The String Coffee Table

Introduction.

Supersymmetric point particles have partition functions which compute the index of the supercharge (a Dirac operator). This index can be thought of as an element in the K-theory over a point.

Supersymmetric strings (heterotic strings, in particular) have partition functions which compute the Witten genus of a manifold. This Witten genus can be thought of as an element in elliptic cohomology over a point.

$$\array{ \mathbf{\text{fundamental object}} & \mathbf{\text{partition function}} & \mathbf{\text{ring of values}} & \mathbf{\text{cohomology}} & \mathbf{\text{group law}} & \mathbf{\text{chrom. filt.}} \\ (-1)-\text{branes} & ? & ? & \text{ordinary} & x\cdot y = x + y & 0 \\ \text{point particles} & index & \mathbb{Z} & K & x\cdot y = x + y + xy & 1 \\ \text{strings} & Witten genus & \text{modular forms} & elliptic & \text{ell. curve} & 2 } \,.$$

The supercharges whose indices are being computed here can be taken to define a background for the superpoint or the superstring. Hence an element of K-theory over a point corresponds to a connected component of the space of all superparticle backgrounds, in a sense.

The sense in which this is true has been made precise by Stephan Stolz and Peter Teichner, and elaborated on by Elke Markert in her thesis

E. Markert
Connective 1-dimensional euclidean field theories
(pdf)

Theorem. The space of 1-dimensional supersymmetric quantum field theories is homotopy equivalent to the K-theory spectrum.

This means in particular, that supersymmetric quantum mechanics knows much more than the K-theory over points - it knows all of K-theory. In string speak, it says that the landscape of superparticle theories is homotopy equivalent to the spectrum of K-theory.

Based on similar observations, Graeme Segal had suggested already twenty years ago, in the last section of

Graeme Segal
Elliptic Cohomology
Séminaire Bourbaki, no. 695
Astérisque 161-162 (1988),

that the space of all string theories is similarly related to elliptic cohomology.

The underlying strategy for making this precise was that of $n$-transport ($\to$).

It is natural to encode the concept of propagation in (supersymmetric) quantum mechanics in terms of functors $$\array{ QM : 1d\mathrm{Cob} &\to& \mathrm{Hilb} \\ (\bullet \overset{t}{\to} \bullet) &\mapsto& (H \overset{\exp(-i t \Delta)}{\to} H) }$$ from 1-dimensional worldlines to Hilbert spaces.

Segal’s famous definition of 2-dimensional conformal field theory

Graeme Segal
The definition of conformal field theory
in U. Tillmann (ed.)
Topology, Geometry and Quantum Field Theory
Lond. Math. Soc. Lecture Note Series 308
Cambridge (2002)

accordingly says, roughly, that 2dCFT is a projective functor on 2-dimensional conformal cobordisms $$\array{ 2dCFT : 2d\mathrm{ConfCob} &\to& \mathrm{Hilb} } \,,$$ and tries to relate the space of all these functors to elliptic cohomology.

Stolz and Teichner notice in

St. Stolz & P. Teichner
What is an elliptic object?
in U. Tillmann (ed.)
Topology, Geometry and Quantum Field Theory
Lond. Math. Soc. Lecture Note Series 308
Cambridge (2002)
(pdf)

that this idea has to be refined in two aspects.

1) It is necessary to concentrate on supersymmetric QFT. This naturally leads to the required grading and prevents the resulting spaces from being topologically trivial.

2) It is necessary to allow string bits. Otherwise excision does not hold, which, in string speak, means something like that otherwise instanton effects lead to nonlocalities.

The last point means, technically, that we consider not 1-functors on 2-cobordisms but 2-functors on 2-paths, of the form $$2d\mathrm{SCFT} : \text{super conformal 2-paths} \to 2\mathrm{Hilb} \,,$$ where $2\mathrm{Hilb}$ is something like a 2-category of 2-Hilbert spaces ($\to$).

The refined version of Segal’s conjecture would hence read

Conjecture. The space of all 2-functors $2d\mathrm{SCFT}$ is homotopy equivalent to the elliptic cohomomolgy spectrum $\mathrm{tmf}$ of topological modular forms.

I’ll explain what all this means in detail below.

Before closing this introduction, I’ll comment on the following question.

How is this related to our previous session on elliptic cohomology and 2-vector bundles? ($\to$)

The index of a Dirac operator takes values in the integers $\mathbb{Z}$. We should think of this integer as a decategorified virtual vector bundle over a point $$\array{ (\mathrm{ker}D,\mathrm{coker}D) \\ \downarrow \\ \bullet } \,,$$ whose fiber is the kernel minus the cokernel of the Dirac operator. $$\begin{aligned} \mathrm{ind} D &= \mathrm{dim}\,\mathrm{ker}D - \mathrm{dim}\,\mathrm{coker}D \\ &= \mathrm{dim}\,\mathrm{ker}D - \mathrm{codim}\, \mathrm{im}D \end{aligned} \,.$$ So we see that here the integers which appear $$\mathbb{Z} = K(\mathrm{pt}) = \mathrm{Groth}(\mathrm{Vect})$$ are really the decategorification of the category of vector spaces.

