Recent developments in heterotic compactifications Eric Sharpe University of Utah & Virginia Tech
Outline Nonperturbative corrections to heterotic strings Potential heterotic swampland, and exotic heterotic CFT constructions to fill the gaps String compactifications on stacks
Part 1: Nonperturbative corrections in heterotic strings (S. Katz, E.S.)
Roughly, two sources of nonperturbative corrections in heterotic strings: * Gauge instantons (& 5-branes) * Worldsheet instantons -- from strings wrapping minimal-area 2-cycles (``holomorphic curves ) in spacetime I ll focus on the latter class.
Worldsheet instantons generate effective superpotential terms in target-space theory. For ex, for a heterotic theory with rk 3 bundle breaking E 8 to E 6, * (27 * ) 3 couplings -- on (2,2) locus, computed by A model * (27) 3 couplings -- on (2,2) locus, computed by B model * singlet couplings -- Beasley-Witten, Silverstein-Witten, Candelas et al
Off the (2,2) locus, there exist analogues of the A and B models. * No longer strictly TFT s, though become TFT s on the (2,2) locus * Nevertheless, some correlation functions still have a mathematical understanding * New symmetries: (0,2) A model on same as (0,2) B model on (X, E) (X, E )
Application: (0,2) mirror symmetry Ordinary mirror symmetry exchanges pairs of topologically-distinct spaces, dualizing quantum-corrected computations into classical computations, and exchanging cohomology. Ex: Quintic Mirror 1 0 0 0 1 0 1 101 101 1 0 1 0 0 0 1 1 0 0 0 101 0 1 1 1 1 0 101 0 0 0 1
(0,2) mirror symmetry * exchanges pairs (space, bundle) * sheaf cohomology exchanged: H 1 (X 1, E 1 ) H 1 (X 2, E 2 ) Very little known. Ex: numerical evidence Horizontal: h 1 (E) h 1 (E ) Vertical: h 1 (E) + h 1 (E ) where E is rk 4 (R Blumenhagen, R. Schimmrigk, A. Wisskirchen, 97)
Ordinary A model g ij φ i φ j + ig ij ψ j D zψ i + ig ijψ j + D zψ i + + R ijkl ψi + ψj + ψk ψl Fermions: ψ ( i χ i ) Γ((φ T 0,1 X) ) ψ+( i ψz) i Γ(K φ T 1,0 X) ψ ı ( ψı z ) Γ(K φ T 0,1 X) ψ+ ı ( χı ) Γ((φ T 1,0 X) ) Under the scalar supercharge, δφ i χ i, δφ ı χ ı δχ i = 0, δχ ı = 0 δψz i 0, δψı z 0 so the states are O b i1 i p ı 1 ı q χ ı1 χ ı q χ i1 χ i p H p,q (X) Q d
`Half-twisted (0,2) model g ij φ i φ j + ih ab λ b D zλ a + ig ijψ j + D zψ i + + F ijab ψi + ψj + λa λb Fermions: λ a Γ(φ E) ψ+ i Γ(K φ T 1,0 X) λ b Γ(K φ E) ψ+ ı Γ((φ T 1,0 X) ) Constraints: Λ top E = K X, ch 2 (E) = ch 2 (T X) States: O b ı1 ı n a 1 a p ψ ı 1 + ψ ı n + λ a 1 λa p Hn (X, Λ p E ) When E = T X, reduces to the A model, since H q (X, Λ p (T X) ) = H p,q (X)
A model classical correlation functions χ i, χ ı For X compact, have n zero modes, plus bosonic zero modes, so < O 1 O m > = X X H p 1,q 1 (X) H p m,q m (X) Selection rule from left, right U(1) R s: p i = i i q i = n Thus: < O 1 O m > X (top-form)
(0,2) model classical correlation functions For Xcompact, we have n ψ+ ı zero modes and r λ a zero modes: < O 1 O m > = X Selection rule: H q 1 (X, Λ p 1 E ) H q m (X, Λ p m E ) i q i = n, i p i = r < O 1 O m > X Htop (X, Λ top E ) The constraint Λ top E = K X makes the integrand a top-form.
