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PoS(TASI2017)015 , https://pos.sissa.it/ c † [email protected] , ∗ , and b , Federico Carta a [email protected] La Caixa-Severo Ochoa Scholar Speaker. We give a brief overviewpactifications. of the We then landscape explain andnumber how techniques of this conjectures used motivates that to the attempt construct to notionholography string characterize in of com- theories the the in context swampland the of swampland. and superstringsstring with review We theory. also the a For compare similar, topological but strings, mucha there simpler sum is case a over of direct a topological definition “quantumidentification of gravitational between topological a foam.” gravity and based In gauge on this theory, bothanother. context, of This which holography points are is defined to independently the a of missing statementdefinition one corner of of in an quantum dualities which ofdual suggests gravity as the rather an search adequate than for substitute relying a (Based direct on on TASI its 2017 strongly lectures given coupled by holographic C. Vafa). † ∗ Copyright owned by the author(s) under the terms of the Creative Commons Department of Physics and Astronomy, Rutgers University Instituto de Física Teórica UAM-CSIC, Jefferson Physical Laboratory, Harvard University c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Theoretical Advanced Study Institute in ElementaryFrontier” Partical Physics (TASI): “Physics at theUniversity Fundamental of Colorado, Boulder Boulder, CO June, 5 – 30 2017 [email protected] T. Daniel Brennan 126 Frelinghuysen Rd., Piscataway NJ 08855, USA Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain Cambridge, MA 02138, USA Email: The String Landscape, the Swampland, andMissing the Corner c a b PoS(TASI2017)015 Cumrun Vafa whose ) 1 . In mid 1990’s, it was ( 1 U ]. 1 1 Recall that D- were discovered only in the late 80’s - early ’90s, and before this moment it was largely In the late 1980’s it was realized that there were five perturbatively consistent, distinct ten- These lecture notes from TASI 2017 give a brief overview of some of the open problems in With this philosophy in mind, we will begin these lectures by reviewing some of what we know The string landscape and the swampland will be the topic of the first two lectures. In the third 1 unknown how to realize gauge groups in Type II theories [ dimensional string theories. Early efforts toas find a low the energy four-dimensional limit standard of string model theoryInitially, with focused the gravity on Heterotic compactifying those string theories theories on various werestandard manifolds. considered model the because most of promising their theories inherent to produce non-abelian the gauge symmetries configurations can be interpreted as defining adual quantum to gravitational a foam. Chern-Simons This theory. is Thefor holographically equivalence of topological these strings. two theories In ismissing the the and case content holography of of is holography full viewedsuggests as string a theory, that substitute a there definition. direct is definition Thedefinition a analogy of of missing with quantum quantum corner topological gravity. gravity strings in is string dualities and that we should1. try Lecture to 1: find The a String direct Landscape 1.1 Review of String Theories string theory. We willsupposed be to generally describe the motivated fundamental by lawsbe the of used philosophy our to universe. study that a String string wide theoryquark-gluon theory array is plasma of or is so aspects physical versatile ultimately of problems that quantum such fields itwork as theories can using in various diverse string topics dimensions. in theory Much condensed has of matter been thethan and recent focused developing on our using understanding its properties ofHere to string solve we theory specific aim as problems to rather aobservable fundamental discuss aspects description topics of of fundamental which our physics. we universe. hope will be usefulabout in string theory bringing and its string possible theory application tothe closer the space universe of to by low describing energy some theories generalities theories about comingthe from “string string landscape.” theory compactifications: this plays isnaturally a called lead key us organizing to investigate principle the in questionlandscape this of or how context. it we This know is a will not. prioriare if The not a set low when energy of coupled theory low to is energythe gravity, in physics is the swampland models called from which the the look “swampland.” landscape consistentconcrete Finding is but predictions simple ultimately of criteria for to our great distinguish universe importance. aswhich are we In aimed will particular at discuss such distinguishing later. the criteria We swampland can review from a lead the number landscape. to of conjectures lecture, which is on aunderstanding somewhat of disjoint quantum topic, gravity. wetheory, review In for critically this which where lecture much we weour more are use current is in the formulation our known, toy of current togravitational example quantum shed theory of gravity a can topological from be new string string formulated light theory. in on terms In what of topological shortcomings a string we non-commutative theory have the in The String Landscape, the Swampland, and the Missing Corner PoS(TASI2017)015 - ) k 1 , 1 1 su- where = = ( R which is / 2 s τ N N ` is related to Cumrun Vafa 1 = S 0 R of R ], since they have isomorphic Lie 6 ) [ 1 32 ( . S 3 and is thus chiral. Heterotic and SO ) 0 by T-duality where , where the radius 0 2 1 1 R S . Physically, M-theory on this interval S = ( × 10 × ] for a review of some aspects of F-theory. 9 M 10 N 11 M M , supersymmetry 10 ) 2 0 and shrink it to zero size, we would expect to get a , , . There is also a conjectured 11D theory called M- 1 gauge group which together give rise to the 10D HE 2 2 2 8 T λ ]. See [ E = ( 9 ]. = , but it is convention to say 2 (HO) 2 3 with Wilson lines turned on N Z . The dualities are generated by the following equivalences ) R / 1 k SUSY operators in the left-handed spin representation of the d-dimensional ) S 32 − M 32 ( p is related to HE on × ( 2 9 SO Z M Spin / 1 is related to Type IIB on S 1 R S × is equivalent to M-theory on × 10 9 10 M M M SUSY operators in the right-handed spin representation. More precisely, in even dimensions, the ]. ): − 8 1 q , indicates that there are 7 ) ]. q 5 , as the complex structure of a compactification torus of some 12D theory. This 12D , p 4 equivariant since Type IIB is self-dual under S-duality. Because of this property, we can , τ is the string length, 3 = ( ) 8 (HE), and Heterotic s Type IIA on theory [ Type IIA coupling constant by ` (whose length is related to Heteroticthe string interval coupling constant) each has of M9-branes which on carry each an end of Type I is related to HO by S-duality (inversion of couplingHO constant), is related to HE on M-theory on Type IIA on Z E , The gauge group is actually N While M-theory appears to be the most central in web of dualities between all of the different Let us now consider Type IIB string theory. It has a complex coupling constant, The five original types of superstring theories are Type I, Type IIA, Type IIB, Heterotic The duality web suggests that all these theories are related by dualities. Let us denote 2 3 • • • • • 2 × ( 8 spin representations of the Lorentz groupsentations. are However, distinguised in by 4D chirality and which 8D,that gives these the we representations left- only are and have related right-handed this by spin chiral complex repre- notation conjugation for and 2D, hence 6D, are and identical 10D. so SL think of algebras [ Lorentz group and theories, in some sense, itSpecifically if doesn’t we see compactify Type M-theory IIB on string theory, even though they are clearly dual. theory is what is referred to as F-theory [ 9D effective theory. However, string dualityLorentz tells symmetry. us This that indicates we that actuallyIIB. M-theory get is Type perhaps IIB not which a has very 10D good way to describe Type pergravity. This theory is highest dimensional theory. The Type IIA has a theory which is not a string theory and has a low energy effective action given by 11D E The String Landscape, the Swampland, and the Missing Corner realized that these five theories weretions, all thus related launching by the different dualities “duality and era” non-perturbative [ comple- supersymmetry and thus is non-chiral. Type IIB has Type I strings are both chiral carrying dimensional Minkowski space by (also see Figure PoS(TASI2017)015 0 is Type I K → con- SUSY N ) K q R + S-dual p Cumrun Vafa -dimensional d ) = ( 32 is some compact ( K SO SUSY 32 and is chiral since N = which has where the volume of 1 , or where Heterotic ) K − . Then we will take the limit 32 charges and is non-chiral SUSY q d , K K × 1 N = R + d × R p d M ( = 8 M SUSY d N = twist w/ Wilson M 1 S 8 E SUSY × 8 N E , . Note that we do not take the limit of ) d q 3 M 2 +

Z p

1 / Heterotic S ( 16 =

M-Theory

SUSY

s

g

for M-theory) and consider the low energy dynamics with energy N 1 S K R has ) q SUSY whereas Type IIB string theory has << , ) p 1 which is adapted for each dimension, we will make use of the notation Type IIA , completely captures the structure of the super Lorentz group. 1 = ( . The relation between the two notations is that in even dimensional spacetime, Planck N l ) = ( 0 , N SUSY (or 2 N , leading to an effective theory on N T-dual K K = ( This graph shows the duality web relating all of the different descriptions of string theory. Here R R

