 4-cycle C 2d instantonsWe and now Chern-Simons discuss terms abetween heterotic Chern-Simons example terms which and perhapspactification charged sheds as objects: some section consider light the on same the heterotic relationship CY com- Anti-self-duality then amounts to The exterior derivative of ( since A tetrad basis for the metric ( ansatz (inspired by the B Self-dual fluxes inIn Taub-NUT this space brief appendixspace. we For will convenience, we derive reproduce the again expression the for Taub-NUT metric the self-dual form of Taub-NUT evaluating the 1-dimensional version of the action ( strings. As we trystring to will stretch be a nucleated.The string anti-stringthe will to Aharonov-Bohm provide a phase, an rendering length equal the enough and charge to opposite lasso contribution the to black hole, an anti- where become gaussian and exactly computable,the giving classical an action overall will factor. give The a term contribution coming of from the form Far from the boundaries, this is a one-dimensional equation with solution JHEP07(2017)123 = F 0 ∧ (C.2) (C.3) (C.1) F dG R ; if we further 4 A , we have ρ ∼ r . As usual, their action and is such that B R . At = 4 r > ρ ) do not cancel out: they leave b . C.2 term allows NS5-branes to fatten. ) , ) 2 2 2 A A F ω ω , tr ) + 6 ) and ( r + 2 ( B 2 ξ is the volume element of the two-dimensional ) vanishes for C.1 dV 2 dV r − ( = 2 )( ξ – 42 – dV )( r also endows heterotic gauge instantons with NS5- 0 r ( ( 2 ξ G ξ F = , and = tr 2 6 F field are just NS5-branes wrapped on F A B ), which permits any use, distribution and reproduction in B or, in other words, ]. The NS5-brane can be regarded as a point-like gauge instanton 2 A CC-BY 4.0 ω 111 This article is distributed under the terms of the Creative Commons , ]; it is possible to overlap a polarized D-brane instanton-anti instanton 110 . 112 4 A is the to Poincar´edual , we argued that charged objects alone are not enough to break the symmetry. is the component of the first Chern class of the gauge bundle along the 2-cycle 2 0 A 6= 0 so that the boundary of the bubble is a particle charged electrically under 4.2 G ω 2 This behavior may seem particular to the heterotic case, but we expect it to hold at So far, we have only fattened the two-dimensional instanton — but it still has a IR- The 10d Chern-Simons term G dual to , as discussed there. In other words, the symmetry-breaking bubble can be constructed ∗ 2 2 . If we are not concerned with solving the Euclidean equations of motion, but only with pair to construct the bubble in the same way. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. section Here, we see thatallows in them the to break presence the of two-dimensional Chern-Simons symmetry. terms theyleast can also fatten, for which type in IIpolarization turn examples. [ The D-brane instantons one gets may fatten due to brane which is precisely oned of the bubbles introducedG in section by fattening and deforming a finite action instanton- anti instanton configuration. In The configuration now consistslonger of divergent. an However, instanton- thea anti gauge component fields instanton along ( pair, and its action is no divergent action. Consider adding now a anti NS5, on top of the NS5, which fattens to where noncompact factor. The smoothhas function the instanton number of a NS5. compactify