The Gopakumar-Vafa Formula

Edward Witten
PiTP, June 19, 2014

In topological string theory, as you have heard in the lectures by
Hirosi Ooguri, one counts the holomorphic maps from a Riemannsurface Σ of genus g (with unspecified complex structure) to atarget space Y . Such a map Φ : Σ → Y determines a class~d ∈ H2 (Y , Z) which measures the “wrapping” of Σ around various
cycles in Y . For each class ~d, we let ag ,~d be the “number” ofholomorphic maps with that homology class where Σ has genus g .
I put the words “number” in quotes because there are many
subtleties in defining this number correctly.

(This talk is based on a paper to appear with M. Dedushenko.)We introduce a set of coupling constants ~t that is dual to thehomology class ~d, and define the genus g topological string
partition free energyX
Fg (~t) =ag ,~d exp(2πi ~d · ~t).
(1)~
d∈H2 (Y ,Z)

The full topological string partition function free energy isF(gst , ~t) =∞X

gst2g −2 Fg (~t).

g =0(The partition function including disconnected contributions is
Z = e F .)(2)

What does topological string theory have to do with physical string
theory? A remarkable answer was proposed by Bershadsky, Cecotti,
Ooguri and Vafa (BCOV; 1993) and confirmed in a hardcalculation by Antoniadis, Gava, Narain, and Taylor (AGNT;
1993). One focuses on Type IIA superstring theory compactified
on R4 × Y where Y is a Calabi-Yau three-manifold. (This case isimportant both in physical string theory – it is used to makesemi-realistic models of particle physics – and topological string
theory, which turns out to be particularly interesting in this case.)In this theory, there are b2 + 1 U(1) gauge fields that come fromthe Ramond-Ramond (RR) sector of superstring theory. One ofthem is simply the RR gauge field that is already present in ten
dimensions and the other b2 arise in Kaluza-Klein reduction of the
RR three-formb2
XC=AI (x) · ωI (y )I =1

where ωI (y ) are harmonic two-forms on Y , normalized to give abasis of H 2 (Y , Z) mod torsion, and AI (x) are U(1) gauge fields on
R4 .

Type IIA on a Calabi-Yau manifold Y has eight unbroken
supersymmetries and the effective action can be described in asuperspace with four bosonic coordinates x µ , four fermioniccoordinates θAi of negative chirality, and four more θ̄Ȧi of positivechirality. (Here all indices A, Ȧ, i take the values 1,2. A and Ȧ arespinor indices of respectively negative and positive chirality in four
dimensions, and the index i is there because Type IIA on a
Calabi-Yau has N = 2 supersymmetry in four dimensions.) A
chiral superfield is a function Ψ(x, θ) that does not depend on thee θ̄) does notθ̄’s, and conversely an antichiral superfield Ψ(x,depend on the θ’s. A generic superfield is a function of all eight
fermionic coordinates.

Of the b2 + 1 U(1) gauge fields, b2 linear combinations are in
vector multiplets, which for N = 2 supersymmetry in four
dimensions are described by chiral superfields
X Λ (x, θ) = X Λ + θΨΛ + θ2 F Λ + . . .
where F Λ is a U(1) gauge field strength. The remaining U(1)gauge field is the “graviphoton,” which is part of the supergravity
multiplet. It also is part of a chiral superfield, but this is not achiral superfield of spin zero. Rather, the anti-selfdual part of thegraviphoton field strength, which I will write as a bispinor WAB (x)µν(= WBA (x) = σABWµν (x)), partly because this is natural in string
perturbation theory, is the bottom component of a chiral superfieldthat itself is a bispinor WAB (x, θ) = WAB (x) + . . . .

In general, in a supersymmetric theory, a term in the effective
action might be a “D-term,” meaning thatRit can be written as anintegral over all of superspace, in our case d4 xd4 θd4 θ̄(. . . ), or an“F-term,” meaning
it cannot be so written. For example, an
R that4
4F-term might be d xd θO, where the operator O has to bechiral (independent of θ̄) so that one can get a supersymmetricresult just by integration over the four θ’s. It usually is only the
F -terms about which one can make nonperturbative statements
and my lecture today is concerned with F -terms.

In the case at hand, for every g ≥ 0, there is a possible F -term
Z
Ig = d4 x d4 θ Fg (X 0 , . . . , X b2 )(WAB W AB )g .
These particular interactions have the remarkable property that,from the standpoint of Type IIA superstring perturbation theory,
Ig is generated only in genus g . The proof of this just follows fromthe low energy supergravity. One transforms the interaction Ig to
the string frame and finds (to simplify slightly) that it is
proportional to gst2g −2 .

