The U-plane Integral, Mock Modularity and Enumerative Geometry

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F22%3A00360743" target="_blank" >RIV/68407700:21230/22:00360743 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1007/s11005-022-01520-7" target="_blank" >https://doi.org/10.1007/s11005-022-01520-7</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s11005-022-01520-7" target="_blank" >10.1007/s11005-022-01520-7</a>

Jazyk výsledku

angličtina

Název v původním jazyce

The U-plane Integral, Mock Modularity and Enumerative Geometry

Popis výsledku v původním jazyce

We revisit the low-energy effective U(1) action of topologically twisted N = 2 SYM theory with gauge group of rank one on a generic oriented smooth four-manifold X with nontrivial fundamental group. After including a specific new set of Q-exact operators to the known action, we express the integrand of the path integral of the low-energy U(1) theory as an anti-holomorphic derivative. This allows us to use the theory of mock modular forms and indefinite theta functions for the explicit evaluation of correlation functions of the theory, thus facilitating the computations compared to previously used methods. As an explicit check of our results, we compute the path integral for the product ruled surfaces X = Sigma(g) x CP1 for the reduction on either factor and compare the results with existing literature. In the case of reduction on the Riemann surface Sigma(g), via an equivalent topological A-model on CP1, we will be able to express the generating function of genus zero Gromov-Witten invariants of the moduli space of flat rank one connections over Sigma(g) in terms of an indefinite theta function, whence we would be able to make concrete numerical predictions of these enumerative invariants in terms of modular data, thereby allowing us to derive results in enumerative geometry from number theory.

Název v anglickém jazyce

The U-plane Integral, Mock Modularity and Enumerative Geometry

Popis výsledku anglicky

We revisit the low-energy effective U(1) action of topologically twisted N = 2 SYM theory with gauge group of rank one on a generic oriented smooth four-manifold X with nontrivial fundamental group. After including a specific new set of Q-exact operators to the known action, we express the integrand of the path integral of the low-energy U(1) theory as an anti-holomorphic derivative. This allows us to use the theory of mock modular forms and indefinite theta functions for the explicit evaluation of correlation functions of the theory, thus facilitating the computations compared to previously used methods. As an explicit check of our results, we compute the path integral for the product ruled surfaces X = Sigma(g) x CP1 for the reduction on either factor and compare the results with existing literature. In the case of reduction on the Riemann surface Sigma(g), via an equivalent topological A-model on CP1, we will be able to express the generating function of genus zero Gromov-Witten invariants of the moduli space of flat rank one connections over Sigma(g) in terms of an indefinite theta function, whence we would be able to make concrete numerical predictions of these enumerative invariants in terms of modular data, thereby allowing us to derive results in enumerative geometry from number theory.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Letters in Mathematical Physics

ISSN

0377-9017

e-ISSN

1573-0530

Svazek periodika

112

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

53

Strana od-do

—

Kód UT WoS článku

000773935800002

EID výsledku v databázi Scopus

2-s2.0-85127236042