/

τ

T 1 2 N << Thus far, we have been implicitly assuming that we are compactifying with compact man- Additionally, as it is useful in classifying the string theories, it also makes sense to classify Since there are so many (possibly infinitely many) manifolds we can compactify on to get We can now try to produce lower dimensional theories and in particular a four dimensional s ` << Type IIB F-Theory served supercharges in 10D, 6D, ordimensions, 2D respectively. Apart from the chirality information in these ifolds. However, even thoughmanifold, there the has actually name been and amanifolds.” significant procedure This amount of of means work that compactification on we “compactifying requires can on also a non-compact study compact string theory on a theory with it has since it has our theories by the amountsupersymmetries of supersymmetry. For our purposes rather than using the number of Lorentz symmetry. which counts the total number ofcompactifications. conserved supercharges, as As this with will not thesupersymmetries. change 10D upon For flat string example, toroidal theories, Type IIA we string will theory also has classify by the chirality of the lower dimensional physics, ittheories. is clear As that physicists we wesuch, generally need we try some will to kind use use Lorentz ofdirections symmetries symmetries organizing which to again are principle for characterize not for this different part purpose. of systems. these the For compact As the direction time to be being, we will take all as this will lead to theleads introduction to of a light higher matter dimensional coming dual from description. the winding states which typically manifold with appropriate dimension with someof characteristic size E theory that describes ourBy universe this by we compactifying mean one that of we these will theories take to a lower spacetime dimensions. of the form Figure 1: the vertical placement corresponds to dimension. 1.2 String Compactification The String Landscape, the Swampland, and the Missing Corner PoS(TASI2017)015 ∞ . In (1.1) (1.2) → 5 S pl × . However, M , ∞ + we may get Cumrun Vafa ∞ R D3-branes in → ) K → N K ( is an appropriate ) K K ( Vol , d Vol units of flux. From the 4 where g R N -dimensions. These can in d − K √ × x d d d , d R AdS d -dimensions which are not coupled to Z d 2 which are decoupled from gravity. This g R space in holography begs the question if − supported by d d pl − 5 M M S √ is our compactification manifold. To leading AdS x × = d K 5 d d . Therefore in the limit , absorbs the non-compactness of the transverse . For the rest of the discussion here we will focus 2 d 3 , 4 − M 1 will be of the form g R d pl Z AdS 2. R d − M ) 1.3.5 where > M √ K d ( x K . However depending on the structure of d 5 × d Vol d d R M ] for some additional details of decoupling gravity. ∼ Z S 12 N G 1 , resulting in a compact geometry with flux. This is the statement π 5 16 space. This has negative curvature that requires background fluxes to be ]. The distinguished role of AdS = spacetime. For example, consider Type IIB with a stack of 13 EH AdS S should be identified with 0, decoupling gravity AdS ) -dimensional Ricci scalar. Comparing to the Einstein-Hilbert action d K → ( N Vol G is the d R Here we assume that we are compactifying to Thus far, the only known stable compactifications to Minkowski space require some amount More generally, the procedure to decouple from gravity is more involved than just taking Non-compact backgrounds are also interesting for the purpose of studying holography. Con- Consider the background to be . Holography tells us that this is dual to direction to become 4 5 9 , 1 + de Sitter spacetime also has similarin properties. string theory However as these we spaces will may discuss be in impossible Section to obtain where of preserved supersymmetry. Therefore, an importantduced part by of compactifying string understanding theory what is theories understanding are what pro- conditions are sufficient and necessary on compactification to Minkowski background. sider string theory in gravity. From this, it is clearQFTs that that studying are non-compact not “compactifications” coupled is to useful gravity. for studying stable. In general this requires string theory to be on a space The String Landscape, the Swampland, and the Missing Corner infinite and the theory has localis normalizable based modes on in the way that thetheory. gravity appears in the low energy effective action for the “compactified” means that and hence normalizable modes of some of theprinciple lead fields to with interesting a interacting finite quantum kinetic systems term in in dimensional manifold with positive curvature toHolography balance tells the us total curvature that in sometimespactification Einstein’s equation. with we can relate non-compactR compactifications to a true com- R of holography - that branes’equivalence interaction between with a gravity QFT leaves living an on imprintbeen the replaced on by branes space flux and which [ its leads gravitational to imprint an where the has we will ignore this subtletywhen for the simplicity. compactified theory In is general, notto we asymptotically be cannot free. able always It to decouple is decouple a sufficient from for gravity. theory See a from [ theory gravity. to be For asymptotically example free/conformal order the gravitational part of the action on non-compact geometry we have endedthe up near with a horizon compact limit one! ina which sense This the happens the D3-brane by transverse worldvolume “zooming” direction direction, in to to the D3-branes looks like PoS(TASI2017)015 Cumrun Vafa . So there is a ) 2 ( whose holonomy holonomy. These ) n 8 SU . However, there are ( ) n → ( SO ) 4 holonomy. These preserve ∼ and hence can be extended ( ) ) Spin 8 7 G . In this case, we can find an ( ( SO ) to n ) ( SO n Spin ( ⊂ SO ⊂ 2 SO ⊂ ) G 7 ( G 6 Spin 5 -dimensional manifold. Generically this action lifts to the n ) are complex 3-folds with SU(3) holonomy. These preserve 3 CY for a ) . As we will see, these will provide an important class of internal n ) ( n 2 ( SO SO ⊂ ) -SUSY. -SUSY. n 1 16 1 8 ( -manifolds are 8-real dimensional with ) SU 7 ( by geodesic translation. - the n-torus. This preserves full SUSY since the holonomy group is trivial. K 2-manifolds are 7-real dimensional manifolds with 3 surface. This is the first non-trivial Calabi-Yau manifold which is a complex 2-fold (4- n -SUSY. -SUSY. -SUSY. 1 1 8 4 2 1 T K real dimensional). In this casetrivial the direction holonomy in is the reduced spinor from bundle and hence a covariantly constant spinor. This preserves preserve Calabi-Yau 4-manifolds (CY4) are complex 4-folds dimensional withpreserve SU(4) holonomy. These G Spin Calabi-Yau 3-manifolds ( Some general classes of compactification manifolds that are useful for string compactifications Note in order to have the reduction of the holonomy group, we must have that this holds for all choices of basepoint. A special class of manifold which has a guaranteed reduction of structure group are Calabi-Yau Since we have seen that the full string theories are all interrelated by a sequence of dualities, This can be described as follows. Pick a tangent vector at any point on the (oriented) manifold 6 • • • • • • . As we parallel transport this vector along any closed path in the manifold, the vector is “pushed to 4D which preserve some amount of SUSY are: manifold for compactification. manifolds. These are Kähler, Ricci flat, complex manifold of real dimension 2 special cases when this groupelement is of reduced the to spin a bundle subgroup whichto all is of fixed by the lift of the action of is given by 1.3 Dualities of Compactified Theories one would expect that their compactifications are also related by dualities. As it turns out, these around” so that when itaction returns of to the an basepoint, element the of spin change bundle of by the spin vector representations can induced be from described the by lift the of K The String Landscape, the Swampland, and the Missing Corner to have supersymmetry for the lowconstant energy theory. spinors This on is the determined by compactificationinvariant the manifold. under number of translation These covariantly along are the globallydependent defined compact of spinors their manifold position which - on are theycovariantly the constant are internal spinors intuitively manifold. can spinors be In thatbundle. translated many are into cases in- a the statement statement about of the existence holonomy of of the tangent PoS(TASI2017)015 32. SUSY = N SUSY Cumrun Vafa N and F-theory, all ) 0 , 2 = ( N 6 ] where we step-by-step reduce the dimension by acting on 14 16. This includes Heterotic and Type I theories in 10D as well as = 32. M-theory in 11 dimensions, F-theory in 12 dimensions, and Type = SUSY N SUSY N Whenever the dimension, number of preserved supercharges, and chiralities . We have already discussed dualities among these, so let us move on to further 2 Z / 32 16 1 S = = SUSY SUSY N N A general principle which helps us in identifying dualities upon compactification is to use Let us start with We will now briefly review some of interconnected web of string dualities with definite Surprisingly, we are aware of no known counter examples. In this sense, dualities in lower “Conjecture”: Consider the case of known dualities in higher dimensions toof build a new ones sort in of lower “adiabatic dimensions. procedure" This [ is based by use in various dimensions. We will not bekey exhaustive examples. in the discussion below, but aim to illustrate some 1.3.1 Adiabatic Principle both sides of a given dualityapplications by a the geometric fibration procedure data such does aswhy compactification. not this However, vary in should adiabatically. most work Therefore,drastically since there at it is every step. is no However, rigorous not amazinglysupersymmetry reason really it is does adiabatic preserved. work in in all any known sense examples as of long the as word some as1.3.2 things change dimensional theories are notexistence hard of to dualities is find, as but Sergiovery Cecotti rather existence puts of are quantum it, systems hard “the of scarcity toone gravity of theory is prevent! rich hard with structures”. to a One arrange In given and symmetry, particular rationale there if the for is we a succeed the good to chance get more we than have landed on the same theory. IIA and Type IIB and all their toroidal compactifications to lower dimensions have of two different compactifications of stringometries theory such match, that there they are are choices dual of descriptions compactification of ge- the same physical theory. dualities with this much supersymmetry.order to We will make sense start of by F-theoryis compactifying we we F-theory. need a need Recall torus at that embedded least in in some our kind compactification manifold of - torus that fibration. We will begin by compactifying F-theory on an The String Landscape, the Swampland, and the Missing Corner relations are so abundant that we can make the following observation: Moreover, except for Type IIBthese in theories 10 are dimensions non-chiral and whose so chiralityall one is would equivalent expect once based compactified on our toboth general the conjecture the that same chiral they dimension. and are non-chiraldimension. And theories further, to we be equivalent expect when (correctly) toroidally that compactified to the same 1.3.3 M-theory on PoS(TASI2017)015 - 2 n 7- T ) , we (1.3) q 2 , T p ( ]. , which is . 9 1 ell S Cumrun Vafa ell ]. Moreover, is equivalent K 2 1 K × S 3 3 [ × K K 2 includes non-trivial . T 4 } 1 T P ∈ z 16 theory. The Heterotic and | 1 3 is dual to is dual to Heterotic ) P = z K and to Type IIA on ( g 1 S to F-theory compactified on an complex SUSY + × , it will have the same dimension and x N B 1 ) z S ell → ( f , is dual to M-theory on K o Z 3 : + T ell ell H 3 32 and hence F-theory on fibers with Wilson lines. These can further be shown to K π K x = 2 1 = 20 and in paritcular we cannot obtain arbitrarily 7 = T S 3, then we find that F-theory on 2 is equivalent to Type IIA. This gives rise to the chain ] for more details. ≤ K y × 1 { g 2 16 SUSY = S T for a discussion of ADE singularities. N = shows that Type IIA on ell is an elliptically fibered manifold. Therefore, we expect 3 rnk 1 K S K 1.4.2 ell 32 of F-theory to an 8d, K 16, but they are 10D theories. So if we reduce them on a is dual to M-theory on = = 3 1 1  16. As it turns out, these theories are dual to each other [ with elliptic K3-fibers over the same base. The physical motivation for this is the . such that S P 8 K / B ] = where g is dual to M-theory on SUSY × SUSY 1 1 15 → N ell N S S which leads to a rank 16 plus an additional 4 coming from winding and X K SUSY : × × 2 N 2 F T π ell ell T . If we compactify F-theory on K K , which are interpreted in Type IIB setup as giving rise to a system of 2 7 T 3 singularities. See Section ]: K 1 dimensional elliptically fibered Calabi-Yau 9 factor on both sides of the 9D duality over a base manifold to construct − 2 n T 18 gauge groups [ as M-theory on F-theory on F-theory on . Generically in the compactification of on = 4 These singularities have a classification given by Kodaira, which mirrors the more well known one for an ADE Additionally, there is an important family of dualities relating F-theory and M-theory. Recall As it turns out this is a special case of a more general which relates Heterotic string theory on Similarly as we discussed M-theory on One can check aspects of this duality by studying singularity structure of the K3 manifold. If we apply this to the case where g 7 8 • • T SUSY Type I theories also have Note that this reduces the on have the same moduli and low energy spectrum. See [ of dualities [ get an 8d theory with to M-theory on going one dimension down on another classification of dimensional Calabi-Yau same as compactification on the K3 fibersmatching again that give rise of to the a system Heterotic of string 7-branes which compactified has on non-abelian gauge the symmetry large rank non-abelian gauge groupsI in compactifitions this on way. This ismomentum charges compatible on with each the circle Heterotic butin we and can the Type have gauge moduli symmetry space enhancementrnk at which certain give points rise to (semi-simple) factors ofthat non-abelian F-theory and groups M-theory with have the up same to a complex fiber degenerates branes which can lead to non-abeliancompactifications. gauge It symmetry turns as out that expected compactification for onassociated Heterotic an or elliptic to K3 Type a manifold I leads toroidal Lie to singularities algebra N these theories to be dual.fiber the This can be derived by the adiabatic principle, where we adiabatically The key to the duality is that the singularities of K3 manifold, which are points where the The String Landscape, the Swampland, and the Missing Corner elliptic K3-manifold. This may be written as a torus fibration over conjectured to be the same as Heterotic on PoS(TASI2017)015 2 × = k , 5 , 1 1 (1.5) (1.4) (1.6) (1.7) k -fold. R 3 SUSY as N CY 2 8 Cumrun Vafa E , × ω -flux, that is the 1 ∧ H 8 3. Before we derive E ω . 3. By the conjecture K ∧ K ω ω  ∧ 2 3 , 2 | 3 0 ω + ˜ H | + 3 manifold become coincident ω 1 2 ) = K ω ∧ whose field strength is given by F d − ( ω , 2 2 B = ) c ) Tr , R Φ − GR 2 ) = ∇ | ( 2 Ω 4 F GR | − TM , + ( Ω Tr which break gauge symmetry. But by turning 2 R YM c 4 Φ )  2 Ω T 8 F ( − Φ -fold gives a six-dimensional theory with ) = , 2 4 1 3 ∧ 24. This means that if there are no five-branes, there ge − F F − ˘ rava-Witten construction by taking M-theory on − ∧ CY ge is the spin connection. This leads to the equations of A 0 - that is the case of compactification with five-branes. ) = √ Tr F − dB ∧ 3 x 8. This can be achieved by compactifying Type I or Het- ( ω 6= − √ A ] 10 = x = Tr TK R d ∧ 3 ( 10 ˜ dH − 2 factors in the ∧ A H represents the second Chern class. This means that if we com- [ d Z c i ) 1 R ) 3 2 SUSY R S 2 10 Z F 1 N g ( ∧ Tr − 2 2 2 10 ( R 1 c κ 1 2 ( dA − 2 Tr ∧ = = 3 A ]. S 17 Tr dH = 24 [ 8 YM = = 2 Ω k 8, we can have these 24 divided between the two factors is the 1-form gauge field and + . The bosonic part of the low energy effective action of the Heterotic string (Heterotic E 1 SUSY A k N × dB Now consider the case where We will now consider the extended example of Heterotic string theory on Now consider the case of In Heterotic string theory there is a 2-form tensor field Compactifying F-theory on an elliptic 8 = E where motion 8. This matches the SUSY and dimensions of Heterotic string theory on This case is actually more clear from the Ho must be 24 instantons livingis in the gauge bundle. Since the structure group of the gauge bundle The String Landscape, the Swampland, and the Missing Corner Wilson lines along the different these off, our compactified theory will exhibitType IIA non-abelian side gauge by symmetry. taking This is the reflected limit in where the non-trivial 2-cycles in the H supergravity) is given by where above, we expect these to be dual. Consider a compactification with only trivial on the gauge bundle, where pactify on a non-trivial manifold, we must have instantons living in thethe gauge duality bundle. with F-theory,This we manifold will has the demonstrate feature some that physical features of this compactification. where and develop singularities as above. We will comment more1.3.4 on this in the next section. erotic string theory on aNow K3 that manifold we and are Type consideringaccount IIA, for non-trivial IIB, the compactifications interplay M-theory of between and the Heterotic F-theory non-trivial string geometry on theory and a gauge we field have fluxes. to case without five-branes (sources of the B-field). This imposes the condition PoS(TASI2017)015 (1.9) (1.8) ) and 2 ] . The 12 this 8 factor Z / . 