43

First let us recall why string theory is attractive. As of now, we have two experimentally verified, but mutually incompatible theories describing fundamental physical phenomena. The standard model of particle physics is a quantum field theory describing all the elementary particle interactions except for gravitation. General relativity is a classical theory ...

29

The problem with this question, for mathematicians, and actually for anyone, is that the term "string theory" is not well-defined, making the question of falsifiability much more complicated.
The most well-defined interpretation of "string theory" would be the superstring in 10 flat space-time dimensions, which is defined by a series expansion. The details ...

26

Monstrous moonshine, the famous relationship between the
dimensions of the irreducible representations of the Monster group and the coefficients in the Fourier expansion of
the $j$-invariant. Borcherds' proof of monstrous moonshine uses the Goddard-Thorn
theorem, which comes out of string theory,
specifically the quantization of the bosonic string.

ag.algebraic-geometry dg.differential-geometry gr.group-theory rt.representation-theory string-theory

18

Here is a purely mathematical reason why we prefer to put $u^{2g-2}$ in our generating function instead of, say, $u^g$.
The generating function you write,
$$\sum_{g,\beta} GW_{g,\beta}(X) u^{2g-2}Q^{\beta},$$
is the generating function for the connected Gromov-Witten invariants of $X$ (I've thrown in another variable that tracks the class $\beta$). ...

ag.algebraic-geometry mp.mathematical-physics gromov-witten-theory string-theory euler-characteristics

16

I want to understand compactifications, Dualities, D-branes, M-branes
etc.
What makes all this hard to learn in a systematic way is that the theory itself is still incomplete and proceeds in parts via educated guesswork.
In principle perturbative string theory is well defined: This says to pick a 2d super-conformal field theory of central charge 15, collect ...

16

Traditional approach. Notice that what is considered in [DMW00; DFM03] and elsewhere to quantize the C-field flux $G_4$ is not just ordinary cohomology, but ordinary cohomology with bells and whistles added as need be:
Foremost there is the half-integral quantization of the $G_4$-flux, mentioned as (3.2) in [DMW00]. Ordinary cohomology may be modified ("...

15

To the best of my knowledge of the literature on this topic, the answer is: not really (a few exceptions appear below). Let me first give a rigorous statement of the problem: the physical equation $\beta = 0$ can be expressed as
$\sum_{k=0}^{\infty} \epsilon^k \beta_k = 0$,
where $\epsilon$ is a physical parameter one imagines is "small," and where $\...

14

It's worth noting this fact. Suppose $L$, $L'$ are even unimodular Euclidean lattices, like $\mathrm{E}_8= \Gamma^8$ or $\Gamma^{16}$ or any direct sum of these, but also the Leech lattice and 23 others in 24 dimensions, and at least 80,000,000,000,000,000 others in 32 dimensions, etc. Let $\Gamma^{1,1}$ be the even unimodular Lorentzian lattice in 2 ...

14

There are two different things which are called "Riemann surface" in the literature.
The modern notion (introduced by Hermann Weyl): complex 1-dimensional manifold.
In older literature this is sometimes called "Abstract Riemann surface".
"Riemann surface spread over the plane" (or over the sphere, or over some other surface). Surface de Riemann etalee in ...

answered Jul 5 '13 at 7:51

Alexandre Eremenko

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14

Let $g \geq 1$ be an integer. Let $\mathcal{M}_{g,1}$ be the moduli space of genus g
Riemann surfaces with one marked point. It is an orbifold (each point comes with an
automorphism group). Let $\chi(\mathcal{M}_{g,1})$ be the orbifold Euler characteristic
of $\mathcal{M}_{g,1}$ (which takes into acount the automorphism groups).
Then Harer and Zagier have ...

14

The partition function should assign to each possible field configuration $\Phi$ (or field history) in your quantum field theory a number $Z(\Phi)$. That is, it should be a function on the collection of fields configurations, and from that function you can derive lots of quantities in the field theory.
It can happen, however, that in order to come up with, ...

14

Although it might seem futile, given how far most of the activity on AdS/CFT is from rigorous mathematics, I think this a good question, provided one is happy, for now, with (very) baby versions of this correspondence.
An example of nontrivial mathematical prediction, using a baby version of AdS/CFT (the Caffarelli-Silvestre extension) is the conformal ...

answered Feb 5 at 18:10

Abdelmalek Abdesselam

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13

The nLab has a page devoted to this question:
ncatlab.org/nlab/show/string+theory+FAQ
I'll be glad to further expand this as need be.

13

Actually I think the idea of the holographic principle is that, as in a holograph, all the information in the 'bulk' is already present at the 'boundary'. So, it claims that any calculation involving bulk observables can be expressed in terms of boundary observables. It may not claim the reverse, though that could often be taken for granted!
In ...