A model -- worldsheet instantons Moduli space of bosonic zero modes = moduli space of worldsheet instantons, M If no zero modes, then ψ i z, ψı z < O 1 O m > M Hp 1,q 1 (M) H p m,q m (M) More generally, < O 1 O m > M Hp 1,q 1 (M) H p m,q m (M) c top (Obs) In all cases: < O 1 O m > (top form) M
(0,2) model -- worldsheet instantons The bundle on induces E X a bundle F (of λ zero modes) on M : F R0 π α E where On the (2,2) locus, where π : Σ M M, E = T X, have α : Σ M X F = T M When no `excess zero modes, < O 1 O m > M Htop (M, Λ top F ) Apply anomaly constraints: Λ top E = K X ch 2 (E) = ch 2 (T X) } GRR = Λ top F = K M so again integrand is a top-form.
(0,2) model -- worldsheet instantons General case: < O 1 O m > ) M H qi (M, Λ pi F H n (M, Λ n F Λ n F 1 Λ n (Obs) ) where ψ j + T M = R 0 π α T X ψ+ i Obs = R 1 π α T X λ a F = R 0 π α E λ b F 1 R 1 π α E (reduces to A model result via Atiyah classes) Apply anomaly constraints: Λ top E = K X ch 2 (E) = ch 2 (T X) } GRR = Λ top F Λ top F 1 Λ top (Obs) = K M so, again, integrand is a top-form.
To do any computations, we need explicit expressions for the space M and bundle F. Luckily for us, the `gauged linear sigma model naturally provides such expressions. Gauged linear sigma models are 2d gauge theories, generalizations of the CP N model, important in string compactifications. The 2d gauge instantons of the gauge theory = worldsheet instantons in IR NLSM
Example: CP N-1 Have N chiral superfields, each charge 1 x 1,, x N For degree d maps, expand: x i = x i0 u d + x i1 u d 1 v + + x id v d where u, v are homog coord s on worldsheet = P 1 Take (x ij ) to be homogeneous coord s on M, then M = P N(d+1) 1
Can do something similar to build F. Example: completely reducible bundle E = a O( n a ) Left-moving fermions are completely free. Expand in zero modes: λ a = λ a0 u n a d+1 + λ a1 u n a d v + Each λ ai O( n a) on M Thus: ( ) F = a H 0 P 1, O( n a d) C O( n a )
Quantum cohomology is an OPE ring. Should be familiar from 4d gauge theories, e.g. Cachazo-Douglas-Seiberg-Witten. 4d N=1 SU(N) SYM 2d susy CP N-1 S N = Λ 3N x N = q W = S(1+log( Λ 3N /S N )) W = Σ(1+log( Λ N / Σ N )) In 2d case, the OPE looks like a mod of classical cohomology rel n, hence `quantum cohomology
Quantum cohomology The OPE corresponds to the corr f ns: < x k > = { q m if k = mn + N 1 0 else Ordinarily need (2,2) susy, but: * Adams-Basu-Sethi conjectured may exist for (0,2) * Katz-E.S. computed, found ring structure * Adams-Distler-Ernebjerg found gen l argument
Quantum cohomology ABS studied a (0,2) theory describing P 1 xp 1 with gauge bundle = def of tangent bundle They conjectured: X 2 = exp(it 2 ) X 2 (ɛ 1 ɛ 2 )X X = exp(it 1 ) (a def of the q.c. ring of P 1 xp 1 )
Quantum cohomology Katz-E.S. checked by directly computing, using technology outlined so far: < X 4 > = < 1 > exp(2it 2 ) = 0 < X X 3 > = < (X X) X 2 > = < X X > exp(it 2 ) = exp(it 2 ) < X 2 X2 > = < X 2 > exp(it 2 ) = (ɛ 1 ɛ 2 ) exp(it 2 ) < X 3 X > = exp(it1 ) + (ɛ 1 ɛ 2 ) 2 exp(it 2 ) < X 4 > = 2(ɛ 1 ɛ 2 ) exp(it 1 ) + (ɛ 1 ɛ 2 ) 3 exp(it 2 ) and so forth, verifying the prediction.
Summary of part 1: There exist (0,2) generalizations of A, B models that compute nonpert corrections in heterotic strings. (S. Katz, E.S.) Part 2: A potential heterotic swampland and exotic heterotic CFT constructions (J. Distler, E.S.)