20 E = 1 , 2 S 3 1  k Cumrun Vafa . Recall that 19 P 2 K × [ ]. / = 3 T n 1 22 F K [ k 1 ± n 2 F T , then we get 1 to P n F ] for a partial review. ]. For the more general splitting . For the case 11 1 17 P [ Heterotic on 1 direction and eject from the M9-brane P 2 × Z 1 . / 1 P ⇐⇒ S 3 1  P and absorbing it on the other side is equivalent K 1 × / n 3 P 3 3 fibered over F  9 K × K CY 1 / is replaced by a Hirzebruch surface P 1 2 P T (whose worldvolume is transverse to the × 2 1 interval, there are M9-branes - each of which support an P 8. This can also be achieved by considering F-theory on an Z 2 2 / Z T 1 = / S ]. In this duality, the transition of an from one 1 ∼ = S 21 3 SUSY 1 supersymmetric theories in four dimensions. We can obtain these 1  N P = CY / the the base manifold. As we will see this can be derived from the previous subsection n N down to a point and the net effect is converting 3 which is elliptically fibered over 1 + 3 P CY 3 12 and 4D theories CY K 4 = is a Calabi-Yau 3-fold which is 2 3 = k . At the ends of the , 2 n CY Z ]. − / SUSY 1 N 18 S 12 3 is a 6D theory with Let us start with the duality of F-theory on K3 and Heterotic string theory on Similarly we can continue down to theories with 4 supercharges. The highest dimension for Now using the duality conjecture, we know that the compactification of Heterotic string theory Here F-theory on × K = 3 can be written as an elliptic fibration 3 1 k to the other has an interestingto geometric the interpretation bulk on corresponds to the blowing F-theoryto up side. blowing a Emitting another point an on instanton which this happens is theories compactifying by Type IYau or 4-folds, Heterotic or strings M-theory on on Calabi-Yau G2to 3-folds, holonomy one F-theory manifolds. another on with Needless Calabi- to suitableon say choices this we of subject expect parameters. which these is There beyond to the hasof be scope been this dual of class a the of large current theories review. and A involves growingdescription F-theory, particularly as literature powerful has it description led is based to oninvolving many relatively supersymmetric simple potential extensions geometrical connections of data. the with This standard supersymmetric model. particle See [ phenomenology gauge group. Instantons in the M9-branesin are the described gauge by theory can dissolved shrink M5-branes. toas These zero M5-branes instantons size into in the the bulk of the elliptically fibered by applying the ‘adiabatic argument’. K comparison between the Higgs branchesF-theory of side Heterotic can be side found and in its [ geometric interpretation on the 1.3.5 on The String Landscape, the Swampland, and the Missing Corner K can also be absorbed by the24 other M5-branes wall for and this dissolved compactification in to it.bulk be [ The stable point spread is out that between there the must two be M9-branes a and total the of Now if we take both sides of the duality and fiber them over a 2-sphere corresponds to a PoS(TASI2017)015 . . c 2 Y Cumrun Vafa so that it appears to c Y , we can decouple gravity even though we do 9 . We can then break supersymmetry by hand by 2 10 T , which describes the local structure of the true compact man- nc Y ]. Given the importance of local singularities we turn to a brief review ]. One drawback of this approach is that this theory will develop a 23 25 ]. 24 [ 2 . In essence we can think of this as “zooming” into a local part of c for a small enough radius of the circle compactification. This problem is unavoidable imposing anti-periodic boundary conditions for fermions.compactification This is [ called the Scherk-Schwarz Y We assume that it is embedded in some grand unified theory. An important difference between compact and non-compact Calabi-Yau manifolds is that com- At the time of writing these notes, supersymmetry had not yet been realized at energy scales Of course we need to understand the full compact geometry in order to understand the structure Given the large number of six-dimensional manifolds, we would like to know something more Since the standard model is asymptotically free 9 1. Compactify a supersymmetric theory on pact manifolds cannot have globalglobal symmetries symmetries while of non-compact the ones standard can.metries model when (decoupled This we from means take gravity) that gravity can the intoture only account, in be Lecture matching approximate beautifully sym- with the first swampland conjec- tested by collider experiments.theory model However, building. supersymmetry Therefore, is in order asomehow to figure fundamental properly out describe component real how physics to of at break string lowtime supersymmetry. energy, which This we is has must deceptively been difficult. a topic For of example, intense consider study the for following a two long cases of it later in this lecture. of the complete standard model, howeversome studying low a energy non-compact approximations version which should we be can sufficient think for of as only probing the local geometry of 1.3.6 Supersymmetry Breaking about what kinds of compactifications are allowed.a Generic large compactifications amount of of this type mattergo can fields have to depending large on number the of matterasymptotically number free. fields of in non-trivial In any homology general theory, this cycles. wecomes means run what As that constraints into we we does the this cannot problem place decouple of onfree gravity. our the theory The theory, be theory or question arbitrary? in being now other not As be- words, it can turns the out non-asymptotically the restrictions are quite strong. The String Landscape, the Swampland, and the Missing Corner It is also worth pointingto out that contradictions not with every observations local and model“swampland” consistency of of and this the hence type string should is landscapewhich permissible. be - does counted These not the as can have part being lead a that consistent in appears UV the completion to with be gravity. a This will consistent be low the energy topic theory of Lecture have gravity in our universe (andrect hence description). have a In compact order internal to manifoldversion study assuming of this just the is the internal the gauge manifold, cor- theoryifold, part of this, we can study a non-compact be so large that we cancontext think is of that it local as singularities a of non-compactaspects CY manifold. of geometry phenomenology. One play One of particularly a the promising key lessonsbe example role one geometrically of learns in encoded this in encoding [ is this and how restricting flavor hierarchy some can PoS(TASI2017)015 Cumrun Vafa this breaks SUSY. But again, ) 2 ( in string theory. Indeed there are ]. SU dS 26 6⊂ Γ space. In order to properly understand the AdS , where 11 Γ / 2 ], inside the Calabi-Yau . The anti-branes C 30 ]. This approach broadly consists in two steps: 1.) sta- 28 ]. Once all moduli are stabilized, we will add anti-D3-branes. It 29 -like vacuum with positive energy, but we are just unable to analyze it yet. ]. ] for related issues in the KKLT scenario. 27 dS 141 studying string theory in thisis background unstable we [ learn we have and hence the system cial holonomy. For examplelocal if orbifold singularity compactify of on the type two complex dimensions and if we have a because we cannot study such aattention to system only and a ignore subset the of radial the modulus, radion or moduli arbitrarily space restrict [ We are now at an impasse. It might be that the KKLT construction consistently breaks SUSY Another important facet of our modern understanding of string theory is the role of singulari- A notable proposal to break supersymmetry and reduce to our universe with positive cosmo- Consider F-theory/IIB supergravity on a Calabi-Yau orientifold with fluxes, with the simpli- 2. We can break supersymmetry by considering compactifications on manifolds without a spe- will then back react onlogical constant, the much non-compact like part how of D3-branes lead spacetime, to causing it to have a positive cosmo- should be noted that those anti-D3-branes willare not all bring other automatically moduli, stabilized as their byat worldvolume the scalars the fluxes. tip of These a anti-D3-branes Klebanov-Strassler are throat then [ wrapped on a cycle KKLT construction, one must haveintractable. full control See [ of the large number of moduli which isand generally reproduces a Since we do not have the toolsto to make analyze any the strong KKLT construction claims in a about realistic a case, KKLT-like it realization is of impossible ties. This is demonstrated in an example from the previous section where Type IIA on K3 is dual to various problems which arise in thismanifold, class of we theories. need In introducing tothat anti-D3-branes take they on do a special compact not care annihilate.anti-D3-branes, of Therefore, we having to really them analyze must well stability haveopen when separated and a introducing closed complete from string both knowledge the moduli. D3-branes ofstabilization D3-branes, and This remains the is so an moduli clearly active space, a area very of including research complex both to problem, date. as the question1.4 of Singularities moduli and Branes logical constant is given by KKLT [ fying assumption of having onlyeffective one Lagrangian, the Kähler complex modulus. structure moduli Due canler moduli to be cannot. stabilized the perturbatively Therefore scale while in order the invariantsuch to Käh- property as stabilize of Euclidean all D3 moduli the instantons one [ must add non-perturbative features This tachyonic behavior and more generallybehavior lack when of we stationary solutions try appears to tomessage. break be supersymmetry. a ubiquitous It would appear that string theory is sending us a bilizing the moduli while preservinganti-D3-branes supersymmetry wrapped on and highly 2.) warped cycles breaking inconstruction the in supersymmetry internal more by manifold. detail. adding We will now review this The String Landscape, the Swampland, and the Missing Corner PoS(TASI2017)015 Cumrun Vafa is the usual translation as a dense open subset X r ) Z ∗ ⊂ C is the zero set of a collection T = ( T B toric variety X Y 12 A can be realized as a toric variety which is a circle fibered 1 C such that the restriction to X X → torus fiber degenerates. So the fiber above points X, Y, and Z will be X : 2 T T . In this example, non-abelian gauge symmetry on the Heterotic side 4 A-cycle respectively. 1 T S , and 2 T ]. In this figure we illustrate the idea of toric varieties. Here the two boundaries (labeled A, B) are This figure is a representation of how 31 B-cycle, full A natural way to encode the topological data of a toric variety is by realizing them as torus Toric geometry is the study of toric varieties. A 1 S Figure 3: over the positive real line. is dual to the K3 manifoldplay an developing integral singularities. part It of understanding is these clearprinciple physical from that theories. this This singularities that is give the just rise an singularitieslocal example to must singularities of can interesting the lead general physical to a phenomena.This deeper in understanding Moreover particular of understanding has quantum led systems, to decoupledin insights from about gravity. up the to existence six of non-trivial dimensions.branes, interacting it quantum Before is systems we helpful to demonstrate do some a of quick the review of interplay toric1.4.1 between geometry. singularities Lightning and Review of Toric Geometry of complex polynomial equations thatand have has an a natural algebraic action torus action Figure 2: where the A- and B-cycles of the Heterotic string theory on the The String Landscape, the Swampland, and the Missing Corner action [ PoS(TASI2017)015 1 3 + C S R (1.10) Cumrun Vafa fibered over A A 3 B T and is related to the B where the number of i A-B (a) (b) σ B i forms a closed polyhedra, B A+B r ∏ toric fan B -coordinate) where the radius = ∂ A r r A ,( B + ∂ R . When 2 fiber. at the origin (right) where the degenerate as , 2 1 X T in the usual way. This allows us to realize r  CP X ) B B / θ , r 13 ( . See Figure i as 3-perpendicular copies, that is a r σ 3 r T B A C . 3 (left) or by gluinga where the radius of the different cycles of the torus are determined 3 . This can be realized as a plane where the complex coordinates S 3 C R degenerating at each end. This can be resolved in two different ways which fiber degenerate. This means that the preimage of a line running diagonally 1 2 . See Figure S r is given by codim T i σ -coordinate) fibered over the positive real line, θ 3 S fiber, we can realize the base of the fibration as a polyhedra where different cycles of This figure shows the two possible resolutions ( left and right) of the (center). In the r T can be exchanged for the real coordinates In this way, we can similarly realize Consider the example of ) ]. ¯ z , z 32 which looks like an octant in by their distance from the 3 axes. corresponds to resolving with an fiber is either the (a) A+B cycle or the (b) A-B cycle in the Figure 4: conifold, the vertical and horizontal(A- line and represent B-cycles) the in locusfrom the in one the axis to base the whereperpendicularly other two embedded in perpendicularly the cycles base represents a 3-sphere since it is a 3-real dimensional manifold with a with a the fiber degenerate along the different boundary components we can encode its data inway the dual in graph which – branes this arise dual in[ graph the is called Type IIB the description of M-theory compactified on a toric variety as a a circle ( ( The String Landscape, the Swampland, and the Missing Corner fibrations. Since we can realize a generic toric variety degenerate cycles on of the circle is given by PoS(TASI2017)015 n n n Z Z / TN 2 , (the ] for n = C (1.11) can be 33 Γ TN , 1 31 blowing up CP , Cumrun Vafa gauge theory 32 ) n ( . More generally 1 SU CP . The gauge theory then . The difference between n is T-dual to Type IIB with 12 , and we can choose a basis for Z n n . This / Cartan matrix. Z Z 2 11 n / ] / C 2 A , 2 \ C 35 1 C , , n × = Z for an example. See [ 34 5 / , n S 2 4 1 α is a finite group. These generally have C R ) 2 ]. There is however, a version of mirror symmetry Dynkin diagram ( T I 1 37 2 2 , O , − n ’s which can be glued together in interesting SU n 2 1 36 = = = A ⊂ BD Γ Γ Γ CP n 14 Γ gauge theory [ !+ Z = ) 1 type singularity in detail. Consider the group n = − Γ 0 ( 1 α − Γ n where SU 0 α A Γ / 2 * * C = Γ ]. different spheres in the fully resolved 38 such that their intersection matrix is given by the s that intersect as in the 1 1 ) n − P n . These are given by Z / 10 2 , the circle fiber asymptotically approaches a finite so that T-duality is well defined whereas the \ C n ( 2 is the is the s the H is a binary , is a binary dihedral group, TN fibered over a finite line segment where the fiber degenerates at the two ends. Using Γ Γ Γ diverges as the distance from the singularity. Therefore, T-duality is not technically defined in NUT centers) which locally has the same singularity structure as Γ Γ 1 n n S . Roughly speaking, Type IIA string theory on Z 2 / 2 C is that in C n Z D-type: A-type: -type: -type: E8-type: / 2 Technically, there are 2 That is they have a classificationThere which is is a identical slight to subtlety that associated of with A-,D-, and this. E-type We Lie technically algebras. need to have a singular Taub-NUT space, Let us study the case of K3 singularities in more detail. The allowed singularities of K3- An important feature of the description of toric varieties as torus fibrations of a base polytope • • • • • C 12 10 11 fiber of coincident D5-branes giving rise to a 1 the homology group which work for this case as well [ can be seen in the Type IIAresults side in by a blowing up collection the of singularity: completely resolving the singularity as one cannot shrink the asymptotic circle to apply T-duality [ S more details. 1.4.2 The McKay Correspondence and Theories of Class singular limit of and manifolds are locally of the form ways to form more complicated resolutions, or by other manifolds. As a toric variety, is that it has a clear geometric interpretation of blow ups. The geometric operation of The String Landscape, the Swampland, and the Missing Corner is a method of resolvingsense, a singularity the by most replacing fundamental a blowsingularities singularity up with can replaces a be a smooth replaced manifold. singularity by with In a a some collection copy of of an ADE classification which is defined by the generator realized as a this, the fundamental resolutiondegenerate on of the singularity base with simply adegenerating replaces cycles line at segment the the where unresolved singular the singularity. fiber point See is Figure where given by cycles a linear combination of the To illustrate these we will consider the acting on n PoS(TASI2017)015 , , st ]. 51 44 with , , leads 41 . (1.12) ) ] as in 4 , 1 50 46 0 [ R , P , 39 40 2 where the × 49 M5-branes S 2 Σ , Cumrun Vafa is the Jacobian = ( N ) 48 X o Z , quotiented by the ( , ) n J N is a multisheeted -fiber degenerate. 1 X 2 Z 47 Ω C T ( ∼ = 0 5-plane on n O H . Here → 2 , C X Σ ]. In addition, this gives a → BD Σ 46 : → , 2 theories of class n π by compactifying 41 Z cycles of the = , ) 13 1 q from , 40 N − ) [ p N C ( ( A C J = → →  ) ell g B Σ Σ K ( type theories by wrapping an / J n : 0 D 15 π = g 2 ]. Compactifying these theories on a Riemann surface T . By the universal covering construction, this space has the same 1 ) 45 7-branes where the C , ) ( 1 q 44 , H ] for a more general review. p , , ( X 44 43 with the same topological twist, where , , . 