The insight of BCOV is that the function Fg (X Λ ) is the
topological string partition function Fg (~t) if one sets t I = X I /X 0 ,I = 1, . . . , b2 . (Technically, here X 0 is the multiplet that
corresponds to the RR 1-form in ten dimensions and
X I , I = 1, . . . , b2 correspond to the vector fields that arise fromthe C -field). This was already known for g = 0 but thegeneralization to g > 0 is quite remarkable. Unfortunately, the only
complete proof that is presently known is rather technical (AGNT).

What did Gopakumar and Vafa (1998) add to this story? Theysuggested that one should view Type IIA on R4 × Y in terms of
M-theory on R4 × S 1 × Y , where S 1 is sometimes called the
M-theory circle. In this correspondence, the string coupling
constant is determined by the radius of the M-theory circle. Since
Ig for given g has a known dependence on gst , its dependence on
the radius of the S 1 is essentially also known and we can calculate
in the region where the S 1 is large and the M-theory description isuseful.

In string perturbation theory, Fg (X Λ ) is computed by counting
superstring worldsheets wrapped on a complex submanifoldΣ ⊂ Y . This corresponds in M-theory to an M2-brane wrapped on
S 1 × Σ. Such an M2-brane can be studied in a Hamiltonian
formalism in terms of states propagating in the S 1 direction. So it
should be possible to compute the F -terms Ig by summing overcontributions of wrapped M2-brane states.

It possibly is intuitively obvious that the only M2-brane states thatcan contribute to an F -term are BPS states and later I will explain
why this is true. For now I give this naive explanation: BPS states
are states that are invariant under some supersymmetries, and they
can generate terms in the effective action that are integrals of
operators that are likewise invariant under some supersymmetries,
in other words F -terms. A state not invariant under any
supersymmetries, when it propagates around the circle, generates
only an operator not invariant under any supersymmetries, in otherwords a D-term.

Moreover, Gopakumar and Vafa had a remarkable idea on how tocompute the Fg . Their proposal was that instead of computing
one particular interactionZ
Ig = d4 x d4 θ Fg (X 0 , . . . , X b2 )(WAB W AB )g ,we should study the whole sum
I=

∞
X
g =0

Z
Ig =

d4 x d4 θ∞X

Fg (X 0 , . . . , X b2 )(WAB W AB )g .

g =0Moreover, they suggested that this sum should be viewed as thesuperspace effective action evaluated in a background in which aconstant anti-selfdual graviphoton field is turned on. This
superspace effective action is supposed to be computed by a
one-loop calculation, summing over one-loop diagrams with BPS
states running around the loop.

This calculation is supposed to be a supersymmetric variant of
Schwinger’s celebrated computation of a one-loop effective action
due to a charged particle in a constant magnetic field. The upshot
then would be to express the interaction I as a sum ofcontributions of BPS states. Moreover, Gopakumar and Vafa
proposed what the resulting formula would look like, and theresulting GV formula has had many applications and enormous
success.

There is also an Ooguri-Vafa (1999) analog of the GV formula. In
this case, one considers Type IIA on R4 × Y with D4-branes on
R2 × L where R2 ⊂ R4 and L ⊂ Y is a Lagrangian submanifold.
The M-theory lift is to R4 × S 1 × Y with M5-branes on
R2 × S 1 × L. Certain chiral interactions in Type IIA on R4 × Y
can be computed in the M-theory description by counting BPS
states that live on the M5-branes. I will not really have time to
explain this.

Before explaining the computation that leads to the GV formula, I
want to first address some general questions about it. Perhaps thefirst question is whether contributions to the interactions Ig canarise by dimensional reduction from a supersymmetric action in fivedimensions. To the extent that the Ig can arise that way, we canonly determine them by knowing the 5d effective action for
M-theory on R5 × Y ; an interaction that is already present in fivedimensions cannot be determined by a Schwinger computationthat involves further compactification from R5 to R4 × S 1 .

However, the Ig do not arise this way; only certain very special
and known terms in I0 and I1 can arise by dimensional reduction
from five dimensions. A not quite complete explanation of thisstatement is that the superfields X Λ contain axion-like modes that
decouple at zero momentum in any interaction that arises bydimensional reduction from five dimensions. But (except for very
special terms in I0 and I1 ) the interactions Ig spoil thisdecoupling, so they cannot arise by classical dimensional reduction.
This means that in principle we can do the calculation that wewant to do.

