Mathematical physicist. Study supersymmetric gauge theories, TQFT, mathematical physics, string theory.

Jan
13
awarded Yearling
Jan
13
awarded Yearling
Jan
13
accepted Finite-dimensional representations of DAHA
Jan
12
revised Finite-dimensional representations of DAHA
edited title
Jan
12
revised Finite-dimensional representations of DAHA
edited tags
Jan
12
asked Finite-dimensional representations of DAHA
Nov
16
comment Reference request for simple $q-$ identities
These are simplest variants of Rogers-Ramanujan (Andrews-Gordon) identities and they are also called quantum dilogarithm identities. You can google these words to reach the original references.
Nov
16
answered Reference request for simple $q-$ identities
Nov
16
revised Quantum cohomology of line bundles over $\mathbb P^N$
added 131 characters in body
Nov
16
revised Quantum cohomology of line bundles over $\mathbb P^N$
deleted 3 characters in body
Nov
16
answered Quantum cohomology of line bundles over $\mathbb P^N$
Nov
10
revised Mathematics of Chiral Rings
edited tags
Nov
9
comment Mathematics of Chiral Rings
Apart from GL twist, GMN also consider linear combinations of two supercharges with the twistor variable (See (4.1) of arxiv.org/abs/1103.2598). However, these two supercharges do not commute. In this example, BPS spectrum jumps at certain parameters of \zeta and central charge. But this is different from what I have asked in this post.
Nov
9
comment Mathematics of Chiral Rings
Yes, what you mention is correct. More generally, the grading can be the fermion number so that it becomes $Z_2$-graded. The GL twisted supercharge is written as $Q=Q_l+tQ_r$ where $Q^2=0$ and $t$ is the twistor parameter. My guess is that there exists a spectral sequence from BPS states at one value of $t$ to those at a different value of $t$. I would like to see this in a simple example, so please let me know if you know something or a good example to work on.
Nov
9
comment Mathematics of Chiral Rings
Thanks to David Ben-Zvi's comment, I have included the condition that the two differentials commute.
Nov
9
revised Mathematics of Chiral Rings
deleted 62 characters in body
Nov
9
comment Mathematics of Chiral Rings
However, $C(A)$ has rank zero! $A^\prime$ is spanned by $e_1$ and $e_2$, which are both in the ideal $\textrm{Im}(d_1)+\textrm{Im}(d_2)$. The chiral ring is empty. Since rk $C(A)$ is not equal (mod 2) to rk $A$, there is no differential d such that $C(A) = H(A,d)$.
Nov
9
comment Mathematics of Chiral Rings
An example makes this easy to see. Let $A$ be of total rank three, generated by $x$ in deg 0 and $e_1$ and $e_2$ in deg 1. Let $d_1(x) = e_1$ and $d_2(x) = e_2$; everything else is zero for degree reasons. In this case, $A$ is in fact a bicomplex, and we can consider various cohomology theories: in order of decreasing size, we have $H(A,d_1)$, $H(A,d_2)$ with respect to single differentials; iterated cohomologies like $H(H(A,d_1),d_2)$; and finally $H(A,d_1+d_2)$ with respect to the total differential. For our example, all of these agree and have one generator in deg 1.
Nov
9
comment Mathematics of Chiral Rings
However, I don't think the theory of spectral sequence apply for chiral rings.
Nov
9
comment Mathematics of Chiral Rings
Thank you very much for your comments. Yes, you are right. In physics, we consider two commuting differentials $d_1$ and $d_2$. I think that I should have included this condition in the definition.
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