13

1) Verlinde's formula: Let $\mathrm{C}$ be a curve of genus $g\geqslant 2$. The Picard group of the Moduli space of rank $2$ bundles on $\mathrm{C}$ with trivial determinant is an infinite cyclic group, with ample generator denoted by $\mathscr{L}$. Verlinde's formula says that $$h^0(\mathscr{L}^{\otimes k})=\sum_{p=0}^{k} \mathrm{S}_{p0}^{-\chi(\mathrm{C})},...

ag.algebraic-geometry dg.differential-geometry gr.group-theory rt.representation-theory string-theory

12

Since the AdS/CFT correspondence links quantum field theory to something as exotic as quantum gravity, I don't think there is any hope for precise mathematical statements coming out of that correspondence in the foreseeable future.
If I interpret the question as "explicit computational consequences of the AdS/CFT correspondence", then perhaps the ...

answered Feb 5 at 12:46

Carlo Beenakker

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10

I am quite late answering this question, even though I followed it when it first appeared, but it must have slipped my mind. Anyway, it's been a while now and nobody seems to have mentioned my favourite (algebraic) reason for this.
In the covariant BRST quantisation of the bosonic string, the space of physical states can be interpreted as the relative semi-...

answered Feb 8 '14 at 16:09

José Figueroa-O'Farrill

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10

I am not an expert in this area but I know that Costello, Gaiotto, Paquette, and others have been studying topological and holomorphic twists of ADS/CFT. Unlike the full correspondence this one seems amenable to mathematical analysis.

9

As noted in Vit's post, the diagonal group $G_D \subset G \times G$ consists of the elements $\{(g,g) : g \in G \}$ with multiplication defined to be $(g,g) \cdot (h,h) = (gh,gh)$. A representation $(R_1,R_2)$ of $G \times G$ thus transforms as the representation $R_1 \otimes R_2$ under $G_D$, where $\otimes$ denotes the tensor product.
Let me illustrate ...

9

The answer is Yes, at least when the ALE space (more precisely the ALF space instead) is of type $A$. Very roughly, one consider Donaldson-Thomas invariants for the same noncompact Calabi-Yau space, but not of rank $1$, of higher ranks instead. (I do not know whether they correspond to Gromov-Witten invariants.) I do not know physics literature, but I wonder ...

8

In massless 2d scalar field theory, the expression $\langle \phi(x)\phi(y) \rangle = -\ln|x-y|$ is best regarded as a mnemonic. (A first hint of trouble: $|x-y|$ has dimensions of length, so you can't just stick it into a logarithm.) Likewise, the expression $e^{-S}$ for the measure. The Euclidean measure is usually supported on functions for which $S$ ...

8

My answer Wiener measure and Bochner Minlos can help defining a free massless real scalar field as a random element of $S_0'(\mathbb{R}^2)$. Namely, take the Schwartz space of rapidly decaying test functions $S(\mathbb{R}^2)$ and consider the subspace $S_0(\mathbb{R}^2)=\{f| \widehat{f}(0)=0\}$ of "charge-neutral" test functions. The bilinear form
$$
B(f,g)=\...

answered Jun 30 '14 at 17:29

Abdelmalek Abdesselam

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8

If the question is set on the level of mentioning important "theorems" or "computations" or "results" which
wouldn‘t have been proved without the development of string theory
i think one could easily build a very-very long list.
Maybe it would be more appropriate to speak about which "theories" wouldn't have been out there (at least in their present ...

ag.algebraic-geometry dg.differential-geometry gr.group-theory rt.representation-theory string-theory

7

Picture changing operators in supergeometry and superstring theory, Alexander Belopolsky (1997).

answered Dec 19 '13 at 12:36

Carlo Beenakker

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7

One of the issues here is that if we define CS by its physics definition (action, fields) and similarly for WZW then it does seem like there is an equivalence. For example, on page 30 of
http://arxiv.org/pdf/hep-th/9904145v1.pdf
we apparently see how to go from a state in CS theory to a correlator in WZW. From a mathematical point of view, this uses ...

6

I was encouraged to post this answer from Physics Stack Exchange to the nearly equivalent question "What experiment would disprove string theory?" here as well.
One can disprove string theory by many observations that will almost certain not occur, for example:
By detecting Lorentz violation at high energies: string theory predicts that the Lorentz ...

6

From a very lowbrow point of view, the question rests on the new ingredients string theory mandates not already embedded on either the Standard Model or General Relativity, namely supersymmetric partners and extra dimensions. The critique then goes that since we currently don't know the energy scales at which these things turn relevant (i.e. the mass or ...

6

Since no one has answered the genus computation, here goes:
The Riemann-Hurwitz formula states that for a map of Riemann surfaces $f : C_1 \to C_2$, that we have
$$
\chi(C_1) = n\chi(C_2) - \deg R
$$
where $n$ is the degree of the map, and $R$ the ramification divisor.
In the case you have, since we are covering $\mathbb{C}$, we are secretly covering $\...

6

Although it’s behind an Elsevier pay-wall, there is one paper which explains in cohomological terms the picture-changing operator in the context of string field theory. If I remember correctly it is a kind of connecting homomorphism. The paper in question is “Picture changing operation and BRST cohomology in superstring field theory” by Francisco Narganes-...

answered Dec 19 '13 at 13:58

José Figueroa-O'Farrill

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