Now, the index of a Dirac operator on loop space in general takes values not on $\mathbb{Z}$, but in the polynomial ring $\mathbb{Z}[\![q, q^{-1}]\!]$ (more precisely, in the ring of integral modular forms in $q$).

Again, the integer coefficients here are really K-classes of a point. In fact, the coefficients $a_k$ in the index $$\mathrm{ind}_\text{Witten}(G) = \sum_{k \in \mathbb{Z}} a_k\, q^k$$ of the heterotic string supercharge $G$ (a Dirac operator on loop space) are the dimension of infinitely many virtual vector spaces, one for each Fourier mode of the string $$\begin{aligned} \mathrm{ind}_\mathrm{Witten}(G) &= \sum_{k \in \mathbb{Z}} ( \mathrm{dim}\,\mathrm{ker}G|_k - \mathrm{dim}\,\mathrm{coker} G|_k ) \, q^k \end{aligned} \,,$$ where $G|_k$ is the restriction of the supercharge to the $k$-th eigenspace of the operator $$P := L - \bar L$$ which generates rigid rotations of (the parametrization of) the string.

It follows, that this loop space index is not a virtual vector bundle over a point, but a virtual vector bundle over the integers. $$\array{ (\mathrm{ker}G, \mathrm{coker} G)&& \cdots & (\mathrm{ker}G|_0, \mathrm{coker} G|_0) & (\mathrm{ker}G|_1, \mathrm{coker} G|_1) & (\mathrm{ker}G|_2, \mathrm{coker} G|_2) & \cdots \\ \downarrow &=& \cdots & \downarrow & \downarrow & \downarrow & \cdots \\ \mathbb{Z} && \cdots & \{0\} & \{1\} & \{2\} & \cdots } \,.$$ Recall ($\to$, $\to$, $\to$) that we can interpret such a virtual vector bundle as a virtual 2-vector in the countably infinite dimensional Kapranov-Voevodsky 2-vector space $\mathrm{Vect}^\mathbb{Z}$.

Warning: Technical details of definitions may be imprecise. For instance, some necessary technical conditions on elliptic curves are missing. Don’t rely on what I write, but check the referenced literature.

Background information.

1) What is a genus?

See for instance

G. Segal, Elliptic Cohomology, or

Gerd Laures,
An introduction to Elliptic Cohomology and Conformal Field Theory
(pdf).

Some comments in the introduction of

Steven Rosenberg
Nonlocal Invariants in Index Theory
(pdf).

also prove useful.

Definition. An $R$-valued genus $\mu$ is a ring homomorphism $$\mu : \Omega_{\mathrm{SO}} \to R$$ from the bordisms ring ($\to$) $$\Omega_{\mathrm{SO}} := \frac{\text{closed oriented manifolds}}{cobordisms}$$ to $R$. The product in $\Omega_\mathrm{SO}$ is the cartesian product of manifolds and the sum is the disjoint union of manifolds.

Examples.

$\bullet$ The Euler characteristic $X \mapsto \chi(X)$ (which is the index of $D = d + d^*$ under the grading by form degree) is close to being a genus, but is not cobordism invariant.

$\bullet$ The signature genus is a genus. It is (as far as I understand) the index of $D = d + d^*$, but under the grading of differential forms as positive or negative, according to the signature of the bilinear form $$(v,w) \mapsto \int_X v \wedge w \,.$$

$\bullet$ The $\hat A$-genus is the index of a Dirac operator coming from a spinor bundle (Atiyah-Singer).

$\bullet$ The elliptic genus has been interpreted (non-rigorously) by Witten as the index of a Dirac operator (the heterotic’s string supercharge, in fact) on loop space.

Bottom line. Genera appear as indices of Dirac operators. As such, they are related to cohomologies of points.

2) What is a generalized cohomology theory?

See for instance

Jacob Lurie
A survey of elliptic cohomology
(pdf)
section 1.1 .

If you feel you need a more gentle introduction, try

John Baez
TWF 149
TWF 151 .

Let $A$ be any abelian group. Given a topological space $X$, there are several equivalent ways to define the $n$th singular cohomology group $$H^n(X,A)$$ with values in $A$. A particularly nice one is to say that this group equals that of homotopy classes of maps from $X$ to the space $K(A,n)$, which is defined to have all homotopy groups trivial except for the $n$th one, which is isomorphic to $A$: $$H^n(X,A) \simeq [X,K(A,n)] \,.$$ For this reason, one says that $K(A,n)$ represents ordinary cohomology.

Even more is true. It is a simple fact that the (based) loop space of $K(A,n)$ is homotopy equivalent to $K(A,n-1)$ $$\Omega K(A,n) \simeq K(A,n-1) \,.$$ A list $E = (E_n)_{n\in \mathbb{N}}$ of topological spaces such that each space is the loop space of the next one $$\Omega E_n \simeq E_{n-1}$$ is called a spectrum. The sequence $E_n := K(A,n)$ is called the Eilenberg-MacLane spectrum, which represents ordinary cohomology.

