K3 metrics from string theory

Shamit Kachru (Stanford) Based on work with Arnav Tripathy and Max Zimet…

arXiv: 1810.10540 arXiv: 2006.02435 (Plus, Max and Arnav have been extending the results and making them into rigorous math while I’ve been doing chair stuff…) 1. Introduction

Calabi conjecture: There is a unique Ricci flat Kahler metric on a Kahler manifold of vanishing first Chern class for each choice of the (cohomology class of the) Kahler form.

Proved by Yau.

The result is famously non-constructive. String theorists are interested because these spaces solve the vacuum Einstein equations relevant in superstring compactification.

More generally, they play a central role in various considerations in duality, differential geometry, and algebraic geometry. Today, I’ll describe some conceptually new approaches to the problem of understanding these metrics in a “Goldilocks” example of such a space.

Our setting will be the simplest compact CY manifold — K3.

We are going to attack the problem of finding its Ricci flat metric in a way that is broadly familiar to aficionados of 3d mirror symmetry: Coulomb side Higgs side Disk instanton sums of algebro- geometic interest Little string on T 3 D2 brane probe

g = g +1 loop + instantons g = gclass class GMN formalism Hyperkahler quotient

Ties to Hitchin question: K3 as infinite dimensional HK quotient?

Accordingly, I will spend a great deal of time developing the quantum geometry of the Coulomb branch, but the true solution emerges from the dual perspective. II. Little string theories and their moduli spaces

A. Simple new theories without gravity

One of the discoveries of the mid 1990s was a zoo of novel theories not previously expected to exist.

Tools of discovery:

Study the low-energy theories on p-branes embedded in the 10d string theory. Study a singular Calabi-Yau and focus on “local physics” at the singularity.

A simple Calabi-Yau singularity is an A-D-E singularity of a K3 surface.

It has a dual description in terms of so-called “NS 5 branes.” By carefully studying the physics of the singular K3 or on the branes, one finds several new theories.

— Famously, there is a (2,0) supersymmetric field theory in 5+1 dimensions. It can be used to explain S-duality.

— Slightly less famously, there is also a theory which reduces to this to low energies but is smaller than the full string theory. It is a so-called “little string theory” living on the NS 5-branes.

The latter is not a quantum field theory, and comes with a scale: the scale of its strings. The simplest (2,0) NS 5 brane theory has many interesting properties.

— Compactified on a 2-torus, it gives rise to a theory which looks - at low energy - like a U(1) gauge theory.

— The (Coulomb branch) moduli space of this theory is the (dual of the) mirror torus. Compactifying on an additional circle, one gets two additional scalars:

— The abelian gauge field present on the Coulomb branch gives a Wilson line on the circle, which is periodic.

— This leaves a 3d photon. But in 3d, via

⇢ fµ⌫ = ✏µ⌫⇢@

one can dualize this to a scalar.

From the M theory picture, you see this theory has a moduli space given by a four-torus. We review this because we will use an analogous, but more complicated, system where there is a little string theory whose moduli space is K3.

B. A theory with K3 as its moduli space

The theory we were talking about has 16 supercharges, or the equivalent of 4d N=4 supersymmetry.

The one we’re interested in is a cousin — the first one discovered — that has 4d N=2 supersymmetry. The salient properties of this theory are:

— it arises on NS 5-branes in the heterotic string, or on D5-branes in the type I string.

— one description: 6d SU(2) gauge theory with 16 doublet hypermultiplets (SO(32) global symmetry). — when compactified on a two-torus, it is (at low energies) a rank 1 N=2 field theory; but its Coulomb branch is a sphere.

This sphere has 24 marked points on it; in a dual picture, think of D3-branes moving on base of elliptic K3. Now, when one compactifies this theory on an additional circle, one gets a 3d theory whose moduli space is a K3 surface! Intriligator

The Wilson line of the gauge field, together with the 3d dual photon, give the elliptic fiber above each point on the base sphere. III. Hyperkahler metrics on Coulomb branches

Let us leave this story for a moment, and discuss some generalities about 4d N=2 field theories.

