# OceanPlan: Hierarchical Planning and Replanning for Natural Language AUV
Piloting in Large-scale Unexplored Ocean Environments

Ruochu Yang1, Fumin Zhang1,2, Mengxue Hou3 This research work is supported by
ONR grants N00014-19-1-2556 and N00014-19-1-2266; AFOSR grant
FA9550-19-1-0283; NSF grants GCR-1934836, CNS-2016582 and ITE-2137798; and
NOAA grant NA16NOS0120028.1 School of Electrical and Computer Engineering,
Georgia Institute of Technology, Atlanta, USA. 2 Department of Electrical and
Computer Engineering, Department of Mechanical and Aerospace Engineering, The
Hong Kong University of Science and Technology, Hong Kong, China. 3 School of
Electrical Engineering, University of Notre Dame, Notre Dame, USA.

###### Abstract

We develop a hierarchical LLM-task-motion planning and replanning framework to
efficiently ground an abstracted human command into tangible Autonomous
Underwater Vehicle (AUV) control through enhanced representations of the
world. We also incorporate a holistic replanner to provide real-world feedback
with all planners for robust AUV operation. While there has been extensive
research in bridging the gap between LLMs and robotic missions, they are
unable to guarantee success of AUV applications in the vast and unknown ocean
environment. To tackle specific challenges in marine robotics, we design a
hierarchical planner to compose executable motion plans, which achieves
planning efficiency and solution quality by decomposing long-horizon missions
into sub-tasks. At the same time, real-time data stream is obtained by a
replanner to address environmental uncertainties during plan execution.
Experiments validate that our proposed framework delivers successful AUV
performance of long-duration missions through natural language piloting.

##  I Introduction

Figure 1: OceanPlan can accomplish AUV missions through natural language
commands in the large-scale unexplored ocean.

AUVs have been actively used in a wide range of marine applications like
hurricane prediction and ocean observation systems [1, 2]. However, it is
usually heavy labor to pilot AUVs in real-world missions given complex
mechanical manuals and mission files. It would be a relief to simplify this
AUV piloting procedure with one single command, preferably in natural
language. Recent upsurge of LLMs offers us a promising option to achieve this
vision. While LLMs are proved to internalize rich knowledge in text formats,
it raises a critical issue of leveraging such knowledge for embodied robot
capabilities in the unknown and dynamically changing physical world [3]. We
keep reflecting on this question: given an abstracted human command ”Search
the aborted warship”, how can we bridge the gaps between overarching LLMs and
physical AUV motions to accomplish the mission? To fulfill our vision, there
are inevitable challenges in the realm of marine robotics. Underwater
localization is widely recognized as a major challenge, since Global
Positioning System (GPS) cannot penetrate seawater. Other terrestrial or
aerial localization methods like map-based localization are unavailable in the
ocean as well. Another challenge is limited information about the ocean
environment. The ocean topography is unknown a priori and highly unstructured.
Geographical terrains and marine objects can pose unexpected collision to
AUVs. Moreover, due to the vast spatial scale of the ocean and limited onboard
battery capacity of the AUV, it is not enough for the planner to generate only
feasible solutions. To ensure mission success, the planner has to offer
energy-efficient strategies.

To address these unique challenges in marine robotics, we design a framework
OceanPlan to accomplish efficient and robust AUV missions as shown in Figure
1. In general, our work belongs to the hierarchical planning category where
there are many similar works. To the best of our knowledge, this is the first
LLM-task-motion planning and replanning framework for natural language
piloting in the AUV domain. By labeling a work as missing a (re)planner, we
imply that it doesn’t explicitly account for that (re)planner. For example,
while every robot needs a motion planner to execute skills, some works assume
that these skills are ready on hand. [4] only uses LLMs to directly call pre-
defined robotic APIs. [5] integrates LLM planning with motion planning but no
task planning and replanning. [6] implements replanning at both task and
motion planning levels without LLMs for human interaction. [7] incorporates
LLM planning to enhance task planning in specific tasks. [8] leverages
replanning to deal with invalid LLM planning. [9] develops a framework of LLM-
motion planning and replanning for robotic manipulators. [10] encompasses all
parts of LLM-task-motion planning, but lack of replanning to handle
uncertainty. Other than simply connecting planners, our framework leverages
their unique advantages to achieve efficient AUV missions targeted to marine
robotics challenges. For robust AUV operation in the uncertain ocean, a
holistic replanner coordinates with all planners based on real-time
environmental feedback. An underwater simulator is necessary because
conducting empirical algorithms or train DRL algorithms on real AUVs is
impractical. The main contributions of this work are summarized as follows:

  * •

We present a hierarchical LLM-task-motion planning and replanning framework
OceanPlan to pilot AUVs in natural language, specifically for long-horizon
missions in large-scale unexplored ocean environments. This hierarchy is key
to achieving planning efficiency with pruned search branches and grounding
robotic plans with refined representations of the real world.

  * •

We instantiate OceanPlan on a photo-realistic ocean simulator HoloEco
featuring various scenes and a polished AUV model EcoMapper. Through
comprehensive simulation, we verify efficiency (due to hierarchical
architecture) and robustness (due to replanning) of our work targeting
inherent AUV challenges.

##  II Related Works

LLMs have exhibited considerable capabilities of interpreting natural language
in the context of real world. Recently, unifying LLMs with robotics has
emerged as a rapidly evolving research topic. Many works focus on LLM planning
pipelines for robotic execution. One common approach relies on prompt
engineering to let LLMs derive a sequential plan towards a user query [4, 7].
[11] harnesses LLM scores and RL-based affordances to select the most provable
skill of completing the overall instruction. [5] utilizes LLMs to create new
code polices for unseen scenarios. [12] guarantees efficiency and optimality
by LLM semantic guidance and LTL consistent guidance. [13] leverages LLMs
specifically for navigation tasks in unknown environments. [14] empowers LLMs
with classical PDDL planners to achieve optimal planning. [15] makes a sharp
point that LLMs are not really planning because they don’t fully understand
the physical world and their ”plans” depend largely on provided prompts. It is
also problematic for LLMs to generate a long-horizon plan at the very
beginning without subsequent updates. [16] proposes dynamical planning to
update future actions based on current and visited scenes. Taking these issues
into consideration, we resort to the following well-established robotic field.

Task and Motion Planning (TAMP) is a vastly investigated field in the robotics
community [17]. Classical TAMP works are established in deterministic and
fully observable space, branching into topics like pick-place planning [18],
manipulation planning [19], path planning [20], and rearrangement planning
[21]. It is a fundamental extension to consider inevitable uncertainty in the
real world [22]. [23] temporally decomposes long-horizon problems into a
sequence of short horizons. [24] develops an interleaved DFS-BnB and MCTS
method to achieve low computation cost and plan optimality. In light of
potential failures in the real world, closed-loop replanning serves as a
suitable solution [25]. Works span from re-prompting LLMs with corrective
instructions [26], receiving real-time environmental feedback [8], leveraging
model-based control like MPC [9], integrating embodied modalities [27], to
hierarchical replanning at both logic and motion levels [6]. Reinforcement
Learning (RL) has been closely linked with motion planning, where a control
policy is directly trained out of robotic action-reward datasets. [28]
illustrates intrinsic connections between RL and planning. [29] proposes a
real-world RL system for fine-tuning locomotion policies of legged robots.
[30] guides indoor robot navigation through a DRL policy conditioned on target
and current images. [31] develops generalized exploration policies over unseen
environments by separately training object localization and navigation
networks.

##  III Problem Formulation

In the 3D ocean world, we denote the AUV physical state as
$s_{t}=[x,y,z,\alpha,\beta,\omega]^{T}\in S$, where $t$ is the timestep,
$x,y,z$ are Cartesian coordinates, $\alpha,\beta,\omega$ are roll-pitch-yaw
angles, and $S$ is the state domain. Denote $u_{t}=[\phi,\psi]^{T}\in U$ as
the AUV control input at timestep $t$, where $\phi$ is the heading angle in
the 2D x-y plane, $\psi$ is the pitch angle to enable vertical movement, and
$U$ is the control domain. We denote the real-world observation as $z_{t}$
obtained from AUV sensors. We denote the human command as $q$, which is an
abstracted text specifying AUV missions. Since planning all the way down from
the abstracted command $q$ to physical AUV control $u_{t}$ is extremely long-
horizon, we decompose this overall problem into three sub-problems.

###  III-A LLM Planning

To streamline planning from the robot intelligence level to the human
intelligence level, we resort to LLM planning. Since the ocean environment is
usually unexplored with open-set objects, we leverage LLMs’ generalized
inferring capabilities trained out of vast amounts of information. We denote
$h$ as a plan heuristic, which is a textual description interpreted by LLMs
about the most probable way of achieving the human command $q$ given semantic
representations of the current environment. Therefore, we formulate the
following LLM planning sub-problem to command AUVs in natural language and
bias the long-horizon search towards this abstracted command in the large-
scale unknown ocean.

Problem 1 (LLM Planning): Given a human command $q$ and the current
observation $z_{t}$, compute a plan heuristic $h$ which is structured into a
symbolic goal state $s_{goal}$ and a symbolic initial state $s_{init}$.

###  III-B Task Planning

We consider that the AUV can perform a set of pre-defined actions, each of
which is described by a set of preconditions and effects. To describe
preconditions and effects using logical predicates, we abstract the AUV
physical state $s_{t}$ into a symbolic state. We also define a set of
abstracted actions $\mathcal{A}$. Each action
$\\{a_{i}\\}_{i=1}^{N}\in\mathcal{A}$ is instantiated with its preconditions
and effects and executed by a series of AUV control inputs $u_{t}$ computed by
a control policy $\pi_{a_{i}}$. Additionally, considering that the LLM planner
is essentially a semantics-based planner which struggles to understand the
physical world, the plan directly generated by the LLM may not be executable
in the real world. Especially when the AUV is navigating in a vast
environment, it is impractical to factorize continuous numerical state-action
space into textual prompts. Therefore, we formulate the task planning sub-
problem as follows.

Problem 2 (Task Planning): Given a plan heuristic $h$ and a set of actions
$\mathcal{A}$, compute a task plan $\Pi=\\{a_{k}\\}_{k=1}^{T}\in\mathcal{A}$,
which transitions the initial state $s_{init}$ to the goal state $s_{goal}$.

###  III-C Motion Planning

We rely on motion planning to execute physical control on the AUV in a
collision-free manner. Due to the unknown ocean environment, the AUV dynamics
will contain uncertainty. Hence we represent the AUV dynamics as an unknown
MDP $M=\langle S,U,P\rangle$, where $S$ is a set of states $s_{t}$, $U$ is a
set of control inputs $u_{t}$, and $P$ is the unknown state transition
function. Further because of no localization system under water, the AUV has
no access to its true state $s_{t}$. We assume that the AUV is equipped with
an on-board camera providing RGB images as observations $z_{t}$ of its
surrounding environment. Therefore, our goal is to ﬁnd a policy based on the
observations, denoted as $\pi(u_{t}|z_{t})$, that maximizes the expected
return $G_{t}=\sum_{k=0}^{\infty}\gamma^{k}R_{t+k+1}$, where $R_{k}$
represents the $k$-step reward and $\gamma$ is the discount factor. Since we
have no explicit distribution of reward $R$ due to the unknown target position
and ocean map, we sample AUV trajectories (action-observation pairs) in the
simulator to learn the reward distribution and thus find the optimal policy.

Problem 3 (Motion Planning): Given an action $a_{k}$ in the task plan $\Pi$,
learn its associated control policy $\pi_{a_{k}}(\cdot|z_{t})$ from the
sampled trajectories. Given the current observation $z_{t}$, compute the
control input $u_{t}$ from the learned control policy
$u_{t}\sim\pi_{a_{k}}(\cdot|z_{t})$.

##  IV Methodology

Figure 2: Hierarchical framework of LLM-task-motion planning and replanning.

The overall framework is supposed to bridge the gaps from an abstracted human
command to physical AUV control while addressing specific challenges in the
ocean. As shown in Figure 2, the framework is composed of four modules: i) A
high-level LLM planner composes a plan heuristic of the human command to guide
subsequent planning; ii) A middle-level task planner creates a feasible plan
given the plan heuristic and predefined actions; iii) A low-level motion
planner computes control inputs to execute the plan based on real-world
observations; iv) A holistic replanner evaluates AUV status and reports
unexpected situations to the corresponding planner for robust execution. This
hierarchy decomposes planning complexity of the overall problem as high-level
planners simplify planning for their low-level partners.