Stated this way, the definition of generalized cohomology is obvious:

Definition. A generalized cohomology is a collection of functors $$\array{ H^n : \mathrm{Top} &\to& \mathrm{AbGrp} \\ X &\mapsto& H^n(X) }$$ from topological spaces to abelian groups, which is represented by a spectrum $E = (E_n)$: $$H^n(X) := [X,E_n] \,.$$

Historically, things were found the other way around. First Eilenberg and Steenrod extracted a list of axioms, the Eilenberg-Steenrod axioms, which are satisfied by ordinary cohomology, and defined a generalized cohomology to be anything satisfying these axioms. Later, Brown proved his representability theorem, which says that every spectrum determines a generalized cohomology theory and every generalized cohomology comes from a spectrum (see Lurie, p.8).

The first interesting example of a generalized cohomology theory, right after ordinary cohomology, is K-theory $$X \mapsto K_n(X) \,.$$ A well known result by Atiyah says that $K_0$ is represented by the space $\mathrm{Fred}(H)$ of Fredholm operators on some seperable Hilbert space $H$ $$K(X) \simeq [X,\mathrm{Fred}(X)] \,.$$

(— to do: what is the entire spectrum? what happens for different flavors of K-theory? —)

Some generalized cohomologies have special properties. In particular, we are used to there being a graded ring structure on the cohomology groups (given by the cap product for ordinary cohomology, and by the tensor product of vector bundles for K-theory).

It is clear that in order for $H^\bullet(X) \simeq[X,E_\bullet]$ to be a ring, the spaces $E_\bullet$ need to have a ring-like structure themselves. Since everything in sight is defined only up to homotopy, there is freedom in having this ring structure defined only up to higher coherent homotopy. There is one natural choice for how to deal with these higher coherencies (see Lurie, p. 9).

Definition. A spectrum $E$ which is a graded ring in this higher coherent sense is called a $E_\infty$-ring spectrum , or simply an $E_\infty$-ring. The “$E$” is supposed to remind you of commutativity, while the “${}_\infty$” reminds us of the fact that everything holds only up to homotopy, which is coherent up to homotopy, which… and so on.

3) What do we need to know about elliptic curves?

If you don’t know anything about elliptic curves, the text

Charles Daney
Elliptic Curves and Elliptic Functions
(html)

is a rather nice brief introduction. Here, I won’t need anything that goes beyond the basic facts stated in that article. In fact, all that is really necessary in order to get a basic idea of elliptic cohomology are these

2 Facts.

1) An elliptic curve over a field $k$ is the collection of solutions in $k x k$ to an equation in $k$ of the form $f(x,y) = A x^3 + B x^2 + C x + D - y^2 = 0 \,,$ where $A,B,C,D \in k$ are constants defining the elliptic curve. For the present purpose, the single most important property of an elliptic curve $S = \{(x,y) \in k\times k | f(x,y) = 0\}$ is that it is naturally equipped with an abelian (algebraic) group structure $S \times S \to S$.

2) Elliptic curves over the complex numbers are precisely the same thing as Riemann surfaces of genus 1, hence the same as 1-loop string diagrams.

This makes it quite plausible that elliptic curves play a central role in a cohomology theory which assigns to a point in the space of 2dSCFTs the corresponding 1-loop partition function.

4) What is the elliptic genus?

See

G. Segal, Elliptic Cohomology

together with

Edward Witten
Elliptic Genera And Quantum Field Theory
Commun.Math.Phys. 109 525 (1987)
(archive)

and

Edward Witten
The Index Of The Dirac Operator In Loop Space.
Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology
Princeton, 1986.
(archive).

I can also very much recommend to have a look at the trascript of a talk

Johannes Ebert
The partition function of CFT’s: Connection to modular forms.
(notes)

given in our winter school on elliptic objects ($\to$).

For our purposes, we are interested in 2-dimensional field theory with $N= (1,0)$-superconformal symmetry, i.e. in the heterotic string. (We could also work with type II superstrings, but these would yield less cohomological information.)

There is a graded Hilbert space $H$ of states, which we can think of as something like the space of sections of a spinor bundle over free loop space.

Represented on $H$ are a Laplace-like operator $$\Delta = L + \bar L$$ and an operator $$P = i(L - \bar L)$$ which generates rigid rotations of (the parameterization) of the loops.

The chiral combinations $$\begin{aligned} L &= \frac{1}{2}(\Delta -i P) \\ \bar L &= \frac{1}{2}(\Delta + i P) \end{aligned}$$ are the 0-modes of two Virasoro algebras.

We say there is $N=(n,m)$ supersymmetry essentially (up to the fact that we are only considering 0-modes at the moment) if there are $n$ mutually (graded-)commuting odd-graded operators $G^{(i)}$ such that $$L = (G^{(i)})^2$$ and $m$ mutually (graded-)commuting odd-graded operators $\bar G^{(i)}$ such that $$\bar L = (\bar G^{(i)})^2 \,.$$ Restricting to $N=(1,0)$ we have a single odd-graded operator $G$ with $L = G^2$.

We are interested in something like the index of this $G$.

For ordinary susy quantum mechanics, we compute the index by means of the partition function over a circle of length $t$ (which turns out to be independent of $t$).

Here we compute the partition function over a torus of length $t$ and twist $s$ as $$\begin{aligned} \mathrm{ind}_\mathrm{Witten} G &= \mathrm{str} \left( \exp(-t \Delta)\exp(-s P) \right) \\ &= \mathrm{str} \left( q^L {\bar q}^{\bar L} \right) \,, \end{aligned}$$ where $q = \exp(-(t+is))$ and $\bar q = \exp(-(t-is))$.