Multiplets:

Vector: , , ,Aµ

˜ ˜ Hyper: Q, Q, Q, Q˜ Moduli space of vacua:

Coulomb branch: special Kahler Higgs branch: hyperKahler On compactification to 3d, the Coulomb branch becomes hyperKahler too.

e.g. for a one-dimensional (complex) 4d Coulomb branch:

* In the F-theory D3 probe picture, fibers degenerate at D7 branes due to “light electrons.” (These electrons arise from the open strings stretched between the D3 and the D7.) How does one find the metric on the 4d and 3d Coulomb branch?

A. The 4d story

Calling the Coulomb branch modulus “a,” for each point on the “a”-plane one has an auxiliary structure: a Seiberg-Witten curve.

In our case of interest, it is of genus 1.

The periods of a preferred meromorphic differential on this curve then determine the geometry: = a a @2 Z ⌧ = F @ @a2 = F @a Zb

The central object is the prepotential F. In terms of F, the Kahler potential is:

@F K =Im a¯ @a ✓ ◆

All states satisfy a BPS bound:

M Z(a; q ,q ) | e m |

States which saturate the bound are in “short” multiplets and are exactly stable. B. The 3d story

Our 4d theory has an action:

At a given point in moduli space, there is an auxiliary torus in the story with Reducing to 3d, the torus becomes physical.

The action for the new scalar fields spanning this torus is given by:

This gives an elliptic fiber E over the old Coulomb branch, with area controlled by The leading large R metric is then given by combining the kinetic terms for the new scalars with those of the 4d Coulomb branch fields.

There are three sources of physical corrections to a naive classical picture:

— one loop corrections in 4d.

— corrections from 4d instantons. — instanton corrections from BPS particles in the 4d theory running around the circle.

Gaiotto-Moore-Neitzke have developed a beautiful story for how to compute the hyperKahler metric on a Coulomb branch given the indices defined in the presence of certain massive probe particles:

F 2⇡RH i✓ X (u, ✓; ⇣)=Tr( 1) e e Q

The trace is taken in the presence of a massive probe particle of charge . Here ⇣ parametrizes the choice of complex structure on the hyperKahler Coulomb branch; there is a sphere of such possible choices.

Now, define the triplet of two forms

“holomorphic symplectic form”

The definition uses the three Kahler forms at the north, front, and right poles of the sphere. It follows from standard results in hyperKahler geometry that

And it is a result of GMN that

Darboux coordinates

where Upshot: if we can compute the X for a basis of the charge lattice, we are in business.

Now lets see how this all fits together in some formulae.

IV. Toy formulae

A. The large radius limit

We should expect from the physics that at large radius of the circle, our formulae simplify. (This is a semiclassical limit from the 3d point of view.) Recall that to compute a line operator index in charge sector , we need to evaluate

In the large radius limit, semiclassical reasoning gives:

For the problem at hand, using the 4d formulae for the Coulomb branch geometry gives the known “semi-flat” metric. That is, there are 24 points with a light electron (in some duality frame), and one finds:

Greene, Shapere, Here: Vafa,Yau

u are heuristically coordinates on the 4d Coulomb branch, and z coordinates on the elliptic fibers: B. Including the instantons

To get a more complete answer, one needs to include the instantons.

In this duality frame, one states the answer in terms of BPS numbers. They are morally more “topological” information than a metric.

Heuristically, one expects: Formally, the Xs solve a Riemann-Hilbert problem written down by GMN. One can show that they solve an integral equation

where

⌦(; u) = BPS count in charge sector

This can be solved by an iterative technique, given the BPS counts. Something rather similar to the Gross-Wilson approximation to the metric will emerge if one works on a smooth elliptic K3, and includes only the instanton corrections to the U(1) gauge theory at each “electron point.”

The most important question is: how do we determine the BPS invariants?

Happily, string duality answers this question: it provides a dual frame, where the metric computation is completely classical! V. The Higgs branch formalism

We got this far by studying the heterotic NS5-brane compactified on a three-torus.

Famously, there is a duality:

Het on T 3 M on K3 $ Wrapped NS5 Transverse M2 $

We can reduce the M-theory side on a transverse circle, turning the M2-brane into a D2-brane of IIA string theory. — The IIA string coupling (which interpolates between D2 and M2 pictures) does not control any corrections to the Higgs branch metric.

— So the Higgs branch metric of the D2 theory will give us the desired K3 metric; this is an analogue of 3d mirror symmetry.