###  IV-A Generalized LLM Planner

The LLM planner retains the basic functionality of interpreting the command,
but emphasizes more on its generalized knowledge to guide the task planner as
a human-like brain, which offers two advantages: i) It can adapt into open
ocean worlds based on its generalized inference rather than manually designed
scene-specific prompts; ii) A reasonable plan heuristic efficiently biases
search directions of achieving the human command in the large-scale ocean.
Considering AUV challenges, we employ the following specific strategies.

Semantic Map: Identifiable objects can be sparse in the large-scale ocean, so
the LLM planner must enrich its knowledge of the world given new observations.
The LLM planner maintains an internal semantic map $\mathcal{M}_{t}$ to
memorize the explored environment so far. We utilize Vision-Language Models
(VLMs) to convert image observations into texts, so that the LLM planner will
be progressively grounded in the mission and reinforce the future plan
heuristic to avoid repeated exploration. For example when searching the
warship, the AUV detects a glider next to it. In the future given a new
command ”Survey the glider”, the LLM planner will prioritize the warship area.

Augmenting XML File: Unlike indoor robotic scenes, ocean environments may not
possess sufficient semantic information for the LLM commonsense reasoning. We
use an XML file $f_{c}$ to augment marine knowledge of the LLM planner. For
instance, it can provide hints like ”Coral reefs usually grow in areas with
ample sunlight”. The plan heuristic could be ”The bright plain on the left is
more likely to grow the coral reef than the dark hill on the right”. Note that
we still don’t hands-on teach the LLM planner how to achieve AUV missions but
just provide related marine knowledge.

We instantiate the current observation $z_{t}$ as agent-view RGB images of the
surrounding environment. Then we use a VLM to associate each image with a text
descriptor like ”a rocky hill on the left”. We score these text descriptors
with the probability of how likely they will complete the command given the
XML file and the current semantic map. The one with the highest probability is
selected as the plan heuristics $h$. By providing query-response pairs with
the LLM planner, the plan heuristic is eventually structured into a formal
task consisting of symbolic initial state $s_{init}$ and goal state $s_{goal}$
for the task planner.

###  IV-B HTN Task Planner

We choose HTN planner [32] as the symbolic task planner and center the actions
$\mathcal{A}$ on marine autonomy. For example, given the task ”Survey the
warship”, the task planner should generate an action sequence $\Pi$ to guide
the AUV to the warship without collision in the complex ocean. As such, we
define the following Boolean-valued predicates:

  * •

navigated ?auv - AUV: if the AUV is navigated towards a certain area.

  * •

env_sensed: if a certain area is sensed.

  * •

detected ?target - object: if the target object is detected.

  * •

captured ?target - object: if the target object is taken photo of.

  * •

approached ?target - object: if the target object is approached at a close
distance.

  * •

reported ?target - object: if photos of the target object is sent to the
pilot.

  * •

replanned: if the replanning signal is sent to the task planner.

The AUV actions are defined as follows:

  * •

navigate(AUV): navigate the AUV towards a certain area.

  * •

sense: take images of a certain area.

  * •

approach(target): move close to the target object.

  * •

capture(target): take photos of a target object.

  * •

report(target): surface to transmit photos and the GPS position of the target
object.

  * •

rescue: surface and send a rescue signal.

Detailed preconditions and effects of each action are presented in Appendix.
We would like to delve deeper into the abstraction nature of our defined
actions. There exist large-scale and unstructured landscapes like canyons and
hills in the ocean, all of which pose collision risks to AUVs. However, these
landscapes will not differ much across different ocean regions and may appear
quite similar in high-dimensional representations like images. Furthermore,
there are much less semantic objects in the ocean compared to indoor
environments. In this sense, the unstructured ocean topography with sparse
semantic information constitutes a challenging factor of controlling AUVs. By
abstracting motions like ”swim through a canyon” or ”bypass a hill” into a
unified action navigate(AUV), we only need to capture the overall landscape
around the AUV through visual observations. In this way, we sidestep
impractical ocean map modeling or precise object detection along the AUV
trajectory.

###  IV-C DQN Motion Planner

The motion planner determines control inputs for the AUV to execute in the
physical world. As presented in Section III-C, we are motivated to leverage
DRL methods to guide AUV motions based on these visual inputs. We select Deep
Q Network (DQN) [33] as our solution, which aims at ﬁnding a direct mapping
(represented by a DNN $\theta$) from the current observation $z_{t}$ to the
Q-function $Q_{\pi}(z_{t},u_{t}|\theta)$ and then uses greedy algorithm to
generate a control policy

$\pi_{a_{i}}(u_{t}|z_{t})\Leftarrow\underset{u_{t}}{\arg\max}Q_{\pi}(z_{t},u_{t}|\theta)$
(1)

associated with an action $a_{i}\in\mathcal{A}$. The control input $u_{t}$ is
drawn from the control policy $u_{t}\sim\pi_{a_{i}}(\cdot|z_{t})$ given the
current observation $z_{t}$. We instantiate $z_{t}$ as the agent-view RGB
image and focus on learning the control policy associated with the
navigate(AUV) action. The control policy plans a primitive motion like ”move
forward” or ”turn left” given the current image, enabling the AUV to safely
explore the specified region.

As mentioned in Section IV-B, we generalize the control policy of the
navigate(AUV) action across diverse landscapes. By leveraging spatial
representations embedded in images, the same control policy can be trained
without differentiating between canyons or hills. DRL approaches typically
take millions of steps to learn composite tasks. Our proposed planning
hierarchy tremendously simplifies the motion planner into short-range
movements. Moreover, our simulator HoloEco enables safe interaction with the
environment so we can freely collect training datasets.

###  IV-D Holistic Replanner

Ocean environments are fairly uncertain and dynamic, so it is essential to
adjust AUV behaviour through replanning. We design a holistic replanner to
trigger replanning at respective planners by simultaneously considering the
plan heuristic, action effects, and AUV states. This hierarchical replanning
at all planners is an intuitive yet effective approach. Low-level issues like
motion drift can be directly addressed by the motion planner without altering
the entire plan. High-level problems should be addressed by the task planner
injecting a corrective action or by the LLM planner re-assessing the
unfinished mission. Given real-world feedback, the symbolic replanning flag is
designed as follows:

$f=\left\\{\begin{array}[]{lcl}\varnothing,&&\text{normal but mission not
done}\\\ 0,&&\text{mission done}\\\ 1,&&\text{LLM replanning required}\\\
2,&&\text{task replanning required}\\\ 3,&&\text{motion replanning
required}\end{array}\right.$ (2)

We instantiate feedback to the replanner as three sensors: an IMU sensor, a
forward laser sensor, and a velocity sensor. Every time the AUV executes the
current control input $u_{t}$ in the real world, the updated observation
$z^{\prime}$ is obtained to analyze the numerical AUV state. Specifically, the
replanner comprises two components: an AUV status evaluator and an environment
monitor.

AUV Status Evaluator: The AUV status evaluator tracks abnormal AUV behavior. A
significant reduction in AUV velocity indicates degeneration of the AUV
mobility. With the replanning flag marked as $f=2$, the task planner
immediately terminates the current plan and implements the rescue action for
assistance. If the IMU sensor detects radical AUV accelerations, the
replanning flag will be triggered as $f=3$, and the motion planner generates a
corrective control input to restore the original moving direction.

Environment Monitor: The environment monitor keeps assessing the surroundings.
If the AUV hasn’t achieved the mission after the current plan heuristic, the
replanning flag is set as $f=1$. The LLM planner re-assesses the environment,
updates the semantic map, and generates a new plan heuristic. Since the
navigate(AUV) action is trained towards unstructured landscapes without
accounting for objects, we employ a forward laser sensor in case of any
collision in front of the AUV. With the replanning flag as $f=3$, the motion
planner controls the AUV away from the collision direction.

##  V Experiments

We evaluate in HoloEco simulator that given one single human command, if the
AUV can efficiently and safely navigate towards the target in the large-scale
unexplored ocean environment.

###  V-A Experimental Setup

To evaluate our system, we build a marine simulator HoloEco upon HoloOcean
[34], which provides high scalability and fidelity for AUV activities in a 3D
ocean environment. It also ensures no damage to AUVs during the DQN training
process. We instantiate three objects ”coral reef, glider, warship” and three
unstructured landscapes ”canyon, hill, plain”. We provide the VLM with
reference images of objects and landscapes so that it can accurately interpret
images for the LLM planner. We instantiate the AUV model as a precise
prototype EcoMapper [35]. This detailed dynamics of the scene and the AUV
presents how real missions can be similarly performed using our method.

###  V-B Mission 1 - Search Aborted Warship

Figure 3: The entire process of EcoMapper searching the aborted warship given
an abstracted human command. Each numbered picture corresponds to a specific
phase of the process.

After we issue an abstracted command ”Search the aborted warship”, OceanPlan
directs EcoMapper to accomplish this comprehensive mission across the wide
unknown ocean. The entire process is shown in Figure 3. Through seven phases
of planning and replanning, EcoMapper successfully locates and reports the
warship after exploring a wide range of the ocean. For example in phase 5,
considering the command and the detected warship image, the LLM planner
generates a plan heuristic ”Directly survey the warship” and structures it
into a formal task as

  * •

s_init = (not (approached warship)) (detected warship) (not (reported
warship))

  * •

s_goal = (reported warship)

To achieve the task, the HTN task planner formulates a plan
[approach(warship), capture(warship), report(warship)] and the motion planner
sequentially executes the actions in the plan. Once the current plan is
executed, the replanner evaluates both mission progress and the AUV status. A
full video is available at the project website
https://sites.google.com/view/oceanplan.

###  V-C Mission 2 - Search Glider near Coral Reef

In this mission, there is a glider working around the coral reef. The AUV
pilot would like to check the glider by issuing an abstracted command ”Search
the glider near coral reef”. In phase 1, the VLM detects a warship and a
glider. The LLM planner updates the semantic map with ”A glider is near the
warship on the right”. The plan heuristic is ”Search the left area for the
other glider near the coral reef”. In phase 2 with few objects, environment
semantic information is extremely sparse. The LLM planner turns to the XML
file in Section V-A for more hints and generates a plan heuristic ”Based on
the coral reef attribute in the XML file, it is likely to grow in the front
plain with ample sunlight”. A full video is available at the project website
https://sites.google.com/view/oceanplan.

###  V-D Mission 3 - Replan

Through three unexpected situations where replanning is initiated at the
corresponding planner, we present robust AUV operation in the unpredictable
ocean. In Mission 1, replanning at the LLM planner has been extensively
presented, so we focus on presenting replanning at task planner and motion
planner. A full video is available at the project website
https://sites.google.com/view/oceanplan.

Situation 1: The replanner detects a significant reduction in AUV speed and
sends a replanning signal to the task planner. The task planner immediately
replaces the current plan with the rescue action so that EcoMapper surfaces
and transmits the rescue signal in time.

Situation 2: The replanner identifies a radical leftward acceleration of AUV,
triggering the replanning flag of the motion planner. The motion planner
responds with a corrective control input ”turn right” to offset the unexpected
leftward acceleration.

Situation 3: The replanner detects that EcoMapper is at a high risk of
colliding with a glider in front of it. The motion planner controls EcoMapper
to move away from the glider.

###  V-E Ablation Studies

We perform two ablation studies of Mission 1 and Mission 2 to evaluate
importance of LLM planner and task planner during long missions in the
unexplored ocean. We carry out 10 simulation runs and average both the
completion time and the success rate. We exclude response time of LLM and VLM.
We aim to claim through quantitative comparison that lack of any component in
our proposed framework will largely depreciate the AUV performance.

Ablation of LLM Planner: In this study, we only use task planner and motion
planner to achieve the same command with the same actions. In each phase, we
manually evaluate the images and form a task for the task planner. We only
evaluate if the target object is in view and don’t consider semantic
information about other objects. If not in view, we randomly choose an
exploring direction. As shown in Figure 4, it takes around three times longer
than the proposed method to achieve Mission 1 and around twice to achieve
Mission 2. We can conclude that it is inefficient to rely solely on the task
planner and motion planner to perform long-horizon planning given an
abstracted command. In absence of heuristics from the LLM planner, the task
planner will take much more time to randomly explore the ocean space in a
brute-force pattern.

Ablation of Task Planner: In this study, we only use LLM planner and motion
planner. We provide the LLM planner with the same actions, but don’t provide
their preconditions and effects or illustrative prompts. The LLM planner
relies on itself to organize the actions to achieve the heuristic. As shown in
Figure 4, the LLM planner generates invalid plans 6 out of 10 runs in Mission
1 and 7 out of 10 runs in Mission 2. For example in Mission 1, for the
heuristic ”Directly survey the warship”, a wrong plan [navigate(ecomapper),
sense, approach(warship), navigate(ecomapper), sense, capture(warship),
navigate(ecomapper), sense, report(warship)] is generated, where the second
navigate(ecomapper) action causes EcoMapper to collide with the warship. We
can conclude that without logical connections introduced by the task planner,
the LLM planner cannot guarantee the plan quality.