There s been a lot of interest recently in the landscape program, an attempt to extract phenomenological predictions by doing statistics on the set of string vacua. One of the problems in the landscape program, is that those string vacua are counted by low-energy effective field theories, and it is not clear that all of those have consistent UV completions -- not all of them may come from an underlying quantum gravity. (Banks, Vafa)
One potential such problem arises in heterotic E 8 xe 8 strings. The conventional worldsheet construction builds each E 8 using a (Z 2 orbifold of a) set of fermions. L = g µν φ µ φ ν + ig ij ψ j + ψ i + + h ab λ b + λ a + The fermions realize a Spin(16) current algebra at level 1, and the Z 2 orbifold gives Spin(16)/Z 2. Spin(16)/Z 2 is a subgroup of E 8, and we use it to realize the E 8.
In more detail: Adjoint rep of E 8 decomposes into adjoint of Spin(16)/Z 2 + spinor: 248 = 120 + 128 left R sector left NS sector So we take currents transforming in adjoint, spinor of Spin(16)/Z 2, and form E 8 via commutation relations. More, in fact: all E 8 d.o.f. are realized via Spin(16)/Z 2
This construction has served us well for many years, but, in order to describe an E 8 bundle w/ connection, that bundle and connection must be reducible to Spin(16)/Z 2. After all, all info in kinetic term h αβ λ α D +λ β Can this always be done? Briefly: Bundles -- yes (in dim 9 or less) Connections -- no. Heterotic swampland?
Reducibility of (R. Thomas) connections On a p-pal G bundle, even a trivial p-pal G bundle, one can find connections with holonomy that fill out all of G, and so cannot be understood as connections on a p-pal H bundle for H a subgroup of G: just take a conn whose curvature generates the Lie algebra of G. Thus, just b/c the bundles can be reduced, doesn t mean we re out of the woods yet.
Reducibility of (R. Thomas) connections We ll build an example of an anomaly-free gauge field satisfying DUY condition that does not sit inside Spin(16)/Z 2. The basic trick is to use the fact that E 8 has an (SU(5)xSU(5))/Z 5 subgroup that does not sit inside Spin(16)/Z 2. We ll build an (SU(5)xSU(5))/Z 5 connection.
E 8 Spin(16)/Z 2 SU(5) SU(5) Z 5
Reducibility of connections Build a stable SU(5) bundle on an elliptically-fibered K3 using Friedman-Morgan-Witten technology. Rk 5 bundle with c 1 =0, c 2 =12 has spectral cover in linear system 5σ + 12f, describing a curve of genus g = 5c 2 5 2 + 1 = 36 together with a line bundle of degree (5 + g 1) = 40
Reducibility of (R. Thomas) connections Result is a (family of) stable SU(5) bundles with c 2 =12 on K3. Holonomy generically fills out all of SU(5). Put two together, and project to Z 5 quotient, to get (SU(5)xSU(5))/Z 5 bundle w/ connection that satisfies anomaly cancellation + DUY.
Lessons for the Landscape E 8 Standard worldsheet construction applies Lowenergy gauge group 0 Standard construction does not apply Statistics on trad l CFT s artificially favors large gps
Alternative E 8 constructions Next, we ll describe alternative worldsheet constructions, starting with 10d flat space. Idea: Replace Spin(16)/Z 2 with other maximal-rank subgroups of E 8, realized as orbifolds of abstract current algebras (since only U(n), Spin(n) have level 1 free field rep s) To be specific, we ll describe ( SU(5) x SU(5) )/Z 5
Alternative E 8 constructions Take current algebras for two copies of SU(5) at level 1, and orbifold by a Z 5 Check: central charge of each SU(5) = 4, so adds up to 8 = central charge of E 8
Check: characters Alternative E 8 constructions χ SU(5) (1, q) = 1 η(τ) 4 χ SU(5) (5, q) = 1 η(τ) 4 χ SU(5) (10, q) = 1 η(τ) 4 q ( m 2 i +( m i ) 2 )/2 m Z 4 m Z 4, m i 1 mod 5 m Z 4, m i 2 mod 5 q ( m 2 i 1 5 ( m i ) 2 )/2 q ( m 2 i 1 5 ( m i ) 2 )/2 Can show χ E8 (1, q) = χ SU(5) (1, q) 2 + 4 χ SU(5) (5, q) χ SU(5) (10, q) (E. Scheidegger) (Kac, Sanielevici) so E 8 worldsheet d.o.f. can be replaced by SU(5) 2
Alternative E 8 constructions Analogous statement for Spin(16)/Z 2 is χ E8 (1, q) = χ Spin(16) (1, q) + χ Spin(16) (128, q) There is a Z 2 orbifold implicit here -- the 1 character is from untwisted sector, the 128 character is from twisted sector. Sim ly, in the expression χ E8 (1, q) = χ SU(5) (1, q) 2 + 4 χ SU(5) (5, q) χ SU(5) (10, q) there is a Z 5 orbifold implicit. Good!: SU(5) 2 /Z 5
Alternative E 8 constructions Another max-rank subgroup: SU(9)/Z 3. Check: central charge = 8 = that of E 8. E 8 conformal family decomposes as [1] = [1] + [84] + [84*] Can show χ E8 (1, q) = χ SU(9) (1, q) + 2 χ SU(9) (84, q) (Note Z 3 orbifold implicit.) So, can describe E 8 w.s. d.o.f. with SU(9)/Z 3.