Since $L = G^2$ is the square of an odd operator, the usual argument applies and we see that this partition function localizes on the kernel of $L$ $$\cdots = \mathrm{str}({\bar q}^{\bar L}|_{\mathrm{ker}L}) \,.$$ We can further simplify this by splitting the supertrace into the contributions coming from the eigenspaces $\mathrm{Eig}(P,-k) = H^k \subset H$ of $P = i(L - \bar L)$ $$\cdots = \sum_{k \in \mathbb{Z}}(q^k) \mathrm{sdim}(\mathrm{ker}L \cap H^k) \,.$$

This power series in $q$ is the index of our loop space Dirac operator. (Had we used full $N=(1,1)$ supersymmetry the remaining sum would have localized itself on the kernel of $\bar L$ and collapsed to a mere integer.)

$\mathrm{ind}_\mathrm{Witten} G$ is a weak integral modular form, which means that

$\bullet$ it is a holomorphic function $f$ on the upper half plane

$\bullet$ with the transformation property $$f(\frac{a\tau + b}{c \tau + d}) = (c\tau + d)^k f(\tau)$$ for all elements of $SL_2(\mathbb{Z})$,

$\bullet$ such that only finitely many terms for negative powers of $k$ are nonvanishing,

$\bullet$ and such that all coefficients are integers.

All weak integral modular forms are combinations of

$\bullet$ the discriminant $$\Delta = q \prod_{n=1}^\infty (1-q^n)^{24}$$ and

$\bullet$ the two Eisenstein series $$\array{ c_4 &=& 1 + 240 \sum_{k \gt 0} \sigma_3(k) q^k \\ c_6 &=& 1 - 504 \sum_{k \gt 0} \sigma_5(k) q^k }$$ with $\sigma_r(k) = \sum_{d|k}d r$. In fact, the ring of weak integral modular forms, $MF_*$ is $$MF_* = \mathbb{Z}[c_4,c_6,\Delta,\Delta^{-1}]/(c_4^3 - c_6^2 - (12)^3\Delta) \,.$$

Bottom line. The index of an ordinary Dirac operator takes values in the integers, which is the $K$-theory of a point. The index of a Dirac operator on loop space takes values in the ring $MF_*$ of weak integral modular forms. This lives in something like the elliptic cohomology of a point.

In fact, for this reason the generalized cohomology appearing here comes from an $E_\infty$-spectrum called $\mathrm{tmf}$, for “topological modular forms”. This is discussed in the next subsection.

5) What is elliptic cohomology?

See G. Segal, Elliptic Cohomology, or

Jacob Lurie
A survey of elliptic cohomology
(pdf)
section 1

Let $H$ be some multiplicative generalized cohomology (i.e. one coming from an $E_\infty$-ring spectrum). It turns out (Lurie, p. 3) that the graded ring associated by $H$ to the space $$K(\mathbb{Z},2) \simeq \mathbb{C}P^\infty \simeq PU(H) \simeq BU(1)$$ is of particular importance.

Since $BU(1)$ is the classifying space for $U(1)$-bundles, the ordinary cohomology group $H^\bullet(BU(1),\mathbb{Z})$ is the ring freely generated, over $\mathbb{Z}$, by the first Chern class $c_1$ of the universal line bundle $U(H) \to PU(H)$.

It is a triviality, but of central importance for what follows, that under the tensor product of two line bundles $L_1 \to X$ and $L_2\to X$, their Chern classes simply add up, in ordinary second cohomology $$c_1(L_1 \otimes_X L_2) = c_1(L_1) + c_2(L_1) \,.$$

This particular property turns out to be modified for other generalized cohomology theories. They will have other group laws, different from the simple one on the right hand side of the above formula.

So let $H$ be any generalized cohomology theory. One finds that its value on $BU(1)$ is always a power series ring $$H^\bullet(BU(1)) = H^\bullet(\mathrm{pt})[\![t]\!]$$ over a single generator $t$ of degree 2 with coefficients in the cohomology ring of the point. Recall that $t$ is hence itself a (class of a) map $$t : BU(1) \to E_2 \,,$$ if $E$ is the spectrum representing the generalized cohomology.

Given any complex line bundle on some space $X$, classified by the class of the map $\phi : X \to BU(1)$, we can pull back the generator $t$ along this map to obtain an element of $H^\bullet(X)$: $$\phi^*t : X \overset{\phi}{\to} BU(1) \overset{t}{\to} E_2 \,.$$ Notice that for ordinary cohomology this is precisely the pullback of the universal first Chern class along $\phi$ to $X$, hence the Chern class $c_1(L) = \phi^* t$ of our complex line bundle on $X$.

For generalized cohomologies $\phi^*t$ is no longer the Chern class of our line bundle - so let’s call it a generalized Chern class.

The whole point of this exercise is that we can now (in principle) compute how these generalized Chern classes of complex line bundles behave under the tensor product operation of line bundles.

In general, they will, unlike the ordinary Chern class, not just add up.