Figure 4: Quantitative results of ablation studies demonstrate that our method
achieves a good balance between efficiency and validity of planning a long-
horizon mission given an abstracted command.

##  VI Conclusion

We propose a hierarchical planning and replanning framework to pilot AUVs
through natural language in the large-scale unknown ocean given major marine
robotics challenges. Given a human command, an LLM planner generates a plan
heuristic, a task planner guarantees a valid plan, a motion planner executes
the plan in the real world, and a holistic replanner ensures robust AUV
operation. In a marine simulator HoloEco, OceanPlan is validated to ground
human commands for effective and safe AUV missions.

## References

  * [1] F. Zhang, D. M. Fratantoni, D. A. Paley, J. M. Lund, and N. E. Leonard, “Control of coordinated patterns for ocean sampling,” _International Journal of Control_ , vol. 80, no. 7, pp. 1186–1199, 2007.
  * [2] D. A. Paley, F. Zhang, and N. E. Leonard, “Cooperative control for ocean sampling: The glider coordinated control system,” _IEEE Transactions on Control Systems Technology_ , vol. 16, no. 4, pp. 735–744, 2008.
  * [3] S. Tellex, N. Gopalan, H. Kress-Gazit, and C. Matuszek, “Robots that use language,” _Annual Review of Control, Robotics, and Autonomous Systems_ , vol. 3, pp. 25–55, 2020.
  * [4] W. Huang, P. Abbeel, D. Pathak, and I. Mordatch, “Language models as zero-shot planners: Extracting actionable knowledge for embodied agents,” in _International Conference on Machine Learning_. PMLR, 2022, pp. 9118–9147.
  * [5] J. Liang, W. Huang, F. Xia, P. Xu, K. Hausman, B. Ichter, P. Florence, and A. Zeng, “Code as policies: Language model programs for embodied control,” in _2023 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2023, pp. 9493–9500.
  * [6] T. Lin, C. Yue, Z. Liu, and X. Cao, “Generalized multi-level replanning tamp framework for dynamic environment,” _arXiv preprint arXiv:2310.14816_ , 2023.
  * [7] I. Singh, V. Blukis, A. Mousavian, A. Goyal, D. Xu, J. Tremblay, D. Fox, J. Thomason, and A. Garg, “Progprompt: program generation for situated robot task planning using large language models,” _Autonomous Robots_ , pp. 1–14, 2023.
  * [8] W. Huang, F. Xia, T. Xiao, H. Chan, J. Liang, P. Florence, A. Zeng, J. Tompson, I. Mordatch, Y. Chebotar _et al._ , “Inner monologue: Embodied reasoning through planning with language models,” in _Conference on Robot Learning_. PMLR, 2023, pp. 1769–1782.
  * [9] W. Huang, C. Wang, R. Zhang, Y. Li, J. Wu, and L. Fei-Fei, “Voxposer: Composable 3d value maps for robotic manipulation with language models,” in _Conference on Robot Learning_. PMLR, 2023, pp. 540–562.
  * [10] D. Shah, M. R. Equi, B. Osiński, F. Xia, S. Levine _et al._ , “Navigation with large language models: Semantic guesswork as a heuristic for planning,” in _7th Annual Conference on Robot Learning_ , 2023.
  * [11] M. Ahn, A. Brohan, N. Brown, Y. Chebotar, O. Cortes, B. David, C. Finn, C. Fu, K. Gopalakrishnan, K. Hausman _et al._ , “Do as i can, not as i say: Grounding language in robotic affordances,” _arXiv preprint arXiv:2204.01691_ , 2022.
  * [12] Z. Dai, A. Asgharivaskasi, T. Duong, S. Lin, M.-E. Tzes, G. Pappas, and N. Atanasov, “Optimal scene graph planning with large language model guidance,” _arXiv preprint arXiv:2309.09182_ , 2023.
  * [13] B. Yu, H. Kasaei, and M. Cao, “L3mvn: Leveraging large language models for visual target navigation,” in _2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , 2023, pp. 3554–3560.
  * [14] B. Liu, Y. Jiang, X. Zhang, Q. Liu, S. Zhang, J. Biswas, and P. Stone, “Llm+ p: Empowering large language models with optimal planning proficiency,” _arXiv preprint arXiv:2304.11477_ , 2023.
  * [15] K. Valmeekam, A. Olmo, S. Sreedharan, and S. Kambhampati, “Large language models still can’t plan (a benchmark for llms on planning and reasoning about change),” in _NeurIPS 2022 Foundation Models for Decision Making Workshop_ , 2022.
  * [16] A. Rajvanshi, K. Sikka, X. Lin, B. Lee, H.-P. Chiu, and A. Velasquez, “Saynav: Grounding large language models for dynamic planning to navigation in new environments,” _arXiv preprint arXiv:2309.04077_ , 2023.
  * [17] C. R. Garrett, R. Chitnis, R. Holladay, B. Kim, T. Silver, L. P. Kaelbling, and T. Lozano-Pérez, “Integrated task and motion planning,” _Annual review of control, robotics, and autonomous systems_ , vol. 4, pp. 265–293, 2021.
  * [18] M. Gualtieri and R. Platt, “Robotic pick-and-place with uncertain object instance segmentation and shape completion,” _IEEE robotics and automation letters_ , vol. 6, no. 2, pp. 1753–1760, 2021.
  * [19] H. Zhang, S.-H. Chan, J. Zhong, J. Li, P. Kolapo, S. Koenig, Z. Agioutantis, S. Schafrik, and S. Nikolaidis, “Multi-robot geometric task-and-motion planning for collaborative manipulation tasks,” _Autonomous Robots_ , pp. 1–22, 2023.
  * [20] M. Burke, K. Lu, D. Angelov, A. Straižys, C. Innes, K. Subr, and S. Ramamoorthy, “Learning rewards from exploratory demonstrations using probabilistic temporal ranking,” _Autonomous Robots_ , vol. 47, no. 6, pp. 733–751, 2023.
  * [21] Y. Ding, X. Zhang, C. Paxton, and S. Zhang, “Task and motion planning with large language models for object rearrangement,” in _2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , 2023, pp. 2086–2092.
  * [22] T. Lozano-Pérez, M. T. Mason, and R. H. Taylor, “Automatic synthesis of fine-motion strategies for robots,” _The International Journal of Robotics Research_ , vol. 3, no. 1, pp. 3–24, 1984.
  * [23] A. Curtis, X. Fang, L. P. Kaelbling, T. Lozano-Pérez, and C. R. Garrett, “Long-horizon manipulation of unknown objects via task and motion planning with estimated affordances,” in _2022 International Conference on Robotics and Automation (ICRA)_. IEEE, 2022, pp. 1940–1946.
  * [24] M. Hou, Y. Li, F. Zhang, S. Sundaram, and S. Mou, “An interleaved algorithm for integration of robotic task and motion planning,” in _2023 American Control Conference (ACC)_. IEEE, 2023, pp. 539–544.
  * [25] C. R. Garrett, C. Paxton, T. Lozano-Pérez, L. P. Kaelbling, and D. Fox, “Online replanning in belief space for partially observable task and motion problems,” in _2020 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2020, pp. 5678–5684.
  * [26] P. Sharma, B. Sundaralingam, V. Blukis, C. Paxton, T. Hermans, A. Torralba, J. Andreas, and D. Fox, “Correcting Robot Plans with Natural Language Feedback,” in _Proceedings of Robotics: Science and Systems_ , New York City, NY, USA, June 2022.
  * [27] D. Driess, F. Xia, M. S. M. Sajjadi, C. Lynch, A. Chowdhery, B. Ichter, A. Wahid, J. Tompson, Q. Vuong, T. Yu, W. Huang, Y. Chebotar, P. Sermanet, D. Duckworth, S. Levine, V. Vanhoucke, K. Hausman, M. Toussaint, K. Greff, A. Zeng, I. Mordatch, and P. Florence, “Palm-e: An embodied multimodal language model,” 2023.
  * [28] R. Munos _et al._ , “From bandits to monte-carlo tree search: The optimistic principle applied to optimization and planning,” _Foundations and Trends® in Machine Learning_ , vol. 7, no. 1, pp. 1–129, 2014.
  * [29] L. Smith, J. C. Kew, X. B. Peng, S. Ha, J. Tan, and S. Levine, “Legged robots that keep on learning: Fine-tuning locomotion policies in the real world,” in _2022 International Conference on Robotics and Automation (ICRA)_. IEEE, 2022, pp. 1593–1599.
  * [30] Y. Zhu, R. Mottaghi, E. Kolve, J. J. Lim, A. Gupta, L. Fei-Fei, and A. Farhadi, “Target-driven visual navigation in indoor scenes using deep reinforcement learning,” in _2017 IEEE international conference on robotics and automation (ICRA)_. IEEE, 2017, pp. 3357–3364.
  * [31] A. Devo, G. Mezzetti, G. Costante, M. L. Fravolini, and P. Valigi, “Towards generalization in target-driven visual navigation by using deep reinforcement learning,” _IEEE Transactions on Robotics_ , vol. 36, no. 5, pp. 1546–1561, 2020.
  * [32] D. Nau, Y. Cao, A. Lotem, and H. Munoz-Avila, “Shop: Simple hierarchical ordered planner,” in _Proceedings of the 16th international joint conference on Artificial intelligence-Volume 2_ , 1999, pp. 968–973.
  * [33] V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski _et al._ , “Human-level control through deep reinforcement learning,” _nature_ , vol. 518, no. 7540, pp. 529–533, 2015.
  * [34] E. Potokar, S. Ashford, M. Kaess, and J. Mangelson, “HoloOcean: An underwater robotics simulator,” in _Proc. IEEE Intl. Conf. on Robotics and Automation, ICRA_ , Philadelphia, PA, USA, May 2022.
  * [35] C. Wang, F. Zhang, and D. Schaefer, “Dynamic modeling of an autonomous underwater vehicle,” _Journal of Marine Science and Technology_ , vol. 20, pp. 199–212, 2015.

## Action Preconditions and Effects

Detailed preconditions and effects of predefined actions are presented in
Figure 5.

Figure 5: Preconditions and effects of predefined AUV actions.

## Action Implementation

We illustrate detailed implementation of the control policies associated with
the navigate(AUV) and approach(target) actions.

###  -A navigate(AUV) action

The control policy of the navigate(AUV) action is a learned stochastic policy
function which takes the current image as input and outputs a primitive
action. We utilize DQN to train the control policy, which requires training
two policies: the behaviour policy and the target policy with the same network
parameters. The behaviour policy collects experiences through interaction with
the simulated environment, while the target policy learns from them to update
its own network. We follow the training pipeline of [31] and instantiate the
training scenario as EcoMapper navigating through a canyon, hill, and plain
safely and quickly.

  * •

Control inputs: turn left, turn right, move forward, move up.

  * •

Observation: the current agent-view RGB image.

  * •

Immediate reward: time penalty -0.1 to encourage shorter trajectories;
collision penalty -0.1 to discourage collision. Goal-reaching reward +10 upon
action completion.

  * •

Terminated condition: The AUV safely navigates through the area. Truncated
condition: Control inputs exceed 30.

  * •

Network architecture: We use Convolutional Neural Network to extract features
of the current image. Next we flatten and feed the features to a Feedforward
Network, which returns the Q value of all control inputs.

The training results in Figure 6 present convergence of both the loss and the
reward. The training hyperparameters are shown in Table I.

TABLE I: DQN training hyperparameters. Hyperparameters | Value
---|---
discount factor | 0.95
replay memory total size | 60000
relay memory batch size | 64
learning rate | 0.005
initial training samples | 2000
target policy updating frequency | 500
training episodes | 5000
epsilon limit of behavior policy | 0.05

(a) Reward over episode.

(b) Huber loss over episode.

Figure 6: DQN training results.

###  -B approach(target) action

The control policy of the approach(target) action is to move the AUV close to
the target object by tracking the target in the image center. The VLM outputs
the relative position of the target with respect to the AUV by comparing the
current image and the target images. The VLM first identifies if the target
object is ’close’ or ’far’ to the AUV. If deemed ’far’, the position is
discretized into five categories: ’left-far’, ’right-far’, ’center-far’, ’top-
far’, and ’bottom-far’. We take into account three classes of objects
’glider’, ’warship’, and ’coral reef’. Following question-answer samples, the
VLM can identify the object’s relative position in the image. The
approach(target) action will not end until the target is detected as ’close’.
Since there is no training for this action, we use a forward laser sensor to
avoid collision with the target object. An illustrative detection result is
shown in Figure 7.