(It also encode an infinite number of BPS state counts, via a “mirror map” to the Coulomb branch picture!)

So, how can we obtain a tractable formulation of the classical Higgs branch of a D2 on K3? Consider a D2 probing an orbifold K3. For D-branes on orbifolds, there is a completely explicit formulation,

including orbifold blow-up modes. Douglas, Moore

Let us work up to the relevant theory in little steps. Example 1: N D2s transverse to C2

Lets start by studying the 3d N=8 theory on D2s probing flat space.

— set dual photon and its real scalar partner to vanish

— break the other three (complex) scalars into an adjoint in the 3d N=4 vector multiplet, and a hyper.

W = g Tr ([U, V ]) [U, V ]=0

D flat : [U, U †]+[V,V †]=0 Solutions are diagonal U,V — up to action of permutation group, which is the Weyl group of the gauge symmetry.

2 Example 2: Quotient C /Z2

We Introductionstart withLittle string a theory covering and K3 3d mirror symmetry theoryHyper-Kähler with quotient a D2BPS spectrum andConclusion its image. Gauge theory (This is the N=2 case of Example 1.) 2 C /Z2 2 I D2-brane probing C /Z2. Worldvolume is obtained by startingOrbifold on C2 coveringaction space imposes with D2-brane projections: and its image and the imposing orbifold projections. I So, starting point is the N = 2 theory from last slide. We then require

U = U , V = U , g = g z z z z z z i↵/2 u+ v+ i✓ e U = , V = , g = e i↵/2 u v e ✓ ◆ ✓ ◆ ✓ ◆ u+ u I µ+ = 0 = ,µR = 0 = 1. ) v+ v ) | | ✓ ◆ ✓ ◆ I U(1): = 1; ↵ = ⇡: (u, v) ( u, v) ⇠ M. Zimet Harvard K3 Metrics Introduction Little string theory and K3 3d mirror symmetry Hyper-Kähler quotient BPS spectrum Conclusion

Gauge theory 2 C /Z2 2 I D2-brane probing C /Z2. Worldvolume is obtained by starting on C2 covering space with D2-brane and its image and the imposing orbifold projections. I So, starting point is the N = 2 theory from last slide. We

Introduction thenLittle string require theory and K3 3d mirror symmetry Hyper-Kähler quotient BPS spectrum Conclusion

Introduction Little string theory and K3 3d mirror symmetry Hyper-Kähler quotient BPS spectrum Conclusion Gauge theory U = z Uz , V = z Uz , g = z gz Gauge theory 2 2 i↵/2 C /Z2 C /Z2 u+ v+ i✓ e U2 = Superpotential, V constraint:= , g = e I D2-brane probing /Z . Worldvolume is obtained2 by i↵/2 IC 2 u / v e 2 D2-brane probing C Z2. Worldvolume is obtained by starting on C covering space✓ with D2-brane2◆ and its image✓ ◆ ✓ ◆ and the imposing orbifoldstarting projections. on C covering space with D2-brane and its image u+ u I So, starting pointI isµand the+ =N the=02 theory imposing from last= orbifold slide. We projections.,µR = 0 = 1. then require ) v+ v ) | | I So, starting✓ point◆ is the✓N=◆ 2 theory from last slide. We I ( ) = ↵ = ⇡ ( , ) ( , ) U = UU, 1V:= U1; , g = : gu v u v z zthen require z z z z ⇠ i↵/2 M. Zimetu+ v+D-term constraint:i✓ e Harvard U = , V = , g = e i↵/2 K3u Metrics v U = z Uz , V e= z Uz , g = z gz ✓ ◆ ✓ ◆ ✓ ◆ u+ u i↵/2 I µ+ = 0 = ,µR =u 0 = 1. v e ) v+ v + ) | | + i✓ ✓ ◆ U✓=◆ , V = , g = e i↵/2 I U(1): = 1; ↵ = ⇡: (u, v) u( u, v) v e ✓⇠ ◆ ✓ ◆ ✓ ◆ M. Zimet Harvard K3 Metrics u+ u I µ+ = 0 Using gauge= freedom: ,µR = 0 = 1. ) v+ v ) | | ✓ ◆ ✓ ◆ I U(1): = 1; ↵ = ⇡: (u, v) ( u, v) ⇠ M. Zimet Harvard K3 Metrics Example 3: Symk(T 4)