For instance, for K-theory one finds the following group law $$c_1(L_1 \otimes_X L_2) = c_1(L_1) + c_1(L_2) + c_1(L_1) c_1(L_2) \,.$$ Most generally, one finds $$c_1\left(L_1 \otimes_X L_2\right) = f\left( c_1(L_1),\; c_2(L_2) \right) \,,$$ where $$f(t_1,t_2) \in H^\bullet(\mathrm{pt})[\![t_1,t_2]\!]$$ is any formal power series with coefficients in the cohomology ring of the point, satisfying the following three axioms

$\bullet$ unit law: $f(v,0) = f(0,v) = v$

$\bullet$ commutativity: $f(u,v) = f(v,u)$

$\bullet$ associativity: $f(u,f(v,w)) = f(f(u,v),w)$.

Such a power series is called a commutative, 1-dimensional formal group law over the ring $H^\bullet(\mathrm{pt})$.

One can go on and on about formal group laws, formal spectra, formal groups and what not, but the important point is the following.

Fact. (see Lurie, p. 7) Under suitable assumptions, over an algebraically closed field, there are only three types of 1-dimensional algebraic groups, corresponding to three types of formal group laws

0) the additive group law: $x \cdot y = x + y$

1) the multiplicative group law: $x \cdot y = x + y + xy$

2) a group law defined by an elliptic curve ($\to$).

Hence we finally have the, now obvious

Definition. (roughly) An elliptic cohomology is any generalized cohomology whose formal group law (which governs the generalized Chern classes of tensor products of complex line bundles) is given by an elliptic curve.

So there are many elliptic cohomology theories. One can do something like taking the direct limit over all of these. The result is represented by an $E_\infty$-spectrum called $\mathrm{tmf}$. Its value over a point is the ring of weak integral modular forms, that arise as the partition function of heterotic strings.

6) How is ellitpic cohomology realized geometrically?

Above we have given the definition of elliptic cohomology. But we would also like to have a nice geometric realization of it. Like we can understand K-theory geometrically as being about classes of vector bundles.

In fact, we can alternatively understand the K-theory spectrum as the space of supersymmetric quantum mechanical systems. That’s the content of the following “warmup exercise”.

It suggests, that the $\mathrm{tmf}$-spectrum for elliptic cohomologies is similarly related to the moduli space of 2-dimensional superconformal field theories.

6a) Warmup: how is the landscape of superpoint theories equal to the K-theory spectrum?

First some sources.

Apart from the the above cited texts by Stolz-Teichner-Markert, information on part a) can be found in a collection of lecture notes from a winter school on elliptic objects.

Sarah Massberg,
The definition of 1-dimensional EFT’s a la Stolz/Teichner.
(pdf),

Juan Wang
From Euclidean Field Theory to K-theory, I
pdf,

Benedikt Plitt
Fredholm Operators and K-theory
(pdf)

and

Moritz Wiethaupt
From Euclidean Field Theory to K-theory, II
(pdf).

Before saying anything about supersymmetric quantum mechanics, consider this standard example, which, in string speak is a point in the geometric phase of the superparticle landscape.

Example. (The Euler characteristic.)

Let $X$ be a compact Riemannian manifold with exterior bundle $\Lambda^\bullet(T^*X)$. Let $H$ be the $\mathbb{Z}_2$-graded (actually even $\mathbb{Z}$-graded) Hilbert space of square integrable sections of this bundle with respect to the Hodge scalar product $$\langle v | w \rangle = \int_X v \wedge \star w \,.$$ From the exterior derivative $d : H \to H$ and its adjoint $d^* : H \to H$ we construct the Dirac operator $$D := d + d^* \,.$$ Time-independent euclidean quantum mechanics of a free (“RR-sector”) superparticle on $X$ could be described by taking $D^2$ to be the corresponding Hamiltonian.

The partition function of this superparticle, for the worldline being a closed path of time length $t$, would be the supertrace $$Z(t) = \mathrm{str}_H \left( \exp\left( - t\, D^2 \right) \right) = \mathrm{tr} \left( (-1)^N \exp\left( - t\, D^2 \right) \right) \,,$$ where $N$ is the operator which measures the degree of a differential form. Due to “supersymmetry” (namely due to the fact that the (euclidean) time-translation operator $D^2$ is the square of an odd-graded operator) all non-vanishing eigenvalues of $D^2$ occur in pairs, one corresponding to an eigenspace in even, the other to one in odd degree.

Therefore, the trace reduces to that over 0-eigenstates, which are precisely the harmonic forms, and we find that the partition function is the Euler characteristic ($\to$) $\chi(X)$ of $X$ $$Z(t) = \sum_i (-1)^i b_i = \chi(X) \,,$$ where $b_i$ are the Betti numbers ($\to$).

$\chi(X)$ is the index ($\to$) of $D$, being the difference between the dimension $\mathrm{dim}_+$ of the space of “bosonic” (forms of even degree) and $\mathrm{dim}_-$, that of “fermionic” 0-modes (forms of odd degree).

A sophisticated way of thinking about this index is as an element of the K-theory ($\to$) over a point, namely the class of a virtual vector bundle over a point, hence a virtual vector space. $$(\mathbb{R}^{\mathrm{dim}_+},\mathbb{R}^{\mathrm{dim}_-})$$

This is one of the standard examples of supersymmetric quantum mechanics. We want to formalize the general notion of supersymmetric quantum mechanics.