Figure 7: On the left side, two target images show the target object ’glider’
at the ’center-close’ position. On the right side, the current image is
identified with the ’glider’ at the ’right-far’ position.


# An Extended Trace Formula for Vertex Operators

Mohammad Reza Rahmati, Gerardo Flores
111Mathematics Subject classification: 17B10, 20C32, 20G05, 22E60, 22E65,
22E70 222Keywords: Fock Space, Infinite wedge representation, Infinite
dimensional Lie algebras, Representation of Lie superalgebras, Vertex
operator, Character formula. 333email<EMAIL_ADDRESS>(M. R. Rahmati),
<EMAIL_ADDRESS>(G. Flores)

###### Abstract

We present an extension of the trace of a vertex operator and explain a
representation-theoretic interpretation of the trace. Specifically, we
consider a twist of the vertex operator with infinitely many Casimir operators
and compute its trace as a character formula. To do this, we define the Fock
space of infinite level $\mathfrak{F}^{\infty}$. Then, we prove a duality
between $\mathfrak{gl}_{\infty}$ and
$\mathfrak{a}_{\infty}=\widehat{\mathfrak{gl}}_{\infty}$ of Howe type, which
provides a decomposition of $\mathfrak{F}^{\infty}$ into irreducible
representations with joint highest weight vector for $\mathfrak{gl}_{\infty}$
and $\mathfrak{a}_{\infty}$. The decomposition of the Fock space
$\mathfrak{F}^{\infty}$ into highest weight representations provides a method
to calculate and interpret the extended trace.

##  1 Introduction

Characters of representations of infinite-dimensional Lie algebras have been
one of the most profound research subjects in the last few decades. Many
questions in mathematics and physics can be expressed by the representation
theory of infinite-dimensional Lie algebras. The character formulas in Lie
theory can be interpreted in various concepts with increasing applications.
Infinite wedge representation provides examples of the representation of
infinite-dimensional Lie algebras that produces such interesting character
formulas. The infinite wedge representation is a fundamental discrete
structure on which many problems in theoretical quantum physics can be
modeled. It provides a general theoretic representation framework that
supports different sampling problems in quantum theory. The infinite wedge
representation is sometimes referred to as the fermionic Fock space.
Interesting character formulas can be extracted from specific operators acting
on the Fock space trace that provide a powerful tool to study, for instamce,
generating series, Faymann integrals, and probability amplitudes in
mathematical physics, [3, 4, 5, 8, 10, 25, 26, 29, 30, 32, 33, 35, 36].

A vertex operator is an operator of an infinite-dimensional Lie algebra that
appears in the form of generating a series of operators. Vertex operators
present a formalism for the linear action on specific infinite-dimensional
vector spaces, such as the fermionic Fock space. The trace of the vertex
operators is related to Schur functions and symmetric polynomials. In this
context, the symmetric functions play a prominent role in connecting
combinatorics to the representation theory of infinite-dimensional Lie
algebras. The theory of vertex operators involves the representation theory of
infinite symmetric group $S_{\infty}$ and the theory of symmetric polynomials;
there are well-known formulas for the trace of vertex operators in terms of
symmetric polynomials.

The computation of the trace of the vertex operators
$Tr(q^{L_{0}}\exp(\sum_{n}A_{n}\alpha_{-n})\exp(\sum_{n}B_{n}\alpha_{n}))$, is
crucial in representation theory. The transformation beneath the trace is
acting on common Fock space $\mathfrak{F}=\mathfrak{F}^{1}$, also called the
infinite wedge representation. One can consider higher Casimir operators
$L_{j},\ j>0$ acting on the Fock space $\mathfrak{F}$. We follow a computation
of Bloch-Okounkov [4] for the character of the infinite wedge representation,
where a product formula is established for the character. Vertex operators
appear in the context of string theory partition functions of CY 3-folds. In
this case, the vertex operator is twisted by one or more Casimir operators,
where the interest is to calculate its trace. We give a natural extension of
the trace to the case where infinitely many Casimir operators appear in the
trace function. Our idea is to use a decomposition of the Fock space of level
infinity into irreducible highest weight representations of Lie superalgebra
$\mathfrak{a}_{\infty}=\hat{\mathfrak{gl}}_{\infty}$. The decomposition breaks
the trace into a sum of the traces on the irreducible components, [5, 6].

###  1.1 Contributions

Based on the computation of the trace formulas for
$Tr(q^{L_{0}}\exp(\sum_{n}A_{n}\alpha_{-n})\exp(\sum_{n}B_{n}\alpha_{n}))$ and
the Bloch-Okounkov result on the character formula for
$Tr\left(\exp(\sum_{j\geq 0}2\pi iL_{j})\right)$, we propose to compute the
following trace:

$Trace=Tr\left(\exp(\sum_{j\geq 0}2\pi
iL_{j})\exp(\sum_{n>0}A_{n}\alpha_{-n})\exp(\sum_{n>0}B_{n}\alpha_{n})\right).$
(1.1)

Our approach is to define the Fock space of level $\infty$ denoted by
$\mathfrak{F}^{\infty}$, as the natural generalization of the Fock space of
finite level $l$. Then, we show the existence of a decomposition,

$\mathfrak{F}^{\infty}=\bigoplus_{\lambda}L(\mathfrak{gl}_{\infty},\lambda)\otimes
L(\mathfrak{a}_{\infty},\Lambda(\lambda))$ (1.2)

where $\lambda$ runs overall generalized partitions, from which a character
formula can be calculated. The $\lambda$ summand is a joint-highest weight
representation of $\mathfrak{gl}_{\infty}$ and $\mathfrak{a}_{\infty}$. We
shall interpret the trace in (1.1) as the trace of an operator acting on the
Fock space $\mathfrak{F}^{\infty}$. The decomposition above in (1.2) gives a
way to express the trace formula as a sum of traces over highest weight
representations of $\mathfrak{gl}_{\infty}\times\mathfrak{a}_{\infty}$, where
we can conduct the computation directly.

###  1.2 Organization of the text

The remainder of this paper is as follows. Section 2 provides basic
definitions on Fock spaces and infinite wedge representation. We also
introduce the vertex operators and character formulas for the vertex
operators. Section 3 explains the problem that we are going to solve in the
paper together with its motivation from physics. Section 4 contains the main
contributions of the paper. Further, we add an application given in Section 5.
Some conclusions are given in Section 6. Finally, the appendix contains the
definition of several infinite-dimensional Lie algebras that are important in
this context.

##  2 Preliminaries

###  2.1 Infinite wedge representation

The Fock space is an infinite-dimensional vector space that appears to
represent certain infinite-dimensional Lie algebras. It provides a systematic
framework to express generating a series of importance in Physics by the trace
of vertex operators [acting on the Fock space]. It also plays a crucial role
in string theory to explain the probabilistic amplitudes.

######  Definition 2.1.

The half infinite wedge or fermionic Fock space $\mathfrak{F}$ defined by:

$\mathfrak{F}=\bigwedge^{\frac{\infty}{2}}V=\bigoplus_{i_{r}\in
1/2+\mathbb{Z}}\ \mathbb{C}.\ v_{i_{1}}\wedge v_{i_{2}}\wedge...,\qquad
i_{j}=i_{j-1}-1/2,\ j\gg 0,$ (2.1)

is the vector space spanned by the semi-infinite wedge product of a fixed
basis of the infinite-dimensional vector space,

$V=\sum_{i\in 1/2+\mathbb{Z}}\mathbb{C}.v_{i},$ (2.2)

i.e., the monomials $v_{i_{1}}\wedge v_{i_{2}}\wedge...$ such that:

  * •

$i_{1}>i_{2}>...$

  * •

$i_{j}=i_{j-1}-1/2$ for $j\gg 0$.

Besides, we have the creation and annihilation operators defined by:

$\displaystyle\psi_{k}:$ $\displaystyle v_{i_{1}}\wedge
v_{i_{2}}\wedge...\mapsto v_{k}\wedge v_{i_{1}}\wedge v_{i_{2}}\wedge...$
(2.3) $\displaystyle\psi_{k}^{*}:$ $\displaystyle v_{i_{1}}\wedge
v_{i_{2}}\wedge...\mapsto(-1)^{l}v_{i_{1}}\wedge
v_{i_{2}}\wedge...\wedge\widehat{v_{i_{l}=k}}\wedge...$

The monomials can be parametrized in terms of partition

$|\lambda\rangle=v_{\lambda}=\lambda_{1}-1/2\wedge\lambda_{2}-3/2\wedge...$
(2.4)

The Fock space $\mathfrak{F}$ is almost a Hilbert space, with respect to the
inner product $\langle\ v_{\lambda},\ v_{\mu}\ \rangle=\delta_{\lambda\mu}$.
Its completion with respect to the norm of the inner product is a Hilbert
space. We can also write this using the Frobenius coordinates of partitions:

$|\lambda\rangle=\prod_{i=1}^{l}\psi_{a_{i}}^{*}\psi_{b_{i}}|0\rangle,\qquad
a_{i}=\lambda_{i}-i+1/2,\ b_{i}=\lambda_{i}^{t}-i+1/2,$ (2.5)

where $(a_{1},...,a_{l}|b_{1},...,b_{l})$ are called the Frobenius coordinates
of $\lambda$. The operator

$C=\sum_{k\in 1/2+\mathbb{Z}}:\psi_{k}\psi_{k}^{*}:$ (2.6)

is called the charge operator, whose action on $\mathfrak{F}$ is:

$C\left(v_{i_{1}}\wedge v_{i_{2}}\wedge...\right)=[(\sharp\ \text{present
positive}\ v_{i})-(\sharp\ \text{missing negative}\ v_{i})]v_{i_{1}}\wedge
v_{i_{2}}\wedge...$ (2.7)

The vectors of $0$-charge are characterized by being annihilated by $C$.
Besides, the energy operator $H$ (Hamiltonian) is defined by

$H=\sum_{k\in 1/2+\mathbb{Z}}k:\psi_{k}\psi_{k}^{*}:$ (2.8)

and satisfies $Hv_{\lambda}=|\lambda|v_{\lambda}$. Now, let define the Boson
operators:

$\alpha_{n}=\sum_{k\in 1/2+\mathbb{Z}}\psi_{k+n}\psi_{k}^{*}.$ (2.9)

They satisfy the commutation relations $[\alpha_{n},\psi_{k}]=\psi_{k+n},\
[\alpha_{n},\psi_{k}^{*}]=-\psi_{k-n}$.

The fermionic Fock space $\mathfrak{F}$ is the Hilbert space generated by a
pair of fermions $\psi^{\pm}(z)$ with components $\psi_{r}^{\pm},\
r\in\frac{1}{2}+\mathbb{Z}$ satisfying the Clifford commutation relations,

$[\psi_{i}^{+},\psi_{j}^{-}]=\delta_{i,-j},\qquad[\psi_{i}^{+},\psi_{j}^{+}]=0,\qquad[\psi_{i}^{-},\psi_{j}^{-}]=0.$
(2.10)

###  2.2 Bosonic Fock space

The Hilbert space $\mathfrak{F}$ can be constructed in two main forms which
are isomorphic (Boson-Fermion correspondence). They are called Fermionic and
Bosonic Fock spaces. The definition 2.1 is usually referred as the fermionic
Fock space. The Bosonic Fock space is a representation defined on the
polynomial ring $\mathbb{C}[x_{1},x_{2},...;q,q^{-1}]$, see [36]. It is not
hard to write a specific isomorphism between $\mathfrak{F}$ and the mentioned
polynomial ring. In this context, the vertex operators act as cetain
differential operators, and the trace of vertex operator appears as generating
a series of the ring of symmetric functions on infinitely many variables. We
review some features of this below.

Consider the coordinate ring of the affine variety $Sym^{k}(\mathbb{C})$,
namely $B_{k}=\mathbb{C}[x_{1},...,x_{n}]/S_{n}$. Write the Hilbert series of
$B_{k}$ as $H_{B_{k}}(q)=\sum_{n}q^{n}h_{n}(B_{k})$, where
$h_{n}(B_{k})=\natural\\{\text{monomials in}\ B_{k}\ \text{of charge}\ k\\}$.
Let also $q$ act as $\mathbb{C}^{\times}$ on the other variables. We
understand that $B_{k}$ is the ring of symmetric functions in variables
$x_{1},...,x_{k}$. It is also generated by Schur functions, i.e.,
$B_{k}=\big{\langle}\ S_{\mu}(x_{1},...,x_{k}),\ |\mu|\leq k\ \big{\rangle}$.
In string theory $B_{k}$ arises as the Hilbert space $\mathcal{H}_{k}$
generated by the Boson oscillators up to charge $k$, with commutation
relations, $[\alpha_{n},\alpha_{m}]=n\delta_{n+m,0}$.