4 associated to the map 58 )) with type theories in a similar way. This corresponds to having orientifolds R 42 , 14 X n B ( × J 57 D ( , 1 Σ singularities, this leads to a six dimensional theory with H 56 g Γ ) = cycles corresponds to a choice of S-duality frame for the Type IIB theory. These / X 2 where ) ( 1 q C , ell H . As we will now explain, this curve can also be realized in Type IIA as the geometry axion, charges and can give rise to non-abelian exceptional groups as studied ]. See [ p ) K ( 55 q . This derives the data of the Seiberg-Witten curve and Seiberg-Witten 1-form, where which is defined as the quotient of all global holomorphic differentials on B.3 , , C 2 supersymmetric systems in 4 dimensions. Using this picture and applying local mirror p X ( 54 conformal systems [ ]. These theories can be constructed for type = , g The Prym variety is the kernel of the induced map Note that this construction can be generalized to The relation between singular geometry and branes is also evident in F-theory. If we take One can then fiber these geometries as in the adiabatic construction and obtain systems with We can also get the If instead of Type IIA, we consider the compactification of Type IIB string theory on a K3- 41 53 action acts as an orbifold. Understanding the interpretation of E-type singularities as branes in 14 13 N , , (with a topological twist) produces the four dimensional 2 space of non-trivial closed 1-cycleshomology on group variety of clear geometric way to study thevalues BPS of spectrum, supersymmetric line operators operators, in surface the operators, associated and four expectation dimensional theory [ this is dual to Type IIB on F-theory on branes are non-perturbative because theyaxio-dilaton are has co-dimension log-type 2 asymptotic behavior. and Into hence their general the these magnetic branes dual are of mutually the non-local due 1.4.3 Branes in F-Theory lower supersymmetry in lower dimensions. In particular fibering these geometries over the curve the Prym variety 52 Z Type IIB requires the use ofIt F-theory is since known the in branes have the dogauge M-theory theories. not setup have or a perturbative the description. Type IIA picture that E-type singularities lead to E-type Appendix 40 on a Riemann surface cover of A choice of in the Type IIB picture and in the geometric picture by replacing of intersecting D4/NS5-branes, which lift to M5 branes in the M-theory lift of Type IIA [ manifold with The String Landscape, the Swampland, and the Missing Corner comes from D-branes wrapping theseall non-trivial become 2-cycles effectively which coincident, upon giving going rise to to a the non-gauge singular symmetry. limit, to symmetry, the Seiberg-Witten curve of these theories can be identified in string theory [ type C PoS(TASI2017)015 × → 4 ) , 1 2 (1.13) R − , 2 − ( ) where we Cumrun Vafa ˜ O B ( : 1 P 15 × 5-branes on 1 fiber, we get a Type ) singularity) by lifting P q 2 n , ]. T p Z . ( / 2 32 2 C NS-charge C q fiber via its dual graph. and 2 2 ˜ T C -cycle. This generalizes the relation ) and ) is an elliptically fibered 2 1 ˜ , C ) K 0 2 ( 1 fiber can be represented by the dual of the − P 2 ,  2 T × where 1 − and similarly for ( 3 P ell B R-charge ( O 16 / K CY direction and form the generically co-dimension 1 p × 5 , -fold given by the affine cone over 6 1 , 3 1 R 2 R CY T -cycle and T-dualize along the ) . 0 5 , 5-branes which have 1 ) ( q , the singular structure of the the , 1 p P ( direction as shown below: × ]. Consider M-theory on 1 2 web 60 P where the 5-branes wrap the R represents the magnetic dual of the form field This is the toric diagram for the ˜ 3 B R . This diagram encodes the data of the singular structure of the 1 ]. One may worry about having these charged objects in a compact space because branes × P Another example of the correspondence between branes and geometry is the construction of This means that after dualizing to Type IIB, we have a system of Here Let us consider an explicit example. Consider M-theory on the non-compact space coincident D6-branes in Type IIA to Taub-NUT geometry (with 59 15 n × ] and thus lead to realization of non-trivial stable charged objects in a compact space. 2 web 1 9 web in the R brane webs [ P toric diagram in Figure Figure 5: IIB theory with On the base 1.4.4 5-Brane Webs . If we compactify and T-dualize on a pair of orthogonal cycles on the source flux and ascharge we in a know compact from space as generalproblem the because theorems flux the has of 7-branes nowhere general source to[ non-abelian go. relativity flux However, we this which cannot F-theory can setup actually have avoids cancel a this with net themselves The String Landscape, the Swampland, and the Missing Corner in [ compactify along the of to M-theory and the relation of Taub-NUT to NS5-branes by T-duality [ PoS(TASI2017)015 Cumrun Vafa ) ) 1 1 , , 1 1 ( ( ) 1 . , 0 π ( 2 YM 4 g = 5 ) ) x 0 0 . , , ∆ 17 1 1 ( ( ) 1 , 0 ( ]. Although it is unknown if all of these compactifications are 61 ) ) SYM where the displacement between the D5-branes (vertical [ 1 1 , , string landscape ) 1 1 2 ( ( ( 272000 SU 2 = N ) gives the Higgs vev and the displacement between NS5-branes (horizontal dis- is at least 10 6 x ) determines the gauge coupling as D 5 x ], there is no compelling solution yet. Due to this huge number of possible choices involved in constructing string vacua, there has One can also combine branes and singularities, by bringing branes to probe singularities and In the previous lecture we briefly reviewed the string theory construction of many low dimen- It has recently been estimated that the number of possible consistent flux compactifications of To complicate matters even more, even if we were to be able to enumerate all of the distinct, In essence, this brane web depicts two D5-branes suspended between two NS5-branes. This 63 , . 62 been a distinct philosophical shiftidentifying “the in correct” string the vacuum community has shifted overone. from the using Instead a past of top-down decade. approach starting to withto a The fully-fledged bottom-up 4D, attitude string many towards theory have andphenomenological studying started features the studying (such compactifications effective down as four supersymmetrictrying dimensional to extensions couple quantum of them the field to gravity. standard theories The model) with common and lore nice then is that because the string landscape is so large, distinct or may be dual descriptionsthat of perhaps the the direct same study theory, of this allstring string large vacua backgrounds number is is of futile. called string This the vacua remarkably suggests large space of inequivalent placement obtain new and interesting quantum systems. A brief review of some examples is given in Appendix 2. Lecture 2: The Swampland sional effective field theories. Asusing string we theory saw, there for model isbackground building, a coming fluxes, very from branes, large the number and choice of etc.particular of choices string the theory Therefore to compactification solution, a make manifold, among when very the relevant enormous question set of is possibilities, how describes toF-theory our to universe. identify 4 which consistent string backgrounds, therechoice is over another. no For known example, top-down whatpact? mechanism forces Why four to did dimensions nature prefer to choose be onenumber the extended specific particular which and string can six background to describing be be[ our constructed com- universe from in the the vast theory? While there have been some suggested ideas displacement A describes the 5D The String Landscape, the Swampland, and the Missing Corner PoS(TASI2017)015 Cumrun Vafa swampland criteria as we started with non-zero G coupled to gravity must have states with ) 1 ( ]. However, it has been conjectured that all these other ]. The rest we believe are ultimately inconsis- 65 since there is no imprint of global charges on a , we will necessarily demand that these theories ]. 24 G 18 16 66 . ] it will do so by emitting particles which carry equal . In the spectrum of the EFT we will have states charged 70 G swampland ] or even loop [ 64 ]. 69 , 68 , 67 , The only way to avoid this seeming contradiction is by forbidding any theory of quantum One could make the argument that there are other theories of quantum gravity different from string theory, such as The motivation for this criterion relies on physics. Suppose we have an EFT coupled It is therefore crucial to understand if a given EFT coupled to gravity lies in the string landscape In this lecture, we aim to argue that this way of thinking is incorrect – that not all consistent 24 16 theories can be attained as a special limit of string theory[ gravity from having global symmetries.about this Remarkably, it criterion, appears as in thatis all string true examples theory because we already usually know, knows global allsymmetries symmetries global of in symmetries the EFTs extra are obtained dimensions, actually but by gauged.compactification such string manifold symmetries compactification This are are arise part gauged since of from diffeomorphisms the of gauge the symmetry of gravity. 2.2 All Charges Must Appear A consistent effective field theory with gauge group U Vasiliev higher spin theories[ number of positive and negative charges under charge and all the charge hashas disappeared come after out. the evaporation of the black hole and no net charge black hole. This process would then violate charge conservation in to gravity which has a global symmetry arise from some string theory compactifications [ which will identify if anconjectured, EFT minimal admits criteria a that string allowslecture us theory we to UV plan exclude completion to a or briefly theory discuss not.rather from ten will the swampland Thus provide string criteria. physical far, landscape. reasoning We we will based havequnatum In be on gravity a this unable realization to to in motivate each provide string of proofs, theory the but on and conjectured [ general criteria. facts The about criteria we present here are based 2.1 No Continuous Global Symmetries An effective field theory coupled to gravity cannot have (continuous or discrete) global symmetries. under such global symmetry.hole. Now The send information a of this state global chargedhole symmetry is under evaporates lost via a by Hawking the global no-hair radiation symmetry theorem. [ inside Thus, when a the black black or the swampland. In order to do so, we would like to identify a complete set of tent. In analogy withstring the theory UV string completion landscape, as we the will call the set of all EFT which do not admit a looking EFTs can be coupled consistentlythe to only gravity known with UV a complete UV theory completion. of Since gravity string theory is The String Landscape, the Swampland, and the Missing Corner it is likely that anygravity consistent can arise looking in lower some dimensional waystring effective from theory field a pretty theory string much theory irrelevant (EFT) for compactification. coupled phenomenological Indeed to questions. this idea would make PoS(TASI2017)015 , ) 1 N ( (2.1) U which ) d ( Cumrun Vafa max gauge group. ) N ( SU ]. Suppose we have a is non-compact and therefore 71 N Z / 2 C . G A 4 19 = . By Hawking’s formula, we know the black hole Q S . 17 charge of the states in that sector. Now, if the theory is not coupled to ) 1 massless gluons in the spectrum. At this level there is no bound on 1 ( − theory coupled to gravity. We will have charged black hole solutions of 2 U ) N 1 . ( Z U . We saw in the first lecture that in this way we can realize an N Z / 2 C This argument is valid as long as charge is large enough so we can truly interpret the object as a black hole. In Consider a A nice discussions of this condition for swampland is given in [ The motivation for this criterion is based on supersymmetric examples. Massless scalar fields Remarkably, string theory seems to “be naturally aware" of this fact, and seems to have ways Notice that this criterion rules out many simple quantum field theories, as for example pure 17 other words, as long as the area of the horizon is much bigger than the Planck scale. Einstein’s equation for any integral charge gauge symmetry in the EFT. Thefor Hilbert each space value of of the the gravity, EFT it will is be possible split that into theeven different maybe spectrum sectors, contains no one only charged states statesgravity. of at some all. specific subset However, of this charges, cannot or happen once the theory is coupled to entropy is given by There are therefore in a lower dimensionalwe EFT compactify are on generated a inmensional Kähler string EFT, manifold theory the in by number of order compactification. scalarsof to will For the preserve generically example compactification some be if proportional manifold. supersymmetry toupper-bound in specific on For the the Hodge the lower possible numbers case hodge di- numbers, of even compact though there CY is manifolds, no there prooffor of seems preventing this. to us be to an getscalar consistent fields. lower An dimensional easy EFT exampleIIA with in on arbitrarily which large we number can see of this light at work is the following. Consider Type The String Landscape, the Swampland, and the Missing Corner arbitrary charge Such an entropy mustmicrostates. have As the a black statistical hole is mechanicsstates charged, should such interpretation exist microstates in as must the be a spectrum charged. sum Therefore all over charged the black hole depends only on the number of spacetime dimensions d. which we can take as largeis as in we no want, contradiction therefore with having thegravity an conjecture is arbitrary stated decoupled high above, in number since the of EFT. gluons. This Maxwell theory coupled tocharged particles. Einstein Note gravity. that at this Suchmany level charged a we states did which theory not should say must belong anything to also about the the spectrum. have masses (perhaps of this massive) 2.3 infinitely Finite Number of Massless Fields A d-dimensional EFT coupledMoreover, the to number of Einstein massless gravity fields is must bounded have from above a by a finite certain number number N of massless fields. PoS(TASI2017)015 . 2 2 = = M (2.2) ∈ N N 0 Cumrun Vafa with no potential. N , therefore putting an zx 2 supergravity may be- ,..., 1 20 = graviphoton in the gravity = ≤ ) i 1 , N ( i N ]. Φ U 72 pure for more details. 1.4.2 . 18 T > ) . p 20 0 , 0 gauge theory to dynamical Einstein gravity, we need p massless scalar fields ) ]). This theory has a ( d N and p, computed by using the metric as the N ]. See Section 24 ( 0 15 SU . In the EFT, the kinetic term for those scalar fields typically M such that of vacua (if non-trivial) is non-compact. In more detail, fix a point p M ∈ M into a compact manifold, i.e., K3. Quite remarkably, what happens in this case is moduli space p N is the distance between p ∃ , Z ) the / p 0 i 2 , i 0 > C Φ p h ( T ∀ is related to geometrical quantities in the compactification manifolds such as for example the To elucidate this criterion, we first need to discuss what we mean by the moduli space metric. This argument was put forward in discussions of C.V. with S. Cecotti and T. Dumitrescu. Notice that this criterion puts a lower bound on the number of possible scalar fields, therefore An interesting consequence of this conjecture is that 4D Note that since we are assuming only Einstein gravity, we exclude more exotic gravity theories However, in order to couple this i 18 i Φ where d Consider an EFT coupled to gravity, with length of the geodesic passing through p and p volumes of some cyclesvarious or their shapes. We will call the algebraic variety parametrized by the pure supergravity from string theory has failed so far. Then Such scalar fields arise generically inh string compactification, and their vacuum expectation values 2.5 The Moduli Space is Non-Compact The moduli space multiplet. Since all chargesobservable, must and be by in the the above spectrum, conjecturefor it the supersymmetric reasons should coupling must be constant be the massless). of expectation Butsupergravity. the value we This graviphoton of have is no a is consistent massless with scalar scalar string field field theory, in (which in pure the sense that attempts of reproducing being complementary to criterion numberin 3. string theory. Again For the examplesuperstrings motivation M-theory is would in that naively 11 this appear dimensions appears toconstant, has to but have no upon be a closer free true inspection free parameter. one parameterof find 10 a out given dimensional that scalar by the field coupling the called constant the choiceeffective is dilaton. the of theories When expectation the whose we value compactify coupling parameters to getwhich go again to related can lower to dimensions be we viewed the end as internal up part with of geometry the of dynamical the degrees of compactification freedom of the theory. 2.4 No Free Parameters A consistent EFT coupled tothe gravity Lagrangian must should be have viewed no as free the parameters. vacuum expectation Every value parameter of a entering field. in long to the swampland (conjectured in [ with infinite number of massless fields such as Vasiliev theory [ The String Landscape, the Swampland, and the Missing Corner upper bound on the number of gluons [ to embed that in order for the compactification to be consistent, it must have PoS(TASI2017)015 (2.5) (2.6) (2.7) (2.3) (2.4) , there ∞ such that → Cumrun Vafa and any real M , which is the T τ ∈ M ]. ) = p p ∈ finite, the geodesic 67 , 0 0 0 p τ p ( will then be given by ˜ R correspond to singular points will have some non-compact of the circle. The Lagrangian for M M , that is as d R , 0 while keeping ... , ) 0 + , ∞ ... i R j ( ) + 0 to be extremely large [ Φ → τ 2 µ log . T in the moduli space, and consider the length τ ∂  i Im 0 T − / τ R α ) Φ to some other point dR τ ˜ − µ R 0 e ( . So in general ∂  21 Im ) R T ( ∼ Z log Φ ]. ( m = log = i j 67 g ∼ R e f f dR = T ˜ L 0 R R e f f Z L . Now, fix a point is larger than ) 0 . We can use this metric to compute distances in the moduli space, Z p , . When we take the limit 2 M ( τ 0. The distance from and to SL . In the limit of infinite distance from p > is a finite manifold where the infinities of 0 p τ M ˆ T ∈ M 0 . This theory has a modulus, which is the radius is compact or not. As it turns out in all known examples from compactifying string 1 0, we will always be able to find a (in general not unique) point R S > M is the metric on , so that and pick a ˆ T 0 i j M g R . Now going to infinity corresponds to going to a singularity where generically extra massless ¯ We can illustrate this criterion in a very easy example. Consider the compactification of a EFT In the previous criterion we saw that for any choice of a starting point Heuristically, we can understand this by considering the point compactification of moduli As an easy example in which this conjecture is realized, we can consider here the case of the Fix a point p M degrees of freedom appear. on a circle number the distance between directions. We want toalong ask one now of what those directions, happens or when equivalently we we take go extremely far away in moduli space of this scalar field, in one lower dimension, will be given by Now, let us see how thejust 1-dimensional, criterion and 5 we and can 6 seeradius immediately apply that to there this are case. infinite distances. In For this example, example, fix a the moduli space is space, will be a tower of states in the EFT whose mass decreases exponentially with T, of the geodesic from moduli space of IIB supergravity. In this case, there is only the axiodilaton modulus The String Landscape, the Swampland, and the Missing Corner takes the form where and ask if theory, the moduli space is non-compact [ combination of the string coupling constantfundamental and domain a of RR 0-form. The moduli space which will be the and we clearly see that this distance is logarithmically divergent. 