Definition. Denote by $1\mathrm{Cob}$ the monoidal category of oriented 1-dimensional Riemannian cobordisms and by $\mathrm{Hilb}$ the category of real seperable Hilbert spaces with bounded operators between them. Then a euclidean 1-dimensional QFT, also known as euclidean quantum mechanics (or as statistical mechanics, for that matter), is a functor $$1d\mathrm{QFT} : 1\mathrm{Cob} \to \mathrm{Hilb} \,,$$ which respects tensor products, orientation involution and the adjunctions on $1\mathrm{Cob}$.

Any such functor is determined by

a) a choice of Hilbert space (“space of states”) $H$

b) a choice of, possibly unbounded, self-adjoint operator $\Delta$, the “Hamiltonian” $$1d\mathrm{QFT}_{(H,\Delta)} ( \bullet \overset{t}{\to} \bullet) \;=\; H \overset{\exp(-t \Delta)}{\to} H \,.$$

Now we want to formulate supersymmetric quantum mechanics in an analogous manner. We will define a category $1\mathrm{sCob}$ of 1-dimensional super-Riemannian cobordisms and take $\mathrm{Hilb}$ to be the category of $\mathbb{Z}_2$-graded seperable (real) Hilbert spaces. Once $1\mathrm{sCob}$ is available, we will make the obvious

Definition. A euclidean 1-dimensional super QFT, also known as euclidean supersymmetric quantum mechanics, is a functor $$1d\mathrm{sQFT} : 1\mathrm{sCob} \to \mathrm{Hilb} \,,$$ which respects tensor products, orientation involution and the adjunctions on $1\mathrm{sCob}$.

Essentially, every such functor is determined by specifying a $\mathbb{Z}_2$-graded Hilbert space $H$, and a self-adjoint odd-graded operator $D$ on $H$, the supercharge, whose square plays the role of the Hamiltonian above.

Given that the space of Fredholm operators is a classifying space for K-theory ($\to$) we may expect that the space of all $1d\mathrm{sQFT}$s is somehow related to K-theory. If we choose our definitions reasonably, then the theorem will say that both spaces are in fact homotopy equivalent.

In order to get there, we first need the category of 1-dimensional super-Riemannian cobordisms.

Definition. The category $1d\mathrm{sCob}$ of $D=1$, $N=1$ super-Riemannian cobordisms has a single object, the superpoint $\bullet = \mathbb{R}^{0|1}$, and a superspace of morphisms $$\mathrm{Hom}(\bullet,\bullet) \simeq \mathbb{R}_+^{1|1} \,.$$ Composition of morphisms is defined in terms of $S$-points (see below) as $$\array{ \mathrm{Hom}(\bullet,\bullet) \times \mathrm{Hom}(\bullet,\bullet) &\to& \mathrm{Hom}(\bullet,\bullet) \\ (\mathbb{R}_+^{1|1} \times \mathbb{R}_+^{1|1})(S) &\to& \mathbb{R}_+^{1|1}(S) \\ (t_1,\theta_1,t_2,\theta_2) &\mapsto& (t_1+t_2+ \theta_1\theta_2, \, \theta_1 + \theta_2) } \,.$$

Here we are working with superspaces in terms of sheaves on the category of $\mathbb{Z}_2$-graded ringed spaces ($\to$).

Briefly, this means the following:

Definition. 1) For us, a representable supermanifold is a manifold $X$ equipped with a sheaf $O_X$ of $\mathbb{Z}_2$-graded rings, equipped with a epimorphism onto the sheaf of continuous functions $C^\infty(X)$ on $X$.

2)In particular, we denote by $\mathbb{R}^{n|m}$ the supermanifold whose underlying manifold is $\mathbb{R}^n$ and whose sheaf of graded rings has as global sections the ring $$O_{\mathbb{R}^{n|m}(\mathbb{R}^n)} := C^\infty(C)\otimes \Lambda^\bullet \mathbb{R}^m$$ of smooth functions with values in the exterior algebra generated by $m$ linear independent differential forms (“Grassmann coordinates”).

2a) We need in particular the super half-line $\mathbb{R}_+^{1|1}$, whose underlying manifold is $\mathbb{R}_+ = [0,\infty)$ and whose ring of global sections is $$C^{\infty}(\mathbb{R}_+)\otimes \Lambda^\bullet \mathbb{R} \simeq C^{\infty}(\mathbb{R}_+) \;\oplus\; \langle \theta \rangle \otimes C^{\infty}(\mathbb{R}_+) \,.$$

3) The category of representable supermanifolds $\mathrm{repSMfld}$ is, in the obvious way, a subcategory of that of ringed spaces.

4) A general supermanifold is a sheaf of sets (thinking of presheaves suffices for our present purposes) on $\mathrm{repSMfld}$, hence a contravariant functor $$\mathrm{SMfld} \to \mathrm{Sets} \,.$$ These functors form the category of general supermanifolds, $\mathrm{SMfld}$.