to two dimensions, theyis become IR instantons divergent. for However,ρ we may fatten thefinding a NS5 finite brane action to deformationa of a U(1) the SO(32) instanton, gauge we instanton field are of configuration allowed radius with to replace the NS5 with A brane charge [ of vanishing size or,The put strings in coupled to another the way, the 4d where JHEP07(2017)123 D ]. Phys. , Phys. Nucl. , 09 (1989) , , (2015) 032 SPIRE B 748 Nucl. Phys. IN , 08 ]. JCAP Phys. Rev. ]. B 325 ][ , , Causality, SPIRE JHEP (2015) 049 SPIRE ]. , IN IN [ Phys. Lett. ][ 04 (2013) 419 , Nucl. Phys. SPIRE (2006) 014 , ]. (2016) 1001 IN Winding out of the swamp: JCAP ][ 10 B 719 , ]. arXiv:1503.04783 SPIRE [ B 913 IN Gravity and global symmetries JHEP ][ SPIRE The string landscape, black holes and , ]. hep-th/0601001 hep-th/0509212 IN ]. [ , ][ Phys. Lett. Natural inflation and quantum gravity ]. , Planckian axions and the weak gravity Transplanckian axions!? (2015) 023 Fencing in the swampland: quantum gravity SPIRE On axionic field ranges, loopholes and the weak SPIRE IN hair Nucl. Phys. IN 10 ]. ]. ][ , arXiv:1504.00659 SPIRE ][ [ – 43 – (2007) 060 /N IN ]. 1 ][ 06 JHEP SPIRE SPIRE , arXiv:1503.07853 IN IN SPIRE [ ][ ][ arXiv:1412.3457 IN Wormholes made without massless matter fields ]. ]. Wormholes and global symmetries (2016) 017 [ [ ]. ]. JHEP Skyrmion black hole hair: conservation of baryon number by ]. ]. , Constraints on string vacua with space-time supersymmetry Symmetries and strings in field theory and gravity 04 Black hole’s hep-th/0605264 On the geometry of the string landscape and the swampland SPIRE SPIRE hep-th/9502069 [ SPIRE SPIRE [ IN IN (2016) 091 SPIRE SPIRE IN IN ][ ][ (1988) 93 IN IN JHEP arXiv:1011.5120 ][ ][ 01 [ , ][ [ (2015) 151303 (2007) 21 On the possibility of large axion moduli spaces Constraints on axion inflation from the weak gravity conjecture ]. B 307 arXiv:1503.07912 arXiv:1503.00795 (1995) 912 JHEP [ [ 114 , The string landscape and the swampland (1990) 387 SPIRE B 766 D 52 IN [ (2011) 084019 arXiv:1503.03886 arXiv:1409.5793 hep-th/0602178 arXiv:1605.00543 arXiv:1203.6575 evading the weak gravity(2015) conjecture with 455 F-term winding inflation? gravity conjecture [ constraints on large field inflation conjecture [ (2015) 020 [ gravity as the weakest force Rev. Lett. Phys. analyticity and an IR obstruction to UV completion black holes and observable[ manifestations Nucl. Phys. Rev. 83 [ 687 B 329 A. Hebecker, P. Mangat, F. Rompineve and L.T. Witkowski, J. Brown, W. Cottrell, G. Shiu and P. Soler, J. Brown, W. Cottrell, G. Shiu and P. Soler, T.C. Bachlechner, C. Long and L. McAllister, T. Rudelius, T. Rudelius, M. Montero, A.M. Uranga and I. Valenzuela, N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, A. de la Fuente, P. Saraswat and R. Sundrum, A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, C. Vafa, H. Ooguri and C. Vafa, G. Dvali and A. Gußmann, T. Banks and L.J. Dixon, T. Banks and N. Seiberg, G. Dvali and C. Gomez, L.F. Abbott and M.B. Wise, S.R. Coleman and K.-M. Lee, R. Kallosh, A.D. Linde, D.A. Linde and L. Susskind, [8] [9] [6] [7] [4] [5] [1] [2] [3] [18] [19] [16] [17] [13] [14] [15] [11] [12] [10] References JHEP07(2017)123 , ]. , ]. 