2.6 New Physics from the Boundaries of Moduli Space length will be approximately PoS(TASI2017)015 (2.8) (2.9) (2.10) Cumrun Vafa but instead of going to 0 R . M , , . R T dR ˜ 1 R − 0 to make this distance as big as we want. So the e R ˜ 22 ˜ R ∼ R Z ∼ 0 m m → ˜ lim R A schematic picture of the moduli space. and therefore we see that we have some fields with mass ) ˜ R ( Figure 6: log grows to infinity, we will have Kaluza-Klein modes, with mass given by ˜ R ∼ T Those extended objects can be for example membranes, strings, etc. Therefore, by this crite- rion one can argue that quantum gravityin cannot the be same a easy theory of example just we particles. used. We will Pick now now show this the same reference point On the other end, which get exponentially light when we go to infinity in This conjecture also implies thehave remarkable fact extended objects that in a its consistent spectrum. theory of quantum gravity must larger radius go to smaller andmoduli space, smaller as radius. We also find that this is another infinite distance in diverges. Therefore, due to criteriongetting 6, exponentially we low. also However, expect such to states have in cannot this be case particle some states states because with all mass the particle and we see that we can always find a suitable criterion number 5 is satisfied.work. Let As the us radius consider the limit of very large radius, to see criterion 6 at The String Landscape, the Swampland, and the Missing Corner PoS(TASI2017)015 . i i Γ g h as a ∈ (2.11) (2.12) γ = h Γ Cumrun Vafa ]. These fixed 67 will be a simply can be decomposed M . If we take Γ with an element -branes. This powerful has no non-trivial loop, Γ p M T ⊂ γ gauge theory. We already know , since ) 1 . Consider now two of these objects ( M q DU . . ] 0 Γ 67 [ / ) = Γ T ]. Here we will present a particular version, just 23 . But since each element of = M 77 ( ] and recently recieved much interest as it is able to Γ , 1 must also be contractible and thus π M 68 76 γ , has a fixed point in , we can identify each loop i 75 g Γ , has fixed points. This means that we can decompose i 74 g ) = , Γ by a group action / 73 ˜ . r T M ( (WGC). There are many inequivalent and more precise versions of this 1 is generated by group elements which act with fixed point [ π ) where each i g Γ where each i g i ∏ have extended gauge symmetries given by the stabilizer in = h M In all known examples in string theory, the moduli space is obtained by quotienting a con- In a consistent EFT coupled to gravity, gravity must always be theThis weakest conjecture force. applies to charged particle states as well as charge product of paths. Eachthe of fixed these point. path components Therefore are we contractible have that since they can be unwinded at Generally the only way to getto one elements is with by fixed the point,the action usual it of identification implies that each loop can be contracted. This is because under (group generated by points in to give the reader thefrom conjecture main 2 idea. that we Suppose needthe to we lightest have have charge charged states a state, in and 4 thetogether, suppose placed spectrum it at of has distance the positive theory. charge Consider then put constraints or completely ruleweak out gravity different conjecture large-field inflation models. This is the so called tractible Teichmüller space conjecture, see for example [ In every known case, connected space. 2.8 The Weak Gravity Conjecture conjecture was originally formulated in [ The String Landscape, the Swampland, and the Missing Corner states will be given by KKradius modes, limit. and those The KKextended modes only objects will which way be can we instead wrap around very can theradius massive circle get limit. in which the light So thus small if become objects lighter our as inin theory we does this the go not limit to small have and the extended small radius we objects,M-theory therefore limit, we when violate do we is the not compactify criterion to have on number any the havegive light 7. circle: some rise states This The at to M2 is all the branes for wrapping light exampleexistence the what string of circle happens extended states become objects in light in in and 10 a consistent dimensions. theory of We quantum thus gravity. 2.7 see that The this Moduli Space conjecture is implies Simply the Connected The first fundamental group of theof closure moduli of space the is moduli simply connected: space is trivial, and therefore the closure PoS(TASI2017)015 0 Q < (2.16) (2.14) (2.15) (2.13) 0 M and positive Cumrun Vafa M ). 2.13 . R N + 2 R . For this black hole to decay via Hawk- P Q 1 2 . . = 1 , = q . There will also be an attractive gravitational 2 1 R M − 2 2 ≤ P r q 2 L − 0 and then P  24 L = ∼ p = 2 N = e m M R m F 2 + . For this to hold, it must happen that the lightest state N e  2 m L F : it is always upheld in string theory constructions. For P 1 2 about the WGC, and prevents us from violating it. ≤ g -dimensional torus. We have an equation relating the allowed = F d 2 and we have that this is the analog of the strict inequality in the m knows 2 1 2 L P a posteriori in the weak gravity conjecture. Consider now the non-supersymmetric Q = M 0. We now have , then when the black hole radiates a particle it will inevitably have q = L > . The claim is that N 2 . The extremality condition implies m r 2 2 p Q m M ∼ g after. Therefore it would violatethus the also extremality violating bound, the developing cosmic a censorshipspectrum conjecture. naked contains singularity The at only and least way one out particle is which to satisfies assume the that bound the ( example, we could try to violatesmaller, this as conjecture by we making know an thatgeometrical internal manifold for size smaller example of and KK the massesclose cycles to are of violating proportional the the to conjecture, compactification but theextra then manifold. degrees inverse the of of size In of some freedom the this becomeappears extra way that light dimension string one is and theory so can this small description get that the breaks down. In some way, it stronger than the gravitation attraction among them. decay. So let us consider the example at an extremal black hole, with mass charge ing radiation, it has to emitwe particles. have But suppose now that for all the states in the spectrum F As we will now show, the (-1) in the left moving sector is related to the inequality in the WGC. In order to illustrate this criterion at work, we consider now an illustrative string theory ex- There will be a repulsive electric force 1. It is true in our universe. The electric repulsion among two2. electrons, forAnother example, motivation is is much 3. Another motivation for the WGC is the fact that all non-BPS black holes should be able to Again, the charge is givenweak by gravity conjecture. Consider first supersymmetric BPS states. which is the analog of states, in which ample. Take Heterotic strings on a masses of string excitations with momentum and winding numbers force in the spectrum satisfies The String Landscape, the Swampland, and the Missing Corner We can motivate this conjecture as follows: PoS(TASI2017)015 Cumrun Vafa dual is known to ]. ], with infinite towers Maxwell theory, with AdS 69 ]: ) 64 1 ( . 69 19 U . The 4 dual, and for that we need to T dual AdS AdS ’s. This case is supersymmetric holography. In 4 T 25 ] or higher spin Vasiliev theory [ 80 , 79 . More precisely in order to find a weak coupling holographic dual we need to 4 T × 3 ]. It was found that if you couple Einstein theory only to a S 78 × 3 Non-supersymmetric AdS/CFT holography belongs to the swampland [ The motivation for this criterion is very simple. We typically get holography by putting branes To illustrate this conjecture let us look at holography in the context of 2D CFT’s. Consider Another piece of evidence that supersymmetric holography with finite number of particles is in the swampland can A very nice application of the WGC was recently discovered in the context of cosmic cen- The equality sign in the Weak Gravity conjecture holds if and only if Let us say immediately that this radical sounding claim is not saying that non-supersymmetric A natural question to ask now is for which states the WGC bound is saturated. The answer is 1. The underlying theory is supersymmetric 2. The states saturating the WGC bound are BPS states. AdS 19 principle, one could think of doingfor the example non-supersymmetric taking analog symmetric of exactly productlimit this of as construction tori by a without CFT. fermions The which problem does arises exist when in we the try orbifold to find the be blow up the singularity associated to coincident perform the blowup of themarginal singularity. deformation of But the as CFT and can soand be we so cannot readily we blow checked will it not the up find and blow a we up are weak modes stuck coupling with are non-supersymmetric the not singularity next to each other in stringis theory that and if then the by branes taking are thedue not near to supersymmetric, horizon WGC. then limit. the In However, repulsion this the between case problem is branes there simply wins is saying over no that attraction way in to the keep non-supersymmetric the setup, branes those close branes to will each repela other. and sigma The fly refined model apart! WGC with target space given by symmetric products of be seen by examining thenot SYK apply. model. However, In we this can caseinfinite ask spectrum there if so is there that an the is model infinite some would tower way violate of the to massless swampland adjust conjectures. states, the So so potential far this in all conjecture attempts the have does SYK been unsuccessful. model in order to truncate this sufficiently strong background electricthe field cosmic you censorship conjecture. can However, develop givenstates) conjecture naked and 2 singularities, the (that WGC thus there are we violating such electrically deduce strong charged that electric there fields. must Taking alsothe this violation be into of account light cosmic resolves enough censorship. the particles naked to singularity be and produced thus by avoids 2.9 Non-Supersymmetric AdS/CFT Holography belongs to the Swampland of particles which do not lead to ordinary theories of gravity. We will not consider these cases. sorship [ holography does not make sense in general.raphy The claim does is not that non-supersymmetric make AdS/CFT holog- sense,versions of provided AdS/CFT, like we in have SYK a [ finite number of particles. Indeed there can be given by a more sharpened version of the WGC, which is the following [ The String Landscape, the Swampland, and the Missing Corner PoS(TASI2017)015 ] ]. dS 86 87 does (2.17) dS . 8 ]. This is vacuum is Cumrun Vafa 69 dS , ]. This places restrictions on 81 SUSY is allowed vacuum is not possible. Of course dS . ⇒ 7 Is Non-SUSY allowed? (2.18) = vacuum is completely unstable, and slowly ⇒ 0 = ]. In this class of models the vacuum by introducing extra ingredients such dS < 85 geometries in a consistent theory of quantum ]. , Λ 0 83 AdS 26 28 geometries in 3D [ > AdS = Λ belongs to the swampland. There are a number of no-go ⇐ AdS dS = does not exist, as it is impossible to define a supersymmetry ⇐ ]. Could it then be possible to have non-supersymmetric dS 84 in string theory: in string theory. For example, an argument by Maldacena-Nunez [ dS dS is of course possible. This can be summarized as: , as in KKLT, LVS, etc. [ dS AdS SUSY is not allowed Non-SUSY is not allowed is the cosmological constant. and quintessence, however it seems that all known examples from string theory which are Λ as anti-D3-branes to make thecosmological cosmological constant takes constant the positive. form given The in scalar Figure potential for the obtained from some uplift of a previously rolling down to the Minkowski case. The potential takes the form given in Figure One may wonder if the opposite situation happens for positive cosmological constant. In this We may conjecture that metastable dS space does not exist as a consistent quantum theory of gravityWe and it have belongs seen to the that swamp- non-supersymmetric AdS/CFT holography lies in the swampland, while The lack of stable non-supersymmetric Answering this question looks difficult for many reasons. There are typically two different . dS ]. The implications of these constraints on neutrino (and Higgs) physics upon compactification 2. Quitessence models. In this class of model the 1. Metastable 82 It is possible therefore that quintessence models are the only ones allowed in string theory. This is shows that in M-theory without strongthis curvature background does not prove it fororientifolds all or backgrounds. higher For stringy exmaple, corrections. itcould may Nevertheless, possibly as this be and well avoided the be by many considering taken similar as no-go mild theorems evidence supporting this last Swampland Conjecture that theorems for constructing case, we know that supersymmetric neutrino mass types and[ ranges. Interesting extensions of this have been recently considered in where algebra in a de Sitterrealized from spacetime string [ theory? not exist as part offor any consistent quantum theory ofcomputationally gravity. under control Even are though of the we quintessence can type. write For the a recent EFT discussion of this see [ land supersymmetric 2.10 dS and the Swampland has also been studied recently in more detail in [ The String Landscape, the Swampland, and the Missing Corner gravity has interesting phenomenologicalrelated implications to related the to fact that neutrino3D depending physics on one [ the may neutrino obtain mass non-supersymmetric types and ranges upon compactification to ways in which we can get PoS(TASI2017)015 Cumrun Vafa T T 400 400 as another motivation case, this is quite hard 20 . dS . dS dS 350 300 does not exist! Arguments based on dS 300 250 200 27 200 The scalar potential for an unstable The scalar potential for a metastable 100 space have been advocated by L. Susskind dS 150 Figure 8: Figure 7: 0 V V 100 0.4 0.2 0.0 1.2 1.0 0.8 0.6 0.4 0.2 0.0 1.2 1.0 0.8 0.6 also defines a time scale, which is about 100 billion years. Why the current age of the Λ Private communication with C.V. This is of course a conjecture, but it is physically well motivated. For example, we live in 20 a universe right now whichconstant is about 14 billion years old. The current value of the cosmological the motivation for our last criterialack of for holographic the duals swampland for that for their lack of existence. universe is so close toto the explain, since Hubble the scale? metastablemodels vacuum If it can we be could are extremely more in long-livedalways naturally whereas the in be in metastable a the of runaway quintessence the situation,present same value. and scale. there If this is Maybe is no this the way points case to it to stabilize should the the be fact observable cosmological that in constant the we to near are the future by the measurement of The String Landscape, the Swampland, and the Missing Corner PoS(TASI2017)015 . R / 2 s (3.1) ` = 0 R Cumrun Vafa becomes useless where x 0 1 R . This is because we S s is periodic with period x >> ` , R i , we can use the duality to map w | s ]. So this is somewhat puzzling for the 0 88 to strings on iwx and the perturbative modes are now e 1 R s << ` S w ∑ R = >> ` i 0 x R | / 2 s ` 28 = , 0 1 to within 5 percent [ R i − p | . is not a useful description of the theory for the case where = 0 0 is related to Fourier transform of momentum states, we can R w x ipx x e π p the momentum modes are very massive and suitable for winding states by: ∑ s 0 x = i x | << ` R are the momentum and winding modes respectively, i w . The example of T-duality demonstrates the idea that in general, there is at most | 0 x is periodic with period 2 0 and ] would suggest that the scalar field has to couple stronger than gravity to some matter x because a single wave packet is made up of a number of very massive winding modes. i p s 89 | 1 for the equation of state for the dark energy which we would conjecture should soon be 21 − The current experimental bounds place One of the important promises of string theory is that it gives a UV complete description of Dualities are a crucial part of our understanding of string theory. In general, a duality relates A good example of a duality in which we have full control over both sides of the theory is One can continue this logic by deducing that the corresponding scalar field responsible for This mapping makes it clear that , and 21 R 6= >> ` π the winding modes. Just as position quintessence picture which suggests no particular reason for it to have this value. define a new notion of position to the dual perturbative description where fields and we already know, bysector, lack that of this violation should of bepicture the in equivalence by principle the finding in dark apparent the sector. violationgenerated visible by of matter It this equivalence would scalar. principle be in interesting the to dark find sector evidence due for to3. such the a Lecture force 3: The Missing Corner quantum gravity including at the Planckknown scale. about Despite how being to of give primary a interest, fundamentalOne there formulation way is of we very quantum can little gravity try arising to from study string quantum theory. gravity is by holographictwo duality. different descriptions of theand utility. same physical They relate system a –another description with each with weak with coupling strong (with different coupling a regimes (withoutthere good a of is perturbation good series). validity no perturbation This physical series) picture systemthey of to with dualities would two tells have descriptions us to that where beorder both by exactly are order weakly the in coupled. perturbation. same description If as there were, they wouldT-duality have in to string match theory. every process This duality relates strings on The String Landscape, the Swampland, and the Missing Corner w quintessence should interact stronglyWGC with [ the dark sector. This is because an extension of the In general, we only havecan a only good make description sense of ofwavelength the particle of string states the states on states. the for circle However, if in the the radius regime is where much larger than the Compton where found! 2 R On the other hand when because we can hardly excitedescribed momentum by modes. In this limit, the theory is more appropriately PoS(TASI2017)015 ) ]. 2 ( 94 (3.3) (3.2) , SU 93 , 92 Cumrun Vafa ]. They showed that which is dependent on 96 ) , u ( 95 τ -fold: 3 , . However, this restricts us to the 4 22 . In this setup it is not correct to say CY C 9 ]. However, it is also possible that all these other . } ⊂ 65 z