The next basic question is whether a supersymmetric backgroundwith a graviphoton field turned on does exist. A “no” answer
would mean we could not get anywhere. In field theory, we could
compute an effective action in an arbitrary background that is notnecessarily supersymmetric or a solution of any classical equationsof motion. We don’t really know how to do that instring/M-theory. But more important, even if we could do a
calculation, we would not learn anything from it. The reason isthat in a non-supersymmetricbackground, in which say there is a
R 4
S with d θ̄ S 6= 0, it is hard to distinguish an F -term
Rsuperfield
4 xd4 θ Φ(x, θ) (for some chiral superfield Φ) from a D-termd
R 4 4 4d xd θd θ̄ SΦ.

So we can only learn about F -terms by expanding about asupersymmetric background. In field theory, one can possibly use abackground that is supersymmetric but not a classical solution, but
it is doubtful that we would understand how to do that instring/M-theory. But at a minimum we need a supersymmetricbackground.

If one turns on an arbitrary linear combination of anti-selfdual
Λ on R4 , one will get no gravitational
U(1) gauge fields FAB
backreaction – since the energy momentum tensor of a selfdual oranti-selfdual Maxwell field vanishes. But generically one will get
scalar backreaction. It turns out that scalar backreaction is
avoided and moreover one gets a supersymmetric background
precisely if one turns on only the graviphoton and not any of thegauge fields that are in vector multiplets.

So a supersymmetric classical solution with the necessary
properties actually exists. It is most simply described as a solution
in a 5d spacetime with Lorentz signature. The solution was called
the supersymmetric Gödel solution by Gauntlett, Gutowski, Hull,
Pakis and Reall (GGHPR, 2003):2µ 2

ds = −(dt + Vµ dx ) +

4
X

1 − ν
(dx µ )2 , Vµ = Tµν
x ,
8

µ=1

− is the constant anti-selfdual graviphoton. To use this inwhere Tµν
M-theory compactification to Type IIA, we need to compactify thet direction but also we want the metric on the circle to by
Euclidean. So we rotate t → iy where y is a periodic variable.

− and the metric
It is not possible to make both the graviphoton Tµν
2
ds real. The lesser evil is to take the graviphoton imaginary sothat the metric is real. The graviphoton being imaginary in the
background is really not a problem since Schwinger’s original
calculation worked fine in either an electric or a magnetic field; animaginary magnetic field in Euclidean space is somewhat like a realelectric field in Lorentz signature.

This background has a remarkable property that was discovered by
GGHPR and also from a different perspective by Berkovits andSeiberg (2003). Naively, if one turns on an anti-selfdual gauge
field, one preserves at most one-half of the supersymmetry, whichin our problem would mean that one preserves only four of theeight supercharges. However, it turns out (GGHPR, BeS) that thesupersymmetric Gödel solution preserves all eight supercharges
that one would have before turning on the graviphoton field. But
the supersymmetry algebra is deformed from what it is before
turning on the graviphoton.

It turns out that the reason that this is important is that tocompute the GV formula, one needs to know the magnetic
moments of the BPS states. The unexpected extra four
supersymmetries, and the deformed supersymmetry algebra,
uniquely determine those magnetic moments.

There are a few more questions we should ask and here is anotherone: Quantum mechanics has wave-particle duality but the rangeof validity of a calculation based on waves can be quite different
from that of a calculation based on particles. Is the Schwinger
calculation that leads to the GV formula supposed to be a
calculation based on particles or on fields?

To answer this question, observe that – to the extent that we dounderstand M-theory – we understand it when the length scales
involved are much greater than the Planck length. This meansthat we should think of the Calabi-Yau manifold Y as being muchbigger than the Planck length, and therefore a massive BPS statecoming from a wrapped M2-brane has a mass much greater thanMPl . We would be wary of using field theory for particles muchheavier than the Planck mass. Moreover, wrapped M2-branes canhave arbitrarily large spin, and for particles of spin > 2 coupled tothe background supergravity multiplet, it is not clear that we havea sensible field theory to use. So we will do a particle calculationfor BPS states that are massive in d = 5, and only use field theory
for the (few) BPS states that are massless in d = 5.

To do the particle calculation, one starts by thinking of theM2-brane wrapped on p × S 1 × Σ ⊂ R4 × S 1 × Y (where p is apoint in R4 and Σ ⊂ Y ) as a sort of instanton:In drawing the picture, I actually ignored Σ and just depicted theM2-brane worldvolume p × S 1 × Σ as a particle worldline
p × S 1 ⊂ R4 × S 1 , with a point p ∈ R4 . This is actually the rightthing to do in the following sense: If the S 1 is sufficiently large,
which is the limit in which our approximations are going to begood, then we can ignore the internal structure of the BPS state
and the existence of the compact manifold Y and just think of the
BPS state as a point particle propagating in R4 × S 1 . We willproceed in that way. We view the particle winding on p × S 1 as asort of instanton in 5d supergravity on R4 × S 1 .