Let us associate monomials to partitions as follows:

$\lambda=1^{\lambda_{1}}2^{\lambda_{2}}....\longmapsto\alpha_{-1}^{-\lambda_{1}}\alpha_{-2}^{-\lambda_{2}}....|0\rangle,$
(2.11)

thus,

$B_{k}\cong\mathcal{H}_{k}=\left\langle\
\alpha_{-1}^{m_{1}}...\alpha_{-k}^{m_{k}}|0\rangle\ |\ m_{1},...m_{k}\geq 0\
\right\rangle,$ (2.12)

and we have the inclusions $\mathcal{H}_{0}\subset\mathcal{H}_{1}\subset...$
corresponding to the nested sequence of Young diagrams of increasing number or
rows. The $\mathbb{C}^{\times}$ induces an action of $Sym^{*}(\mathbb{C})$
such that the functions $S_{\mu}(x_{1},...,x_{k})$ are eigen-functions with
eigenvalues $q^{|\mu|}$ of the action of $q^{L_{0}}$ on $\mathcal{H}$, where

$L_{0}=\sum_{n>0}\alpha_{-n}\alpha_{n}.$ (2.13)

Then, we can write:

$H_{B_{k}}(q)=Tr_{\mathcal{H}_{k}}q^{L_{0}}=\sum_{|\mu|\leq
k}q^{|\mu|}=\prod_{n=1}^{k}(1-q^{n})^{-1}.$ (2.14)

The generating function of these series becomes:

$G(t,q)=\sum_{n=0}^{\infty}t^{k}H(B_{k})(q)=\sum_{k}t^{k}Tr_{\mathcal{H}_{k}}q^{L_{0}}=\sum_{\mu}s_{\mu}(t)s_{\mu}(1,q,...).$
(2.15)

We can generalize this argument by considering the coordinate ring of the
affine variety $B_{k_{1},...,k_{n}}=Sym^{k_{1}}(\mathbb{C})\times...\times
Sym^{k_{n}}(\mathbb{C})$, and define the generating series
$G(t_{1},...,t_{n};q)=\sum_{k_{1},...,k_{n}}t_{1}^{k_{1}}...t_{n}^{k_{n}}H_{B_{k_{1},...,k_{n}}}(q)$.
The ring $B_{k_{1},...,k_{n}}$ is generated by the product of the Schur
polynomials
$B_{k_{1},...,k_{n}}=\big{\langle}S_{\mu_{1}}(x_{1,1},...,x_{1,k_{1}}).\
.\dots.\ S_{\mu_{n}}(x_{n,1},...,x_{n,k_{n}});\ l(\mu_{j})\leq
k_{j}\big{\rangle}$. The above ring is isomorphic to the Hilbert space spanned
by the Bosonic operators,

$\mathcal{H}_{k_{1},...,k_{n}}=\left\langle\prod_{j=1}^{n}(\alpha_{-1}^{j})^{n_{j,1}}...(\alpha_{-k_{n}}^{j})^{n_{j,k_{n}}}|0\rangle\right\rangle.$
(2.16)

In the new terminology, the Hilbert series can be written as:

$H_{B_{k_{1},...,k_{n}}}(q)=Tr_{\mathcal{H}_{k_{1},...,k_{n}}}q^{L_{0}},$
(2.17)

where $L_{0}=\sum_{j=1}^{n}\sum_{r>0}\alpha_{-r}^{j}\alpha_{r}^{j}$ is the
charge operator. We obtain
$G(t_{1},...,t_{n},q)=\prod_{j=1}^{n}G(t_{j},q)=\prod_{j=1}^{n}\prod_{i}(1-q^{-i}t_{j})$.
The classical Boson-Fermion correspondence is an isomorphism between two
representations of the Heisenberg algebra, namely the Bosonic Fock space and
the fermionic Fock space. The Boson-Fermion correspondence is a basic result
in mathematical physics. There are various applications of this
correspondence. It provides an explicit way of comparing expressions for
$q$-dimensions of representations, through which new combinatorial identities
are derived by computing characters of representations in two different ways.

###  2.3 The Fock space of level $l$

The Fock space of level $l$ denoted by $\mathfrak{F}^{l}$ is the Fermionic
Fock spaces on $l$ pairs of fermions $\psi_{r}^{\pm,j},r\in 1/2+\mathbb{Z},\
j=1,...,l$. Let denote $\widehat{C}^{l}$ be the Clifford algebra on the
components. $\mathfrak{F}^{l}$ is a simple $\widehat{C}^{l}$-module generated
by $|0\rangle$, such that $\psi_{r}^{\pm,j}|0\rangle=0,\ r>0$. The
$\frac{1}{2}\mathbb{Z}_{+}$-gradation of $\mathfrak{F}^{l}$ is given by the
eigenvalues of the degree operator $d$ such that $d|0\rangle=0,\
[d,\psi_{-r}^{\pm,j}]=r\psi_{-r}^{\pm,j}$, where any graded subspace is
finite-dimensional. We shall use normal ordering notations defined by,

$\displaystyle:\psi_{r}^{+,j}\psi_{s}^{-,k}:$
$\displaystyle=\begin{cases}-\psi_{s}^{-,k}\psi_{r}^{+,j},\qquad s=-r<0\\\
\psi_{r}^{+,j}\psi_{s}^{-,k}\end{cases}$ (2.18)
$\displaystyle:\psi_{r}^{-,j}\psi_{s}^{+,k}:$
$\displaystyle=\begin{cases}-\psi_{s}^{+,k}\psi_{r}^{-,j},\qquad s=-r<0\\\
\psi_{r}^{-,j}\psi_{s}^{+,k}\end{cases}$
$\displaystyle:\psi_{r}^{+,j}\psi_{s}^{+,k}:$
$\displaystyle=\psi_{r}^{+,j}\psi_{s}^{+,k}$
$\displaystyle:\psi_{r}^{-,j}\psi_{s}^{-,k}:$
$\displaystyle=\psi_{r}^{-,j}\psi_{s}^{-,k}$
$\displaystyle:\psi_{r}^{\pm,j}\phi_{s}^{+,k}:$
$\displaystyle=\psi_{r}^{\pm,j}\phi_{s}^{+,k}.$

Define the operators $e_{ij}^{*}$ and $e_{*}^{ij}$ by:

$e_{ij}^{*}=\sum_{k=1}^{l}:\psi_{1/2-i}^{+,k}\psi_{j-1/2}^{-,k}:,\qquad
e_{*}^{ij}=\sum_{r\in 1/2+\mathbb{Z}}:\psi_{-r}^{+,i}\psi_{r}^{-,j}:$ (2.19)

With the above set-up, we have the following:

  * •

The map $\mathfrak{a}_{\infty}\longrightarrow End(\mathfrak{F}^{l}),\
E_{ij}\longmapsto e_{ij}^{*}$ defines a representation of
$\mathfrak{a}_{\infty}$.

  * •

The map $\mathfrak{gl}_{l}\longrightarrow End(\mathfrak{F}^{l}),\
E^{ij}\longmapsto e_{*}^{ij}$ is a representation of $\mathfrak{gl}_{l}$.

  * •

The above actions of $\mathfrak{gl}_{l}$ and $\mathfrak{a}_{\infty}$ on
$\mathfrak{F}^{l}$ commute, [see [6] sec. 5.4]. Thus, we have the relation

$[e_{ij}^{*},e_{*}^{rs}]=[\sum_{u\in
1/2+\mathbb{Z}}:\psi_{-u}^{-,i}\psi_{-u}^{-,j}:\ ,\
\sum_{l=1}^{\infty}:\psi_{1/2-r}^{+,l}\psi_{1/2-s}^{-,l}:]=0.$ (2.20)

We use the generalized partitions as the weights of representations of
$\mathfrak{gl}_{l}$, i. e., the partitions $\lambda$ of the form
$\lambda=(\lambda_{1}\geq\lambda_{2}\geq...\geq\lambda_{i-1}\geq\lambda_{i}=...=\lambda_{j-1}=0\geq\lambda_{j}\geq...\geq\lambda_{r})$.
Also, for the weight of representations of $\mathfrak{a}_{\infty}$ we denote
$\Lambda(\lambda)=\Lambda_{\lambda_{1}}+...+\Lambda_{\lambda_{l}}=l\Lambda_{0}+\sum_{i}\lambda_{i}^{\prime}\epsilon_{i}$,
where,

$\lambda_{i}^{\prime}=\begin{cases}\ |\ \\{\ j\ |\lambda_{j}\geq i,\ i\geq 1\
\\}\ |\\\ -|\ \\{\ j\ |\lambda_{j}<i,\ i\leq 0\ \\}\ |\end{cases}.$ (2.21)

Notice that we are using the notation in Appendix A, in particular eq. (6.4).

Let’s define,

$\begin{aligned}
\varpi_{m}^{+,j}=\psi_{-m+1/2}^{+,s}...\psi_{-3/2}^{+,s}\psi_{-1/2}^{+,s}\\\
\varpi_{m}^{-,s}=\psi_{-m+1/2}^{-,s}...\psi_{-3/2}^{-,s}\psi_{-1/2}^{-,s}\end{aligned},\qquad\varpi_{0}^{+,s}=1,$
(2.22)

A joint highest weight vector in $\mathfrak{F}^{l}$ associated to $\lambda$
with respect to the standard Borel
$\mathfrak{gl}(l)\times\mathfrak{a}_{\infty}$ is

$v_{\lambda}=\varpi_{\lambda_{1}}^{+,s}\varpi_{\lambda_{2}}^{+,s}...\varpi_{\lambda_{i}}^{+,s}\varpi_{-\lambda_{j}}^{-,s}\varpi_{-\lambda_{j+1}}^{+,s}...\varpi_{-\lambda_{l}}^{+,s},$
(2.23)

whose weight with respect to $\mathfrak{gl}(l)$ is $\lambda$, and with respect
to $\mathfrak{a}_{\infty}$ is $\Lambda(\lambda)$, [see [6] sec. 5.4].

######  Theorem 2.2 (See [6] sec. 5.4).

We have the decomposition

$\mathfrak{F}^{l}=\bigoplus_{\lambda}L(\mathfrak{gl}(l),\lambda)\otimes
L(\mathfrak{a}_{\infty},\Lambda(\lambda)).$ (2.24)

We may state other kind of dualities by considering different vertex
operators. In order to illustrate the fermionic operators, let
$V=(\mathbb{C}^{l}\otimes\mathbb{C}^{\infty})\bigoplus(\mathbb{C}^{l*}\otimes\mathbb{C}^{{\infty}*})$,
where $\mathbb{C}^{\infty}$ is a vector space with basis
$w_{r},r\in-1/2-\mathbb{Z}_{+}$, the dual indexed is given by $w_{-r}$, and
$\mathbb{C}^{l}$ has basis $v^{+,i}$ with dual $v^{-,i}$. Then, we may
illustrate $\psi_{r}^{\pm,i}=v^{\pm,i}\otimes w_{r}$.