/ 2 1 φ ]. + 66 u Tr 29

− ]. See Figure 2 1 2 x 96 SYM. However, we can UV complete both theories, the = , effective theory, by embedding them into string theory. + ) u ) z 2 supersymmetric gauge theory [ 95 1 ( ( ) theory with coupling parameter = 2 ) U ( SU 1 ( vw { SU U 2 ] or even [ = 64 ]. In this duality, the instantons of the Type IIA side are N theory is mirror dual to Type IIA with D-branes which describe an 39 ) 1 ( U . s g SYM theory and the theory is not UV complete, just by field theory reasoning, it is clear that this must be ) ) 1 2 ( ( U SU 2 SYM theory [ 2 One could make the argument that there are other theories of quantum gravity different from string theory, such These facts seem to suggest that string theory is a complete theory even though we only have a While dualities are an integral part of our understanding of string theory, the fact that there are In order to have a complete understanding of string theory, we need to have to go beyond per- = = 22 N This is evident from mirror symmetry where Type IIB on the which describes a N a low energy effective theory for the Since the theories could be attained as a special limit of string theory[ computed in the Type IIBreduces to mirror the decription Seiberg-Witten solution as [ period integrals of a holomorphic 3-form which as Vasiliev higher spin theories [ perturbative understanding of it. To furthercomplete illustrate theory, the let idea us that consider string Seiberg-Witten theorylow theory. should energy In be 1994, dynamics seen Seiberg of as and a Witten solved for the While there has been tremendoustheory successes as along a Chern-Simons this theory, direction, thereclosed such is string still as field much formulating theory. that open is string unknown such as a good description of the theory can be described bythe a vev of the vector multiplet scalar that the Type IIB side is only an ‘effective’ description of the physics and the Type IIA side is the many dual descriptions does not meanprovides that a string good theory theory itself of isfact quantum an is gravity effective arguably theory. that the String is only theory perturbatively complete well defined quantum to theory all of orders gravity and in turbative description and find ato full accomplish non-perturbative this. description. Some There of have thesegrees approaches been of include many string success. attempts field The theory idea whichwould has of replicate had string-field all varying theory de- of is the to scatteringwe create could amplitudes a hope of to spacetime full have quantum multiple string dual field theory.limit descriptions theory which If of that would such string allow us a theory to theory and study were the unify strong to the coupling exist different formulations. For a brief overview, see [ The String Landscape, the Swampland, and the Missing Corner one useful description of a physicalsingled system out for globally each point as in the parameterthis best philosophy space description; the and sometimes “Democracy no of there description Theories.” is is Ofbut no course the we good physics can description. use will either We lookdefine description will in things very call all which complicated ranges may in look non-local terms with of respect the to that wrong variable. variables and we may have to regime of small PoS(TASI2017)015 N N / (3.5) closed . This is 1 P ) ) 1 1 Cumrun Vafa × . Here the , , 1 5 1 1 ]. ( ( P side, is of course 102 ) AdS 1 , 0 ( AdS . ) ) 0 0 , , fixed , (3.4) 1 1 ( ( N limit: string perturbation in 1 ]. 2 YM g N 97 ) , 1 , = 0 32 . ( ]. In this correspondence Type IIB string λ 2 0 13 ) ) α 1 1 space , , N 1 1 5 s ( ( , -fold given by the affine cone over g 3 π 30 AdS 4 SYM theory on the boundary of ]. Because of this, he suggested that perhaps ∞ CY ) = 98 ]. In taking the limit N → 5-branes (right)[ ( 98 ) N 4 AdS q L , SU p , a non-perturbative definition of the 4 ] for a review. This is very reminiscent of the summation in ( , = 101 becomes a sum over ribbon graphs which have the topology of , 0 N N 100 / → gauge theory [ , in principle ) YM 99 N g ( is dual to SU 5 has no complete definition. The fact that the CFT side, i.e. the non-perturbative S × 5 AdS limit of AdS (Left) This is the toric diagram for the N As it turns out, ‘t Hooft’s intuition was correct. This can be exactly realized in string theory We can now ask the question if using this AdS/CFT correspondence gives a non-perturbative Now we discuss a way to define quantum gravity which is called holography. But in order to dual to a Type IIB brane web with given in the context of celebrated AdS/CFT correspondence [ definition of string theory. The motivationof for SYM this is theory, that for we example candual theory give by a in lattice non-perturbative definition regularization, whereas the holographic quantum gravity string theory would be a solution to strongly coupled Yang-Mills theory definition of SYM, gives true. But this may bethat not the very gravity useful side for is deeper weakly questions coupled of is quantum big gravity. corresponds In to fact when the the regime SYM is strongly coupled. theory on the perturbation theory inRiemann 1 surfaces. See [ string theory over worldsheet topologies. ‘tthe Hooft boundaries realized this of and the thoughtsmooth Riemann that closed at surfaces surfaces strong without (or coupling boundaries really [ the ribbon diagrams) could close up to form Figure 9: do so, we will firstthe need large to take a brief historical detour. In the mid 1970s ‘t Hooft was studying The String Landscape, the Swampland, and the Missing Corner ‘real’ definition. The Type IIAtheory. and We thus Type see IIB this are as both another example on of the ‘democracy same of footing theories’. in terms of defining a in the SYM theory relates to the size of the the This duality has beenmatches checked to all very orders rigorously in in the expansion the on the large SYM theory. For more details see [ PoS(TASI2017)015 ) 1 ( (3.6) U using 1 S Cumrun Vafa . B side. This is counter to the AdS to the gauge theory question at strong . ]. Topological string theory can be seen S − side, the democracy of theories is violated: e 107 theory in the SW example discussed above. , ) 1 solution ( AdS 106 ∑ U , 31 top. and geo. 105 ∼ QG Z ]. This leads naturally to the idea that quantum gravity should 104 , 103 . AdS ]. A review of basic aspects of topological strings is given in Appendix side. In a sense, gravity is always an ‘effective theory’ rather than fundamental theory. 109 In general we have no idea about what description will lead to the correct sum over geometries One case where we have special insight on how to give a non-perturbative description of This is analogous in the context of T-duality to defining the physics of a boson on After this very long introduction to the problem, we find ourself back at the beginning: we Despite this, the the AdS/CFT correspondence has given us a lot of insight into strongly cou- , AdS 108 as a restriction to[ a special, supersymmetric subspace of the Hilbert space of full string theory be equivalent to summing over all spacetime topologies and geometries: and topologies. We only do know thatscale there fluctuations should to be produce some a mechanism smoothcome that space from washes out some at the new lower fundamental Planck energies. principle, Ittry rather seems than or from that AdS/CFT. some this This duality description such lack must “the as of missing mirror knowledge corner" symme- in of our describing understanding the of gravity string theory. side quantum3.1 mechanically Introduction is to Topological String Theory quantum gravity is in topologicalstring string theory theory. that was This introduced can by Witten be [ thought of as a sort of toy model of winding modes when the spacecan is give much us some larger very than usefultell the insights us string into directly scale. the how non-perturbative While regime tothe the of describe string AdS/CFT it. theory, duality it Some does have not This argued is that analogous perhaps to the there example is of no the direct effective definition of However, we saw in that case there is a complete string theory behind the would be effective The String Landscape, the Swampland, and the Missing Corner In fact ‘t Hooft was tryingcoupling to and use not the string other theory way as around! a theory. Moreover if there is no direct definition of the CFT side would be viewed as more fundamental than the want to know fundamentally, what is quantumof gravity? the It metric. should From describe a the brief quantumof analysis fluctuations of the the metric standard at Einstein-Hilbert action, thetopology we Planck of see scale the that spacetime fluctuations should [ become very violent, leading to potential changes in the pled CFT by usinganswering the many questions semi-classical we gravitational have picture. aboutlocality the starting bulk. from However, the This it is boundarywinding is because theory, modes not it similar in is to a the very how good hard T-dualitythe it example to tool most is discuss from for difficult interesting bulk before. to phenomena, describe such Thiswalls, locality as makes and using what it additionally suggests happens difficult that to withto the answer black understand AdS/CFT the some hole CFT correspondence of evaporation side, should or rather bequantum with then gravity used attempting fire- in in to use order the to CFT try side in order to define and study fundamental idea of a duality as well as to all the other known examples. PoS(TASI2017)015 s t X g (3.9) (3.7) (3.8) , then (3.11) (3.10) a , and q X in Cumrun Vafa gets mapped t d . A string field theory M ∗ . , T with degree  g  ⊂ A is independent of the complex A with local coordinates worldsheet to the free energy, ∧ X , U ∧ M g A 4 X A × → C ∧ ⊂ 3 ∧ , Σ A R A : a Lagrangian submanifold. The local U 2 3 # i 2 3 -fold } ⊂ 2 ∼ = , 3 φ 0 X − + + a g 2 s A U = CY ⊂ F ∗ g ¯ dq ∂ 2 ) ) T ∧ t on the B-model side is captured by the period M ∧ ∧ x ( A ( g a 0 0 A F F  32  dp W g ∑ Tr a Tr + " ∑ 2 M y ∧ . The A-model only depends on Kahler structure of Z = exp X + J Ω  s X = uv g 1 Z so that in a local patch 2 Z   is the ‘number’ of curves of genus M s ≡ { is the contribution of the genus ∗ g 1 d g = X ) 2 T n t · . Here we say analytically continued Chern-Simons theory because  ) ] shows that the branes will induce an analytically continued Chern- D-branes wrapped on d brane N = − N ( . S ( S 110 X U exp = d g G n is of the form d In the topological A-model, these boundary components will map to Lagrangian In the topological B-model, these boundary components will map to holomorphic ∑ M : : ) = t are the fiber coordinates on the trivialized ( a g p F B-model As discussed there, we have two types of topological strings A-model or B-model, which we Consider a stack of A-model We can also make sense of open topological string theory by considering that have computation following [ with gauge group we have that the level is generically non-integer. Simons theory on the worldvolume of the D-branes given by submanifolds of the target space. Consequentlyon this will the induce worldvolume holomorphic of Chern-Simons the theory wrapped D-branes submanifolds of the target space. This is because the map will take to be on a Calabi-Yau 3-fold The String Landscape, the Swampland, and the Missing Corner and B-model depends only on themirror symmetry. complex The structure. path-integral Moreover for they the are A-model3-folds. is related The restricted to partition to one function be another is on given by holomorphic as maps to CY where structure and hence the boundary component must also be invariant. structure near denotes the Kahler parameter, the Kähler form is of the form where Consider compactifying Type IIB string theory on the is the string coupling constant.to The complex B-model deformation is parameters. the mirror Moreover of this computation and integrals of 3-forms on boundary components. In this case the boundary components map to branes in the target space. PoS(TASI2017)015 C 1 by -form (3.12) (3.16) (3.13) (3.15) (3.14) = p N which is a , this result Cumrun Vafa 3 L ) such that X CY 23 ]. -form flux. In addition, such that we can link it p linking any holomorphic L 114 , Y ) are dual to Liouville theory on the 113 W , ]. 114 112 , , 39 113 . , , 2 theory which is broken to an ) -form: the Kähler 2-form and holomorphic . . 48 , p [ = duality and holography. s Φ , s 111 0 ( [ times the normal direction in N 0 S ) Ng = we again see that there will be branes wrapping N W Ng . 3 , then the integral 2 = Φ L = X 33 y ( = M Tr k M s k C + 1 Ω W g Z C Y Z uv Z = S 0 we get locally a conifold geometry which can be reolved. (since together they have co-dimension 1 in duality works in this context and how it relates to geometric C looks like ) = N x 3 ( D-branes on 0 N W CY is a trivial in the absence of branes : ) x C ( above. As it turns out, this theory is exactly the 2D CFT that describes the vertex operators W SW Σ . Locally . Since 3 ]. In topological string theory we would expect a similar behavior. In both the L \ 13 CY X ]. Consider the topological A-model with a real codimension 3 Lagrangian sub- ⊂ Holography in Topological String Theory 3 115 0 by nature of being a Kähler form. However, once we wrap branes on N such that if we wrap M is the number of D-branes wrapped on = M N dk It is interesting to note that these matrix models (with suitable choices of Now we discuss how the large If we consider the topological B-model on One of the key features of D-branes in string theory is that they source Topological string theory is a very powerful tool and a good first step towards understanding Similarly in the B-model, the D-branes are wrapped on holomorphic 2-cycles. Following the 23 manifold Riemann surface given by corresponding to brane insertionsphysics and of further the is corresponding four-dimensional 2D theory CFT of class associated to the AGT correspondence which describes the transitions [ The theory in this case is described by a matrix model with action given by topological A- and the B-model, there is only a single field of the theory. In the A-model, D-branes wrap a Lagrangian 3-cycle the holomorphic 2-cycles given by the degenerate 2-spheres given by blowing up 3-form respectively. This means that the branes in each theory must support the respective The String Landscape, the Swampland, and the Missing Corner For each critical point of We can wrap branes around it.giving This a leads superpotential to to a the 4D adjoint field Tr 3.2 Large string theory in its fulltheories, generality. it also In gives addition a clear to manifestation giving of tools large for studyingthey exact can quantities often in be 4D exchangedas for in a AdS/CFT different [ background geometry supported by their sourced flux with a homologically trivial 2-cycle same argument, we find that there is a homologically trivial 3-cycle where 2-cycle is non-trivial in since changes to count the flux of the D-branes PoS(TASI2017)015 2 is N S on has ) ) 3 2 S (3.17) N S ∗ ( T ]. As an U ( ∂ Cumrun Vafa 115 . Now if we 3 . This large 4 M -folds by gluing is the bigger this 3 . In other words D-branes wrapped 2 shrinks and ]. N S 2 at the zero section of CY N 3 S (b) S 1 117 P resolution (b) of the singular with = 1 and Figure 3 2 P S S D-branes on ∗ 10 T N so that the boundary = 3 where 3 R . The bigger the 1 2 P CY S → and so the D-branes on it give is a finite ) . 3 ] as in Figure 1 s S . Now wrap 3 − ( Ng is given by resolution (a) with an 116 M ∗ O 3 3 34 = S S T ⊕ k ∗ duality in A-model topological string [ ) 2 ∼ T S 1 N Z 3 − in ( 3 CY O S links with the 2 S 3 C , then we have open topological strings ending on the D-branes. As al- 3 . The geometric transition underlying this holographic duality is exactly the M 1 (a) P at infinity. The . This means we will have a geometric transition where in the fiber of the cotangent bundle we find s 3 . 3 S ) 3 . 3 Ng S 3 M × S M 2 = Here the holographic duality replaces an ; S ) k D-branes on 2 N ]. Therefore, this holographic duality is indeed true. And as in the conifold transition, this S ( N R However, these branes back-react on the geometry. By integrating the Kähler class over an The normal direction to the base Now we can ask if there is any definition of the theory on the closed string side which is, U 115 ( becomes. This is the content of large [ k 3 2 now a finite size leading tothis a geometry bundle. So we endbut up with with finite topological size A-model onphysical the manifestation resolved conifold of without the any conifold branes, transition [ size: given by duality can be checked bystring computing side both which sides involves independently.orders considering The in holomoprhic partition the maps function perturbative to of expansionS the closed with resolved the conifold Chern-Simons agrees perturbative expansion to for all Figure 10: ready discussed, the effective theory on these branes is given by complex Chern-Simons theory wrap surrounding example consider the topological A-model onon the the conifold base conifold. holographic duality can be generalizedtogether building to blocks all by toric using the geometries technology sitting of inside the topological3.3 vertex [ Missing Corner for from the target space point of view, a non-perturbative theory of gravity? We will show that this is The String Landscape, the Swampland, and the Missing Corner cotangent space and can be written as This means that we can interpret D-branes as sourcing the volume for the the D-branes can be replacedS by giving finite size to this CS PoS(TASI2017)015 are ] 3 (3.19) (3.18) (3.20) (3.21) (3.22) (3.23) (3.24) Chern C . A key 124 th X ) → 1 Σ Cumrun Vafa − , : s g g ( X − e = is the q 1 − since the techniques used g c 3 , C . Now in the sum over these k k n , and = ) g ∧ n 1 , we must implement a quantization . Chern classes are differential forms given q k c . F . ∧ − + 24 k 1 . 1 3.18 Z 3 ( k 1 is non-compact, all maps as a sum over changing spacetime topolo- ∈ CY = ∧ 3 , ∞ n Z k . N ¯ C gauge bundle) associated to the top holomorphic form 2 s Ω ∏ . 1 ∧ 0 ) g 3.18 ∧ 1 0 k , ( = 6= 3 Ω U = 35 k ) = which we can without loss of generality take to be ) Z CY 2 3 1 i N Z 3 . Since dk M s − 3 2 s = 3 g C Z g 1 CY g c C h ( ) = g = ], the classical action for the A-model is given by [ Σ k ( Vol 2 S over the moduli space 2 s M M 1 123 g g Z Z , 2 = M ]. − g cl 122 s 2 33 → S g g ∑ H ( ], the partition function of this theory has been computed to be -folds [ 3 exp 121 , CY ) = s 120 g , ( Z is the moduli space of Riemann surfaces with genus 119 , g M gauge bundle which is classified by its first Chern class 118 The Hodge bundle is the line bundle (equivalently a Now we can ask if there is a target space or quantum gravitational formulation of this result Now we want to try to reinterpret the result of Consider the topological A-model on , ) 24 1 (which has a phase redundancy). This bundle has a metric whose associated Kahler function is ( 109 Ω constant maps – i.e. they map to a point in gies. Note that we are summing over the moduli space of Kähler classes of the manifold condition which has a compatible connection withby a wedge generically products non-trivial curvature of the curvature of a bundle that encode topological data of the bundle. where we sum over allquantum possible gravity? geometries and As topologies itBatalin-Vilkovisky as formalism turns [ we out would there expect is. from a By theory using of a string field theory computation using the the origin. Using various[ arguments using topological string dualities and Chern-Simons theory generalize to other where class of the Hodge bundle This can be viewed as which we can think about as coming from a cosmological constant term. feature of Kähler forms is that they are closed forms In a sense, we can then interpret them asline bundles the we curvature have of to integrate a over line the non-trivial bundle classes – such the that field strength of a However, as it turns out, in order to reproduce the results of The String Landscape, the Swampland, and the Missing Corner indeed the case and showyet how to another recover way. the In full other partitionthis words, topological function we setup. in fill We topological the will string missing proceed theory corner by in computing of the what example the of quantum gravity means in U PoS(TASI2017)015 is ∈ s ` F s g (3.28) (3.25) (3.27) (3.26) . When for = ` p F ` s Cumrun Vafa g ∼ ` = k ]. 33 gauge field. In this case we now have . In this way we can translate the sum . , 1 gauge fluxes) or equivalently the blown ) F and that spacetime geometry fluctuations 1 ) , , ( 3 1 CP L ∧ form should be in the class n 3 ( ) 3 U z L k F 2 U ( ch n 2 ) 3 z ∧ 3 is a smooth quantum geometry, and 1 ( n 1 s F CY δ z ` 3 36 s Z 3 n s g ∼ , CY g 2 ` n Z , . In the line bundle interpretation, this corresponds to s 1 = ] = n g k S a [ i = 3.24 n ∑ S as curvature of a F . Here we interpret ) Z Here we demonstrate the changing spacetime geometry depending on energy scale ; is the third Chern character of a line bundle over the different components of the blown 3 3 , spacetime is that of classical geometry, ch CY s ( ) By performing blow ups of this geometric singularity at origin, the singular line bundles are Now we can rewrite the action as Now we are summing over Kähler classes with singularities. Without loss of generality, we 1 , 1 ( ` >> ` the “quantum foam" with violently changing topology given by fluctuating blow ups. Figure 11: where the D-branes are wrapping a Lagrangian 3-cycle over singular Kähler classes to actual changes in spacetime topology [ up geometry amounts to counting the number ofas sections the of number these of line bundles terms which in can the be realized polynomial replaced with smooth line bundlesblowing that up have a non-trivial sufficient curvature number oncurvature of the has times, a blown we single up can unit geometry. of in By charge fact for make each any blown line up bundle smooth so that the should be sourced purely by D-branes. This suggests that we should rather take summing over singular line bundles localizedthere. over the origin since the curvature is only non-trivial The String Landscape, the Swampland, and the Missing Corner This quantization of the Kähler form implies that the H that where up geometry. Summing over the line bundles (that is can take these singularities to be at– the giving origin rise – arising to from the D-branes wrapping non-trivial a integral collapsed 3-cycle PoS(TASI2017)015 3 + Z ∈ (3.29) ) 3 n , . Then we 2 ) Cumrun Vafa n 0 , , . Deleting the 1 s 0 g n , ( − 0 e ( = q . s , then we can blow up by l , etc. When we take all of s 2 3 + g q Z ... , correspond to + a 2 q 3 + q + 1 = n ]. This physically corresponds to a sum over all of ) 37 n in the sum are constrained based on the blown up 33 q 3 n − , 2 1 1 n ( , 1 1 n as giving an octant of a lattice = a ∞ n 3 n ∏ , 2 n , ) = s 1 g n ( Z . This gives a satisfactory realization of how smooth geometry emerges in the limit of 11 1, when we look at scales much bigger than the Planck scale of Specifically viewing the We have now seen that using topological string theory as a toy model of full string theory The authors would like to thank the organizers of the Theoretical Advanced Study Institute Another important facet of understanding the role of branes and singularities in string theory << s in the complement of the deleted set of points [ taking 3D Young diagrams andwould removing have points the starting restriction from that the the non-vanishing corner coefficients near of geometry. these contributions and sum them we end up with provides many promising results.theories, Besides it also giving has tools many to othertype study holography. properties We we four have know dimensional seen to that quantum holdthere topological field in may string full theory be string suggests an theory that independent such include as complete the a AdS/CFT full definition sum string over of theory spacetime thescales. topologies gravity Topological but side, string still theory which give thus rise will stronglyunderstanding to suggests in of that a particular quantum indeed smooth in- gravity. there spacetime is geometry a at missing large corner in our Acknowledgements summer program and the Universityto of thank Coloardo the Boulder. Simons TDBhospitality Foundation and while and CV writing the would this additionally organizers paper.grant like of DOE-SC0010008 TDB the to is 2017 Rutgers supported Simonsship University. by Summer of the The Workshop the U.S. for international work Department programme of of(MINECO/FEDER “La F.C. Energy EU) Caixa-Severo is Ochoa", under of supported and the through the “CentroAdvanced grants a Grant de FPA2015-65480-P fellow- SPLE Excelencia under Severo contract Ochoa” ERC-2012-ADG-20120216-320421.supported Programme, in The part research and by of the NSF CV grant ERC is PHY-1067976. A. Branes Probing Singularities is how they interact when a brane probes a singularity (when a brane transversely intersects a the ways in which the fluxno can points be gives “distributed” a among contribution different 1.origin blown and up Deleting one geometries. the of So origin the deleting give three a points contribution next to of it, gives the contribution of 3 thus reproducing the perturbativetum closed gravitational string foam answer realization in ofdescription a the of rather same the novel partition quantum function way.geometry gravity filling depending This the side. on is missing which scale The corner the we in quantum quan- consider.in the This foam Figure scale gives dependent a view of different spacetimeg description is shown of the The String Landscape, the Swampland, and the Missing Corner where the non-vanishing coefficients PoS(TASI2017)015 , and 126 ) (A.2) (A.1) , i where N n ( Z 125 U / 2 Cumrun Vafa C of fundamental decomposes as × ]. These fields i 6 F Φ M 125 4 SYM theory to leading . Under this embedding, the = . The worldvolume theory of ) n , N Z ( -symmetry group, resulting in a N N / R 2 U ) = C N i → ( , N , n action. This means that they generically U i i n ρ ∑ Z . Consider a spacetime ) : representation. Z n x ( ι ) Z ) 1 i / ( + action so that a generic field 2 , i Φ n C N 1 ( 1 Z 38 ) − i = i U n M representations depending on the embedding of the N ( × n ] on branes probing orbifold singularities where a stack ) U i ) = Z x N i ( vector multiplet. This leads to a cyclic quiver with funda- ( 125 ∏ ) Φ U i → N singularity breaks SUSY by half since rotational symmetry is ( ) n U N Z ( / U 2 C action acts on the fields by global gauge transformations coupled with n is the flavor symmetry. This will lead to a collection Z 2 theory. F ]. G = 129 , N are determined by the embedding of where } 128 ] and has recently been revived as a possible extension of the story of theories of class are irreducible representations. This means that we now have a collection of fields i , F i N ρ G { 129 127 , , by projecting onto the different irreducible representations. → , 1 D3-branes probes an orbifold singularity 4 vector multiplet is broken into a sum over vector multiplets with gauge group ) n . Generically this means that the gauge symmetry is broken x 128 + We can additionally couple the D3-branes to D7-branes so that the D3-D7 strings introduce More precisely, the Intuitively, since the fields living on the D3-branes come from oscillations of strings stretching We will first consider the work of [ n ( 130 Z = , ) [ i Z : N ( f fundamental hypermultiplets in the worldvolume theory of the D3-branes [ bifundamental hypermultiplets of the N where the where the Φ an R-symmetry transformation. This means thatof the gauge group is broken down to the commutant decompose as irreducible representations of the in the perpendicular directions, they are acted on by the order. The presence ofno the longer a symmetryfour-dimensional along this direction, thus reducing the the D3-branes are transverse toa and localized free at stack the of origin of D3-branes is given by four dimensional will similarly decompose asι a sum over hypermultiplets coupled to each mental hypermultiplets as displayed in the quiver below: of 127 The String Landscape, the Swampland, and the Missing Corner singularity). There has been a large amount of research on thisS in the past twenty years [ PoS(TASI2017)015 or ) D3- 1 N ± , . These g N D5-branes Γ ( / Cumrun Vafa 2 N C 1 gauge theory since = 5-brane stacks freeze out ) N ]. Now consider a stack of 1 ± 1 quiver gauge theory. As an , with singularities. The allowed 126 D6-branes wrapping non-trivial N = X ( 3 2 N F F -fold with transverse D3-branes, we N 3 fibers [ -fold 2 3 3 2 CY N N where the dual graph describes where the T 5 CY 1 theory. 4 1 4 1 F F N N = 39 N ] for more details. 5 N N N 125 5 N F F -branes give rise to a four-dimensional ) . See [ 0 ˆ g , D3-branes probing a toric singularity. The worldvolume theory is given N fiber degenerate. The idea of a brane tiling is that we have ( N 2 T 12 is a toric variety with degenerating Y branes which results in a four-dimensional which is partitioned into “tiles” by transversely intersecting NS5-branes whose an- ) 0 2 , T N where ( C 5-branes. This tiling creates a 3-colored torus where no two adjacent tiles are of the same × ) This brane tiling arises physically as follows. Consider the mirror dual of the stack of We can additionally consider branes probing more complicated singularities. We will con- Consider a stack of 0 Y , N different cycles of the singularities for these manifolds areity toric – that isD3-branes locally probing they this locally singularity. lookof like As the a singularity in toric restricts the singular- thetheory. case fluctuations Note of of that the orbifold since D3-branes singularities we andare are above, generically hence considering the studying modify Type a IIB structure their four-dimensional on worldvolume a wrapping a the combination of NS- andmassless D-type degrees boundary of conditions freedom. on Then the westacks get of bifundamental chiral multiplets byexample, diagonally see touching Figure branes probing the toric singularity. This is given by a stack of In general, there are other types of orbifold singularities which are of the form sider a wide class whichCalabi-Yau will manifold. illustrate Consider the Type general IIB method on for a analyzing generic generic singularities of a gle on the torusdiagram. is These determined pieces by together the form a degenerate tiling cycle of of the the torus where external each components tile of is the a stack toric of by a quiver gauge theory thatlarity arises can from be what described by is a called toric a diagram brane as tiling. in Recall Figure that the toric singu- The String Landscape, the Swampland, and the Missing Corner theories will generally leadassociated to affine quivers Lie algebra which are are of the form of the Dynkin diagram of the ( color. Then the stacks of PoS(TASI2017)015 (A.4) (A.5) (A.6) (A.3) (A.7) to the s , which w W -fiber of is C ) . See Figure Cumrun Vafa y , W x , black are stacks C ( ) 0 ∈ , s N w ( -plane from the . , . q 0 | W y u p | = x / . ) ) s u q y , γ , W s p x ( C ( c