5) By the Yoneda embedding ($\to$), every representable supermanifold is a general supermanifold $$\array{ \mathrm{repSMfld} &\to& \mathrm{SMfld} \\ (X,O_x) &\mapsto& \mathrm{Hom}_{\mathrm{repSMfld}}(--,(X,O_X)) }$$ which associates to each representable supermanifold $S = (|S|,O_S)$ the set of $S$-points $$X(S) := \mathrm{Hom}_{\mathrm{repSMfld}}(S,X) \,.$$

We reobtain the physicist’s way of thinking about supermanifolds in terms of Grassmann coordinates as follows:

Fact. The $S$-points of $\mathbb{R}^{n|m}$ are in bijection with ordered tuples $$(t_1,t_2,\cdots, t_n,\; \theta_1,\theta_2,\cdots, \theta_m)$$ of $n$ even and $m$ odd global sections of $S$.

In particular, if $S$ is the superpoint, $S = \mathbb{R}^{0|1}$, then an $S$-point of $\mathbb{R}^{n|m}$ is precisely one ordinary point in $\mathbb{R}^n$ together with precisely one 1-form on $(\mathbb{R}^{m})^*$.

Remark. Stolz and Teichner in their work, and in particular Elke Markert in her thesis (based on ideas by H. Hohnhold), go through the trouble of deriving the above stated super-$\mathrm{Hom}$-spaces of the category $1d\mathrm{sCob}$, including their composition law, from “first principles”, i.e. from just demanding the super-Riemann property and implementing it consistently in the arrow-theoretic setup. When the general-abstract-nonsense-dust has settled, what remains is the simple composition law stated above, known otherwise as ordinary super-translation.

As a nice example for these constructions, and also because it is central for the following constructions, consider the following

Observation. Every $\mathbb{Z}_2$-graded vector space $V = V^\mathrm{ev}\oplus V^\mathrm{odd}$ gives rise to a general supermanifold by defining its $S$ points as $$(S,O_S) \mapsto (V \otimes O_S(S))^\mathrm{ev} := V^\mathrm{ev}\otimes O_S(S)^\mathrm{ev} \oplus V^\mathrm{odd}\otimes O_S(S)^\mathrm{odd} \,.$$ (In fact, the supermanifold defined this way is representable ($\to$), but that need not concern us here.)

It follows, that we can regard the category of $\mathbb{Z}_2$-graded Hilbert spaces as a category whose objects and $\mathrm{Hom}$-spaces are in fact supermanifolds (albeit possibly infinite dimensional ones). In particular, we have

Observation. Given a $\mathbb{Z}_2$-graded Hilbert space, the vector space $\mathrm{Hom}(H,H)$ of bounded graded operators on that Hilbert space is a supermanifold whose superpoints (i.e $S$-points for $S = \mathbb{R}^{0|1}$) are pairs $(A,B)$ consisting of one even graded and one odd graded operator.

This, then, allows us to make precise what we mean by a functor $$1d\mathrm{sQFT} : 1\mathrm{Cob} \to \mathrm{Hilb} \,,$$ namely a functor on categories enriched over supermanifolds. This simply means that on morphisms it is a map of supermanifolds $$\array{ 1d\mathrm{sQFT} : \mathrm{Hom}_{1d\mathrm{scob}}(\bullet,\bullet) \simeq \mathbb{R}^{1|1} &\to& \mathrm{Hom}_\mathrm{Hilb}(H,H) \\ \mathbb{R}^{1|1}(S) &\to& (\mathrm{End}_H\otimes S)^\mathrm{ev} \\ (t,\theta) &\mapsto& A(t) + \theta B(t) \,, }$$ where $A(t)$ is an even-graded and $B(t)$ and odd-graded endomorphism of $H$.

Being maps of supermanifolds, these functors form a supermanifold themselves. Hence it makes sense to talk about the topology of the space of 1-dimensional super QFTs.

With this supermachinery in hand, we can now state the theorem that we are after, together with the two lemmas that prove it.

Theorem.([Stolz/Teichner] (simplified version)) The space of $N=1$ supersymmetric 1-dimensional euclidean QFTs is homotopy equivalent to $\mathrm{KO}_0$, the 0th space in the $\Omega$-spectrum of real K-theory.

Remark.

i) For the purposes of this presentation, I have suppressed a certain additional “Clifford grading” that Stolz-Teichner put on the super cobordisms. If one includes this, one finds spaces of supersymmetric 1D QFTs for all grades $n\in \mathbb{N}$, and that these are homotopy equivalent to the $n$-th space of the $\Omega$-spectrum of $\mathrm{KO}$-theory.

(Personally, I should remark that, while introducing this additional grading leads to the desired result, it is otherwise not naturally motivated and looks a little ad hoc to me. It is slightly better motivated in the 2-dimensional case (superstrings) that we are ultimately interested in, since there it is related to the Pfaffian line bundles coming from the string’s path integral.

In the end, however, it seem to me that we should figure out how this grading arises naturally. I think I heard rumours that it would be implied by simply using higher supersymmetry on the worldvolume. But typing this, I realize that I may or may not have dreamed this rumour.)

ii) With just slightly more effort one also gets complex K-theory. But I will not discuss that. See p. 38 in Stolz-Teichner.

The above theorem follows from two lemmas.