WeIntroduction can useLittle the string technology theory and K3 3d developed mirror symmetry byHyper-Kähler Taylor quotient (’96) toBPS spectrum Conclusion

Gauge theorycapture the dynamics on a compact space. k Sym T 4 T 4 = R4/⇤

4 4 The k branesI Compact lift to an exampleinfinite number [Taylor ’96].on theT covering= R /⇤, space. so starting point is 3d = 8 U(k 4) theory. Chirals U, V are N 1 Matter fields:matrices: U ; , i, j = 1,...,k, m, n ⇤. We’ll leave the i, j im jn 2 indices implicit, so Umn denotes a k k matrix. (and similar for V) ⇥ I Orbifold projections:

Lets leave the U(k) gauge indicesu implicit, and so label v U(m+`)(n+`) = Umn + ` Imn , V(m+`)(n+`) = Vmn + ` Imn the fields as e.g. Umn . `u = `1 + i`2 , `v = `3 + i`4

Only Un U0n and Vn V0n are independent: ⌘ ⌘ u Umn = Un m if m = n, Unn = U0 + n I. 6

M. Zimet Harvard K3 Metrics Introduction Little string theory and K3 3d mirror symmetry Hyper-Kähler quotient BPS spectrum Conclusion

Gauge theory Symk T 4

4 4 I Compact example [Taylor ’96]. T = R /⇤, so starting point is 3d = 8 U(k 4) theory. Chirals U, V are N 1 matrices: U , i, j = 1,...,k, m, n ⇤. We’ll leave the i, j im;jn 2 indices implicit, so U denotes a k k matrix. Then the orbifoldmn projections⇥ yield: I Orbifold projections:

u v U(m+`)(n+`) = Umn + ` Imn , V(m+`)(n+`) = Vmn + ` Imn

`u = `1 + i`2 , `v = `3 + i`4

Only Un U0n and Vn V0n are independent: ⌘ ⌘ u Umn = Un m if m = n, Unn = U0 + n I. 6

M. Zimet Harvard K3In Metrics eventually deriving the low-energy dynamics, one would integrate out the massive (n>0) modes. This is not so interesting in the torus case…

But in the analogous construction for K3, the analysis of the (blown-up) orbifold just involves integrating out the massive modes to find the low-energy action for the zero modes. Introduction Little stringExample theory and K3 3d mirror4: symmetryKummerHyper-Kähler K3 quotient orbifoldBPS spectrum Conclusion Gauge theory K3 We now repeat the previous construction, with two simple modifications: 4 4 I Now, realize K3 as resolution of T /Z2; i.e., orbifold R by 4 ⇤, and then by Z2, or equivalently by D 4 = Z o Z2. Z 1 — replace[Ho-Wu the ’98, torus Ramgoolam-Waldram with its 2 ’98,quotient Greene-Lazaroiu-Yi ’98] 4 — study the U [(2( ) ) theory with projections I Start with U(2)1gauge theory and then impose Z2 projections:

U n = z Unz , V n = z Vnz , g n = z gnz

The resulting theory has 58 parameters: 16 x 3 FI parameters M. Zimet Harvard K3parameterized Metrics by choices of points in ⇤ / 2 ⇤ , and ten metric parameters for the covering torus. Integrating out the “massive” U n ,V n ,n> 0 and writing the kinetic term for the remaining low-energy scalars yields the K3 metric!

— checked at lowest non-trivial order in blow-up modes in arXiv:2006.02435

— checked that a suitable Poisson-resummation of the resulting formulae yields a match to the Coulomb branch picture, and allows one to read off BPS spectrum of e.g. SU(2) gauge theory with four flavors

— used by Tripathy and Zimet to reproduce (and predict new) BPS counts of Minahan-Nemeschansky theories Much further work — both to formulate expansions around other loci in moduli space, and to prove analysis results about convergence etc. of the resulting expansions — is under way by Tripathy, Zimet (Fredrickson, Mazzeo, …).

The resulting BPS invariants — in the Coulomb branch picture — have interpretations in enumerative geometry (as counts of holomorphic discs). These should constitute predictions for algebraic geometry.

Thanks for your attention.