degenerates at Q i | → ∈ P 1 s ∈ y u are unfixed coefficients describing the ) S | w ∂ q . ) ∑ / , q p , u } = ( p 0 ( ) s } → c y , -fiber degenerates. Here the w s ) = x γ ) = x ( , ’s where y (

= 1 s , charge. In going from the brane tiling to the quiver, we P P -fibration. This Riemann surface has an embedded S x 40 x ) y ( 1 γ uv w ∂ 1 S P where β Σ , { − w , W fibers are given by the action , → = N C 1 ( y where the , 0 ) = S ∈ 0 y Σ fiber over the straight line in the W , s 3 ) x C w s -fold as a ( T y 3 ) = , P ∈ , y s { ⊂ , 0 x : x CY which is defined by 2 x ( ( 0 π T P = α P Σ w = + → s x -fold given by uv w 3 CY fibers are locally given by -fold where the three -plane, -fold. We can more conveniently write this as a double fibration over 2 3 3 T W for more details on how this is related to the toric diagram CY CY charge and grey are stacks with 13 This figure shows how to pass from toric diagram (left) to brane tiling (center) to quiver gauge ) that degenerates at points 1 are the coordinates of the toric diagram and s + w , Q Σ N Therefore, we can view the ( → , . 1 s given by a Riemann surface This is a toric These fibers degenerate at See Figure we will call the moduli of the 14 where The D6-branes, which areformed the by mirror the image pre-image of of a the D3-branes, wrap the closed 3-cycles that are associate a quiver node withcorresponding each to white adjacent tile faces. and then attach bifundamental hypermultiplets between nodes 3-cycles in the dual S with origin. Here the theory (right). In going from thediagram toric give diagram the to angles the brane ofThis tiling, the leads the to cycles normal the on of tiling the the in external torus the legs center where of where the the the toric NS5-branes white are intersect stacks the of stack branes of with charge D5-branes. Figure 12: The String Landscape, the Swampland, and the Missing Corner PoS(TASI2017)015 ]. = -fold. 101 N 3 CY Cumrun Vafa w described above. See Figure 2 T where we additionally include a sum X whose intersection matrix is the adjacency (right) is related to the data of the toric diagram → 0 Γ Σ Σ : φ 41 Σ along a graph 0 Σ fibered over the origin. Σ ] for more details. 129 , This figure illustrates the mirror dual of D3-branes probing a toric singularity of a This figure shows how the Riemann surface 128 -plane intersecting the , W supersymmetric non-linear sigma model 127 In this appendix, we will review some basics of topological string theory following [ These D6-branes intersect on ) 2 . In this way, the quiver gauge theory is derived from first principles by living on the intersecting , 2 over the worldsheet topologies in the path integral, thereby coupling to 2D quantum gravity. We Under mirror duality, the D3-branes becomeover D6-branes the wrapping 3-cycles which are shown here as fibered Topological string theory is the reductiontheory of invariant under string deformations theory of to the the worldsheet( topological metric. sector: This can the be part formulated of from the a Figure 14: D6-branes under the mirror symmetrySee map [ of the D3-branes probing the Calabi-Yau singularity. B. Review of Topological Strings 15 Figure 13: matrix of the associatedcanonical quiver way which gauge leads exactly theory. to the data This of intersection the brane graph tiling of can be “untwisted” in a The String Landscape, the Swampland, and the Missing Corner (left). The Riemann surface associated to the brane tiling is the thickened dual graph to the toric diagram. PoS(TASI2017)015 R ) 1 ( (B.3) (B.1) (B.2) (B.4) times U E ) × 2 L ( ) Quiver . 1 Cumrun Vafa ( SO o ¯ ` U D ψ × ¯ j C E ) ψ k 2 B vector symmetry. Re-

.

( daec Map Adjacency i ψ V i A ) F SO 1 A ψ ¯ ( ` η ¯ jk U i + R i . This projects onto the diagonal φ z CD and ε Brane Tiling D i B 2 , φ AB T ¯ representations determine the spin of η ) ε ¯ i E ¯ AB − ) are the left and right chiral symmetries Φ ε symmetry shifts the spins of the particles 2 , ,  i i ( ) R 2 ¯ j , ¯ R j B L Φ ) ¯ − ) ( SO 1 ψ K 1 ¯ ( ∂φ k A 2 i ( = ¯ topological twist axial symmetry and ψ U ∂ z K i ¯ ¯ A i j ` B U 2 ¯ 42 k A ∂φ d ψ ) Γ z δψ 1 θ Twist Map = ( 4 AB D ¯ j ¯ j ε A d i U ¯ ψ Σ G + Z ¯ j , AB ¯ F ε = i i )( A S k B + ψ ¯ j ψ Tiling B ¯ j φ A η Σ ¯ z ψ D AB i jk i ε Γ φ z non-linear sigma model with target space given by a Kähler manifold is = -type symmetry. Since the AB D i ) R ε ¯ j 2 ) -fold. i , + δφ 3 1 i 2 G ( F n CY U ( z