Alternative E 8 constructions To make this useful, however, we ll need to fiber the current algebras over general base spaces. To that end, we next describe fibered WZW models. (J Distler, ES; J Gates, W Siegel, etc)
Fibered WZW models S = k 2π First, recall ordinary WZW models. Σ Tr [ g 1 gg 1 g ] ik 2π B d 3 yɛ ijk Tr [ g 1 i gg 1 j gg 1 k g ] Looks like sigma model on mfld G w/ H flux. Has a global G L xg R symmetry, with currents J(z) = g 1 g J(z) = g g 1 obeying J(z) = J(z) = 0 -- realizes G Kac-Moody algebra at level k
Fibered WZW models Let P be a principal G bundle over X, with connection A. Replace the left-movers of ordinary heterotic with WZW model with left-multiplication gauged with A. 1 α k 4π k 2π Tr Σ Σ ( gij α φ i α φ j + ) NLSM on X Tr ( g 1 gg 1 g ) ik d 3 yɛ ijk Tr ( g 1 i gg 1 j gg 1 k g ) 12π B (( φ µ )A µ gg 1 + 12 ) ( φµ φ ν )A µ A ν Gauge left-multiplication WZW
Fibered WZW models A WZW model action is invariant under gauging symmetric group multiplications, but not under the chiral group multiplications that we have here. Under A µ g hg ha µ h 1 + h µ h 1 the classical action is not invariant. As expected -- this is bosonization of chiral anomaly.... but this does create a potential well-definedness issue in our fibered WZW construction...
Fibered WZW models In add n to the classical contribution, the classical action also picks up a quantum correction across coord patches, due to right-moving chiral fermi anomaly. To make the action gauge-invariant, we proceed in the usual form for heterotic strings: assign a transformation law to the B field. Turns out this implies k ch 2 (E) = ch 2 (T X) Anom canc at level k If that is obeyed, then action well-defined globally.
Fibered WZW models The right-moving fermion kinetic terms on the worldsheet couple to H flux: where i 2 g µνψ µ + D zψ ν + D z ψ µ + = ψµ + + φµ ( Γ ν σµ Hν σµ) ψ σ + To make fermion kinetic terms gauge-invariant, set H = db + (α ) (kcs(a) CS(ω)) k ch 2 (E) = ch 2 (T X) Anomaly-cancellation
Fibered WZW models Demand (0,2) supersymmetry on base. Discover an old faux-susy-anomaly in subleading terms in α Off-shell susy trans in ordinary heterotic string: δλ = iɛψ µ + A µλ -- same as a chiral gauge transformation, with parameter iɛψ µ + A µ -- b/c of chiral anomaly, there is a quantum contribution to susy trans at order α -- appears classically in bosonized description (Sen)
Fibered WZW models (0,2) supersymmetry: One fermi-terms in susy transformations of: NLSM Base: WZW fiber: Quantum: 1 α k Σ Σ Σ (iαψ ı ) φ µ φ ν (H db) ıµν (iαψ ı ) φ µ φ ν CS(A) ıµν (iαψ ı ) φ µ φ ν CS(ω) ıµν H = db + α (kcs(a) CS(ω)) for susy to close
Fibered WZW models Massless spectrum: In an ordinary WZW model, the massless spectrum is counted by WZW primaries, which are associated to integrable rep s of G. Here, for each integrable rep R of the principal G bundle P, we get an associated vector bundle. E R Massless spectrum = H (X, E R ) for each R.
Fibered WZW models Massless spectrum: Example: G = SU(n), level 1 Here the integrable reps are the fundamental n and its exterior powers. Massless spectrum: H (X, Λ E) (Distler-Greene, 88)
Fibered WZW models Elliptic genera: These fibered WZW constructions realize the `new elliptic genera of Ando, Liu. Ordinary elliptic genera describe left-movers coupled to a level 1 current algebra; these, have left-moving level k current algebra. Black hole applications?
