28
votes
Mathematical uses of string theory
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. ...
22
votes
Accepted
Freeman Dyson's approach to string theory
Dyson's A walk through Ramanujan's garden gives the background of this comment: He explains that the "seeds from Ramanujan's garden have been
blowing on the wind and have been sprouting all over ...
- 163k
19
votes
Accepted
In Gromov-Witten theory, why is the string coupling constant weighted by $2g-2$?
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^...
- 5,880
18
votes
Accepted
In M-theory, what can hypothesis H tell us that quantization in ordinary cohomology cannot?
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 ...
- 19.2k
16
votes
Mathematician trying to learn string theory
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 ...
16
votes
Anomaly in QFT physics v.s. determinant line bundle
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 ...
- 261
15
votes
Does $SO(32) \sim_T E_8 \times E_8$ relate to some group theoretical fact?
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 ...
- 20.6k
15
votes
Mathematical uses of string theory
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 ...
15
votes
Accepted
What are "branes", and why do they form a category?
Let me start by putting your questions into a bit more context. Kapustin and Witten's story occurs within string theory, a theory of 1-dimensional extended objects. Strings may be "closed," ...
- 238
14
votes
Accepted
Mathematical predictions of AdS/CFT
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 ...
- 20.9k
11
votes
Mathematical predictions of AdS/CFT
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 ...
- 163k
10
votes
Accepted
Do all $\mathcal{N}=2$ Gauge Theories "Descend" from String Theory?
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, ...
- 2,781
10
votes
Freeman Dyson's approach to string theory
I don't think it would have convinced Feynman because he didn't like the rabbit hole that string theory seemed to be going down. That instead of trying to explain some phenomenon, that they were ...
- 101
9
votes
Accepted
What does it mean to take the diagonal of the group $SU(2) \times SU(2) $?
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 $(...
- 424
9
votes
Mathematical predictions of AdS/CFT
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 ...
- 1,444
8
votes
Mathematical uses of string theory
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 ...
7
votes
What is the relation between BRST quantization and gauge fixing quantization
I do not think that it makes sense to say that the gauge-fixing is a special case of BRST quantization:
$\rightarrow$ The gauge-fixing procedure is actually a normalization technique and it is ...
- 7,468
7
votes
Accepted
Why is the inertia stack of a smooth Deligne-Mumford stacks called inertia?
I think of the word "inertia" in "inertia stack" as representing the same idea as the "inertia" in "inertia group" (which presumably came first). This latter group typically comes up when one has a ...
- 9,418
6
votes
Accepted
Incorporating Divisors (D4-branes) into Donaldson-Thomas Theory?
The original definition of DT invariants ( https://arxiv.org/abs/math/9806111 ) works for any ch such that there is no strictly semistable objects. Later, this was generalized by Joyce and Song to ...
- 6,750
6
votes
In Gromov-Witten theory, why is the string coupling constant weighted by $2g-2$?
Here are two ways how physicists think about the string coupling constant:
1) In usual quantum field theory defined by quantization of a classical field theory, the partition function is defined by a ...
- 6,750
6
votes
Navier-Stokes fluid dynamics, Einstein gravity and holography
The first point to make is that the fluid/gravity correspondence relates the general theory of relativity to relativistic fluid dynamics. I don't see how the usual non-relativistic Navier-Stokes ...
- 163k
6
votes
Accepted
Degree-3 curves on the Calabi–Yau quintic
The physicists Candelas, De La Ossa, Green, Parkes predicted the virtual number $n_d$ of rational curves of degree $d$ on a quintic threefold for any $d\geqslant 1$. The numbers $n_d$ are defined by ...
- 2,729
5
votes
Accepted
How to construct the mirror partner of a blowup?
You can find a lot of such examples in the paper of Abouzaid-Auroux-Katzarkov: https://link.springer.com/article/10.1007/s10240-016-0081-9.
Basically, they studied the case when $X$ is $(\mathbb{C}^\...
- 3,035
5
votes
Accepted
Manifolds with negative dimension – Definition, References
Smooth manifolds of negative dimension are defined in derived geometry.
Recall that if A→M and B→M are two transversal submanifolds
of codimension a and b respectively,
then their intersection C is ...
- 33.7k
5
votes
Mathematical uses of string theory
I recall that Richard Wentworth's first paper on precise constants in bosonization formula (which is part of his PhD thesis?) extensively used computational methods from bosonic string theory. I am ...
5
votes
Accepted
Supersymmetry charge $Q$ as anti-linear and anti-unitary operator
Suppose you are given a super Hilbert space $\mathcal{H} = \mathcal{H}_0 \oplus \mathcal{H}_1$, with bosonic and fermionic subspaces $\mathcal{H}_0$ and $\mathcal{H}_1$ respectively. Define a new ...
- 51.4k
4
votes
Why does bosonic string theory require 26 spacetime dimensions?
Here is a purely mathematical statement where 26 appears. Introduce for every integer $d \geq 4$, $k \geq 0$ , the degree $k$ polynomial $P_k^{(d)}(x)$ coefficients of the expansion
$\frac{1}{(1-2xy+...
- 6,750
4
votes
Gromov-Witten and integrability.
Let $X$ be a smooth projective variety. Actually, as long as the quantum cohomology of $X$ is semisimple, the partition function of the (descendent) GW-invariants of $X$ is always identified with a ...
- 41
4
votes
The Unreasonable Effectiveness of Physics in Mathematics. Why ? What/how to catch?
If you accept Max Tegmark's view that "our physical world is an abstract mathematical structure" (per https://arxiv.org/abs/0704.0646), then it would follow that the "effectiveness" of physics in ...
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