Mirror Symmetry = ( are chiral superfields with lowest component 2 is the Euclidean symmetry and charge. − d } i Σ R E N we can write the action as ) ) Z -symmetry for this procedure: Φ This graph shows the flow of deriving the worldvolume theory of D3-branes probing a toric 1 2 } R { ( ( i − F U = , SO i A Toric Diagram S Now to make the theory topological, we must perform the topological twist. There are two Let us see now in some more detail the construction of topological string theory. The theory ψ , i φ where This action is supersymmetric under the transformations where the call that the classical bosonic symmetry of the worldsheet theory respects is the Käler metric of{ the target manifold. Writing out the components of the chiral superfield as described by the action choices of by their of a general While this is defined for general3-fold Calabi-Yau manifolds, for we the will remainder restrict of to this the review. case of a Calabi-Yau some worldsheet the different fields, this means that twisting by a Figure 15: singularity inside a The String Landscape, the Swampland, and the Missing Corner then make this theory topological by performingcomponent a of the 2D Lorentz (or rather euclidean rotation in euclidean string theory) PoS(TASI2017)015 (B.8) (B.6) (B.7) (B.5) (B.9) (B.10) (B.11) so that A F , Cumrun Vafa V F , ¯ j , R . F χ o ν ` − − , χ , , D L − ¯ k i µ F A V Q χ ρ F F j ν = + ρ µν + − A ¯ i − µ g , J . F J ρ + ` ¯ z ` − = ¯ = k ¯ j ρ Q , ¯ j ¯ i j B A φ , , R χ = ν E E . , k 2 1 J ∂ J R i χ − − Q F respectively. Similarly we can topo- , , , ` φ ¯ + j , ¯ ¯ i i + − R µ k + 0 ,  k i ¯ z ¯ j ∂ ψ ψ L L ν R ρ ) B = A j F F , , g 1 = = i µν . E µ E } + ( χ ¯ ¯ i i i z ) ε = ) √ F k i 0 i ¯ z jk 1 B-twist: χ ρ U χ 1 V ( F ( , Γ -folds have a trivial canonical bundle, we can µν j = + F 3 g ¯ U j U Q − χ } 2 1 i i { ¯ z φ 43 jk : CY → , Q , , → ν F A Γ − , ∂ ) A i i V − ) 1 − ) Q i χ φ ( i 1 − + + 1 { , , , , ν µ ( φ ( χ ¯ U i i z − − + ¯ ∂ z D U ∂ U ¯ j ψ ψ Q i × µ D 2 µν . Since we are twisting the worldsheet Euclidean group by a × 2 ρ χ × V g _ + = = , ) = E E i i ¯ z ¯ E j µν ) 1 + ) , i ) } ( ρ χ ] = g 2 ] = i ¯ 2 z i i 1 + ¯ ( z G ( ( U ρ − φ F Q , n , , U SO g ∼ = SO Q = Q Q R symmetries are often combined into vector and axial symmetries by √ { [ [ ) z Q 1 R 2 , ( d L ) U Σ variables are of zero spin, but are grassmann variables, they can be identified 1 Z B-twist: ( i A-twist: × χ L U = ) A A-twist: 1 ( S U are the generators of the Lie algebra for is the generator of R _ , , . This is related to the fact that since L E F J X Topologically twisting, in addition to making fields bosonic, changes the spin of the super- In the case of the topological A-model we can write the action as where chiral symmetry, we shift the spins of the fields of the model by their charge under charge operators. This allows ustopological to supercharge define a pair of scalar supercharges for each model called the the fields are all bosonicthese topological (although twists they are may referred to be as Grassmann the fields). A- and B- The models theories respectively. resulting from Now since there are no bosonic symmetry currents, the supercharges must be nilpotent logically twist by projecting Now the supersymmetry transformations take the form of identify the chiral spin bundle with the bundle of holomorphic differential forms and hence the with differential forms – andspace their products with general wedge product of 1-forms on the target where where have made the identification Note that since the changing basis of the generators The String Landscape, the Swampland, and the Missing Corner respectively. The PoS(TASI2017)015 . X (B.14) (B.16) (B.15) (B.17) (B.12) (B.13) Cumrun Vafa , with the de Rahm  , ) ) i i Q θ o φ ¯ ¯ j z ν ∂ D F ¯ i j µ + F ρ ¯ j ¯ j i + η G ¯ . ¯ j z i , , , k ¯ z ) ¯ ¯ i φ j D − ¯ ¯ φ k j ρ ¯ j ¯ z i ν j i η z ¯ F F χ ∂ θ ∂ ¯ ¯ j j ρ , we can associate the supercharge G k ¯ i z i ¯ µ i ` ( − η = i ρ z ρ G ] = ¯ ¯ j η j k η ¯ i ¯ i ¯ z j ρ } k ¯ i ¯ ` χ z ρ φ Γ = j ¯ ` ( + ( , − ρ i ¯ j i − } , , i and ν η i  R ¯ Q j k , i F , ¯ θ [ } G ` Q = ¯ j j , µ + φ φ i i { V z i z } i ρ = , ¯ i µν S Q R ∂ θ ρ i i 1 2 i ¯ g { z F δ Q θ with the additional identification φ − , δ F , ¯ D { z + ) 44 i ¯ ∂ j , Q − ν − = θ = { ¯ j i z F ¯ 0 z + ) can be identified with 1-forms on the target space , B.10 i ¯ i i j µ , ρ φ D µν z θ ρ ν φ φ T TFT , z ¯ ] = z ∂ − D 1 2 i i ¯ 0 i S ∂ ¯ j i ∂ µ ¯ i  η φ χ = ¯ ρ j , = η φ = i z z } + } i µν } Q G ¯ ¯ ∂ i i D i z [ ¯ j g F ¯ χ j ¯ i η j ρ , µν i i , , G = G G ( Q gg Q i ¯ Q ¯ z i h { n { { ρ g √ z η (and hence z 2 − 2 √ i d z d χ 2 Σ ]. The above analysis tells us that the A-model deals with the de Rahm ]. Σ Z d Z . Σ = 131 1 2 Z d 132 B , = = S . A B ¯ 101 ∂ V V , = 31 B Q Note that the action of these topologically twisted theories can be written as Thus far we have described some of the general features of these twisted quantum field theo- Similarly the B-model can be written as This general analysis uses what is called the Mathai-Quillen formulation of a topological non- for the A- and B-model respectively. Since the stress energy tensor is defined by where ries. In order to show thatof these supersymmetric theories operators are topological is we invariant under need to a show deformation that of the the expectation metric. value cohomology of the Calabi-Yau manifoldand whereas hence the B-model the deals A-model withinvariant is Dolbeault under invariant cohomology under Kähler complex deformations. structureKähler moduli, deformations Since one while mirror would the symmetry (correctly) B-model expectdetails exchanges that see is complex [ the structure A- and and B- models are mirror dual. For more The String Landscape, the Swampland, and the Missing Corner non-chiral spin bundle with the cotangent bundle.differential operator In this way, we can identify where we have again defined the fields as in Using the B-twisted supercharge, these fields satisfy the supersymmetry relations Again since we have that Therefore from the supersymmetry relationsoperator relating linear sigma model [ PoS(TASI2017)015 . ) X ( p (B.21) (B.18) (B.19) (B.22) H ] for more 107 to the zeros Cumrun Vafa , 106 , -forms in p 105 0 (B.20) localizing = such that i ) } should be associated with i i µν O χ . In this case we can integrate { G i n ∀ O 0 ... = 1 i O -closed operators of the form , ( O . , p i Q δ δ i , h } χ ∀ . In this case the expectation value reduces } i V = ... , in field space – hence as the BRST charge. This means that we can O µν 1 i i Q G where χ { V Q µν , p 0 i } x i G Q d 45 , we can see that the should be associated with primitive tangent vectors = δ ,..., { O d i 1 z n i { } i M ρ = O α Z O B.11 ... , = µν 1 = T Q α O S { h O a supercharge and we want to compute the expectation value of = . Therefore, since the expectation value of a product of any (su- i 1) and the Q n 25 + acts on the operators O and Q ... 1 V -exact, we have that this expectation value vanishes order by order by O h δ µν δ g . δ µν G 1). Therefore, we should identify generic ] for more details on localization. − 133 Recall that in the A-model the different fields can be associated with : We are ignoring the subtleties coming from boundary contributions in field space. See [ Note that this similar, but distinct to the idea of localization. In the case of localization, we So far, we have only introduced topological field theories since we have not yet implemented Now consider the expectation value of a collection of operators . See [ 25 V details. associate the ghost number of a given operator with the level ofA-model the operator as a differential form. From the supersymmetry transformations to the path integral integral over the zero-locus of again have that a sum over worldsheet topologies.give insight However, into even understanding the the topological associated topological fieldlevel string contribution. theory theory As since expectation they in values represent quantum the fieldthe tree theory, theory understanding is the crucial anomalies to and ourwe ghost understanding can structure of interpret of these bosonic, Grassmann theories. supercharge In these topologically twisted theories Now since the action is integrating by parts in field space persymmetric) operators is invariant under variations of the metric, these theories are topological. for some potential function some collection of supersymmetric operators by parts in field space so that of B.1 Correlation Functions in Topological Field Theory primitive 1-forms (ghost charge (ghost charge The String Landscape, the Swampland, and the Missing Corner we have that in these cases for some tensor By use of the Ward identities,the we worldsheet have metric that the variation of their expectation value with respect to PoS(TASI2017)015 , then (B.27) (B.25) (B.26) (B.28) (B.24) (B.23) invari- ) X ( which are 2 which repre- b ) ) Cumrun Vafa Z Z ; ; chiral symme- . in the A-model X ,..., 2 ) X ( 0 unless ) 1 i ( 2 1 2 ¯ z | 3 2 ( z H | α = . Because the zeros H dzd + U i i ≡ ∈ O 1 , k , ( 26 2 ∈ Gromov-Witten B α α β , = ω O σ O ) i Q 2 1 S ) Z ... i α Z ds 1 ; α , α , where O ( X ) = h ( β O i )) -fold so that the superselection I 2 Z h t 3 ) ; X H Z ( ; X 1 ( X . CY ∈ c ( 2 i ( 2 0 ∑ , ∗ H H α . . These fields couple differently to the ∈ φ is a = . . This theory has a ∈ β g ] are an example of Σ i Σ i ] X 6 i β X Z S β S , [ ¯ Q 2 i [ ∂φ 0 ) + − , n β 3 and , N 6 ) 0 α 1 is a scalar on the worldsheet. X ) + X 3 -exact, we have that the expectation value will ( N = ∩ = i g ∑ 2 i α i 2 b , Q χ β 46 α − β ∑ α -form on Z = O 1 p ∩ 2 ( . These X 1 β d ) = α 27 t α 2 . In our case, deg β ( ( has non-trivial expectation values. → 0 Z X # 1 = 2 F 1 k C Σ i = S ∑ i , = α : α i } β = i O t dim i φ Z is a closed t { i Σ i = is a vector and β n p , according to their ghost charge. However, this symmetry can be 3 i e 0 deg i α d µ i ) z N 1 ρ dx 1 X ρ O ( k 2 = = 2 i ∏ ∑ i ∧ α b and ) = O i ... and = 1 Σ α i α ∧ ( β , each of which represents a different instanton sector χ 1 O β A i Q I h V dx p i ,..., 1 i . Since we have that the action is α 2 S is the (complexified) Kähler class = is the genus of = g α ω are given exactly by the holomorphic maps Σ Let us consider the expectation value of three rank 2 operators The complexified Kähler class is a shift of the “normal” Käher class by an additional We can see this more explicitly by using the Fubini-study metric of the 2-sphere The calculation of the leads to the selection rule that A 27 26 V where the different instanton sectors arethe classified image of by the holomorphic genus 0 2-cycles worldsheet.putation in Because of of the the partition selection function rules above,2-cycles. will this essentially To means be reduce that explicit, to the if com- the wewe triple pick have intersection a that of basis the of associated 2-cycles and ants. In fact, these areworldsheet) what determine the prepotential (that is the free action from the genus zero with of localize to zeros of sents coupling the theory to a non-trivial B-field. worldsheet topology because This tells us that only the case of where rule becomes anomalous because of the non-trivial topology of The String Landscape, the Swampland, and the Missing Corner where try which rotates the PoS(TASI2017)015 , i 3 Σ α in a O ) 2 (B.37) (B.36) (B.33) (B.35) (B.34) (B.29) (B.30) (B.31) (B.32) α X ( O 3 1 such that . α H 1 Cumrun Vafa , O 2 Ω h h . The instanton ∧ B 3 . ¯ j V is the genus of ¯ z ,..., ) , d 0 g Z ) ; , ∧ = X 2 α ¯ j ( TX b ¯ fields according to their z ) q , q d , i a Λ . p θ , ( ∧ , 0 i 1 X ¯ j H ( η ¯ z . which induces an isomorphism with = p ¯ , ∂ ∈ i d b ) which constrains the expectation ) 3 q H . i b , j Ω B ∀ 2 Z i B q Σ ∈ , 1 , j , ∩ i Ω dz a ) θ q X a a , j Ω ∧ ( g B A B 3 x ) 3 ∂ ... ¯ j ( Z i p 1 -form -exact rank (1,1) operators ∂ ... j − , ) ) 0 = q 3 = θ 1 0 Q ∧ ∧ b − p ( α , ¯ = i a 1 d ( j A 3 d ( i 2 ... 2 η F ( ¯ j i φ ∩ H ) dz = ∧ ... d a 2 i 1 ¯ 1 i ∧ j α A α → 47 p ( q η x ¯ ∂ i ) 1 , and choose a local symplectic basis for q 1 ¯ ¯ z j j i p ∂ i ¯ i X ) d ∑ p , ¯ q i X 1 ,..., ∧ ,..., , 1 α = Λ 1 Ω j ¯ dx i ( . This means that we can write the expectation value , i a ... for X X α α A ∧ X b Ω . Z p ( ∧ Z a s . = ¯ d 1 ∂ ... δ → c ¯ i i = = ¯ z ∑ α H ∧ Σ 3 = a d M : 1 -closed operators are generically of the form O : which we will denote by ¯ q z i α b j p i ¯ i ∈ J Q B Ω ∧ . dx φ ,..., J q 2 ,..., ∩ j 1 p 1 j Ω ¯ i ¯ i α a α ,..., A ∧ ,..., 1 1 j ¯ i 1 = α α ˆ α X to vanish identically unless = containing Z i . n → s α . α = c α are the holomorphic and anti-holomorphic components of i O . 3 ) M α ... X α 1 C O q α 2 Now consider the B-model. Now instead of de Rahm cohomology, the B-model relies , α O α : h dim O p 1 ( = α Since we are on a CY 3-fold, there is a unique Fix a complex structure Now let us consider the expectation value of three d O h between Now define the periods of the CY manifold This means that there are no non-trivial instanton sectors in the topological B-model. This rewriting allows us to encodeunique the holomorphic data 3-form purely in terms of the topological data encoded by the in the B-model. Again we have thatsectors the are expectation described value localizes by to the the constant zeros maps of by contracting with the free indices of local patch in and where or alternatively which is canonically isomorphic to Again we have anghost anomalous charge. chiral However, symmetry this which timewe rotates since have the they a have selection the rulevalue same that of worldsheet only and couples spacetime to properties, the topology of where The String Landscape, the Swampland, and the Missing Corner B-model on Doubeault cohomology. So PoS(TASI2017)015 is a will z Z (B.38) (B.39) (B.40) log = ]. The com- Cumrun Vafa F -fold produces 3 109 , . manifold becomes CY ] 3 c 108 A [ CY c . These are only defined ` prepotential a to denote the special pro- z c . This means that the vector ∑ . a , s , which is given by the triple z . ) c c = 1 z 3 , M ∂ ˆ 0 1 α b of the theory, without loss of gen- F z . 3 s ∂ . = ( ∂ a c i . z α , ∂ ) M t c O . ( ` g ] b a b , which is the genus zero contribution to the instead of the . We can write this as F . Therefore, they can individually be thought 0 m ) F 2 . A deg a a 3 [ s a F − . t z b ˆ n c g α , 2 s . Now we see that the free action m c a , g ∩ 2 ∑ M b 48 0 , b ∑ 2 − 2 1 a ∑ ˆ ∞ g = where α ∑ 2 s g = = i prepotential computed above in the A- and B-twisted g = ∩ 3 i 2 1 0 α 0 3 ˆ = ˆ α ) = F F α α O t ) ( 2 ( g O α Σ 2 F ( α O χ 1 , O − s α 1 ), the physical data is encoded in the scale invariant projection of g α O ] Ω h O a h A Theories [ a 2 2 theory. In the limit that the volume of the is exactly the n as projective coordinates parametrizing ) a = a = t parametrizes the complex moduli ∑ z ( ) 0 = . N F Z weighted by N a 1 , . z g . ˆ α s X . Σ ( c ) 1 , M 2 ( form a fiber of a vector bundle over H a F are projective coordinates and the function a z A salient difference between the A- and B- topological strings is that the A-model depends It is well known that compactifying Type IIA or Type IIB string theory on a In order to construct topological string theory we have to couple our theory to 2D topological Since Using this we can define the generating function which is called the In the literature we often use the coordinates . Now the expectation value 0 only on Kähler structure whereasthat the mirror B-model symmetry depends exchanges on complex theA- and and complex B-models. Kähler structure. This geometry, Since turns mirror we outIIA symmetry to and have exchanges be Type IIB. a the direct consequence of the mirror symmetryB.3 between Type Relation to 4D associated with the rescaling of F intersection of the associated 2-cycles # putation of these terms is still an active field of research. Since free energy, is a homogeneous polynomial in terms of these coordinates (a rescaling of the jective coordinates which have removed thepatchwise scaling on dependence of the generically have the form a four-dimensional B.2 Topological Strings and Mirror Symmetry gravity. This means thatsheet in topologies the path integral we have to include a sum over the different world- The String Landscape, the Swampland, and the Missing Corner erality, we can take the as functions of the topological field theories. These highererally genus relying terms on are generically mirror very symmetry difficult and to the compute holomorphic gen- anomaly equations [ space of As it turns out here PoS(TASI2017)015 2 + R ] 28 (B.43) (B.44) (B.42) (B.41) (B.45) (B.46) . + 114 ]. Hence manifold R , 3 ]. 96 Cumrun Vafa , 2 theory can 113 CY 113 , 95 = 112 N , , 39 2 Weyl multiplet ) 2 SUSY QFT. Many of 2 . = is the self-dual part of the − = , g Ω , + 2 constant, we decouple from + i + N F j 4 F B 0 by the nature of the residue N F Z 2 + 2 gauge theory [ C YM = . R ∧ g = = u )( i + i i is the dx } ⊂ F F a 0 N ( ∧ , i j g W = τ } F dy 4 0 2 x ) d ∧ , 4 x = d ( 0 Z u 2 du ) Z x W Ω ∼ i ( = 0 ) = A + i 0 while keeping Z 2 49 a + W j ( y dx F g → = + ∧ F + i s 2 ∧ f g z y 2 v ,` + dy uv { i s ∂ ] for more details. F g W ∧ ) = 31 ) 0 ) = θ , F du Σ 4 x Z j , , are the terms in the topological string free energy, then the low ∂ y i xd i Y = ) , ( 4 t ∂ v F 3 ( i ( , d Ω z g x u H 4 F ( 2 1 Z are the vector multiplet fields, f d 2 SUSY QFT. Many SUSY operators in the 4D i { 2 theory from compactifying Type II string theory on the = ∼ a Z 0 = = F 0, manifold as suggested in the previous section. For example, the partition EFT 0, we find that the effective action is given by: 3 N S 6= N g = CY g where Σ over the 3-cycles in ⊂ γ Ω That is the multiplet with highest component given by the field strength of the gravitational multiplet: For the case In fact, the topological string computes exactly the low energy effective action of the vector If we consider the topological string theory on a CY-3fold of the form [ Recall that in the B-model the prepotential is given in terms of the periods of the holomorphic 28 which is exactly the same form as the Seiberg-Witten low energy effective action [ then the holomorphic 3-form is of the form Y. Specifically we find that if energy effective theory will go as for the case where is the self-contraction of the self-dualfield part strength of of the the Riemann graviphoton. tensor and See [ multiplet for the 4D This is of courseSieberg-Witten reminiscent theory and (and the is instanton partition in function fact calculation exactly of Nekrasov. the generalization of) the relation between the genus-0 term gives us exactly the prepotential for the 4D In this case, the period integrals reduce to integrals localized at The String Landscape, the Swampland, and the Missing Corner infinitely large (or is non-compact) and theorem. That is to1-cycle say, the period integrals reduce to integrals over a disc whose boundary is a gravity producing a 3-form be computed in the associatedon topological the string associated theory given by taking topological string theory the SUSY operators in these theoriescan can be be interpreted realized clearly as BPS inexpectation branes the value in using topological the the string full techniques picture, string of theory topological giving which string us theory. the ability to calculate their function of the topological string computes exactly that of the 4D PoS(TASI2017)015 , , (1996) (B.47) 54 , ]. Nucl. Phys. B471 , Cumrun Vafa Int. J. Theor. (2010) , from first prin- 60 ) Mod. Phys. Lett. , (1995) 329–334 ]. SW (2008) 483–610, Σ , Phys. Rev. Lett. 87 (1995) 85–126 Nucl. Phys. , , B357 SW λ to integrable models. B443 hep-th/9602022 ]. ) = ( S ,[ Σ . , Les Houches Phys. Lett. , , ydx ydx ( hep-th/0012104 Σ Nucl. Phys. Ann. Rev. Nucl. Part. 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