Summary of part 2: There exist perturbative vacua which cannot be described by trad l heterotic worldsheets; there exist alternative worldsheet constructions which describe them -- no swampland. (J. Distler, E.S.) Part 3: String compactifications on stacks (S. Hellerman, T. Pantev, E.S.,...)
Stacks Stacks are a mild generalization of spaces. One would like to understand strings on stacks: -- to understand the most general possible string compactifications -- they often appear physically inside various constructions
Stacks What is a stack, roughly? We can cover with coordinate charts, just like a manifold. So, locally, looks just like a space. Globally, on a manifold, across triple overlaps, the coordinate charts close to 1. On a stack, across triple overlaps, the coordinate charts need only close up to an automorphism. Locally, just like a space -- difference is global.
Stacks How to make sense of strings on stacks concretely? Every* (smooth, Deligne-Mumford) stack can be presented as a global quotient [X/G] for X a space and G a group. To such a presentation, associate a G-gauged sigma model on X. (* with minor caveats)
Stacks If to [X/G] we associate ``G-gauged sigma model, then: [C 2 defines a 2d theory with a symmetry /Z 2 ] called conformal invariance [X/C ] defines a 2d theory w/o conformal invariance Potential presentation-dependence problem: fix with renormalization group flow (just as with derived categories)
Stacks Potential problems / reasons to believe that presentation-independence fails: Deformations of * Deformations of stacks = physical theories * Cluster decomposition issue for gerbes These potential problems can be fixed. (ES, T Pantev) Results include: mirror symmetry for stacks, new Landau-Ginzburg models, physical calculations of quantum cohomology for stacks, understanding of noneffective quotients in physics
Quantum cohomology Ex: Quantum cohomology ring of CP N is C[x]/(x N+1 - q) Quantum cohomology ring of Z k gerbe over CP N with characteristic class -n mod k is C[x,y]/(y k - q 2, x N+1 - y n q 1 ) -- so, even though isn t a space, one can still make sense of these sorts of notions
Toda duals Ex: The ``Toda dual of CP N is described by the holomorphic function W = exp( Y 1 ) + + exp( Y N ) + exp(y 1 + + Y N ) The analogous duals to Z k gerbes over CP N are described by W = exp( Y 1 ) + + exp( Y N ) + Υ n exp(y 1 + + Y N ) where Υ is a character-valued field (ES, T Pantev, 05)
Mirrors to stacks More generally, there exists a notion of toric stacks (Borisov, Chen, Smith, 04) which allows us to realize sigma models on many stacks in terms of simple 2d gauge theories. Standard mirror constructions now produce discretevalued fields, a new effect, which ties into the stacky fan description of (BCS 04). (ES, T Pantev, 05)
Gerbes We now believe that for (banded, abelian) G-gerbes on spaces X, (2,2) strings cannot distinguish between -- the gerbe itself -- disjoint union of copies of X, with ``flat B fields H 2 (X, G) G U(1) H 2 (X, U(1)) (Comes up in mirror constructions; meaning of ``fields valued in roots of unity; quantum cohomology implications) (S. Hellerman, A Henriques, T Pantev, E.S., M Ando 06)
Gerbes This result can be applied to understand GLSM s. Example: P 7 [2,2,2,2] At the Landau-Ginzburg point, have superpotential a p a G a (φ) = ij φ i A ij (p)φ j * mass terms for the φ i, away from locus {det A = 0}. * leaves just the p fields, of charge -2 * Z 2 gerbe, hence double cover
Gerbes Ex, cont d P 7 [2,2,2,2] The GLSM realizes: branched double cover of P 3 where RHS realized at LG point via local Z 2 gerbe structure + Berry phase. (A. Caldararu, J. Distler, S. Hellerman, T. Pantev, E.S., in progress) Non-birational twisted derived equivalence Physical realization of Kuznetsov s homological projective duality
Gerbes Although (2,2) models decompose into a disjoint union, (0,2) models do not seem to in general. Prototype: O(1) P [k,k,,k] `` O(1/k) -- understanding of some of the 2d (0,4) theories appearing in geometric Langlands program -- genuinely new string compactifications Another lesson for the landscape: many more string vacua may exist than previously enumerated.
Future directions (0,2) mirror symmetry non-kahler compactifications new heterotic string compactifications (eg gerbes), & new GLSM understanding supercritical strings