To get the effective action, we have to integrate over the moduli of
the instanton. The simplest case is that the only moduli are theones that have to exist because of translation invariance andsupersymmetry. The instanton inevitably has at least four bosonicmoduli associated to translation symmetry. These moduli are thecoordinates x µ of the point p ∈ R4 . If the particle is not BPS,
then its worldline is not invariant under any of the eightsupersymmetries and therefore there will be eight fermionic moduli
generated by the broken supersymmetries. I will call them θAi andθȦj , A, Ȧ, i, j = 1, 2, where A and Ȧ are indices of negative or
positiveRchirality. Thus for a non-BPS particle the effective action
will be d4 xd4 θd4 θ̄ O where O is the instanton amplitude without
integrating over the moduli that are generated by spacetime
symmetries (but after integrating over other moduli, if there areany). In other words, a non-BPS particle will generate a D-term.

Let us see instead what happens for a half-BPS particle that isinvariant under the four supersymmetries Q Ȧi . Such a BPS
particle comes from an M2-brane wrapped on a complex Riemannsurface Σ ⊂ Y . For such a particle we only have four fermionicAi
moduli θAi generated
R 4 4by the Q and therefore the effective action
is going to be d xd θ O, where O is whatever we get before
integrating over the moduli generated by symmetries. In otherwords, a half BPS particle will generate an F -term. These are theparticles that contribute to the GV formula.

The BPS particles that have no moduli except the ones forced by
the symmetries are massive hypermultiplets; they come from an
M2-brane wrapped on an isolated genus 0 curve Σ ⊂ Y . Let us
analyze the contribution of such a particle. First we need tounderstand its mass. The Kahler form of Y is
X
ω=
hI ωI
I

where ωI , I = 1, . . . , b2 are a basis of H 2 (Y , Z) and hI are Kahler
moduli. (I will take some minor shortcuts to avoid
R too many
details.) The charges of the BPS state are qI = Σ ωI . We
abbreviate q1 , . . . , qb2 as ~q . The area of Σ is the central chargeXζ(~q ) =qI h II

and (in units in which the M2-brane tension is 1) the mass of the
BPS state isM(~q ) = ζ(~q ).
(We don’t take the absolute value since a holomorphically wrappedM2-brane has ζ > 0.)

The 5d theory has gauge fields AI and corresponding magnetic
fields F I = dAI . In the graviphoton background, these are
F I = hI T −where T − is the graviphoton. (I did not write this formula before
because I wrote the supersymmetric Gödel solution only for pure
5d supergravity without vector multiplets. To embed this solution
in a model with vector multiplets, one needs this formula.) The
effectiveP
magnetic field seen by a particle of charges ~q is
F (~q ) = I qI F I , which in the graviphoton background for a BPS
particle isX
F (~q ) =qI hI T − = ζ(~q )T − = MT − .
I

The fact that the effective magnetic field seen by the BPS state
depends only on its mass is very important in getting the GV
formula.

Now let us discuss the action for the BPS particle. We can make anonrelativistic approximation since the worldline of the BPS state
will be fluctuating only slightly around the static orbit p × S 1 . We
include a constant −Mc 2 for the rest mass. We also have thed Ainonrelativistic kinetic energy 12 M ẋ 2 for bosons, and iM
2 ψAi dt ψfor the fermions. So the minimal action that we can write is!
Z
M X µ 2 iM
d Ai(ẋ ) +
I0 = dt −M +
ψAi ψ
.
2 µ
2
dt
It turns out that this is enough since any higher order terms are“irrelevant” when the radius R of the M-theory circle is large anddon’t contribute to the F -terms.

One can verify the supersymmetry of the action. The conserved
momenta P µ that generate translations of x µ are the canonicalmomenta p µ :P µ = p µ = M ẋ µ
They commute[P µ , P ν ] = 0.
The supercharges are
Q Ai = Mψ Ai ,Q Ȧi = γµAȦ Pµ ψAi

and are obviously conserved. The Hamiltonian is
H=M+

P2
.
2M

It is straightforward to verify (the appropriate nonrelativistic limitof) the supersymmetry algebra.

Now let us turn on the graviphoton field. There is one
R obvious µterm that we need to add to the action:acoupling
dt Aµ (~q )ẋ
P
to the background gauge field A(~q ) = I qI AI . As we have seen,− so the term wethis field has the constant field strength MTµν
have to add is
Z
M
− µ ν
I1 =
dtTµν
x ẋ .
2
The minimal possible action is thusZ
I = I0 +I1 = M

dt

1
−1 +
2



dx µ
dt

2

1 − µ νi
+ ψAi ψ̇ Ai + Tµν
x ẋ
2
2

It turns out that this is the complete supersymmetric action,
modulo irrelevant terms of higher dimension.