###  2.4 The character of Vertex operators

The operators of the form:

$\Gamma_{+}(x)=\exp(\sum_{n\geq
1}\frac{x^{n}}{n}\alpha_{n}),\qquad\Gamma_{-}(x)=\exp(\sum_{n>0}\frac{x^{n}}{n}\alpha_{-n})$
(2.25)

are called vertex operators. They are adjoint with respect to the natural
inner product. We have a commutation relation:

$\Gamma_{+}(x)\Gamma_{-}(y)=(1-xy)\Gamma_{-}(y)\Gamma_{+}(x).$ (2.26)

Also, we have,

$\Gamma_{+}(x)v_{\mu}=\sum_{\lambda\supset\mu}S_{\lambda/\mu}(x)v_{\lambda}.$
(2.27)

Vertex operators provide powerful tools to express partitions. For example, we
can write,

$\displaystyle\Gamma_{+}(1)|\mu\rangle=\sum_{\lambda\supset\mu}|\lambda\rangle$
(2.28)
$\displaystyle\Gamma_{-}(1)|\mu\rangle=\sum_{\lambda\subset\mu}|\lambda\rangle.$

Besides, we may write the McMahon function as:

$\displaystyle Z$ $\displaystyle=\sum_{\text{3-dim partitions}}q^{\natural\
boxes}=\langle(\prod_{t=0}^{\infty}q^{L_{0}}\Gamma_{+}(1))q^{L_{0}}(\prod_{t=-\infty}^{-1}\Gamma_{-}(1)q^{L_{0}})\rangle$
(2.29)
$\displaystyle=\langle\prod_{n>0}\Gamma_{+}(q^{n-1/2})\prod_{n>0}\Gamma_{-}(q^{-n-1/2})\rangle.$

We may divide a 3-dimensional partition into slices of two-dimensional
partitions, for instance, along the diagonals. In this way, the vertex
operator divides into the multiplication of many vertex operators of the
slices,

$Z(\\{x_{m}^{\pm}\\})=\langle...\prod_{u_{i}<m<v_{i+1}}\Gamma_{+}(x_{m}^{-})\prod_{v_{i}<m<u_{i}}\Gamma_{-}(x_{m}^{+})...\rangle=\langle\prod_{u_{0}<m<u_{n}}\Gamma_{-\epsilon(m)}(x_{m}^{\epsilon(m)})\rangle.$
(2.30)

In this way, one can obtain product formulas such as

$Z(\\{x_{m}^{\pm}\\})\prod_{m_{1}<m_{2}}(1-x_{m_{1}}^{-}x_{m_{2}}^{+}).$
(2.31)

For example, we may choose

$\displaystyle\\{x_{m}^{\pm}\\}$
$\displaystyle=\\{t^{i}q^{v_{i}}|i=1,2,...\\}$ (2.32)
$\displaystyle\\{x_{m}^{\pm}\\}$
$\displaystyle=\\{t^{j-1}q^{-v_{i}^{t}}|j=1,2,...\\},$

and we get,

$Z_{\lambda\mu\nu}(t,q)=\langle\prod_{u_{0}<m<u_{n}}\Gamma_{-\epsilon(m)}(x_{m}^{\epsilon(m)})\rangle.$
(2.33)

The partition function can also be read by putting a wall on the distance $M$
along with one of the axis. Then, using commutation (2.26) we have expressions
of the form,

$\displaystyle Z$
$\displaystyle=\langle\prod_{0<m<\infty}\Gamma_{-}(x_{m}^{+})\prod_{-M<m<0}\Gamma_{+}(x_{m}^{-})\rangle$
(2.34)
$\displaystyle=\prod_{l_{1}=1}^{\infty}\prod_{l_{2}=1}^{M}(1-x_{l_{1}-1/2}^{+}x_{-l_{2}+1/2}^{-})^{-1}(\langle\prod_{-M<m<0}\Gamma_{+}(x_{m}^{-})\prod_{0<m<\infty}\Gamma_{-}(x_{m}^{+})\rangle,$

where the last factor in parentheses is equal to $1$, and we obtain a product
formula.

The following theorem is another instant of generating series which arise from
the trace of vertex operator.

######  Theorem 2.3.

[36] We have the following formula for the trace of a vertex operator acting
on $\mathfrak{F}=\bigwedge^{\frac{\infty}{2}}V$:

$Tr\left(q^{L_{0}}\exp(\sum_{n}A_{n}\alpha_{-n})\exp(\sum_{n}B_{n}\alpha_{n})\right)=\prod_{n}\sum_{k}\sum_{l=0}^{k}\frac{n^{l}A_{n}^{l}B_{n}^{l}}{l!}q^{nk}{k\choose
l},$ (2.35)

where $L_{0}$ is the charge operator, and
$q^{L_{0}}|\lambda\rangle=|\lambda||\lambda\rangle$.

###### Proof.

Denote the operator in the trace by $T$. We have the isomorphism:

$\bigwedge^{\frac{\infty}{2}}V=\bigotimes_{n=1}^{\infty}\bigoplus_{k=0}^{\infty}\alpha_{-n}^{k}|0\rangle,$
(2.36)

which implies:

$\displaystyle Tr(T)$
$\displaystyle=\prod_{n=1}^{\infty}Tr\left(T|_{\bigoplus_{k=0}^{\infty}\alpha_{-n}^{k}|0\rangle}\right)$
(2.37) $\displaystyle=\prod_{n}\sum_{k}\langle
0|\alpha_{n}^{k}\Big{|}q^{L_{0}}e^{A_{n}\alpha_{-n}}e^{B_{n}\alpha_{n}}\Big{|}|\alpha_{n}^{k}|0\rangle\rangle$
$\displaystyle=\prod_{n}\sum_{k,l,m}\frac{A_{n}^{l}B_{n}^{m}}{l!m!}q^{n(l-m+k)}\langle
0|\alpha_{n}^{k}\Big{|}\alpha_{-n}^{l}\alpha_{n}^{m}\Big{|}|\alpha_{n}^{k}|0\rangle\rangle$
$\displaystyle=\prod_{n}\sum_{k,l}\frac{A_{n}^{l}B_{n}^{l}}{l!l!}q^{nk}\langle
0|\alpha_{n}^{k}\Big{|}\alpha_{-n}^{l}\alpha_{n}^{l}\Big{|}|\alpha_{n}^{k}|0\rangle\rangle$
$\displaystyle=\prod_{n}\sum_{k}\sum_{l=0}^{k}\frac{n^{l}A_{n}^{l}B_{n}^{l}}{l!}q^{nk}{k\choose
l}.$

∎

In the following, we present another calculation due to Bloch, and Okounkov
[4] of the trace of a representation on the infinite wedge space defined in
2.1. This is the base of our main result 4.3 given in Section 4.

######  Theorem 2.4.

[4] Let $\mathfrak{F}=\bigwedge^{\frac{\infty}{2}}V$ be the Fock space on a
fixed basis of $V=\bigoplus_{j\in\mathbb{Z}}\mathbb{C}v_{j}$. Then, the
character of $\mathfrak{F}$ is given by

$ch(\mathfrak{F})=\prod_{n\geq
0}(1+y_{0}y_{1}^{n+1/2}y_{2}^{n+3/2)^{2}}....)(1+y_{0}^{-1}y_{1}^{n-1/2}y_{2}^{-(n-1/2)^{2}}....),$
(2.38)

where $y_{j}=e^{2\pi i\tau_{j}}$.

###### Proof.

The associated character is the trace of the operator,

$y_{0}^{L_{0}}y_{1}^{L_{1}}....y_{n}^{L_{n}}...=\exp(\sum_{j}2\pi
i\tau_{j}L_{j}),$ (2.39)

where $L_{i}$ are commuting operators which act on $\mathfrak{F}$ by

$L_{j}\longmapsto\sum_{n\in 1/2+\mathbb{Z}}(n-\frac{1}{2})^{j}E_{n,n}.$ (2.40)

The operators $L_{j}$ mutually define a representation of

$H=\mathbb{C}L_{0}\oplus\mathbb{C}L_{1}\oplus...\longrightarrow
End(\mathfrak{F}).$ (2.41)

We compute the action of the operators $\exp(2\pi i\tau_{j}L_{j})$ on the
basis elements. Thus, we have,

$\displaystyle
L_{j}.\left(\psi_{-i_{1}}\psi_{-i_{2}}...\psi_{-i_{l}}\psi_{-j_{1}}^{*}...\psi_{-j_{k}}^{*}|0\rangle\right)=(\sum_{a=1}^{l}(i_{a}-1/2)^{j}-$
$\displaystyle\sum_{b=1}^{k}(-j_{b}-1/2)^{j})$ (2.42)
$\displaystyle\left(\psi_{-i_{1}}\psi_{-i_{2}}...\psi_{-i_{l}}\psi_{-j_{1}}^{*}...\psi_{-j_{k}}^{*}|0\rangle\right).$

We also have,

$\displaystyle\exp(2\pi i\tau_{j}L_{j})\psi_{-n}|0\rangle$
$\displaystyle=\left(y_{0}y_{1}^{n-1/2}y_{2}^{(n-1/2)^{2}}...\right)\psi_{-n}|0\rangle$
(2.43) $\displaystyle\exp(2\pi i\tau_{j}L_{j})\psi_{-n}^{*}|0\rangle$
$\displaystyle=\left(y_{0}^{-1}y_{1}^{n-1/2}y_{2}^{-(n-1/2)^{2}}...\right)\psi_{-n}^{*}|0\rangle.$

The claim of the theorem follows from the isomorphism:

$\mathfrak{F}=\bigwedge^{\frac{\infty}{2}}V=\bigotimes_{n}(1+\psi_{-n})(1+\psi_{-n}^{*})|0\rangle.$
(2.44)

∎

##  3 Problem statement

Motivated by the two calculations of the trace of vertex operator presented in
the previous section, i.e.,

$Tr\left(q^{L_{0}}\exp(\sum_{n}A_{n}\alpha_{-n})\exp(\sum_{n}B_{n}\alpha_{n})\right),\qquad
Tr\left(\exp(\sum_{j\geq 0}2\pi iL_{j})\right),$ (3.1)

where $\alpha_{n},\ n\in\mathbb{Z}$ are Boson operators; $L_{0}$ is the energy
operator; and $L_{j},\ j>0$ are certain Casimir operators, [36, 4]. Thus, we
propose to compute the following trace:

$Trace=Tr\left(\exp(\sum_{j\geq 0}2\pi
iL_{j})\exp(\sum_{n>0}A_{n}\alpha_{-n})\exp(\sum_{n>0}B_{n}\alpha_{n})\right).$
(3.2)

Therefore, we pose the following questions:

  * •

How can one compute the trace in terms of the former traces?

  * •

What is the representation theory interpretation of that?

  * •

In case that the coefficients $A_{n},B_{n}$ are suitably chosen, what is the
Physical interpretation of the trace in terms of string theory partition
functions?

The character can be studied from different points of view. A direct way to
calculate it could be to expand the exponentials inside the trace, apply basic
commutation rules between the operators, and then compute the matrix elements.
Some formulas in Lie theory, such as the Backer-Campbell-Hausdorff formula or
the Wick formula, can also be helpful for the calculation. Although this
method can bring computational insights toward the above question, it hits
Adhoc complexity and difficulties. One may expand the operators in the trace
both in terms of Bosonic operators $\alpha_{\pm n}$, and also fermionic
operators $\psi_{j},\ j\in\frac{1}{2}+\mathbb{Z}$.

##  4 Main results

In order to interpret and compute a trace formula for (3.2) we make the
following definition that is a natural generalization of the Fock space of
level $l$, defined in 2.3.

######  Definition 4.1.

(Fock space $\mathfrak{F}^{\infty}$) Consider the fermionic fields
$\psi_{r}^{\pm,j},\ r\in 1/2+\mathbb{Z},\ j\in\mathbb{Z}$ satisfying the
natural Clifford commutation relations,

$\displaystyle=\delta_{i,j}\delta_{r,-s}I$ (4.1)
$\displaystyle[\psi_{r}^{+,i},\psi_{r}^{+,j}]$
$\displaystyle=[\psi_{r}^{-,i},\psi_{s}^{-,i}]=0.$

Set also $\widehat{C}^{\infty}$ be the Clifford algebra on these fields. By
definition $\mathfrak{F}^{\infty}$ is a simple $\widehat{C}^{\infty}$-module
generated by $|0\rangle$, such that $\psi_{r}^{\pm,j}|0\rangle=0,\ r>0$.

Next, we express a duality of Howe-type for the pair
$(\mathfrak{gl}_{\infty},\mathfrak{a}_{\infty})$. In other words, the Fock
space $\mathfrak{F}^{\infty}$ is a representation of both the Lie algebras
$\mathfrak{gl}_{\infty}$ and, $\mathfrak{a}_{\infty}$. Moreover,
$\mathfrak{F}^{\infty}$ decomposes to the sum of their irreducible
representation. Next, we are ready to present our first main result.

######  Theorem 4.2.

(Main Result. ($\mathfrak{gl}_{\infty},\mathfrak{a}_{\infty})$-Howe duality)
There exists a decomposition,

$\mathfrak{F}^{\infty}=\bigoplus_{\lambda}L(\mathfrak{gl}_{\infty},\lambda)\otimes
L(\mathfrak{a}_{\infty},\Lambda(\lambda)),$ (4.2)

where $\lambda$ runs over all generalized partitions. Besides, there is a
character formula,

$ch(\mathfrak{F}^{\infty})=\prod_{i}^{\infty}\prod_{j}^{\infty}(1+y_{j}x_{i})(1+y_{j}^{-1}x_{i}^{-1}),$
(4.3)

where $x_{i},y_{j},\ i,j\in\mathbb{N}$ are variables.