Lemma 1. ([Stolz-Teichner, prop. 3.2.6 & lemma 3.2.14])

a) 1-dimensional supersymmetric euclidean field theories are in bijection with maps of supermanifolds from $\mathbb{R}_+^{1|1}$ to the superspace of self-adjoint graded Hilbert-Schmidt operators $$[1d\mathrm{sCob},\mathrm{Hilb}] \simeq \mathrm{Hom}( \mathbb{R}^{1|1},\; HS^\mathrm{sa}(H) ) \,,$$ which respect the super semigroup structure on $\mathbb{R}_+^{1|1}$.

b) The latter space is homotopy equivalent to that of graded $C^*$-algebra morphisms from continuous functions to compact operators $$\mathrm{Hom}( \mathbb{R}^{1|1},\; HS^\mathrm{sa}(H) ) \simeq \mathrm{Hom}_{C^*}(C_0(\mathbb{R}), K(H)) \,.$$

Explanation and sketch of proof:

The first statement expresses essentially just functoriality. Under composition, the morphisms of $1d\mathrm{sCob}$ realize the supertranslation semigroup $\mathbb{R}_+^{1|1}$, and our $\mathrm{sQFT}$ has to respect that. In fact, the QFT functor is realized by the obvious susy generalization of the bosonic propagator: given any suitable odd-graded operator $D$ (the supercharge, or Dirac operator), the QFT functor acts on $S$-points of supercobordisms as $$\array{ 1d\mathrm{sQFT} &:& \mathrm{Hom}_{1d\mathrm{sCob}}(\bullet,\bullet) &\to& \mathrm{Hilb} \\ && \mathbb{R}^{1|1}(S) &\to& (\mathrm{End}(H))(S) \\ && (t,\theta) &\mapsto& \exp(-t D^2) + \theta D \exp(-t D^2) \,. }$$ Notice how the right hand side in the last line is indeed a combination of even graded operator times even sections of $S$ plus an odd graded operator times an odd graded section of $S$, as it should be.

It is an elementary exercise in supersymmetry reasoning to check that this assignment is indeed a supergroup homomorphism.

The second statement boils down essentially to a slight refinement of this situation. We take the algebra $C_0(\mathbb{R})$ of continuous functions on the real line to be graded in the sense that even functions (satisfying $f(x) = f(-x)$) are assigned even grade and odd functions are assigned odd grade. Regarded as a superalgebra this way, we can, as above, define a supergroup homomorphisms $$\array{ \mathbb{R}^{1|1}(S) &\to& (C_0(\mathbb{R}))(S) \\ (t,\theta) &\mapsto& \exp(-t x^2) + \theta x \exp(-t x^2) \,. }$$ Notice how the right hand side is indeed an even function plus an odd function times an odd section.

In conclusion, this tells us that the space of supersymmetric quantum mechanics is homotopy equivalent to the space of algebra homomorphisms from continuous functions on the real line to compact operators. This space, however, is well known to be homotopy equivalent to the classifying space of (real) K-theory.

Lemma 3.([Higson-Guentner]) The space $$\mathrm{Hom}_{C^*}(C_0(\mathbb{R}), K(H))$$ is homotpy equivalent to the 0th space in the $\Omega$-spectrum of real K-theory.

Now some examples.

The most accessible examples, like the one already given above, come from the geometric phase of the theory, where our superparticles are goverened by operators that act on Hilbert spaces of sections of spinor bundles over some manifolds (as opposed to being completely abstractly defined odd graded operators on some abstract Hilbert space).

Example. Given a vector bundle with spin structure, we can define the Thom class […]

6b) How does one expect the landscape of superstring theories to be equal to the spectrum of elliptic cohomology?

Unfinished notes.

The entire point of the above exercise of reformulating K-theory in terms of supersymmetric quantum mechanics is that this lends itself to the expected generalization from point particles to strings.

$\to$

Expected Definition. A 2-dimensional superconformal field theory is a 2-functor $$2d\mathrm{SCFT} : scP_2 \to 2\mathrm{Hilb}$$ from superconformal-2-paths to graded 2-Hilbert spaces.

[…]

The heterotic string partition function and the Witten genus.

[…]

Now examples.

As before for the superparticle, we are interested in examples from the geometric phase, where our supercharge comes from an operator that acts on sections of some stringy generalization of a vector bundle with spin structure. We expect there to be stringy analogs of the Thom class and the Euler class in this sense.

Definition. A 2-vector bundle with connection is a 2-functor from 2-paths to $\mathrm{Mod}_C$, for $C$ some monoidal category.

Example.([L2B]) (surface transport in abelian gerbe (line 2-bundle))

$\bullet$ Points are sent to the category $\multiscripts{_A}{\mathrm{Mod}}{}$, for $A$ morita equivalent to $\mathbb{C}$. Hence $A$ is an algebra of compact operators and we get a bundle of compact operator - a $\mathrm{PU}(H)$-bundle.

$\bullet$ Morphisms are sent to bimodules.

$\bullet$ 2-Morphisms are sent to bimodule homomorphisms.

Example.(Stolz-Teichner) (surface transport in string gerbe ($\hat \Omega \mathrm{Spin}$-2-bundle $\to$))

$\bullet$ Points are sent to the category $\multiscripts{_A}{\mathrm{Mod}}{}$, for $A$ a type $\text{III}_1$ von Neumann algebra factor.

$\bullet$ Morphisms are sent to bimodules.

$\bullet$ 2-Morphisms are sent to bimodule homomorphisms.

[…]