09 , ]. JHEP JHEP , SPIRE , , Phys. IN (2015) 188 SPIRE [ (2017) , ]. JHEP IN SPIRE , 10 ][ IN superconformal ]. ][ D 95 SPIRE IN JHEP = 2 arXiv:1610.01533 ]. ]. ][ , , SPIRE N ]. IN ][ SPIRE SPIRE hep-th/0507215 The Phys. Rev. IN IN , , ]. SPIRE ]. ]. ][ ][ IN arXiv:1701.06553 Can gravitational instantons ][ [ Relaxion monodromy and the weak arXiv:1607.06814 SPIRE SPIRE SPIRE ]. [ IN IN [ Generalized global symmetries What is the magnetic weak gravity IN arXiv:0808.0706 ]. ][ [ ][ SPIRE Gravity waves and linear inflation from IN arXiv:1509.06374 SPIRE ]. Warping the weak gravity conjecture [ ][ Sharpening the weak gravity conjecture with Weak gravity strongly constrains large-field Evidence for a lattice weak gravity conjecture (2017) 097 ]. IN F-term axion monodromy inflation (2017) 1700011 ][ arXiv:1610.08836 arXiv:1512.00025 02 ]. [ [ – 44 – 65 SPIRE arXiv:1506.03447 ]. SPIRE [ (2010) 046003 IN hep-th/9501106 IN The weak gravity conjecture in three dimensions Weak gravity conjecture and extremal black holes ][ , SPIRE (2016) 140 ][ JHEP arXiv:0803.3085 arXiv:1412.7541 Monodromy in the CMB: gravity waves and string inflation IN , [ SPIRE [ D 82 02 ]. ][ IN (2017) 109 (2016) 020 An axion-induced SM/MSSM Higgs landscape and the weak ]. ]. ][ (2015) 108 02 04 arXiv:1509.07049 Non-supersymmetric AdS and the swampland SPIRE [ Fortsch. Phys. JHEP 12 SPIRE SPIRE arXiv:1504.03566 , , IN [ IN IN (2016) 183 [ [ ][ Phys. Rev. JHEP JHEP , (2008) 106003 , , 03 JHEP arXiv:1412.5148 arXiv:1606.08438 , Large-field inflation with multiple axions and the weak gravity conjecture [ [ Weak gravity conjecture and effective field theory Trouble for remnants (2016) 402 arXiv:1404.3040 D 78 [ (2016) 128 JHEP Three-form gauging of axion symmetries and gravity On natural inflation and moduli stabilisation in string theory ]. , arXiv:1608.06951 02 [ B 759 (2015) 172 (2016) 159 SPIRE IN arXiv:1508.00009 Phys. Rev. axion monodromy bootstrap 02 (2014) 184 conjecture for axions? gravity conjecture [ arXiv:1611.06270 really constrain axion inflation? 025013 gravity conjecture 10 arXiv:1606.08437 Lett. axion inflation JHEP [ dimensional reduction E. Silverstein and A. Westphal, L. McAllister, E. Silverstein and A. Westphal, G. Dvali, D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, F. Marchesano, G. Shiu and A.M. Uranga, A. Hebecker, P. Henkenjohann and L.T. Witkowski, L. Susskind, C. Beem, M. Lemos, P. Liendo, L. Rastelli and B.C. van Rees, H. Ooguri and C. Vafa, W. Cottrell, G. Shiu and P. Soler, A. Hebecker, P. Mangat, S. Theisen and L.T. Witkowski, P. Saraswat, A. Herraez and L.E. Ib´a˜nez, M. Montero, G. Shiu and P. Soler, B. Heidenreich, M. Reece and T. Rudelius, K. Kooner, S. Parameswaran and I. Zavala, B. Heidenreich, M. Reece and T. Rudelius, Montero,L.E. A. Ib´a˜nez,M. Uranga and I. Valenzuela, E. Palti, B. Heidenreich, M. Reece and T. Rudelius, D. Junghans, [38] [39] [40] [36] [37] [33] [34] [35] [31] [32] [28] [29] [30] [26] [27] [23] [24] [25] [21] [22] [20] JHEP07(2017)123 , , B ]. ]. 102 JHEP B 378 D 45 , ]. Phys. SPIRE , SPIRE IN Phys. Rev. IN [ , Phys. Rev. charged Axionic black Phys. Lett. SPIRE , ][ Fortsch. Phys. , Cambridge , IN , Zp Phys. Rev. ][ Nucl. Phys. ]. , Lect. Notes Phys. , gr-qc/9811079 ]. , , Phys. Rev. Lett. , (1988) 2823 SPIRE Discrete gauge symmetries ]. IN SPIRE 61 IN ][ ’s in type-I and type IIB Pair creation of dilaton black hep-th/9808139 ][ [ SPIRE The powers of monodromy U(1) ]. IN correspondence arXiv:1211.5317 ][ [ ]. 