!
.

Once one turns on the graviphoton, one has to modify thetranslation generators P µ , which no longer coincide with thecanonical momenta p µ
P µ = pµ −

M −µν
T
xν
2

and no longer commute−[Pµ , Pν ] = iMTµν.

This is not surprising for a charged particle in a constant magnetic
field. We can still use the old formulas for the supercharges
Q Ai = Mψ Ai ,Q Ȧi = γµAȦ Pµ ψAi ,and they are obviously still conserved. But since the definition ofPµ has changed, the algebra they generate is deformed – inprecisely the way seen in supergravity.

Now let us evaluate the path integral for fluctuations around theparticle trajectory p × S 1 . The answer is going to be schematically√Z
det DF44d xd θ exp(−Icl ) · √,DBwhere Icl is the classical action and det DF0 , det DB0 are fermionicand bosonic one-loop determinants with zero-modes removed. As
is usual in instanton physics, we remove the zero-modes from thedeterminants and replace them with an integral over thecorresponding moduli or collective coordinates x µ and θAi .

One term in the classical action is 2πRM, where 2πR is thecircumference of the M-theory circle. However, once wecompactify the time direction to a circle, we should allow for the
possibility that the gauge fields AI can have monodromies
exp(−2πiαI ) around the circle. The parameters αI are important
moduli in the Type IIA description. They are the real parts (atθ = 0) of the chiral superfields X I that we discussed before thatare in vector multiplets, or more exactly they are the real parts of
Z I = X I /X 0 . In the presence of these monodromies, a particle ofcharges qI propagating around the circle gets a phaseexp(−2πiqI αI ). It turns out that the productexp(−2πiqI αI ) exp(−2πRM) is none other thanexp(−2πiqI Z I ).
So our answer is going to be√Z
det DF4
4
Id xd θ exp(−2πiqI Z ) √
.
DB

(3)

We still have to evaluate the determinants, but they are more orless the simplest functional determinants one will see. The
fermionic determinant is completely trivial, since the fermions arefree (DF = id/dt), and the bosonic determinant is a classicalexample since DB = D1 D2 where D1 = −d/dt andD2 = d/dt + T − , where T − is a constant matrix. After a little
work one gets a result that is part of the GV formula:Z
T2
d4 xd4 θ exp(−2πiqI Z I )sinh2 (πRT )q− −µν
where T = Tµν
T.

One has to add further contributions from the case that the BPS
particle wraps k times around the circle for k > 1. In thepreceeding calculation, this multiplies the classical action by k. It
also multiplies R by k, since the particle is effectively going oncearound a circle of circumference 2πkR. Finally, one has to divide
the result by k to account for the k-fold cyclic symmetry among
the different sheets of the particle orbit. So the result is that thecontribution to the GV formula of a massive BPS hypermultiplet of
charge ~q isZ

d4 xd4 θ

∞
X
1
k=1

k

exp(−2πikqI Z I )T2.
sinh2 (πkRT )(It takes some discussion to justify ignoring the self-interactions of
the BPS particles for k > 1, since after all the interactions between
M2-branes are strong.)We still have to go on and analyze the contribution to the GV
formula of other BPS states. For massive BPS states that are notin hypermultiplets, the idea is the same, though some details aredifferent. One uses supersymmetry – notably the fact that thegraviphoton background preserves eight rather than four
supercharges – to determine the appropriate Hamiltonian for thesemore general massive BPS states. Then one computes in much theway as I have described, with a few minor twists.

For those (few) BPS states that are massless in five dimensions,the “instanton” point of view does not work well, but there is adifferent simplification. These modes just come from 11d
supergravity, and unlike the massive BPS states that come fromwrapped M2-branes, they have spins ≤ 2 and we do have a goodfield theory that describes them, namely 11-dimensional
supergravity. Actually, since these modes are zero-modes along Y ,one can do the calculation in 5d supergravity.

One just needs to compute one-loop determinants for masslessfields in the 5d graviphoton background. It is convenient to do thiscomputation by making a Kaluza-Klein reduction along the
M-theory circle, to reduce to a sum of one-loop determinants of 4d(massive) BPS states with different values of the Kaluza-Klein
momentum. Here the calculation one has to perform really is (asum over spins and masses of) Schwinger’s classic calculation for a
4d charged particle of mass m in a constant magnetic field F .