###### Proof.

In (2.19), we replace the operators $e_{ij}^{*}$ by,

$e_{ij}^{*}=\sum_{k=-\infty}^{\infty}:\psi_{1/2-i}^{+,k}\psi_{j-1/2}^{-,k}:\
,\qquad i,j\in\mathbb{Z}.$ (4.4)

The map $\mathfrak{a}_{\infty}\longrightarrow End(\mathfrak{F}^{\infty}),\
E_{ij}\longmapsto e_{ij}^{*}$ defines a representation of
$\mathfrak{a}_{\infty}$. Also, let define the operators $e_{*}^{ij}(n)$ by

$e_{*}^{ij}=\sum_{r\in
1/2+\mathbb{Z}}:\psi_{-r}^{+,i}\psi_{r}^{-,j}:,\qquad(r\in 1/2+\mathbb{Z},\
i,j\in\mathbb{Z}).$ (4.5)

The map $\mathfrak{gl}_{\infty}\longrightarrow End(\mathfrak{F}^{\infty}),\
E^{ij}\longmapsto e_{*}^{ij}$ is a representation of $\mathfrak{gl}_{\infty}$.
We need to check that the action of $\mathfrak{gl}_{\infty}$ and
$\mathfrak{a}_{\infty}$ commute. That is,

$[e_{ij}^{*},e_{*}^{rs}]=[\sum_{u\in
1/2+\mathbb{Z}}:\psi_{-u}^{-,i}\psi_{-u}^{-,j}:,\sum_{l=1}^{\infty}:\psi_{1/2-r}^{+,l}\psi_{1/2-s}^{-,l}:]=0.$
(4.6)

A joint highest weight vector in $\mathfrak{F}^{\infty}$, associated to a
generalized partition $\lambda=(\lambda_{1},...,\lambda_{j})$ w.r.t the
standard Borel of $\mathfrak{gl}_{\infty}\times\mathfrak{a}_{\infty}$, is
$v_{\lambda}=\varpi_{\lambda_{1}}^{+,s}\varpi_{\lambda_{2}}^{+,s}...\varpi_{\lambda_{i}}^{+,s}\varpi_{-\lambda_{j}}^{-,s}\varpi_{-\lambda_{j+1}}^{+,s}...\varpi_{-\lambda_{l}}^{+,s}$.
Where whose weight w.r.t. $\mathfrak{gl}_{\infty}$ is $\lambda$ and w.r.t.
$\mathfrak{a}_{\infty}$ is $\Lambda(\lambda)$. By applying any root vector of
$\mathfrak{gl}_{\infty}$ and $\mathfrak{a}_{\infty}$ to $v_{\lambda}$, it
produces two identical $\psi_{\bullet}^{\bullet,\bullet}$ in the resulting
monomial. As in the finite case any irreducible representation of
$\mathfrak{gl}_{\infty}$ appears in $\mathfrak{F}^{\infty}$ and the
multiplicity free decomposition
$\mathfrak{F}^{\infty}=\bigoplus_{\lambda}L(\mathfrak{gl}_{\infty},\lambda)\otimes
L(\mathfrak{a}_{\infty},\Lambda(\lambda))$ follows.

The character of $\mathfrak{gl}_{\infty}$ on $\mathfrak{F}^{\infty}$ is the
trace of the operator $\prod_{i}x_{i}^{\epsilon_{*}^{ii}}$. The character of
$\mathfrak{a}_{\infty}$ on $\mathfrak{F}^{\infty}$ is the trace of
$\prod_{j}y_{j}^{\epsilon_{jj}^{*}}$. Therefore, $ch(\mathfrak{F}^{\infty})$
is the trace of the product of the two operators. Calculating the trace of the
operator
$\prod_{i}x_{i}^{\epsilon_{*}^{ii}}\prod_{j}y_{j}^{\epsilon_{jj}^{*}}$ on both
sides of (4.2) we get:

$ch(\mathfrak{F}^{\infty})=\bigoplus_{\lambda}ch\left(L(\mathfrak{gl}_{\infty},\lambda)\right)\otimes
ch\left(L(\mathfrak{a}_{\infty},\Lambda(\lambda))\right)=\prod_{i}\prod_{j}(1+y_{j}x_{i})(1+y_{j}^{-1}x_{i}^{-1}).$
(4.7)

∎

The duality in Theorem 4.2 enables us to compute the trace formula (3.2) by
computing it on each summand. Specifically, we have the following.

######  Theorem 4.3.

(Main result) We have the following formula for the trace in (3.2)

$\displaystyle Trace=\sum_{\lambda}\prod_{n\geq
0}(1+y_{0}y_{1}^{n+1/2}y_{2}^{n+3/2)^{2}}....)(1+y_{0}^{-1}y_{1}^{n-1/2}y_{2}^{-(n-1/2)^{2}}....)$
(4.8) $\displaystyle\prod_{r\geq
1}y_{r}^{p_{r}(\lambda)}\sum_{\mu\prec\lambda}S_{\lambda/\mu}^{(A_{n})}(x_{1},x_{2},...)S_{\lambda^{t}/\mu}^{(B_{n})}(x_{1},x_{2},...),$

where $x_{i}$ and $y_{i}$ are independent variables, and

$p_{r}(\lambda)=\sum_{l}(\lambda_{l}-l+\frac{1}{2})^{r}+(-1)^{r+1}(l-\frac{1}{2})^{r}=\sum_{l}(m_{l}+\frac{1}{2})^{r}+(-1)^{r+1}(n_{l}+\frac{1}{2})^{r}$
(4.9)

holds, where $(m_{1},...,m_{s}|n_{1},...,n_{s})$ are Frobenius coordinates of
$\lambda$. The effect of the coefficients $A_{n},B_{n}$ is absorbed in the
variables $x_{1},x_{2},...,$, [see the proof for the explanation on dependence
to the coefficients $A_{n},B_{n}$].

######  Remark 4.4.

The dependence of the above trace to the coefficients $A_{n}$ and $B_{n}$ is
somehow formal. The contribution to the trace coming from
$Tr\left(\exp(\sum_{n>0}A_{n}\alpha_{-n})\exp(\sum_{n>0}B_{n}\alpha_{n})\right)$
is given in the last sum appearing in (4.8), where the effect of $A_{n},B_{n}$
is absorbed in the variables $x_{1},x_{2},...$, [see [36] page 11 and 24, or
[25] pages 25 and 70 for the notation].

###### Proof.

(proof of Theorem 4.3) Let us denote,

$\mathcal{L}=\exp(\sum_{j}2\pi iL_{j}),\qquad
T=\exp(\sum_{n}A_{n}\alpha_{-n})\exp(\sum_{n}B_{n}\alpha_{n}).$ (4.10)

According to Theorem 4.2, we need to compute:

$\sum_{\lambda}Tr\left(\mathcal{L}\big{|}_{L(\mathfrak{gl}_{\infty},\lambda)}\right)Tr\left(T\big{|}_{L(\mathfrak{a}_{\infty},\Lambda^{\mathfrak{a}_{\infty}}(\lambda)}\right).$
(4.11)

We first compute the factor relevant to $\mathcal{L}$. Consider
$v_{\lambda}=|\lambda\rangle$, the vector of weight $\lambda$. We have the
formula:

$\mathfrak{F}^{\lambda}=\bigotimes_{n}(1+\psi_{-n})(1+\psi_{-n}^{*})|\lambda\rangle.$
(4.12)

By lemma 5.1 in [4], we also have:

$\displaystyle\exp(2\pi iL_{j})\psi_{-n}|\lambda\rangle$
$\displaystyle=\left(y_{0}y_{1}^{n-1/2}y_{2}^{(n-1/2)^{2}}...\prod_{r\geq
1}y_{r}^{p_{r}(\lambda)}\right)\psi_{-n}|\lambda\rangle$ (4.13)
$\displaystyle\exp(2\pi iL_{j})\psi_{-n}^{*}|\lambda\rangle$
$\displaystyle=\left(y_{0}^{-1}y_{1}^{n-1/2}y_{2}^{-(n-1/2)^{2}}...\prod_{r\geq
1}y_{r}^{p_{r}(\lambda)}\right)\psi_{-n}^{*}|\lambda\rangle,$

therefore

$Tr\left(\mathcal{L}\big{|}_{L(\mathfrak{gl}_{\infty},\lambda)}\right)=\prod_{n\geq
0}(1+y_{0}y_{1}^{n+1/2}y_{2}^{n+3/2)^{2}}....)(1+y_{0}^{-1}y_{1}^{n-1/2}y_{2}^{-(n-1/2)^{2}}....)\prod_{r\geq
1}y_{r}^{p_{r}(\lambda)}.$ (4.14)

On the other hand, it is well known that,

$Tr\left(T\big{|}_{L(\mathfrak{a}_{\infty},\Lambda^{\mathfrak{a}_{\infty}}(\lambda)}\right)=\sum_{\mu\prec\lambda}S_{\lambda/\mu}^{(A_{n})}(x_{1},x_{2},...)S_{\lambda^{t}/\mu}^{(B_{n})}(x_{1},x_{2},...),$
(4.15)

where $S_{\lambda/\mu}^{(A_{n})}$ is the skew Schur function $S_{\lambda/\mu}$
specialized to the case in which the symmetric power functions $p_{n}$ equal
$nA_{n}$. Similarly, $S_{\lambda/\mu}^{(B_{n})}$ is the skew Schur function
$S_{\lambda/\mu}$ specialized to the case in which the symmetric power
functions $p_{n}$ equals $nB_{n}$, [see [36] page 11 and 24 for the notation].
∎

##  5 Application

The string theory partition function of toric $CY$ $3$-folds can be formulated
from their tropical diagram by basic combinatorial rules of the topological
vertex, [17], [18], [19], [20], [21]. These partition functions can also be
described combinatorially by certain vector fields acting on the tropical
diagram that fix the vertices. The action of a symmetry group preserves the
action, a unitary group $U(N)$, called the gauge group. In this context, one
computes another generating series related to the topological one. The
equality of these two series has been shown in many cases. In this way, one
may consider tropical diagrams that are more complicated, especially when
there are many cells. According to the known formulations of the generating
series under consideration, this motivates more twists in the corresponding
vertex operator. Thus one may ask what happens if we apply infinitely many
twists in the vertex operator. Although finding the corresponding geometric
object that produces such a tropical configuration may raise the question, we
expect that the associated diagram can be explained according to some limiting
procedure in the real world. Next, we present an instance of how this can be
presented. The partition function of $U(1)$ theory can be written in the form:

$Z(\tau,m,\epsilon)=Tr\left(Q_{\tau}^{L_{0}}\exp\left(\sum_{n\geq
1}\frac{Q_{m}^{n}-1}{n(q^{n/2}-q^{-n/2})}\alpha_{n}\right)\exp\left(\sum_{n\geq
1}\frac{Q_{-m}^{n}-1}{n(q^{n/2}-q^{-n/2})}\alpha_{-n}\right)\right).$ (5.1)

Using the commutation relation of $\alpha_{\pm}$ it can be written as follows,
cf. [19, 20, 21],

$Z(\tau,m,\epsilon)=\prod_{k}(1-Q_{\tau}^{k})^{-1}\prod_{i,j}\frac{(1-Q_{\tau}^{k}Q_{m}^{-1}q^{i+j-1})(1-Q_{\tau}^{k}Q_{m}q^{i+j-1})}{(1-Q_{\tau}^{k}q^{i+j-1})}.$
(5.2)

Besides, the partition function in (5.1) can be generalized to

$Z(\tau,m,\epsilon,t)=Tr\left(Q_{\tau}^{L_{0}}e^{\sum_{n}t_{n}L_{n}}\exp\left(\sum_{n\geq
1}\frac{Q_{m}^{n}-1}{n(q^{n/2}-q^{-n/2})}\alpha_{n}\right)\exp\left(\sum_{n\geq
1}\frac{Q_{-m}^{n}-1}{n(q^{n/2}-q^{-n/2})}\alpha_{-n}\right)\right).$ (5.3)

Also, in the limit $m\mapsto 0$ we obtain,

$Z(\tau,m=0,\epsilon,t)=Tr\left(Q_{\tau}^{L_{0}}e^{\sum_{n}t_{n}L_{n}}\right).$
(5.4)

We can write the partition function in terms of the Gromov-Witten potentials:

$Z(\tau,m,\epsilon)=\exp\left(\sum_{g\geq 0}\epsilon^{2g-2}F_{g}\right),$
(5.5)

where

$e^{F_{1}}=\prod_{k}(1-Q_{\tau}^{k})^{-1}\left(\frac{(1-Q_{\tau}^{k})^{2}Q_{m}^{-1})(1-Q_{\tau}^{k}Q_{m})^{2}}{(1-Q_{\tau}^{k})^{4}}\right)^{1/24}.$
(5.6)

A natural motivation is to investigate the interpretation of the trace in 5.3.
One way to proceed with this is to think of (5.3) as a limit of the case when
finitely many $L_{j}$ exists in the trace function. The case with finitely
many $L_{j}$ naturally appears in the topological string partition functions
associated to toric CY 3-folds where the web diagram has an $M\times N$ cell
structure, where $M,N\in\mathbb{N}$.