2 SPIRE IN (1999) 112 /CF T SPIRE -form formulation of axion potentials from Phys. Rev. Lett. 3 ][ 3 IN , ]. Anomalous ]. ]. ]. ]. ][ Euclidean black hole vortices (2013) 138 arXiv:1605.08092 Quantum hair on black holes AdS arXiv:1106.4169 [ B 542 [ ]. String universality in ten dimensions 04 – 45 – ]. SPIRE On the SPIRE SPIRE SPIRE SPIRE hep-th/9309075 IN [ IN IN IN IN ][ species solution to the hierarchy problem Topological mass generation in four dimensions SPIRE ][ ][ ][ ][ Semiclassical wheeler wormhole production JHEP SPIRE IN N , (2017) 087 IN Winter School of Theoretical Physics: Fields and Geometry (2011) 113 ][ Instantons for black hole pair production hep-th/0511175 ]. ][ Nucl. Phys. [ 02 Cosmological production of charged black hole pairs nd , 12 The weak gravity conjecture and the axionic black hole paradox A natural framework for chaotic inflation arXiv:1006.1352 ]. [ (1994) 2909 SPIRE IN JHEP [ JHEP gr-qc/9504015 , SPIRE , [ D 49 IN hep-th/0609074 Quantized flux tubes in Einstein-Maxwell theory and noncompact internal arXiv:0811.1989 arXiv:0706.2050 arXiv:1405.3652 [ string vacua String theory. Volume 1: an introduction to the bosonic string [ [ [ (2006) 081602 hep-th/9112065 hep-th/9201059 [ [ 96 = 1 Lectures on black holes and the (2010) 071601 Black holes and large- ]. N Phys. Rev. (1995) 2254 , , talk given at the 22 105 , (1991) 146 (2008) 193 [ (2010) 528 (2014) 123 = 4 SPIRE IN February 17–March 1, Karpacz, Poland (1986). 256 holes D 52 [ spaces 58 755 holes and a Bohm-Aharonov effect for strings (1992) 175 (1992) 2762 arXiv:1702.06130 University Press, Cambridge U.K. (1998). in D-brane models 09 Lett. branes in flux compactifications D Rev. Lett. (2009) 121301 D-brane instantons D. Garfinkle and A. Strominger, F. Dowker, J.P. Gauntlett, D.A. Kastor and J.H. Traschen, P.M. Branoff and D.R. Brill, G.W. Gibbons, G. Dvali, P. Kraus, R.B. Mann and S.F. Ross, S.R. Coleman, J. Preskill and F. Wilczek, F. Dowker, R. Gregory and J.H. Traschen, A. Hebecker and P. Soler, J. Polchinski, M.J. Bowick, S.B. Giddings, J.A. Harvey, G.T. Horowitz and A. Strominger, L. McAllister, E. Silverstein, A. Westphal and T. Wrase, A. Adams, O. DeWolfe and W. Taylor, M. Berasaluce-Gonzalez, P.G. Camara, F. Marchesano and A.M. Uranga, L.E. Rabad´anand A.M. Ib´a˜nez,R. Uranga, M. Berasaluce-Gonzalez, Soler L.E. and Ib´a˜nez,P. A.M. Uranga, N. Kaloper and L. Sorbo, E. and Garc´ıa-Valdecasas A. Uranga, G. Dvali, R. Jackiw and S.-Y. Pi, [59] [60] [57] [58] [54] [55] [56] [52] [53] [49] [50] [51] [47] [48] [44] [45] [46] [42] [43] [41] JHEP07(2017)123 , , , ]. Class. , (1977) Int. J. , , string 17 Nucl. Phys. Rev. 5 Phys. Lett. SPIRE , , , S IN B 120 ][ ]. (1998) 253 ]. (2006) 044013 International 2 (1983) 2019. (1996) 3296 th SPIRE SPIRE 77 D 74 IN IN Commun. Math. Phys. D 28 Nucl. Phys. ][ Phys. Rev. Lett. ][ , , , Extra natural inflation ]. ]. hep-th/9804170 [ ]. Phys. Rev. ]. SPIRE ]. , Phys. Rev. IN SPIRE , ][ Phys. Rev. Lett. IN ]. SPIRE , talk given at the 14 , SPIRE ][ SPIRE ]. hep-th/9608086 IN hep-th/9803131 IN Adv. Theor. Math. Phys. (1998) 002 [ IN ][ , ][ Entropy, area and black hole pairs ][ ]. SPIRE 05 IN SPIRE ][ IN instantons Sphalerons, merons and unstable branes in AdS – 46 – ][ SPIRE D sphalerons and the topology of string configuration D JHEP (1996) 521 IN (1998) 505 [ , hep-th/0301218 ][ 2 [ hep-th/0004131 [ gr-qc/0610018 Absence of ferromagnetism or antiferromagnetism in AdS/CFT correspondence and symmetry breaking limit of superconformal field theories and supergravity hep-th/9711200 Hypermultiplet moduli space and string compactification to B 388 [ hep-th/0001143 [ Duality, axion charge and quantum mechanical hair ]. ]. ]. N ]. Nonperturbative effects in AdS in five-dimensions x [ Boundary conditions and new dualities: vector fields in AdS/CFT hep-th/9905104 Summing up ]. [ hep-th/0606113 SPIRE SPIRE SPIRE ]. SPIRE [ (2003) 221302 IN IN IN IN gr-qc/9409013 SUSY Yang-Mills Phys. Lett. [ [ (2000) 086007 ][ ][ ][ The large- (2007) R171 SPIRE The asymptotic safety scenario in quantum gravity: an introduction (1999) 1113 (2000) 021 90 , Quark confinement and topology of gauge groups IN There are no Goldstone bosons in two-dimensions SPIRE Topology in the Weinberg-Salam theory 24 = 4 [ (1999) 89 38 03 Critical phenomena for field theorists IN ]. [ D 62 Anti-de Sitter space, thermal phase transition and confinement in gauge Anti-de Sitter space and holography D (2006) 085 Black holes with quantum massive spin-2 hair Adv. Theor. Math. Phys. , (1991) 355 11 SPIRE JHEP (1995) 4302 B 556 , IN [ (1973) 259 hep-th/9802150 hep-th/9608079 hep-th/0605295 School of Subnuclear Physics:July Understanding 23–August the 8, Fundamental Erice, Constitutents Italy of (1976). Matter Quant. Grav. Phys. Rev. [ Theor. Phys. theories space JHEP theory and (1966) 1133 31 Phys. three-dimensions Phys. Rev. Lett. one-dimensional or two-dimensional isotropic Heisenberg models 429 [ D 51 B 254 [ S. Weinberg, M. Niedermaier, E. Witten, J.M. Maldacena, E. Witten, J.A. Harvey, P.and Hoˇrava P. Kraus, N. Drukker, D.J. Gross and N. Itzhaki, D. Marolf and S.F. Ross, T. Banks and M.B. Green, N.S. Manton, S.R. Coleman, I.R. Klebanov and E. Witten, N. Arkani-Hamed, H.-C. Cheng, P. Creminelli and L. Randall, N.D. Mermin and H. Wagner, A.M. Polyakov, H. Ooguri and C. Vafa, N. Seiberg and S.H. Shenker, L.M. Krauss and S.-J. Rey, G. Dvali, S.W. Hawking, G.T. Horowitz and S.F. Ross, [79] [80] [77] [78] [74] [75] [76] [71] [72] [73] [69] [70] [67] [68] [64] [65] [66] [62] [63] [61] JHEP07(2017)123 , B 7(7) (2005) E B 153 10 ]. Adv. Theor. ]. , ]. Phys. Lett. ]. (1998) 181 , ]. (1998) 027 JHEP Class. Quant. SPIRE , , SPIRE IN Nucl. Phys. 11 [ SPIRE (2010) 009 (2011) 265 , IN ]. SPIRE IN SPIRE B 532 ][ (2007) 087 IN 12 ][ IN [ ][ (2009) 067 11 JHEP SPIRE B 694 , 06 IN JHEP ][ , Open string multi-branched and JHEP , Nucl. Phys. JHEP , ]. ]. , ]. arXiv:0801.0988 Phys. Lett. hep-th/0004134 , ]. , [ arXiv:1507.06793 arXiv:0704.0777 arXiv:0706.3359 SPIRE SPIRE supergravity SPIRE [ [ , supergravity ultraviolet finite? IN IN untwisted IN Minkowski 3-forms, flux string vacua, axion [ SPIRE ][ ]. 5 ][ = 8 Nondecoupling of maximal supergravity from the IN S = 8 arXiv:1606.00508 ][ N × (2000) 006 – 47 – [ N supergravity 5 Nongeometric flux compactifications SPIRE Is 06 (2015) 119 IN Axion induced topology change in quantum gravity and = 8 AdS (1988) 890 ][ 12 (2007) 041601 ]. N String theory and particle physics: an introduction to string Quantization of four form fluxes and dynamical neutralization , Cambridge University Press, Cambridge U.K. (2007). JHEP 99 ]. ]. ]. ]. How to get masses from extra dimensions (2016) 062 , ]. ]. hep-th/9707042 B 306 JHEP SPIRE 09 arXiv:0808.4076 , Genus two partition functions of extremal conformal field theories IN [ SPIRE SPIRE SPIRE SPIRE arXiv:0707.3437 ][ SPIRE SPIRE [ IN IN IN IN IN IN ]. ][ ][ ][ ][ JHEP ][ ][ , Constraints on extremal self-dual CFTs hep-th/0611086 [ , Cambridge University Press, Cambridge U.K. (2012). (1998) 115 [ Lectures on gauged supergravity and flux compactifications Nucl. Phys. 1 SPIRE , Phys. Rev. Lett. Monster symmetry and extremal CFTs The ultraviolet finiteness of Massive string theories from M-theory and F-theory Three-dimensional gravity revisited Branes, instantons, and Taub-NUT spaces , IN (2007) 029 Gravity and strings (2008) 214002 [ Dynamics of multiple Kaluza-Klein monopoles in M and string theory 08 25 hep-th/0508133 (2007) 265 [ arXiv:0707.4073 hep-th/9811021 hep-th/9803061 arXiv:0902.0948 arXiv:1009.1643 arXiv:1009.1135 JHEP [ K¨ahlerpotentials of the cosmological constant string theory stability and naturalness [ 085 [ Math. Phys. [ Grav. phenomenology constraints on counterterms in [ [ (1979) 61 superstring 644 M.R. Gaberdiel, D. Gaiotto, D. Gaiotto and X. Yin, F. Carta, F. Marchesano, W. Staessens and G. Zoccarato, R. Bousso and J. Polchinski, E. Witten, S.B. Giddings and A. Strominger, S. Bielleman, L.E.I. Ib´a˜nezand Valenzuela, J. Shelton, W. Taylor and B. Wecht, M.J. Duff, H. L¨uand C.N. Pope, E. Witten, T. Ort´ın, C.M. Hull, H. Samtleben, L.E. Ibanez and A.M. Uranga, A. Sen, N. Beisert, H. Elvang, D.Z. Freedman, M. Kiermaier, A. Morales and S. Stieberger, R. Kallosh, M.B. Green, H. Ooguri and J.H. Schwarz, Z. Bern, L.J. Dixon and R. Roiban, J. Scherk and J.H. Schwarz, [99] [96] [97] [98] [94] [95] [92] [93] [89] [90] [91] [86] [87] [88] [84] [85] [82] [83] [81] [100] [101] JHEP07(2017)123 2 02 2) , , , = (2 JHEP ]. , N ]. ]. superconformal , SPIRE 3 Small black holes ]. IN = 2 SPIRE SPIRE ][ IN IN Extremal N ][ ][ SPIRE (1996) 541 IN ][ ]. Commun. Num. Theor. Phys. B 460 , An extremal SPIRE hep-th/9910053 IN Three-dimensional AdS gravity and [ ][ gravity and CFT spectroscopy arXiv:0708.3386 ]. arXiv:1603.08524 ]. [ D [ 3 Nucl. Phys. , SPIRE arXiv:1507.00004 Chiral gravity, log gravity and extremal CFT SPIRE [ (1999) 022 IN ]. ]. IN – 48 – ][ [ 12 (2007) 022 Small instanton transitions in heterotic M-theory (2016) 023 11 SPIRE SPIRE arXiv:0903.4573 08 IN IN [ JHEP Bootstrapping pure quantum gravity in AdS ][ ][ , Poincar´eseries, ]. (2014) 505 ]. (2015) 495401 JHEP JHEP , , m 257 SPIRE hep-th/0001133 A 48 SPIRE [ IN = 8 [ IN c (2010) 064007 ][ Remainder formula and zeta expression for extremal CFT partition Dielectric branes J. Phys. arXiv:0805.4216 arXiv:1407.6008 D 81 Small instantons in string theory , [ [ Prog. Math. (2000) 045 , 05 conformal field theories and constraints of modularity D hep-th/9511030 arXiv:1610.05814 [ JHEP field theory and near-extremal CFTs (2015) 080 functions extremal CFTs at 2 (2008) 743 Phys. Rev. R.C. Myers, E. Witten, B.A. Ovrut, T. Pantev and J. Park, N. Benjamin, E. Dyer, A.L. Fitzpatrick and S. Kachru, N. Benjamin, E. Dyer, A.L. Fitzpatrick, A. Maloney and E.J.-B. Perlmutter, Bae, K. Lee and S. Lee, C.A. Keller and A. Maloney, F.L. Williams, M.R. Gaberdiel, S. Gukov, C.A. Keller, G.W. Moore and H. Ooguri, A. Maloney, W. Song and A. Strominger, S.D. Avramis, A. Kehagias and C. Mattheopoulou, [112] [110] [111] [107] [108] [109] [105] [106] [103] [104] [102]