##  6 Conclusions

An extension of the calculation of the trace of the vertex operator with
infinitely many Casimirs is presented based on a duality of Howe type for the
pair $(\mathfrak{a}_{\infty},\mathfrak{gl}_{\infty})$. The trace formula
applies to an extension of the topological string theory partition function of
CY 3-folds.

## Appendix: The Lie algebras $\mathfrak{a}_{\infty},\mathfrak{c}_{\infty}$
and $\mathfrak{d}_{\infty}$

We introduce the three infinite-dimensional Lie algebras whose representations
are crucial in string theory. The reference for this appendix is [6] Section
5.4, where all the materials discussed can be found there with more details.

(1) Lie Algebra $\mathfrak{a}_{\infty}$ : Let
$\mathfrak{a}_{\infty}=\widehat{\mathfrak{gl}}_{\infty}=\mathfrak{gl}_{\infty}\oplus\mathbb{C}K$
with the braket,

$[X+cK,Y+dK]=[X,Y]^{\prime}+\tau(X,Y)K.$ (6.1)

The function $\tau(X,Y)=Tr([J,X]Y)$ is called a cocycle function, where
$J=\sum_{j\leq 0}E_{jj}$ and $[.,.]^{\prime}$ is the bracket of
$\mathfrak{gl}_{\infty}$. We have the degree gradation
$\mathfrak{gl}_{\infty}=\bigoplus_{j}\mathfrak{gl}_{\infty}^{j}$, where $j$
runs over integers and it is called $\mathbb{Z}$-principal gradation. The
degree of $E_{ij}$ would be $j-i$. Besides, we have a decomposition:

$\mathfrak{a}_{\infty}=\mathfrak{a}_{\infty}^{+}\oplus\mathfrak{a}_{\infty}^{0}\oplus\mathfrak{a}_{\infty}^{-},\qquad\mathfrak{a}_{\infty}^{\pm}=\bigoplus_{j>0}\mathfrak{gl}_{\infty}^{\pm
j},\ \mathfrak{a}_{\infty}^{0}=\mathfrak{gl}_{\infty}^{0}\oplus\mathbb{C}K.$
(6.2)

The root system of $\mathfrak{a}_{\infty}$ is

$R=\\{\epsilon_{i}-\epsilon_{j}\ |i\neq
j\\},\qquad\epsilon_{i}(E_{ii})=\delta_{ij},\ \epsilon_{i}(K)=0.$ (6.3)

The set $\Pi=\\{\epsilon_{i}-\epsilon_{i+1}\ |i\in\mathbb{Z}\\}$ is a
fundamental system for $\mathfrak{a}_{\infty}$ with corresponding co-roots
$\\{H_{i}^{\mathfrak{a}}=E_{ii}-E_{i+1,i+1}+\delta_{i,0}K\\}$. We denote by
$\Lambda_{j}^{\mathfrak{a}}$ the $j$-th fundamental weight of
$\mathfrak{a}_{\infty}$, i.e.
$\Lambda_{j}^{\mathfrak{a}}(H_{i}^{\mathfrak{a}})=\delta_{ij},\
(i\in\mathbb{Z}),\ \Lambda_{j}^{\mathfrak{a}}(K)=1$. A straight forward
computation gives:

$\displaystyle\Lambda_{j}^{\mathfrak{a}}$
$\displaystyle=\Lambda_{0}^{\mathfrak{a}}-\sum_{i=j+1}^{0}\epsilon_{i}\qquad
j<0$ (6.4) $\displaystyle\Lambda_{j}^{\mathfrak{a}}$
$\displaystyle=\Lambda_{0}^{\mathfrak{a}}+\sum_{i=1}^{j}\epsilon_{i}\qquad\ \
j\geq 1.$

The irreducible highest weight representation of $\mathfrak{a}_{\infty}$ of
the highest weight $\Lambda$ is denoted by $L(\mathfrak{a}_{\infty},\Lambda)$.

(2) Lie algebra $\mathfrak{c}_{\infty}$: Let
$V=\bigoplus_{j\in\mathbb{Z}}\mathbb{C}v_{j}$ be the vector space generated by
the vectors $v_{j}$, where $E_{ij}v_{j}=v_{i}$. Consider the symmetric
bilinear form,

$C(v_{i},v_{j})=(-1)^{i}\delta_{i,1-j},\forall i,j\in\mathbb{Z}.$ (6.5)

Set $\mathfrak{c}_{\infty}=\bar{\mathfrak{c}}_{\infty}\oplus\mathbb{C}K$,
where

$\bar{\mathfrak{c}}_{\infty}=\\{X\in\mathfrak{gl}_{\infty}\ |\
C(X(u),v)+C(u,X(v))=0\\}.$ (6.6)

A fundamental system for $\mathfrak{c}_{\infty}$ is given by
$\\{-2\epsilon_{1},\epsilon_{i}-\epsilon_{i+1};\ i\geq 0\\}$ with simple co-
roots:

$\displaystyle H_{i}^{\mathfrak{c}}$
$\displaystyle=E_{ii}+E_{-i,-i}=E_{i+1,i+1}-E_{1-i,1-i},$ (6.7) $\displaystyle
H_{0}^{\mathfrak{c}}$ $\displaystyle=E_{00}-E_{11}+K.$

The $j$-th fundamental weight for $\mathfrak{c}_{\infty}$ is defined by the
same as the case for $\mathfrak{a}_{\infty}$ and explicitly written as
follows,

$\Lambda_{j}^{\mathfrak{c}}=\Lambda_{0}^{\mathfrak{c}}+\sum_{i=1}^{j}\epsilon_{i},\qquad
j\geq 1.$ (6.8)

(3) Lie Algebra $\mathfrak{d}_{\infty}$: Define the Lie algebra
$\mathfrak{d}_{\infty}=\bar{\mathfrak{d}}_{\infty}\oplus\mathbb{C}K$ where,

$\bar{\mathfrak{d}}_{\infty}=\\{X\in\mathfrak{gl}_{\infty}|D(X(u),v)=D(u,X(v))\\},$
(6.9)

and where $D(v_{i},v_{j})=\delta_{i,1-j}$. It has the fundamental system
$\\{\pm\epsilon_{1}-\epsilon_{2},\epsilon_{i}-\epsilon_{i+1},\ i\geq 2\\}$
with simple co-roots:

$\displaystyle H_{i}$
$\displaystyle=E_{ii}-E_{-i,-i}-E_{i+1,i+1}-E_{1-i,1-i},$ (6.10)
$\displaystyle H_{0}$ $\displaystyle=E_{00}-E_{-1,-1}-E_{22}-E_{11}+2K.$

## References

  * [1] Alexei Borodin, Andrei Okounkov, and Grigori Olshanski. Asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc., 13(3):481–515 (electronic), 2000.
  * [2] Alexander I. Bufetov. On the Vershik-Kerov conjecture concerning the Shannon-McMillan-Breiman theorem for the Plancherel family of measures on the space of Young diagrams. Geom. Funct. Anal., 22(4):938–975, 2012.
  * [3] J. Bryan, M. Kool and B. Young, Trace identities for the topological vertex. Sel. ath. New Ser. 24, 1527-1548 (2018).
  * [4] S. Bloch, A. Okounkov, The Character of the Infinite Wedge Representation, Advances in Mathematics Volume 149, Issue 1, 1-60 (2000)
  * [5] S. Cheng, N. Lam, Infinite-Dimensional Lie Superalgebras and Hook Schur Functions, Communications in Mathematical Physics volume 238, 95-118 (2003)
  * [6] S. Cheng, W. Wang, Dualities and Representations of Lie Superalgebras, GSM, 144, AMS, (2012)
  * [7] S. J. Cheng and W. Q. Wang, The Bloch-Okounkov correlation functions at higher levels. Transformation Groups 9, 133-142, (2004).
  * [8] R. Dijkgraaf, G. Moore, R. Plesser, “The partition function of 2d string theory,” Nucl. Phys. B394, 356 (1993)
  * [9] A. Dembo, A. Vershik, O. Zeitouni. Large deviations for integer partitions. Markov Process. Related Fields, 6(2):147–179, (2000).
  * [10] A. Dembo, A. Vershik, and O. Zeitouni. Large deviations for integer partitions. Markov Process. Related Fields, 6(2):147–179, 2000.
  * [11] A. Eskin and A. Okounkov. Pillowcases and quasimodular forms. In Algebraic geometry and number theory, volume 253 of Progr. Math., pages 1–25. Birkhäuser Boston, Boston, MA, 2006\.
  * [12] Alex Eskin and Andrei Okounkov. Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Invent. Math., 145(1):59–103, 2001\.
  * [13] A. Eskin and A. Okounkov. Pillowcases and quasimodular forms. In Algebraic geometry and number theory, volume 253 of Progr. Math., pages 1–25. Birkhauser Boston, Boston, MA, (2006).
  * [14] A. Eskin, A. Okounkov. Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Invent. Math., 145(1): 59–103, (2001)
  * [15] R. Gopakumar, C. Vafa, M-Theory and Topological Strings I, hep-th/9809187 (1998)
  * [16] R. Gopakumar, C. Vafa, M-Theory and Topological Strings II, hep-th/9812127 (1998)
  * [17] B. Haghighat, A. Iqbal, C. Kozcaz, G. Lockhart and C. Vafa, M-Strings, Commun. Math. Phys. 334, no. 2, 779 (2015)
  * [18] A. Iqbal and A. Kashani-Poor, “Instanton counting and Chern-Simons theory,” Adv. Theor. Math. Phys. 7, 459-499 (2003)
  * [19] A. Iqbal, A. Kashani-Poor, SU(N) Geometries and Topological String Amplitudes, Preprint(2018)
  * [20] A. Iqbal, C. Kozcaz, K. Shabbir, Refined Topological Vertex, Cylindric Partitions and the $U(1)$ Adjoint Theory, Nucl. Phys. B 838, 422 (2010)
  * [21] A. Iqbal, C. Kozcaz, C. Vafa, The Refined topological vertex, Communications in Mathematical Physics 298 (3), 757-785, (2010).
  * [22] V. G. Kac. Infinite-dimensional Lie algebras. Cambridge University Press, Cambridge, third edition, 1990.
  * [23] Masanobu Kaneko and Don Zagier. A generalized Jacobi theta function and quasimodular forms. In The moduli space of curves (Texel Island, 1994), volume 129 of Progr. Math., pages 165–172. Birkhäuser Boston, Boston, MA, 1995.
  * [24] B. F. Logan and L. A. Shepp. A variational problem for random Young tableaux. Advances in Math., 26(2):206–222, 1977.
  * [25] I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications.
  * [26] Sho Matsumoto. Correlation functions of the shifted Schur measure. J. Math. Soc. Japan, 57(3):619–637, 2005.
  * [27] Davesh Maulik and Andrei Okounkov. Quantum groups and quantum cohomology. Preprint. arXiv:1211.1287
  * [28] H. Nakajima, Instantons on ALE spaces, quiver varieties and Kac-Moody algebras, Duke Math. J. 76(2): 365-416 (November 1994).
  * [29] H. Nakajima, K. Yoshioka Instanton counting on blowup. I. 4-dimensional pure gauge theory Inventiones mathematicae volume 162, 313–355 (2005)
  * [30] Richard P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.
  * [31] M. Szalay and P. Turán. On some problems of the statistical theory of partitions with application to characters of the symmetric group. I. Acta Math. Acad. Sci. Hungar., 29(3-4):361–379, 1977\.
  * [32] A. M. Vershik. Statistical mechanics of combinatorial partitions, and their limit configurations. Funktsional. Anal. i Prilozhen., 30(2):19–39, 96, 1996.
  * [33] A. M. Vershik and S. V. Kerov. Asymptotic of the largest and the typical dimensions of irreducible representations of a symmetric group. Functional Analysis and Its Applications, 19:21–31, 1985. 10.1007/BF01086021.
  * [34] Wang, W.: Dual Pairs and Infinite Dimensional Lie Algebras, In: Recent Developments in Quantum Affine Algebras and Related Topics, (eds. N. Jing and K.C. Misra), Contemp. Math. 248 (1999) 453–469.
  * [35] E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121(3): 351-399 (1989).
  * [36] R. Rıos-Zertuche, An introduction to the half-infinite wedge, preprint (2014).